statistical methods in analyzing the shape of...
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STATISTICAL METHODS IN ANALYZING THE SHAPE OF
MAXILLARY DENTAL ARCHES FOR DENTAL
APPLICATIONS
NORLI ANIDA BINTI ABDULLAH
FACULTY OF SCIENCE
UNIVERSITY OF MALAYA
KUALA LUMPUR
2016
STATISTICAL METHODS IN ANALYZING THE SHAPE OF
MAXILLARY DENTAL ARCHES FOR DENTAL APPLICATIONS
NORLI ANIDA BINTI ABDULLAH
THESIS SUBMITTED IN FULFILLMENT
OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
FACULTY OF SCIENCE
UNIVERSITY OF MALAYA
KUALA LUMPUR
2016
ii
UNIVERSITY OF MALAYA
ORIGINAL LITERARY WORK DECLARATION
Name of Candidate: Norli Anida Binti Abdullah (I.C No: 851121-14-6336)
Registration/Matric No: SHB090010
Name of Degree: Doctor of Philosophy
Title of Project Paper/Research Report/Dissertation/Thesis (“this Work”):
STATISTICAL METHODS IN ANALYZING THE SHAPE OF MAXILLARY
DENTAL ARCHES FOR DENTAL APPLICATIONS
Field of Study:
Applied Statistics
I do solemnly and sincerely declare that:
(1) I am the sole author/writer of this Work;
(2) This Work is original;
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dealing and for permitted purposes and any excerpt or extract from, or
reference to or reproduction of any copyright work has been disclosed
expressly and sufficiently and the title of the Work and its authorship have
been acknowledged in this Work;
(4) I do not have any actual knowledge nor do I ought reasonably to know that
the making of this work constitutes an infringement of any copyright work;
(5) I hereby assign all and every rights in the copyright to this Work to the
University of Malaya (“UM”), who henceforth shall be owner of the
copyright in this Work and that any reproduction or use in any form or by any
means whatsoever is prohibited without the written consent of UM having
been first had and obtained;
(6) I am fully aware that if in the course of making this Work I have infringed
any copyright whether intentionally or otherwise, I may be subject to legal
action or any other action as may be determined by UM.
Candidate’s Signature Date:
Subscribed and solemnly declared before,
Witness’s Signature Date:
Name:
Designation:
iii
ABSTRACT
This study aimed to propose shape features and statistical shape models to
develop a novel shape discrimination procedure for the maxillary dental arch with
important applications in dentistry. Standardized digital images of randomly selected
dental casts were obtained and the image calibration and registration were attended to
enable comparison of shape of the dental arches. A collective teeth positions from the
digital images were proposed as a novel shape feature of the dental arch. Each tooth
position is established from origin defined from stable anatomical landmarks. The mean
shape category obtained from clustering method and the probability distribution of each
shape category was further investigated to provide better statistical inference for the
shape models of the dental arch. A modified COVRATIO statistics which incorporates
the problem of small sample size and minimal model assumptions was then proposed as
a discrimination method of shape and compared to the linear discrimination method.
The proposed shape discrimination method was then used to determine suitable arch
shape and indicate natural teeth positions. The results from this study show that
multivariate normal and multivariate complex normal shape models together with the
use of the proposed discrimination method can be used to discriminate shape of the
dental arch. Consequently guides to determining suitable impression trays and
predicting teeth positions for the edentulous patients (patients with all teeth missing) are
provided. Verification of the proposed guides show that 91.42% of the sample studied
indicates appropriate fitting to the resultant impression trays, and the original teeth
positions (with an average error of 0.95 mm for each tooth position) were adequately
estimated by 80% of the arches studied. The presented statistical methods may be
beneficial in assisting inexperienced dentists and dental laboratory technicians to choose
the most appropriate impression tray and to determine natural teeth positions for the
Malaysian population.
iv
ABSTRAK
Kajian ini mencadangkan ciri-ciri serta model bentuk statistik untuk
menghasilkan prosedur diskriminasi yang baru bagi bentuk arkus pergigian pada rahang
atas dengan aplikasi penting dalam bidang pergigian. Imej digital yang seragam dari
acuan pergigian dipilih secara rawak dan penentukuran serta padanan imej dilakukan
untuk membolehkan perbandingan bentuk arkus pergigian. Kedudukan gigi secara
kolektif dari imej-imej digital dicadangkan sebagai ciri-ciri bentuk arkus pergigian yang
baru. Setiap kedudukan gigi ditentukan daripada penanda anatomi yang stabil. Nilai
purata setiap kategori bentuk dianggarkan daripada kaedah kelompok dan taburan
kebarangkalian setiap kategori bentuk seterusnya disiasat untuk memberikan inferens
statistik yang lebih baik untuk model bentuk arkus pergigian. Statistik COVRATIO
yang diubahsuai untuk disesuaikan dengan masalah saiz sampel yang kecil dan andaian
model yang minimum kemudiannya dicadangkan sebagai kaedah diskriminasi bentuk
serta dibandingkan dengan fungsi diskriminasi linear. Kaedah diskriminasi yang
dicadangkan ini kemudiannya digunakan untuk menentukan bentuk arkus dan
kedudukan gigi asal. Hasil daripada kajian ini menunjukkan bahawa model bentuk
multivariat normal dan multivariat kompleks normal dengan kombinasi kaedah
diskriminasi yang dicadangkan boleh digunakan untuk membezakan bentuk arkus
pergigian. Ini seterusnya membolehkan dua panduan dicadangkan dalam pemilihan
ceper impresi yang sesuai dan meramalkan kedudukan gigi untuk pesakit edentulus
(pesakit dengan ketiadaan gigi). Kajian pengesahan untuk panduan-panduan yang
dicadangkan menunjukkan bahawa 91.42% daripada sampel yang dikaji menunjukkan
pemilihan ceper impresi yang sesuai, dan 80% daripada arkus pergigian yang dikaji
boleh menganggarkan kedudukan gigi asal (dengan ralat purata 0.95 mm untuk setiap
posisi gigi). Kaedah statistik yang dicadangkan boleh dimanfaatkan dalam membantu
v
doktor gigi dan juruteknik makmal pergigian yang tidak berpengalaman untuk memilih
ceper impresi yang sesuai dan memudahkan anggaran kedudukan gigi asal untuk
penduduk Malaysia.
vi
To my honey, Hafiz
To my awesome buddies, Iqbal & Yati
vii
ACKNOWLEDGEMENTS
Praise to the Almighty, for giving me the strength to complete this work and
blessed me with loving people around me.
My greatest gratitude goes to my respected supervisors Assoc. Prof. Dr. Omar
Mohd Rijal and Assoc. Prof. Dr. Zakiah Mohd Isa for their guidance throughout my
study. Their supervision has taught me to appreciate knowledge by heart and never
giveup on your dreams.
Special thanks to Assoc. Prof. Dr. Yong Zulina, Dr. Ali Zaid, Dr. Mamun,
Professor Dr. Hanif, Professor Dr. Imon, Professor Dr. Rao and Professor Dr. Sahar for
their willingness to read through my thesis and suggest necessary improvements. To my
friends Iqbal, Yati, Omar, Adia, Adzhar, Rany, Dela, Siti, Zanariah, Faizol, and
Hafrizal, Kuna, Katz, Laili, Maz and Meksu, thank you for always being there, listening
and motivating me throughout these years – you guys are the best. Not forgetting the
staff at the Center of Foundation Study in Science, Institute of Mathematical Science
and Dean Office of Faculty of Science for their assistance right up to the completion of
this thesis.
My heartfelt appreciation goes to my beloved husband, for his continuous love
and the shoulder that I always look for to cry on. To mak, abah, mama, papa, my
siblings, and my favourite uncle Pak itam, thank you for your continuous prayers which
have driven me to complete this thesis. Not forgetting my two boys, Hamza and Aqeel,
that would never fail to make me smile everyday.
To my late sister, Maria, I miss you so much and may Allah place you in a
garden of paradise with the righteous ones.
viii
TABLE OF CONTENTS
ABSTRACT iii
ABSTRAK iv
ACKNOWLEDGEMENT vii
TABLE OF CONTENTS viii
LIST OF FIGURES xv
LIST OF TABLES xx
LIST OF SYMBOLS AND ABBREVIATIONS xxv
LIST OF APPENDICES xxvii
CHAPTER 1: INTRODUCTION 1
1.1 General Introduction: Shape Analysis 1
1.2 Relevance of Shape in Dentistry 2
1.3 Issues Related to Shape Analysis of the Dental Arch in Dentistry 3
1.3.1 Discriminating Shape of the Dental Arch 3
1.3.2 Shape Feature of the Dental Arch 3
1.3.3 Statistical Shape Model of the Dental Arch 4
1.3.4 Issues with Stock Impression Trays 5
1.3.5 Issues with Rehabilitating the Edentulous Patients 5
1.4 Objectives of the Study 6
1.5 Thesis Outline 6
ix
CHAPTER 2: LITERATURE REVIEW 9
2.1 Shape Analysis from Digital Images: Image Acquisition and Storing 9
2.2 Quantitative Description of the Dental Shape from 2D Images 12
2.2.1 Multivariate Morphometrics 12
2.2.2 Boundary Morphometrics 13
2.3 Discrete Fourier Transform (DFT) 18
2.3.1 Derivation of DFT 19
2.3.2 Application of DFT in 2D Shape Analysis 20
2.4 Shape Classification from Digital Images 22
2.4.1 Shape Alignment: Comparing two or more Arch Shapes 22
2.4.2 Arch Shape Classification 24
2.5 Statistical Shape Model of the Dental Arches 26
2.6 Discrimination of Shape 33
2.7 Multivariate Normal Distribution and its Tests 34
2.7.1 Mardia’s Multivariate Skewness and Kurtosis Tests 35
2.7.2 Doornik and Hansen Omnibus Test 36
2.7.3 Royston Test 38
2.7.4 Henze-Zirkler Test 40
2.8 Multivariate Complex Normal Distribution (MVCN) 41
2.8.1 The Univariate and Multivariate Complex Random Variables and
Distributions
42
2.8.2 Properties of the MVCN Distribution 43
2.8.3 Parameter Estimation and Hypothesis testing of the MVCN
Distribution
44
2.9 Missing Values Analysis 47
2.9.1 Data Augmentation (DA) Algorithm 47
x
2.9.1.1 Definition 47
2.9.1.2 Application of DA in the Missing Values Problem 49
2.9.2 Expectation maximization (EM) Algorithm 50
2.9.2.1 Definition 50
2.9.2.2 EM for Regular Exponential Families in Missing
Values Problem
51
CHAPTER 3: EXPLORATORY DATA ANALYSIS OF SHAPE
FEATURE AND INVESTIGATION OF CATEGORIES OF THE
DENTAL ARCH SHAPE
54
3.1 Introduction 54
3.2 Data Collection 54
3.2.1 Selection of Samples 55
3.2.2 Dental Impression and Stone Cast Making 55
3.2.3 Cast Preparation 56
3.2.4 Image Acquisition, Shape Alignment and Calibration of
Measurements
57
3.3 Shape Feature of the Dental Arch 59
3.4 Properties of the Shape Feature 62
3.4.1 Approximation of Circular Normal to Linear Normal
Distribution
64
3.4.2 Results on Approximation of Circular Variables of the Shape
Feature
66
3.5 Dental Arch Symmetry and Dimension Reduction 69
3.5.1 Test of Arch Symmetry 70
3.5.2 Results for Test of Symmetry 72
xi
3.6 Categories of Dental Arch Shape 74
3.6.1 Clustering Method 75
3.6.2 Definitive Number of Clusters 81
3.6.2.1 Principal Component Analysis 81
3.6.2.2 Dunn’s Validity Index 83
3.6.3 Results on Categories of Dental Arch Shape 84
3.7 Discussion 86
CHAPTER 4. SHAPE MODEL AND DISCRIMINATION OF THE
DENTAL ARCH: AS A GUIDE IN DETERMINING APPROPRIATE
IMPRESSION TRAY FOR ORAL DIAGNOSIS AND TREATMENT
PLANNING
89
4.1 Introduction 89
4.2 Shape Models of Maxillary Dental Arch 90
4.2.1 A Simulation Study on Performance of Multivariate Normality
Tests for Small Sample Size
90
4.2.1.1 Type I Error Rates 91
4.2.1.2 Power of Test 95
4.2.1.3 Summary of Simulation Results for Testing
Multivariate Normality
103
4.2.2 Multivariate Normal Distribution of Categories of Shape 104
4.2.3 Test of Separation between the Shape Models 109
4.2.3.1 Test for Equality of Covariance Matrices 109
4.2.3.2 Investigating Separation of Mean Shape using the
Hotelling T2 test
110
4.2.4 Verification of the Arch Shape Models 111
xii
4.3 Application of Shape Models to Impression Tray Design 113
4.3.1 Fabricating Three Impression Trays 113
4.3.2 Verification of the Fabricated Impression Trays 114
4.4 Discrimination of the Dental Arch Shape 117
4.4.1 Proposed )( iCOVRATIO as Discrimination Method 118
4.4.2 Simulation Study for Comparing Performance of LDF and
)( iCOVRATIO
120
4.4.3 Results for Discrimination of Dental Arch Shape 124
4.5 Investigating Dental Arch Shape with Missing Tooth: A Simulation
Study on Performance of Data Augmentation and Expectation
Maximization
125
4.6 Application of Shape Discrimination 128
4.6.1 A Proposed Guide for Determining Appropriate Impression Tray
for Patients (Without and with Missing Tooth)
128
4.6.2 Verification of the Proposed Guide for Determining Appropriate
Impression Trays
129
4.7 Discussion 130
CHAPTER 5. SHAPE MODEL AND DISCRIMINATION OF THE
DENTAL ARCH: AS A GUIDE IN ESTIMATING NATURAL TEETH
POSITIONS ON COMPLETE DENTURES FOR THE EDENTULOUS
132
5.1 Introduction 132
5.2 Shape model of the dental arch using Fourier Descriptor (FD) 133
5.2.1 The Ability of FD in representing Dental Arch Shape 133
5.2.2 Selecting Number of FD Terms as Shape Feature 136
5.2.3 Categories of Arch Shape using FD Shape Feature 139
xiii
5.2.4 Probability Distribution of Shape Categories using FD 141
5.2.5 Test of Separation between MVCN Shape Models 148
5.2.5.1 The Proposed Hotelling T2 Test for MVCN
Distribution
149
5.2.5.2 Simulation Study on Performance of the Hotelling T2
for MVCN Distribution
150
5.2.5.2.1 Type I Error Rates 151
5.2.5.2.2 Power of Test 153
5.2.5.3 Results on Test of Separation between MVCN Shape
Models of Dental Arch
155
5.3 Discrimination of Shape for MVCN Model 156
5.3.1 LDF for MVCN Distribution 157
5.3.2 Proposed MVCNiCOVRATIO )( for MVCN Distribution as
Discrimination Method
158
5.3.3 Simulation Study on Performance of LDF and
MVCNiCOVRATIO )(
159
5.3.4 Results for Shape Discrimination of Dental Arch using FD 164
5.4 Linking Anatomical Landmarks to the MVCN Shape Models of the
Dental Arch on Edentulous Arch
165
5.4.1 Shape model of Anatomical Landmarks 166
5.4.2 Verification of the Anatomical Landmarks Model 168
5.5 Application of Shape Discrimination for MVCN model 170
5.5.1 A Proposed Guide to Teeth Positioning on
Complete Dentures
170
5.5.2 Verification of the Proposed Teeth Positioning Guide for the
Edentulous
174
xiv
5.6 Discussion 177
CHAPTER 6: CONCLUDING REMARKS 178
6.1 Conclusion 178
6.2 Limitations of the study 180
6.3 Direction for Future Research 181
6.3.1 Exploring other Applications of Shape Model and
Shape Discrimination Procedure in Dental Problems
181
6.3.2 Extension to Mandibular and 3D Shape of the Dental Arches 181
6.3.3 Multivariate Complex Normal with Relation Matrix 182
6.3.4 Comparing Shape Discrimination using COVRATIO and
Bayesian Methods
182
6.3.5 Regression Ideas for Shape Models and Discrimination 183
REFERENCES 184
LIST OF PUBLICATIONS AND PAPERS PRESENTED 198
APPENDICES 199
xv
LIST OF FIGURES
Figure 2.1: Maxillary dental cast of a dentate subject 9
Figure 2.2: Each pixel in a digital image can be indicated by a Cartesian
coordinate with origin (1,1) at the left top of the picture.
10
Figure 2.3: Cartesian axes were established using teeth as landmarks. 11
Figure 2.4: Common linear measurements used to represent the dental arch. 13
Figure 2.5: Bezier curve defined by four control points of which has starting
point ),,( 00 yx directional points ),( 11 yx and ),( 22 yx and end point
),( 33 yx .
16
Figure 2.6: Two copies of the same shape of a hand. The picture on the right
was rotated roughly at 45 anti clockwise from the vertical axis.
24
Figure 3.1: Dental impression made in an impression tray. 55
Figure 3.2: Dental plaster was poured onto the impression to obtain the
stone dental cast.
56
Figure 3.3: Method used to make the base of the stone cast parallel to the
occlusal plane.
57
Figure 3.4: Two metal rulers positioned on a plane parallel to the occlusal
plane enable measurements to be calibrated. Shape alignment is
achieved by the creation of the Cartesian plane define from
anatomical landmarks which were marked with crosses.
58
Figure 3.5: Selected points on the teeth used to represent the dental arch
shape
59
Figure 3.6: Computation of angle and distance of the central incisor tooth
from the geometrical Cartesian origin made on the digital
images.
62
xvi
Figure 3.7: Differences of linear and circular measurement. 63
Figure 3.8: The arithmetic mean points the wrong way. 63
Figure 3.9: QQ circular normal plot for RRRR wwww 4321 and ,, . 68
Figure 3.10: Raw data plot for RRRR wwww 4321 and ,, . 68
Figure 3.11: Chi square plot for testing 16-variate normal distribution. 72
Figure 3.12: Dendrograms showing number of clusters at different cut-off
levels obtained from complete linkage method.
79
Figure 3.13: The pyramid of cluster boxes where each cluster is represented
by the vector of means.
80
Figure 3.14: Investigation of separation of clusters using the first two
principal components indicates the existence of three distinct
groups of dental arches in the sample studied.
82
Figure 3.15: Mean shapes of )3,2,1( kGk 85
Figure 3.16: Boxplot of each shape feature for )3,2,1( kGk 85
Figure 4.1: Examples of different sets of generated MVN data for p = 2. 93
Figure 4.2: Example of generated multivariate t distribution for df = 10 and
83
38Σ .
97
Figure 4.3: Example of generated )87,36(U data for p = 2. 97
Figure 4.4: Example of a multivariate lognormal distribution generated for
0μ and 2IΣ .
98
Figure 4.5: Empirical power for MS, MK, DH, HZ and Royston test statistic
against multivariate t distribution.
99
Figure 4.6: Empirical power for MS, MK, DH, HZ and Royston test statistic
against multivariate t distribution.
100
xvii
Figure 4.7: Empirical power for MS, MK, DH, HZ and Royston test statistic
against lognormal distribution.
102
Figure 4.8: Mean location of a particular tooth and its variations. 107
Figure 4.9: Mean shape and variation of 1G . 107
Figure 4.10: Mean shape and variation of 2G . 108
Figure 4.11: Mean shape and variation of 3G . 108
Figure 4.12: (a) An example of commercial impression tray with space
around the tray to allow for variation of arch sizes and adequate
thickness of alginate impression material. (b) Mean shape v
was indicated by solid line. The broken line indicates 5mm
added to the perimeter of the arch shape. (c) The light cure
acrylic resin tray.
114
Figure 4.13: Some of the mal-aligned arches in the 40 samples used for
verification of the three fabricated trays.
116
Figure 4.14: A small ruler was placed near the teeth (for image calibration)
while capturing the 2D image of the patient’s dental arch by
using an intraoral camera.
129
Figure 5.1: An example of a particular dental cast which shows the hamular
notches (HN) and incisive papilla (IP) that were used to establish
the Cartesian axes. Twenty one selected points as illustrated in
the diagram were used as the shape boundary.
134
Figure 5.2: The ability of 2, 3, 4, 5, 6, 19, 20 and all 21 FD terms in
representing the dental arch shape.
135
Figure 5.3: The magnitude plot for 2 casts as an example. The first 6 and the
last 2 terms shows higher contribution in representing the
boundary of the arch shape.
137
xviii
Figure 5.4: The Procrustes distances between the original 21 points and their
estimated positions using q-FD terms gradually levels off at q=8.
138
Figure 5.5: Plot of the original arch shape oix and the shape boundary q
ix
approximated using q = 8 FD terms.
139
Figure 5.6: Mean shapes of 21 ˆ,ˆ AA and 3A using 8 FD as shape feature. 141
Figure 5.7: Variation of the k-th point from its mean. 146
Figure 5.8: Mean shape and variation for shape model )1(A . 147
Figure 5.9: Mean shape and variation for shape model )2(A . 148
Figure 5.10: Mean shape and variation for shape model )3(A . 148
Figure 5.11: Examples of the generated complex normal data for 4p and
.4p
153
Figure 5.12: Examples of the generated complex normal data for different
values of 1θ and
2θ .
154
Figure 5.13: Examples of the generated complex normal data for 4p and
.4p
163
Figure 5.14: Edentulous cast showing the incisive papilla (IP) and hamular
notches (HN).
165
Figure 5.15: Brief outline of steps in the construction of complete dentures. 170
Figure 5.16: Proposed guide for construction of complete dentures. 172
Figure 5.17: Smallest Procrustes distance PD = 9.7520 mm between the
original and estimated teeth position. Red dots indicate the
estimated teeth position (sample N110).
175
Figure 5.18: An average Procrustes distance PD = 17.4664 mm, between the
original and estimated teeth position. Red dots indicate the
estimated teeth position (sample N130).
176
xix
Figure 5.19: Largest Procrustes distance PD = 26.1236 mm, between the
original and estimated teeth position. Red dots indicate the
estimated teeth position (sample N117).
176
xx
LIST OF TABLES
Table 2.1: Summary of literature for shape feature and shape model of the
dental arch.
29
Table 3.1: The sample correlation coefficient r, and the test statistic for
1;0 H to test the reliability of measurements taken at three
different times. The lower and upper critical values are -2.0141
and 2.0141 respectively when α = 0.05 was used.
60
Table 3.2: Mean, standard deviation (SD), minimum and maximum values
of the angular measurements (in degrees).
61
Table 3.3: Mean, standard deviation (SD), minimum and maximum values
of the length measurements (in mm).
61
Table 3.4: The test statistic for the Watson U2 and Kuiper’s tests and
estimated concentration parameter κ. All the variables show
non-significant results and indicate normality.
69
Table 3.5: Mean and variance of angular variables using circular and linear
statistic.
69
Table 3.6: Results of Kolmogorov-Smirnov (KS) goodness-of-fit test for
testing normality of residuals. Critical value of KS statistic used
is 0.1743.
73
Table 3.7: Test statistic for the Hotelling one-sample T2 test. 73
Table 3.8: Number of clusters formed using the available linkage methods
(M1 to M7) at selected percentage of similarity levels.
78
Table 3.9: Validation of number of clusters using Dunn’s index. 83
Table 3.10: Ethnic and gender homogeneity for each cluster. 84
Table 3.11: The anterior and posterior width and length for each of the shape
xxi
category. 86
Table 4.1: Empirical Type I error rate (in percentage) against the MVN
distribution with dimension p = 2 for different sets of
parameters.
94
Table 4.2: Empirical Type I error rate (in percentage) against the MVN
distribution with dimension p = 4 for different sets of
parameters.
94
Table 4.3: Empirical Type I error rate (in percentage) against the MVN
distribution with dimension p = 8 for different sets of
parameters.
95
Table 4.4: The proportion of observations satisfying
)())(())(( 21 pii kk vvSvv in each shape category .kG
Arbitrary values of 0.7 and 0.6 ,5.0 were used giving
3441.7)5.0(28 , 8.3505)6.0(2
8 and 9.5245)7.0(28 .
104
Table 4.5: The T values for Kolmogorov-Smirnov test with sample size
,111 n .14,22 32 nn
105
Table 4.6: Test statistic to investigate multivariate normality using Doornik
and Hansen (DH) test. Lower and upper 2.5% critical values are
6.9076 and 28.8453 respectively.
105
Table 4.7: Test statistic to investigate multivariate normality using Henze-
Zirkler (HZ) test.
105
Table 4.8: Test statistic to investigate multivariate normality using
Royston’s test.
106
Table 4.9: Hotelling 2T test for comparing two multivariate means. 111
Table 4.10: Doornik and Hansen (DH) test to investigate multivariate
xxii
normality of the shape categories using 122 casts. Lower and
upper 2.5% critical values are 6.9077 and 28.8454 respectively.
112
Table 4.11: Henze-Zirkler (HZ) test to investigate multivariate normality of
the shape categories using 122 casts.
112
Table 4.12: Royston’s test to investigate multivariate normality of the shape
categories using 122 casts.
112
Table 4.13: p-value of the Hotelling two sample 2T test for comparing 3
clusters using 47 casts and 122 casts.
113
Table 4.14: Amount of plasticine thickness in the impression tray and
number of cast which fit the fabricated trays C1, C2 and C3
when impression of 47 control samples was taken.
115
Table 4.15: Amount of plasticine thickness in the impression tray and
number of cast which fit the fabricated trays C1, C2 and C3
when impression of 40 test samples was taken.
116
Table 4.16: Misclassification probability for LDF and )( iCOVRATIO when
different sample size, dimension, mean vector and covariance
matrices were considered.
123
Table 4.17: Misclassification probability when 47 control casts were re-
assigned into either one of the population of the shape model
using the LDF and )( iCOVRATIO .
124
Table 4.18: The misclassification probability using DA and EM for missing
values imputation when the number of missing values 20 n .
127
Table 4.19: The misclassification probability using DA and EM for missing
values imputation when the number of missing values 40 n .
127
Table 4.20: The misclassification probability using DA and EM for missing
xxiii
values imputation when the number of missing values 60 n . 127
Table 4.21: Plasticine thickness in the impression tray and number of casts
(n=35) which fit the fabricated trays C1, C2 and C3.
130
Table 5.1: Mean, standard deviation (SD) and range of the FD terms 140
Table 5.2: Test statistic to investigate bivariate normality of each
ˆ
ˆ,,
ˆ
ˆ,
ˆ
ˆ
2
2
1
1
q
q
d
c
d
c
d
c using HZ test in the 3 shape categories. LCV
and UCV are the abbreviation for lower and upper critical value
respectively.
142
Table 5.3: Test statistic to investigate bivariate normality of each
ˆ
ˆ,,
ˆ
ˆ,
ˆ
ˆ
2
2
1
1
q
q
d
c
d
c
d
c using DH test in the 3 shape categories. LCV
and UCV are the abbreviation for lower and upper critical value
respectively.
143
Table 5.4: Test statistic to investigate bivariate normality of each
ˆ
ˆ,,
ˆ
ˆ,
ˆ
ˆ
2
2
1
1
q
q
d
c
d
c
d
c using Royston’s test in the 3 shape categories.
The critical values were denoted in the bracket as (Lower 2.5% ,
Upper 2.5%).
143
Table 5.5: Empirical Type I error rate against equal means for different sets
of parameters.
152
Table 5.6: Empirical power of test against unequal means for different sets
of parameters.
155
Table 5.7: Hotelling 2T test for comparing two MVCN mean. 156
Table 5.8: Misclassification probability of LDF and MVCN
iCOVRATIO )( for
xxiv
MVCN model when different sample size, dimension, mean
vector and covariance matrices were considered.
161
Table 5.9: Misclassification probability when 47 control casts were re-
assigned into either one of the population of the shape model
using LDF and MVCNiCOVRATIO )( .
164
Table 5.10: The T2 test for comparing two MVCN mean for anatomical
landmarks.
166
Table 5.11: Mean, standard deviation (SD) and range of the FD terms for the
anatomical landmarks.
167
Table 5.12: Misclassification probability when 47 control casts was
discriminated using the anatomical landmarks ),,( 321 iiii LLLL
of the dental arch.
169
Table 5.13: Misclassification probabilities when 40 test casts were
discriminated using anatomical landmarks ),,( 321 iiii LLLL and
teeth location iii
i aaa 821ˆ , ,ˆ ,ˆˆ A .
169
Table 5.14: Group membership when the anatomical landmarks of the 35
test cast was assigned using MVCNiCOVRATIO )( .
175
Table 5.15: The number and percentage of casts according to the Procrustes
distance (PD) intervals indicating sum squared of difference
between the estimated and original teeth position.
175
xxv
LIST OF SYMBOLS AND ABBREVIATIONS
AHC Agglomerative hierarchical clustering
ANOVA Analysis of variance
c.d.f Cumulative distribution function
CI Confidence interval
DA Data augmentation
d.f Degrees of freedom
DFT Discrete Fourier Transform
DH Doornik and Hansen
EFF Elliptical Fourier function
EM Expectation maximization
FD Fourier descriptor
FS Fourier series
HN Hamular notches
HZ Henze-Zirkler
i.i.d Independent and identically
IP Incisive papilla
KS Kolmogorov-Smirnov
LDF` Linear discrimination function
m.l.e Maximum likelihood estimates
mm Millimeters
MS Mardia’s skewness
MK Mardia’s kurtosis
MVN Multivariate normal
MVCN Multivariate complex normal
xxvi
PC Principal component
PD Procrustes distance
SD Standard deviation
SW Shapiro-Wilk
xxvii
LIST OF APPENDICES
Page
Appendix A Ethics approval 198
Appendix B Examples of Research Data for
),,,,,,,,( 44332211
RRRRRRRRR lwlwlwlwv iii aaa 821ˆ , ,ˆ ,ˆˆ A and
),,( 321 LLLL
199
Appendix C Summary of works on dental arch symmetry 201
Appendix D MATLAB Program for Determine Appropriate Impression
Tray for Dentate Patients
205
Appendix E MATLAB program for Estimating Natural Teeth Positions
on Complete Dentures for the Edentulous
209
1
CHAPTER 1: INTRODUCTION
1.1 General Introduction: Shape Analysis
The word shape is commonly used to refer to the external form of an object.
Describing, comparing, classifying, recognizing and discriminating the shape of natural
and man-made objects are of interest in many disciplines. In physical anthropology,
forensic anthropology and archaeology, investigation of bodily appearance of living and
dead humans and animals was carried out to study the evolution of primates
(O’Higgins, 2000), identifying body shape or structure of an athlete according to a
particular type of sports (Maas, 1974), to differentiate skeletal of the apes (White, 1960)
and shape of the human head (Lacko et al., 2015). Other fields of research that involve
the study of shape are biology (classification of plant leaves according to their species,
discrimination of honey bee based on wing shape and species identification of fish)
(Bruno et al., 2008; Yu et al., 2014; Charistos et al., 2014), medicine (detecting
osteoporosis from dental radiographs, predicting 3D surface model of knee from 2D
image of knee joint for computer-aided knee surgery and discrimination of liver for
diagnosis of liver cirrhosis) (Allen et al., 2007; Tsai et al., 2015; Uetani et al., 2015),
security (fingerprints recognition) (Dass & Li, 2009) and computer vision analysis
(automatic trademark image search system to ensure uniqueness of each new registered
trademark) (Eakins et al., 2003).
Advances in technology particularly in image acquisition techniques have
facilitated the collection of data for shape information. The information was then used
to model the object of interest. A popular approach for shape modelling is using
statistical shape model, which finds the mean shape and shape variations of the training
2
data from the principal component analysis (Allen et al., 2007; Lacko et al., 2015; Tsai
et al., 2015). The shape models were usually incorporated into computers and used as a
model-based tool to consistently predict, compare or recognize shapes by iteratively
deforming the initial shape set from the shape model to fit the new shape feature (also
known as an active shape model).
Another approach for shape modelling is the Procrustes analysis, which analyzes
the distribution of a random sample of a set of shape feature that must be optimally
superimposed so that all shapes are comparable. This approach is usually used to
describe differences of shape or how shape is related to size (O’higgins and Dryden,
1993; Sampson, 1983; Dryden & Mardia, 1998).
The development of statistical shape model is essentially dependent on how the
shape feature was quantified. Quantitative description of an object is called
morphometrics. Basically, an object can be seen as a set of measurements (multivariate
morphometrics) or a boundary (boundary morphometrics). The multivariate
morphometrics though considered a traditional approach is, however, still commonly
used and useful for exploring the main characteristics of the shape and also for
discriminating shape (Lee et al., 2011; Scanavini et al., 2012; Uetani et al., 2015).
1.2 Relevance of Shape in Dentistry
The development of shape analysis in dentistry is not as extensive as in
biomedical engineering and other associated fields. However, works related on arch
shape analysis are increasingly sophisticated in the following applications:
In orthodontics, information about normal arches are used in designing and
manufacturing preformed arch wires to correct arches with irregularities in tooth
positions and in the maintenance phase of the corrected teeth in the new position
(Kairalla et al., 2014; Lee et al., 2011). In restorative dentistry, successful complete
3
dentures requires that the artificial teeth be placed where the natural teeth were, so that
natural function and aesthetics are restored.
Orthodontic and prosthetic treatments are completed on casts made from
impressions of the dental arches. The selection of stock trays to be used by dentists
when making impressions for the casts will be facilitated if the tray design and size are
as close as possible to the patient’s dental arch (Wiland, 1971; Yergin et al., 2001).
In forensic dentistry, bite mark patterns found either on the victim of an assault
or homicide, or inflicted by a deceased victim on the assailant may be compared with
the dental arch shape and dentition of either victim or assailant for the purpose of
identifying or exclusion of suspect or victim (Blackwell et al., 2007; Bush et al., 2011).
1.3 Issues Related to Shape Analysis of the Dental Arch in Dentistry
From the review of the literature, a few issues related to the analysis of dental arch
shape were identified as follows:
1.3.1 Discriminating Shape of the Dental Arch
This is important when choosing suitable available pre-formed arch wire and
stock impression tray for a particular patient (Yergin et al., 2001; Beale, 2007; Lee et
al., 2011), and discriminating an assailant’s identity through the bite marks from a few
suspects (Bush 2011; Sheets et al., 2013). To date, shape discrimination based on
statistical methods has not been used to address these applications relevant to dentistry.
1.3.2 Shape Feature of the Dental Arch
Collective shape features such as linear distances and angles between teeth,
ratios of two distances, perimeter, area and points on the teeth were typically used to
describe dental arch shape and size. The shape features were obtained from manipulated
pixels of digital image that correspond to an established Cartesian coordinate system.
4
However, some of these shape features together with the methods of analysis may be
inferior, as they only indicate size, but not the shape of the arch (Maurice & Kula, 1998;
Nojima et al., 2001; Šlaj et al., 2003; Tong et al., 2012). Moreover, as most of the
proposed Cartesian coordinate axes were established from teeth, the shape feature and
reconstruction of shape may not be possible if the teeth are lost (Kasai et al., 1995;
Triviño et al., 2008; Mikami et al., 2010; Lee et al., 2011). Hence, applications related
to the shape of the dental arch may be restrictive.
1.3.3 Statistical Shape Model of the Dental Arch
A statistical shape model should include a mean shape and variability (Hufnagel,
2001). Most studies on dental arches only determined the “average” shape of the arch.
Information on how the sample arches deviated from the established average arch shape
was not mentioned (Arai & Will, 2011; Ferrario et al., 1994; Lee et al., 2011; McKee &
Molnar, 1988; Raberin et al., 1993).
Only three studies considered the shape variation (Sampson, 1983; Wu et al.,
2012; Elhabian & Farag, 2014). These shape models were defined as a single ideal arch
shape, with relatively large variations from their respective mean shapes. Other studies
have however, showed that more than one category of arch shape exists when possible
cluster of arches were investigated or assigned to a specific arch form (Lee et al., 2011;
Nakatsuka et al., 2011; Triviño et al., 2008; Zia et al., 2009). Clinical experience shows
that a single impression tray would not fit all patients’ arches. The existence of a single-
ideal arch shape model in a particular population is therefore unrealistic and categories
of shape model should be proposed instead.
5
1.3.4 Issues with Stock Impression Trays
Stock impression trays are available in various sizes and shapes to accommodate
different mouths. In many instances the trays require modification as they do not
provide for variations found in patients’ mouths (Beale, 2007; Bomberg et al., 1985). It
was shown that some commercially available stock impression trays may only be
suitable for a particular patient population (Yergin et al., 2001).
There also appears to be no scientific basis for the design of stock impression trays
(Yergin et al., 2001). Specifications for the commercialization of stock impression trays
were made without any reference to the average dimensions and variability of the
human dental arch size and shape. Further, a guide using statistical methods to assign a
suitable tray for a particular patient has never been proposed.
1.3.5 Issues with Rehabilitating the Edentulous Patients
Treating the edentulous patient may be difficult and require years of clinical
observation and experience, especially when pre-extraction records (diagnostic casts or
photographs) are not available to be used as guides for indicating the positions of the
natural teeth (Bissasu, 1992; Faigenblum & Sharma, 2007). With the loss of teeth,
alveolar bone resorption follows and this further complicates the process of
redeveloping the natural arch shape and determining the position of the natural teeth.
Many studies have provided guides to the size of the artificial teeth, based on
observations of the size of the natural teeth and facial dimensions (Abdullah, 2002; Isa
et al., 2010). However, guides for reconstruction of arch shape and precise location of
each tooth in the edentulous arch remains a valid endeavour (Chu, 2007). A guide using
statistical methods has never been proposed to assist inexperienced dentists and dental
laboratory technicians in locating the natural teeth positions.
6
1.5 Objectives of the Study
The aim of this study was to develop a statistical shape model and subsequently
a discrimination procedure of shape for the maxillary dental arch with respect to:
a. Designing stock impression trays and selecting suitable impression trays for
patients.
b. Reconstructing of arch shape and predicting natural teeth positions for
edentulous patients.
To achieve these aims, the following objectives were set:
1. To propose novel shape features derived from stable anatomical landmarks
to enable reconstruction of the arch shape even when all teeth are lost.
2. To investigate the properties of the shape features of the dental arch,
particularly arch symmetry and shape categories as means of shape variation.
3. To develop statistical categories of shape model of dental arches by
investigating the probability distribution of each shape category and its
distinction.
4. To propose a shape discrimination procedure with minimal model
assumptions.
5. To propose guides in assisting inexperienced dentists and dental laboratory
technicians to choose the most appropriate stock tray for dentate or partially
edentulous patients and to estimate natural teeth positions for edentulous
patients.
1.6 Thesis Outline
This thesis is organized as follows:
Chapter 2 is a review of the literature on quantitative description, shape
classification and statistical shape model of the dental arch from 2D digital images.
7
Tests on multivariate normal distribution, introduction to multivariate complex normal
distribution and COVRATIO statistics are also reviewed to provide possible statistical
shape model of the dental arch and discrimination method for shape. An introduction to
the analysis of missing values is provided for a missing tooth, so that a complete shape
feature of the dental arch can still be obtained.
Chapter 3 reviews the properties of the shape feature, particularly categories of
dental arch shape. A novel shape feature v which indicate the teeth positions by a
collective of distances and angles from digital images was proposed and exploratory
data analysis of the shape feature was carried out. The linear property of the angular
variables used as shape feature was investigated. Further, a test of dental arch
symmetry reduced the dimensions of the shape feature. Exploration on categories of
dental arch shape was then carried out using the agglomerative hierarchical clustering
method. Investigation of the best linkage method to provide realistic clusters of arch
shape was carried out. These clusters represent shape categories and were then verified
using principal component analysis and the Dunn’s validity index.
A particular probability distribution of the dental arch as statistical shape model
should explain the mean shape and variation in each shape category. Chapter 4 proposes
statistical shape models of dental arches by investigating the probability distribution of
each shape category. A simulation study comparing tests of multivariate normal
distribution which performs best when the data mimics the shape feature and the sample
population was carried out. Three multivariate normal distributions define the shape
model of the dental arch. Test casts were used to validate the existence of the MVN
shape models. Application of the shape models was demonstrated in the design of three
impression trays. Subsequently, discrimination procedure for the shape of the maxillary
dental arch using the established statistical shape models was developed. A modified
COVRATIO statistics was proposed as a discrimination method of shape and compared
8
to the linear discrimination method. Using this knowledge, a guide for determining
appropriate impression tray for patients (without and with a missing tooth) was
proposed to facilitate the procedures of oral diagnosis and treatment planning.
Chapter 5 proposes a more precise shape feature that provides sufficient detailed
information about teeth positions on the dental arch using the Fourier descriptor (FD).
The ability of 8 FD terms in estimating all teeth positions was demonstrated. Using
these FDs, the 3 categories of shape established from v were verified and tested for
multivariate complex normality as its shape model. A hypothesis testing for two sample
means from MVCN based on the Hotelling 2T test was derived and employed to
confirm the existence of the 3 MVCN shape models. Then, a shape discrimination
procedure for the maxillary dental arch using the established statistical shape models for
MVCN was developed using the modified COVRATIO statistics. Further, 3 anatomical
landmarks which remain stable in the edentulous arch were linked to the 3 categories of
MVCN shape models. The application of the knowledge about the shape discrimination
for MVCN model together with the relation between the MVCN shape model and
anatomical landmarks was demonstrated in a guide for estimation of natural teeth
positions on complete dentures. Verification of the proposed guide for estimation of
natural teeth positions was carried out by comparing the original teeth positions of
dentate patients with the estimated teeth positions using the proposed guide.
Chapter 6 concludes the study. Limitations of the study and possible future
directions were also presented.
9
CHAPTER 2: LITERATURE REVIEW
2.1 Shape Analysis from Digital Images: Image Acquisition and Storing
The shape of the dental arch is usually represented by the curve composite
structure of the natural dentition. The arch curve is commonly indicated by a series of
landmarks on the arches in some biologically-meaningful way (Dryden & Mardia,
1998).
To analyze the dental arch of a particular patient, a replica of his oral structure
was cast in dental stone or plaster (Figure 2.1). An image of the dental cast was then
acquired and stored for analysis. Two common approaches may be used to acquire the
2-dimensional digital image of the dental cast. The dental cast was either:
a. Photographed using a digital camera (Henrikson et al., 2001; Lestrel et al.,
2004; Mikami et al., 2010) or
b. Scanned using a scanner (Lee et al., 2011; Taner et al., 2004; Wellens, 2007).
Rulers or calibration sheets were included when casts were being photographed or
scanned to ensure careful control of magnification.
Figure 2.1: Maxillary dental cast of a dentate subject.
10
Each digital image consisting of numerical presentations and images are stored
in an array of real or complex numbers and the elements of such a digital array are
called image elements, picture elements or pixels (Gonzalez & Woods, 2002). Each
pixel can be referred as a Cartesian coordinate. Establishment of the Cartesian axes and
subsequently its origin may be carried out as follows:
a. The origin of the Cartesian axes is usually taken as the upper left of the image,
as illustrated in Figure 2.2 (Henrikson et al., 2001; Lestrel et al., 2004; Taner et
al., 2004).
b. Landmarks on the dental arch may be used to establish the x and y Cartesian
axes and subsequently the origin (Kasai et al., 1995; Lee et al., 2011; Mikami et
al., 2010; Triviño et al., 2008). For instance, a line joining the cusp tip of the
molar teeth was regarded as the x axis and a perpendicular line to the x axis
which passes through the center of two central incisor teeth was regarded as the
y axis (Figure 2.3).
)3,3()2,3()1,3(
)3,2()2,2()1,2(
)3,1()2,1()1,1(
Figure 2.2: Each pixel in a digital image can be indicated by a Cartesian coordinate
with origin (1,1) at the left top of the picture. Source: www.ashleymills.com.
11
Figure 2.3: Cartesian axes established using teeth as landmarks.
Source: Kasai et al. (1995).
The pixel coordinate of the selected landmarks representing the dental arch with
respect to the established Cartesian axes will be subsequently transformed to the actual
coordinate with the scale calibrated from rulers or calibration sheet. This procedure was
usually carried out using custom-made program in any available programming language
such as Delphi (CodeGear, Scotts Valley, Calif) and available digitised software such as
DigitizeIt 1.5.7 (I. Bormann, Bormisoft, Germany) (Lee et al., 2011; Wellens, 2007).
With the actual coordinate of the arch shape, shape feature such as linear
distances between landmarks and angles between two landmarks can be easily obtained
by manipulating the pixel coordinates (Henrikson et al., 2001). Other studies on the
other hand used the coordinates as the shape feature (Triviño et al., 2008; Sampson,
1983). The literature on quantitative description of the dental arch shape is reviewed in
the following section.
12
2.2 Quantitative Description of the Dental Arch Shape from 2D Images
The following subsections review multivariate and boundary morphometrics
approaches to dental arch size and shape.
2.2.1. Multivariate Morphometrics
Extraction of several features of the arch shape such as linear distances between
landmarks, angles between two landmarks, ratios of two linear distances, perimeter and
area of the dental arch measured from the coordinates of the landmarks were typically
used to describe dental arch shape and size. Among the distances used were tooth to
tooth distance, such as the intercanine distance and intermolar distance and the length of
the perpendicular line that passes through the midline of the central incisors (Burris &
Harris, 2000; Hao et al., 2000; Lee et al., 2011). Ratios of these linear measurements
were also used to provide a general idea of the arch shape (Ferrario et al., 1993; Nojima
et al., 2001; Raberin et al., 1993).
Besides distances, angular measurements such as the distolateral angle at the
intersection of lines formed by a cusp tip of a tooth and the midpalatal raphe, perimeter
and area of the arch were also collectively used to indicate arch shape (Figure 2.4)
(Burris & Harris, 2000; Cassidy et al., 1998; Nakatsuka et al., 2011; Scanavini et al.,
2012). These j-th measurements were then defined as a (1 x j) vector and analyzed
using multivariate analysis. This approach is known as the “multivariate
morphometrics”.
Several studies have used the above features collectively to describe shape;
however they were not analyzed simultaneously using a multivariate approach (Maurice
& Kula, 1998; Nojima et al., 2001; Šlaj et al., 2003; Tong et al., 2012). These studies
may therefore only indicate size, but not the shape of the arch. It is known that
distances, ratios, and angles obtained from landmarks may not be the best method to
13
represent arch shape as they only represent the general form of the arch and only
capture certain features of shape. However, this approach may utilize available
multivariate techniques such as cluster analysis, which has successfully unravelled the
problems of shape classification and is still commonly used in biological research
(Costa & Cesar, 2009; Dryden & Mardia, 1998).
Figure 2.4: Common linear measurements used to represent the dental arch.
Source: Nakatsuka, et al. (2011)
2.2.2 Boundary Morphometrics
Instead of extracting certain features about shape, the coordinates of the selected
landmarks were used to indicate the boundary of the shape. These boundary coordinates
were then fitted to the following mathematical curves:
i. Polynomial Function
The n-th order of a polynomial function is given as
nn xaxaxaaxf 2
210)( , (2.1)
where naaa ,,, 10 are constant coefficients and n is a non-negative integer
which indicates the order of the function. The 2-nd order polynomial
function was used to predict the shape of the anterior (frontal) arch in an
edentulous patient (Preti et al., 1986). The 3-rd, 4-th and 6-th order
polynomial functions on the other hand, were shown to follow the shape
14
of the whole arch (Shrestha, 2013; McKee & Molnar, 1988). The
coefficients may describe the characteristics of arch shape. For the 4-th
order polynomial, the odd numbered terms ( 31 and aa ) define the left-
right asymmetry of the arch and the even numbered terms measure the
taperedness ( 2a ) and squareness ( 4a ) (Burris & Harris, 2000; Lu, 1966).
Higher degree of polynomials were shown to provide more precise
representations of the arch whereby they may capture any off-aligned
teeth on the arch and give a wavy, rather than smooth polynomial
(Triviño et al., 2008).
ii. Conic Section
Conic section is a general function which comprises of a family
of the simplest plane curves which include circle, ellipse, parabola or
hyperbola given as
022 FEyDxCyBxyAx , (2.2)
where FBA ,,, are the constant coefficients.
The eccentricity, e, of a conic section (a measure of how far it
deviates from being circular) determines the types of conics, whereby if
e=0 corresponds to circle, e < 1 being ellipses, those with e=1 being the
parabolas and e > 1 being hyperbolas (Sampson, 1981). This measure
was used to quantify the shape of the conics, specific for each dental arch
(de la Cruz et al., 1995).
The limitation of this function in quantifying arch shape is that it
provides a symmetry curve, whereas naturally, the arch shape may be
asymmetrical (Henrikson et al., 2001).
15
iii. Cubic Spline
The cubic spline is a spline constructed from piecewise of the 3rd
order polynomials which passes through a series of points called knots. This
curve will insert a touch of individuality to each arch shape as the curve is
forced to pass through the knots, therefore generating different curve
configurations and does not necessarily produce a symmetric curve. This
curve appears to be an ideal means for representing dental arch form as it
may more adequately reflect the actual shape of the arch (BeGole, 1980;
BeGole & Lyew, 1998).
iv. Beta Functions
An empirical function derived from beta function based on two
parameters: arch length, D and molar width, W, was proposed to indicate
the shape of the arch, given as:
8.08.0
2
1
2
10314.3
W
x
W
xDy . (2.3)
Since the beta function is symmetric, therefore this function will
inherently produce a symmetrical curve, which may not always be the
case in all arch shapes (AlHarbi et al., 2008; Braun et al., 1998).
16
v. Bezier Cubic
A cubic Bezier curve is defined by four control points of which has
starting point ),( 00 yx , directional points ),( 11 yx and ),( 22 yx and end point
),( 33 yx . The Bezier curve is given as:
)}(),({)( tytxtR , (2.4)
where 001
2
120
3
2103 )()2()33()( xtxxtxxxtxxxxtx and
001
2
120
3
2103 )()2()33()( ytyytyyytyyyyty , are the
cubic equations defined in the interval 10 t . As increasing values for t
are supplied to the equations, the curve starts at ),( 00 yx going
towards ),( 11 yx and arrives at ),( 33 yx from the direction of ),( 22 yx (Figure
2.5) .
If the Bezier curve is fitted to a set of points representing the arch shape,
it will only pass through the first and last points. This is a shortcoming since
the other points will only provide directional information and do not lie on
the curve. Henceforth, the Bezier curve may only be an approximation of the
dental arch shape, but not an accurate shape.
Figure 2.5: Bezier curve defined by four control points of which has starting point
),,( 00 yx directional points ),( 11 yx and ),( 22 yx and end point ),( 33 yx .
Source: http://math.fullerton.edu
y
x
17
vi. Fourier Series
A Fourier series (FS) represents a function as an infinite sum of sines and
cosines signals of various amplitudes and frequencies. The FS of periodic
continuous-time function with frequency 0 and period 0
0
2
T is given by
tuctuba
tf u
u
u 0
1
00 sincos
2)(
, (2.5)
where the Fourier coefficients are:
0
)(2
00
Tdttf
Ta , (2.6)
0
00
cos)(2
Tu dttutf
Tb , (2.7)
0
00
sin)(2
Tu dttutf
Tc , (2.8)
for u =1, 2, …. (Oppenheim et al., 1983).
The FS may be decomposed into separate components or harmonics. If
the limit of (2.5) is changed, it can be simplified to the finite form
N
u
u
N
u
u tuctuba
tf
1
0
1
00 sincos
2)( , (2.9)
where N is the maximum harmonic number. It is clear that the term 0a
corresponds to the average value of the original function along the period
],[ , and ua and ub are coefficients of the sine and cosine functions
respectively (Costa & Cesar, 2009). Therefore, the amplitude for the u-th
harmonic (u=1,…,n) may be obtained as
22uuu cbA , (2.10)
which can be considered as a measure of the influence (or weight) that each
harmonic has on the curve. The first 4 Fourier harmonics were shown to well
reproduce the maxillary dental arch (Kasai et al., 1995; Mikami et al., 2010).
18
Another study used all 14 Fourier harmonics and concluded that FS
precisely express the form and size of dental arches, compared to using the
fourth-grade polynomial function (Valenzuela et al., 2002).
vii. Elliptical Fourier Function
The elliptical Fourier function (EFF) was developed from the
conventional FS which defines x and y points on a curve in 2 dimensions as
separate functions of a third variable t as follows:
N
n
n
N
n
n ntcntbAtx
11
0 sincos)( , (2.11)
and
N
n
n
N
n
n ntentdBty
11
0 sincos)( , (2.12)
respectively, where the estimated coefficients are given in algebraic form,
instead of integral (Kuhl & Giardina, 1982).
The EFF was shown to fit the dental arch (Lestrel et al., 2004; Lestrel,
2008). An increase in the number of harmonic gives better representation of the
dental arch shape. With 24 harmonics, a small mean residual of 0.10 mm was
obtained between observed data and the fitted data using EFF.
Although representation of the dental arch using FS is also favourably
satisfactory, the EFF was claimed easier to imply, in the sense that its
coefficients are generated algebraically, instead of the integral solutions. This
makes computations simpler and faster.
2.3 Discrete Fourier Transform (DFT)
The DFT is another approach which is based on the FS and is increasingly
important in the study of shape. It has been widely used for vision and pattern
19
recognition applications but was never been applied for representing and modelling the
human dental arch.
The following section briefly explains the fundamental theory on DFT and its
application in 2D shape analysis.
2.3.1 Derivation of DFT
Using the Euler formula ,sincosexp ji the FS for periodic continuous-
time signal in (2.5) can be expressed in complex form of
u
u tjuatf 0exp)( , (2.13)
where
0
00
exp)(1
Tu dttjutf
Ta . (2.14)
are the coefficients of the FS for periodic continuous-time function (Kreyszig, 2007;
Oppenheim et al., 1983).
In the case when function f(t) is a discrete-time function, the FS for periodic
continuous-time function in (2.13) and (2.14) can be extended to discrete-time function
(Oppenheim et al., 1983). The FS for periodic discrete-time function s(k) at period
0
2
N is defined as follows
, 2
exp
exp)( 0
Nu
u
Nu
u
N
kjua
kjuaks
(2.15)
where
Nk
uN
kjuks
Na
2exp)(
1. (2.16)
Consequently, a one period (finite-duration) signal can be constructed from the periodic
signal in (2.15) by letting
20
.10 interval theoutside ,0)( N-nks (2.17)
The coefficients of FS at the interval of one period is given by
1,...,1,0for 2
exp)(1
1
0
NuN
kjuks
Na
N
k
u
. (2.18)
The set of complex coefficients ua defined in (2.18) comprise the DFT of )(ks
(Oppenheim et al., 1983).
Using (2.15), the original finite-duration signal can be recovered from its DFT
by
2
exp )(
1
0
N
u
uN
kjuaks
, (2.19)
1,...,1,0for Nk .
2.3.2 Application of DFT in 2D Shape Analysis
The basic underlying idea of the DFT in 2D shape analysis is by letting the 2D
data points of a boundary be transformed as a 1 dimensional data of
)()()( kjykxks , (2.20)
for k = 0, 1,…, N-1 and )]1(),1([)],...,1(),1([)],0(),0([ NyNxyxyx are the anti-clockwise
sequence of N coordinate points on the xy plane.
The complex coefficients ua in (2.19) are also known as the Fourier descriptors
(FD) of a boundary in shape analysis. Let )(')(' ujyux denotes the complex form of .ua
Rewrite (2.18) as
21
.2
sin)(2
cos)()(')('
,2
sin2
cos)()(')('
, 2
exp)(1
1
0
1
0
1
0
1
0
N
k
N
k
N
k
N
k
u
N
kuksj
N
kuksujyux
N
kuj
N
kuksujyux
N
kjuks
Na
(2.21)
From (2.21), the real and imaginary parts of ua are
, )('2
sin)()('
and )('2
cos)()('
1
0
1
0
uyN
kuksuy
uxN
kuksux
N
k
N
k
(2.22)
respectively. The physical interpretation of ua can be explained by writing (2.19) as
a
b
barrbaN
uka
N
ukjuy
N
ukux
N
ukjj
N
uka
N
ukjaks
N
u
au
N
u
N
u
u
N
u
u
u
1
221
0
1
0
1
0
1
0
tan and
where,cossincos ,)2
(cos
,)2
sin()(')2
cos()('
,)2
sin()2
cos(
,)2
exp( )(
(2.23)
where
22 )(')(' uyuxau , (2.24)
and
)('
)('tan 1
ux
uyua encode the amplitude and phase of )(ks respectively (Costa &
Cesar, 2009).
Since the amplitude ua (also known as the FD or DFT of )(ks ) also expresses the
size implicating the synthesized signal, we can evaluate how ua affects the boundary.
The ua may be regarded as quantification of contribution or weight in representing the
boundary of the arch shape (Mikami et al., 2010). Therefore, any u-th term may be
selected and used as an approximation of the boundary as follows
22
terms}{
2exp)()(ˆ
thuUN
ukjuaks
. (2.25)
The larger the total number of FD terms becomes, the closer the shape of the
boundary will be to the original one. One of the advantages of using FD is that it is a
reversible linear transformation which retains all the information in the original
boundary of the object (Keyes & Winstanley, 1999). This important feature is desirable
when precise shape and its boundary point are required.
Computation of DFT was usually done using fast Fourier transform, which uses
the Butterfly algorithm (Cooley & Tukey, 1965) for removing redundancies in the DFT
calculation, therefore allowing faster computation time (Costa & Cesar, 2009).
2.4 Shape Classification from Digital Images
Many applications involving 2D shape analysis require comparing, matching
and classifying the quantified shapes. The first step in achieving these tasks requires
alignment or registration between two or more associated shapes from the images, so
that post variation may be eliminated and the shapes are comparable, therefore yielding
meaningful results. This section explores how dental arch shape is aligned from digital
images and classified.
2.4.1 Shape Alignment: Comparing two or more Arch Shapes
Shape alignment of the dental arch shape may be carried out in two different
ways:
1. Indirect Shape Alignment by Establishing Cartesian Coordinate System
The shape of the arch was indirectly aligned when establishment of
Cartesian coordinate system was carried out (Kasai et al., 1995; Lee et al.,
2011; Taner et al., 2004) (see Section 2.2). Any measurements or
23
coordinates collected with respect to this origin are comparable with each
other.
2. Procrustes Alignment
If the standardizations above is not done, coordinates representing each
arch shape may be aligned using the Procrustes transformation (Banabilh et
al., 2006; Schaefer et al., 2006).
Let the 2D coordinate )]1(),1([)],...,1(),1([)],0(),0([ NyNxyxyx of two
dental casts be denoted in )2( N matrix 1x and 2
x . Alignment of two
shapes 1x and 2
x can be carried out by moving the points 2x relative to points
1x until their residual sum of squares
N
r
rrrr
1
1212 )()'( xxxx . (2.26)
is minimal (Mardia et al., 1979). Movement of 2X relative to points 1
X
through rotation and translation can be carried out by
' 1*2bxAx rr , (2.27)
where Nr ,...,1 , *2rx is the points after transformation, 'A is a Procrustes rotation
of )( pp orthogonal matrix and b is the translation factor. Therefore, the
goodness of fit measure (or superimposition) of 2 shape boundaries 1x and 2
x
can be obtained by solving
n
r
rrrrR
1
12122 )()'(min bAxxbAxxbA,
. (2.28)
The estimates of A and b are found by least squares estimation method given as
'ˆ VUA , (2.29)
and
12ˆ xA'xb , (2.30)
24
respectively, where V and U are the orthogonal )( pp matrices and Γ is a
diagonal matrix of non-negative elements obtained using the singular value
decomposition theorem by writing
'21UVX'X . (2.31)
The shape alignment of 2x relative to points 1
x can be done by
substituting the estimates of A and b in (2.27).
3. Bookstein’s Alignment
Bookstein’s shape alignment is the earliest and the simplest form of
shape alignment. It uses geometry shape with linearized spaces (Bookstein,
1986). It was defined in such a way that the first 2 landmarks for all
configurations were set to (a,0) and (b,0) as baseline, where a and b are
arbitrary constants. An example of the baseline was (0,0) and (1,0) in order
to align the coordinate boundary of the dental arch (Sampson, 1983).
Once the baseline has been defined, the remaining coordinates were
translated, rotated and rescaled according to the baseline. Bookstein admits
that there is a problem when using this approach for shape alignment. This
method works well only if the variations of the landmarks are small.
Figure 2.6: Two copies of the same shape of a hand. The picture on the right was
rotated roughly at 45 anti clockwise from the vertical axis. Source: Stegmann and
Gomez (2002).
25
2.4.2 Arch Shape Classification
Once the shapes of a particular interest were aligned, the arch shape
classification was carried out in two approaches according to how the shape was
quantified:
A. Shape Classification using Multivariate Morphometrics
Arch shape categories may be investigated using the k-means
clustering method from which the ratio of arch width and length were
used to quantify the arch shape (Raberin et al., 1993). Two to 8 shape
categories were considered and the homogenous sizes within each
classification were examined. Five categories of dental arch were
deemed appropriate after employing the analysis of variance and the
mean values of ratios of arch length and width in each category were
calculated. Then, the polynomial function was fitted to these mean
values to indicate 5 shape (and size) models of the dental arch.
Other clustering methods besides k-means were also used to
explore possible categories of the dental arch. The partitioning around
medoids method formed 3 categories of arch shape: narrow, middle and
wide (Lee et al., 2011).
The agglomerative hierarchical clustering method together with
analysis of variance (ANOVA) shows that the 4 categories of arch shape
(and size) were significantly different (Nakatsuka et al., 2011). The
mean shape of these shape categories were then indicated by the average
of linear and angular measurements made from the casts of the
corresponding categories.
26
B. Shape Classification using Boundary Morphometrics
Three categories of shape, determined by factor analysis were established
by McKee and Molnar (1988). Mean slopes (of a series of third degree
polynomial fitted on the arch) for each shape category were graphically
illustrated to indicate the 3 mean shapes. Arch shape variation in each arch
category was summarized heuristically as having steeper slope or flared at a
particular region of the arch.
Besides using factor analysis, dental arches were divided into 3
categories of sizes as defined from fixed constants of polynomial equation
(Preti et al., 1986). They were categorized as small, medium and large. The
disadvantage of this kind of arch category is that, they are purely varied to
the arch size only, with similar polynomial shape.
2.5 Statistical Shape Model of the Dental Arches
A summary of the literature which involves the study of the dental arch shape is
presented in Table 2.1. In general, the shape of the dental arch was described as either a
single-ideal or in categories of arch shape, according to the application of the study.
Most of the works only provided respective mean arch shapes. However, there appears
to be little work done to define the probability distribution of arch shape in order to
demonstrate the variation of the shape from its mean shape.
Only three studies provided the shape distribution of the dental arches. Work
carried out by Sampson (1983) defined shape model of the dental arch with mean and
variation using Bingham distribution from four coefficients of the conic arch. The
density of the four-variate Bingham distribution is given as
ΛγYYγΔ
ΛYYΛΛY
2124
1
14
2
21
1
214
4
- of tionparameriza , exp2
1
exp2
1);(
j jTT
jj
T
K
K
(2.32)
27
where TECBA ,,,Y denotes the coefficients of conic section
0)( 22 EyCyBxyxxA , (2.33)
which passes through (0,0) and (1,0) in the yx, plane and TΓΔΓΛ
21 is the spectral
decomposition of the symmetric matrix Λ where 41 ,, Γ is the matrix of
eigenvectors and )( 41 diagΔ are the corresponding eigenvalues (Sampson,
1983).
The appropriateness of conic arch for the Bingham distribution was assessed by
investigating the asymptotic properties of )4,...,1( jTj Yγ whereby they should be
asymptotically normally distributed as j 4 . Further test of the adequacy of the
Bingham model was carried out after obtaining the approximate maximum likelihood
estimators of the distribution.
Graphical characteristic of the distribution model for the arch shape was
examined using the modal arch, as a reflection of the average shape in the sample. The
50% and 95% confidence region for the population model arch was constructed as an
envelope to show variations in the model.
Wu et al. (2012) on the other hand built the statistical shape model using
principal component (PC) analysis to study the arch shape variations. The vector
TMzMyMxzyx DDDDD ],,,,,,[ 111 X of selected n points on the gingival contour of each
individual tooth in 3 dimensions where M = Ln and L is the number of teeth were used
as shape feature of the dental arch. All X were then represented as a weighted linear
combination of the PCs and the mean shape was given as
K
iii pb
1XX , (2.34)
where ip is the unit eigenvector corresponding to the eigenvalues i of the covariance
matrix, and ib are the weights that define the shape parameters of a deformable model
that are linearly independent and follow a normal distribution of a null mean and
28
variance i . It was found that 96% of the total amount variation present in the training
set can be described by the first 25 PCs and 68% by the first 3 PCs which in turn gives
3 standard deviations from X . The same approach was carried out by Elhabian and
Farag (2014) for jaw reconstruction.
These studies have defined a single ideal arch shape model, with relatively large
variations from the respective mean shape. This type of shape model does not allow
shape to be discriminated. Additionally, clinical experience shows that a single
impression tray would not fit all patients’ arches. The existence of a single-ideal arch
shape model in a particular population is therefore invalid and categories of shape
model should be proposed instead.
Table 2.1 summarizes the literature for shape feature and shape model of the
dental arches whereby in general mean shape were demonstrated. However, information
regarding the variations from the mean shapes established is lacking.
29
Table 2.1: Summary of literature for shape feature and shape model of the dental arch
Authors Applications of
the study
How shape of the
dental arches was
described
The shape
feature used
Mean shape
demonstrated
Variability
model
demonstrated
Shape
distribution
demonstrated
Sampson
(1981)
Orthodontic: evaluate
changes before and
after treatment
Single ‘ideal’ shape
of conic arcs
coefficient of
conic arcs
1 mean shape Variability is
measured as
concentration
parameter Λ of
the Bingham
distribution
The coefficients
of conic arcs are
distributed as the
Bingham
distribution
Preti et al.
(1986)
Prosthetic: prediction
of arch shape and teeth
position for edentulous
patients
3 types of shapes For arch shape:
coefficient of
parabolic
3 means of arch
shape
No No
Raberin et al.
(1993)
Orthodontic::
Classification of arch
forms for easier
clinical practice
5 forms of arch
shape
Ratios of arch
width and
length.
5 mean shapes No No
Kasai et al.
(1995)
Not stated. Investigate
suitability of Fourier
harmonic in describing
dental arch form and
estimate contribution
of genetic factors to
observed variability
Single ideal shape of
Fourier series
Fourier
coefficient
No. each arch was
fitted to FS
Not related Not related
30
BeGole and
Lyew (1998)
Orthodontic:
evaluate changes
before and after
treatment
Single ‘ideal’
shape of conic
spline
Coefficients of
conic spline
No No No
Braun et al.
(1998)
Orthodontic:
treatment planning
Single ‘ideal’
shape of beta
function
Coefficients of
beta function
1 Mean of each
coefficient of beta
function
No No
Ferrario et al.
(1999)
Orthodontic:
treatment planning
2 types of shape Linear distances 2 Mean of each
linear distances
No No
Burris and
Harris (2000)
Orthodontic:
treatment planning
2 types of shape Linear distances
(arch width and
length), arch
parameter,
arch area,
4th order polynomial
curve fitted to 16
dental landmarks
2 Mean of each
linear distances
No No
Henrikson et
al. (2001)
Orthodontic:
evaluate changes
before and after
treatment
Single ‘ideal’ shape
of Sampson (1981)
conic arc
Coefficients of
conics
No No No
Nojima et al.
(2001)
Orthodontic:
Fabricate preformed
arch wire
3 types of shapes Linear distance:
inter canine and
molar, length from
origin to intercanine
and intermolar lines
Ratio: canine and
molar arch width
and length ratio
3 Mean of each
linear distances
No No
31
Yergin et al.
(2001)
Develop an
automated technique
for the selection of
an appropriate tray
for the patient
Not related –
matching segmented
tray and arch for
matching.
2D pixel coordinate Not related Not related Not related
Gu et al.
(2002)
Orthodontic:
Fabricate preformed
arch wire
3 types of shapes
grouped by vector
quantization, VQ
2D pixel coordinates 3 typical shapes
extracted from VQ
No No
Lestrel et al.
(2004)
Orthodontic:
treatment planning
Single ‘ideal’ shape
of elliptical fourier
function, EFF
Coefficients of EFF 1 Mean of each coeff
of EFF
No No
Taner et al.
(2004)
Orthodontic:
evaluate changes
before and after
treatment
Single ‘ideal’ shape
of Bezier arch curve
Coefficients of
Bezier
No No No
Wellens
(2007)
Orthodontic:
evaluate changes
before and after
treatment
Single ‘ideal’ shape
of hyperbolic cosine
curve, HCC
Coefficients of HCC No No No
Triviño et al.
(2008)
Orthodontic:
Fabricate preformed
arch wire
8 types of shapes by
assigning arches to
specified values of
coefficient in
ascending order (all
possible values).
Coefficient of 6
degree polynomial
8 mean shapes No No
32
Mikami et al.
(2010)
Clinical purpose
related to studies of
morphological of the
dental arch
4 categories of
shapes, preliminary
classified visually,
then, Fourier series
were fitted to each of
the category and
essential factors that
affect the arch shape
categories were
investigated using
the amplitude of
different harmonics
Coefficient of FS 4 mean shapes
obtained from 4
coefficient
(maxillary) and 6
coefficients
(mandibular)
No No
Bush et al.
(2011)
Forensic: investigate
uniqueness of human
dentition for
bitemark analysis
Not related – finding
matches between
samples using
Procrustes distance
2D and 3D pixel
coordinate from 14
selected landmarks
on the teeth
Not related Not related Not related
Lee et al.
(2011)
Orthodontic:
Fabricate preformed
arch wire
3 types of shapes Linear distances 3 mean shapes No No
Nakatsuka et
al. (2011)
Orthodontic:
treatment planning
4 types of shapes
obtained from cluster
analysis
Linear and angular
variables
4 mean shapes Variation in each
cluster was described
using PCA.
No
Wu et al.
(2012)
Prosthetic: contour
reconstruction for
the partial edntulous
Single ‘ideal’ shape
of a collective B-
spline curve from
contour of all teeth
3D pixel coordinates
from the B-spline
curve
1 mean shape from
training set
Eigenvector and
weight that define the
shape parameter from
training set
No
Elharbian and
Farag (2014)
Surgery: jaw
reconstruction
Single ‘ideal’ shape
of a collective thin-
plate spline from
contour of all teeth
3D pixel coordinates
from thin-plate
spline
1 mean shape from
training set
Eigenvector and
weight that define the
shape parameter from
training set
No
33
2.6 Discrimination of Shape
Another important goal in the analysis of shape is to discriminate categories or
classes of shapes to enable classification of a new shape. A literature survey shows no
work on dental arch shape discrimination has been done, however work on
discrimination of animal morphology and human anatomy are increasingly
sophisticated, especially in discriminating the normal and abnormal shape of human
anatomy for medical diagnosis (Mukherjee et al., 2013; Uetani et al., 2015; Higashuira
et al. 2012; Pilgram et al., 2006).
Most of the existing studies represented the shape variability of normal and
abnormal anatomy using statistical shape model which was constructed by the principle
component analysis. Then, discrimination of normal and diseased anatomy shape were
carried out using support vector machine, whereby a hyperplane in a hyper dimensional
space was constructed based on the two closest points of two convex hull from training
examples of the respective categories (Bennett and Bredensteiner, 2000). However, the
major drawback of support vector machine are the high training time complexity,
extensive memory requirements and limited to binary classification (Kang et al., 2015;
Yu et al.,2010). Multiclass SVM has been developed and due to the complexity of the
method, its applications have been integrated with image features such as colour or
grayscale histogram and texture features to improve discrimination (Hu et al. 2014;
Zhang and Wu 2012).
Uetani et al. (2015) on the other hand did not use the support vector machine,
but used the linear discriminant function for discriminating shape of normal and
abnormal livers. The weight of the discriminant function was determined from the
learning data by the method of least squares. A relatively good classification on
abnormal livers was obtained using this method, with 84% classification accuracy.
34
2.7 Multivariate Normal Distribution and Its Tests
The multivariate normal distribution is useful in practical for two reasons. First,
it serves as a natural population model in many natural phenomena as many real
problems fall naturally within the framework of normal theory. Secondly, it serves as an
approximate sampling distribution for many multivariate statistics due to a central limit
property, regardless of the form of the parent population (Johnson & Wichern, 1992).
Shape analysis using multivariate morphometrics often involve multivariate
response data. Many of the analyses often necessitate procedures such as discriminant
analysis, multivariate ANOVA and multivariate regression which assume multivariate
normality (MVN). However, establishing a multivariate normal distribution is relatively
difficult, especially when the dimension of the data gets higher. More than 50 tests of
MVN were proposed in the literature for detecting departures from MVN. In general,
four categories of MVN test were identified:
1. Procedure based on graphical plots and correlation coefficient
2. Goodness-of-fit test which is an extension from the univariate method
3. Tests based on measure of skewness and kurtosis
4. Consistent procedures based on the empirical characteristic function.
Although a large number of MVN tests exist in the literature, they are seldom
used by practicing statisticians (Looney, 1995). This is due to the lack of readily
available software to conduct such tests and the reluctance to use the procedures
because of the little information about the quality and the power of the procedures. Most
work done in evaluating the power of MVN tests were carried out when developing new
tests and comparing them to several other tests with limited range. Often, the
simulations only consider the situation for data with dimension of p = 2 (bivariate
normal) and some studies simply considered data with components that were identically
and independently distributed from a common univariate distribution.
35
Mecklin and Mundfrom (2003) examined the power of 8 promising tests of
MVN via an extensive simulation study. In their study, realistic data were simulated
with small sample size and higher dimension up to p = 5 were considered using
different multivariate distribution ranging from multivariate normal to severe departures
from normality. Though the study showed that the Henze-Zirkler is the best procedure
to test MVN, it was concluded that no single method is sufficient for testing the MVN.
It was suggested that multiple approaches of MVN tests should be carried out and it is
best to start with simpler tests such as the univariate Kolmogorov-Smirnov, ellipsoid
test and graphical approach (for example the chi square plot). Following this, it is
recommended to use the multivariate measures of skewness and kurtosis by Mardia
(1970) and the Henze and Zirkler’s empirical characteristic function approach (Mecklin
& Mundfrom, 2003).
The following subsections review some of the basic and advance tests of
multivariate normal distribution which was recommended by many studies.
2.7.1 Mardia’s Multivariate Skewness and Kurtosis Test
The most commonly used test for MVN test is the Mardia’s multivariate
skewness and kurtosis test (Mardia, 1970). The test statistic for Mardia’s test of
skewness and kurtosis are
6
,1 pnb
A , (2.35)
and
,/)2(8
)2(,2
npp
ppbB
p
(2.36)
respectively, where ,)()'(1
3
1 1
1
2,1
n
i
p
j
iipn
b xxSxx
,)()'(1
2
1
1,2
n
i
iipn
b xxSxx p = 4
is the dimension of ix , n = 47 and S is the sample covariance matrix. It is known that
36
A is asymptotically a chi-square probability distributed with 6
)2)(1( ppp degrees
of freedom and B has a standard normal distribution (Mecklin & Mundfrom, 2003). If
the test rejects the null hypothesis, then ix is not following MVN distribution.
MVN tests based on measures of skewness and kurtosis have been demonstrated
to have relatively low power towards rejecting the alternative hypothesis (Horswell &
Looney, 1992; Mecklin & Mundfrom, 2004). Simulation studies by Horswell and
Looney (1992) and Mecklin and Mundfrom (2004) confirm the relatively low power of
these tests. The tests have no power against the multivariate t distribution, which is a
mild deviation from normality and symmetric.
Nevertheless, the power of Mardia’s skewness and kurtosis tests is higher when
compared to Small (1980) and Srivastava (1984) tests, which are also based on the
measure of skewness and kurtosis. The power of performance for Mardia’s skewness is
considerably high, at almost 100% power of rejecting MVN as the sample size increases
(Mecklin & Mundfrom, 2004). This shows that Mardia’s tests are relatively good
indicators of multivariate normal distribution. However other tests should be supplied to
support the finding.
2.7.2 Doornik and Hansen Omnibus test
To improve upon power of test based on skewness and kurtosis, some authors
have attempted to combine measures of skewness and kurtosis into a single `omnibus'
test statistic. Mardia and Foster (1983) derived six omnibus statistics; however it was
found that this statistic lacked power (Horswell & Looney, 1992).
Doornik and Hansen (2008) proposed an extension of the omnibus univariate
test based on skewness and kurtosis by Shenton and Bowman (1977). Simulation study
demonstrated that this test possessed good power properties and was suggested to be
37
used when Mardia’s test of multivariate skewness and kurtosis was considered as a
supportive result.
Let ),,(' 1 nxxX be a )( np matrix of n observations on a p-dimensional vector
with sample mean )( 11
nn xxX and covariance )(1XXS n where
),,( 1 XxXxX n
. The observations were transformed into
XVHHΛR1/2
, (2.37)
where ),,(' 1 nRR R is a )( np matrix of n observations on a p-dimensional vector, V is
a matrix such that pIVV 1 with elements on the diagonal:
2/12/111 ,, ppSSdiag V , (2.38)
VSVC is a correlation matrix, Λ is the matrix with eigenvalues of C on the diagonal, H is
the corresponding eigenvectors such that pIHH and CHHΛ .
The univariate skewness and kurtosis of each transformed n-vector of
observations are given as 2/3
2
31
m
mb and
22
42
m
mb respectively, where
n
i
ii RRnm
1
1 )( and )( 11
nRRnR .
The Doornik-Hansen multivariate omnibus test statistic is
)2( ~ 22211 pZZZZEp
, (2.39)
where ),,( 1111 pzzZ
and ),,( 2212 pzzZ
are the transformation for skewness 1b
and kurtosis 2b to standard normal. Each transformed n-vector of observations for
skewness are:
])1(log[ 2/121 yyz , (2.40)
38
where
2/12
1)2)(6(2
)3)(1)(1(
n
nnby
,
2/12 )log(
1
, 2/12 )1(21 and
)9)(7)(5)(2(
)3)(1)(7027(3 2
nnnn
nnnn . As for kurtosis, its transformation is given as
2/13/1
2 )9(9
11
2
z , (2.41)
where kbb 2)1( 12 , cba 1 , 12
)3131137)(7)(5( 23
nnnnnk
6
)52)(7)(5)(7( 2
nnnnnc ,
6
)7027)(7)(5)(2( 2
nnnnna and
)415)(1)(3( 2 nnnn .
2.7.3 Royston Test
The Shapiro and Wilk (1965) test was shown to be among the most powerful
tests for detecting departures from univariate normality (Srivastava & Hui, 1987).
Royston (1983) proposed an extension of the Shapiro-Wilk (SW) goodness of fit test to
MVN distribution. A revision of Royston (1992) MVN test for approximation of
coefficients requires the calculation of the SW test to correct the specification of the null
distribution. The revised version of the Royston’s MVN test was shown to have good
Type I error control and power against rejecting non MVN distributions (Farrell et al.,
2007).
Suppose ),,( 1 pxxX is the p-variate random vector to be examined for
multivariate normality consisting of j-th random sample ),,( 1 njjj xx x of n cases. Let
nyy 1 be an ordered sample jx of univariate data. The SW test is defined as
2
2
)(
)(
yy
yaW
i
ii
, (2.42)
39
where ),,( 1 naa a is such that ii yan 2/1)1( is the best linear unbiased estimate of the
standard deviation of the iy , assuming normality (Royston, 1983). A revised
approximation of a is given as
ii ma ~~ 2/1 , (2.43)
for , where )/()(~ 14
38
1 nimi , is the normal cumulative
distribution function (c.d.f), )~2~21/()~2~2~~( 21
221
2 nnnn aammmm , naa ~~
1 , 12~~
naa ,
5432 706056.2434685.4071190.2147981.02221157.0~ xxxxxca nn
543211 582663.3682633.5752461.1293762.0042981.0~ xxxxxca nn
nx , and ii mc ~)~~(12 mm (Royston, 1992).
The W statistic could be transformed to an approximately standard normal
variate, z. A simpler normalize transformation for the W statistic is given as
wz , (2.44)
where 32 0006714.00205054.039978.05440.0 nnn , )1ln(ln Ww ,
n459.0273.2 , and )0020322.0062767.077857.03822.1exp( 32 nnn , for
114 n . As for 200012 n , 32 0038915.0083751.031082.05861.1 xxx ,
)1ln( Ww , )0030302.0082676.04803.0exp( 2xx and nx ln (Royston, 1992).
Suppose pzz ,,1 of the transformed W statistic have been obtained as above,
with ),...,1( pjz j correspond to the j-th component from ),,( 1 pxxX . The Royston’s
MVN test based on the Shapiro-Wilk is defined as
eGH , (2.45)
)5for ( 2,,3 nni
40
where pkG
p
j
j
1
,
2
1 )(2
1
jj zk , is the normal c.d.f, cp
pe
)1(1 ,
)( 2 ppcc ij is an estimate for the average correlation among the jk ,
),( jiij kkcorrc is the correlation matrix of k. Approximation of ijc , is given as
)1(1ˆv
cij , (2.46)
where is the correlation coefficient, 715.0,5 ,
32 0018034.0015124.021364.0 xxv and nx ln . The Royston’s MVN test H was
shown to approximately distributed as chi-square distribution with e degrees of
freedom (Royston, 1983).
2.7.4 Henze-Zirkler Test
The Henze-Zirkler (HZ) test is recommended as a formal test of MVN (Mecklin
& Mundfrom, 2003). The HZ test is based on a nonnegative functional distance
that measures the distance between two distribution functions
dtttQtPQPd )(|)(ˆ)(ˆ|),( 2
, (2.47)
where the characteristic function of the multivariate normal distribution )(ˆ tP and the
empirical characteristic function )(ˆ tQ are the Fourier transformations of P and Q
respectively, is the weight or kernel function of ),0( 2
pp IN and β is the smoothing
parameter (Henze & Zirkler, 1990).
The closed form for the Henze-Zirkler statistic is
,)21(||||)1(2
exp1
)1(2
||||2
exp1
1
2/2
22
22/2
1,
2
2,
n
i
p
i
p
n
ji
jin
Yn
YYn
T
(2.48)
41
where )4/(1)4/(1
4
12
2
1
p
p
np
is the smoothing parameter, )()'(|||| 12ddSdd
jiji YY
and )()'(|||| 12ddSdd
iiiY . The limiting distribution of ,nT is a lognormal
distribution with the following mean and variance:
22
4
2
2
2/2
)21(2
)2(
211)21(1)]([
ppppTE p , (2.49)
,)(2
)2(
)(2
31)(4
)21(4
)2(3
)21(2
21)21(2)41(2)]([
2
84
2/
42
8
22
4
22/2
w
pp
w
pw
ppppTVar
p
pp
(2.50)
where )31)(1()( 32 w (Henze & Zirkler, 1990). Rejection of the null hypothesis
indicates non-normality distribution.
2.8 Multivariate Complex Normal Distribution (MVCN)
The complex random variables often appear as actual data in areas such as in
geophysics, circular analysis, communications, signal processing and shape analysis.
The random variables come in pairs of the form x + jy, where x is the real part and y is
the imaginary part. The introduction of a complex-number-based theory can often lower
the dimension involved and subsequently simplify mathematical models.
Modelling complex random variables has been indirectly applied in the field of
physics, which examine the arrangement of atoms in solids. However Wooding (1956)
and Goodman (1963) are among the first who explicitly introduced the multivariate
distributions of complex random variables which is also called the “multivariate
complex normal distribution”. Although work to develop the theory of the multivariate
complex normal distribution has been carried out, little applications of the distribution
can be found in the literature (Pannu et al., 2003).
42
The following section introduces the complex random variables and complex
normal distribution together with their properties. Then, a review on the parameter
estimation and test of separation between two vectors from multivariate complex
normal distribution is presented.
2.8.1 The Univariate and Multivariate Complex Random Variables and
Distributions
Let U and V be real random variables. The random variable iVUX where
X is taking values in C (field of complex numbers), may be regarded as a univariate
complex normal random variable by the following definition:
Definition 1. A univariate complex random variable is a complex random variable
iVUX such that the distribution of T
V
UX
has a bivariate normal distribution
(Anderson et al., 1995, p. 5; Giri, 2003, p. 86).
Using the above definition, the univariate complex random variable of complex
random variable X may be investigated by proving that T
V
UX
has bivariate normal
distribution.
The p-variate complex random vector on the other hand is defined as follows:
Definition 2. Let kXX be a p-dimensional vector, where kX for pk ,...,1 is a
complex random variable such that the real 2p-vector ),( kk VUY has a 2p-variate
normal distribution (Andersen et al., 1995; Giri, 2003, p. 86).
Now, we consider the definition of the univariate complex normal distribution
which is given as follows:
Definition 3. The univariate complex random variable X is distributed as univariate
complex normal distribution ),( 2CN with its probability density function given as
43
,)var()var(
)()(exp
))var()(var(
1
)var(
*))((exp
)var(
1
*))((exp
1)(
22
22
VU
vu
VU
X
xx
X
xxxf X
(2.51)
where j is the mean of x and *)( x is the conjugate and transpose of
)( x (Andersen et al., 1995, p. 20; Giri, 2003, p. 86).
The definition of the multivariate complex normal (MVCN) distribution on the
other hand is as follows:
Definition 4. A complex random p-vector kXX with values in Cp has a multivariate
complex normal distribution if, for b Cp , each Xb* has a univariate complex normal
distribution (Giri, 2003, p. 91).
The probability density function of the MVCN distribution is given as
(Andersen, et al., 1995, p. 26; Giri, 2003, p. 86):
.)()(exp)det()( 1*1θxHθxHxX pf , (2.52)
where θ is the mean vector, H is a Hermitian matrix which correspond to a real matrix
being symmetry and *)( θx is the conjugate and transpose of ).( θx We write
),(~ HθX pMVCN (Andersen et al., 1995).
2.8.2 Properties of the MVCN Distribution
There is a close relationship between real and complex normal distribution
established by the isomorphism, : pp 2RC . For ),( ~ HθX pMVCN , it holds that
HθX
2
1, 2~ pN
kVkU
, (2.53)
44
where ,,,1 pk (Andersen et al., 1995, p. 25). The distribution of the pp complex
random matrix XXW* follows a complex Wishart distribution with parameters H and
n , denoted by ),(~ nCWp HW and it holds that
),,2
1(~ 2 nW p HW (2.54)
for the real Wishart distribution (Andersen et al., 1995, pg. 40).
Further, most of the properties of the multivariate complex normal distribution are
relatively similar to the real multivariate normal distribution. Let pCc , it holds that
),*,*(~ HccθccX pMVCN (2.55)
and
).,(~ 2121 nnCWp HWW (2.56)
(Andersen et al., 1995, p. 23 & 43).
2.8.3 Parameter Estimation of the MVCN distribution
The parameters θ and H can be estimated using the maximum likelihood
method analogous to real multivariate normal distribution (Goodman, 1963). Let
)( kXX be a p-dimensional random vector, where kX for pk ,,2,1 is a complex
random variable kkk jVUX . Therefore, for n independent realisation X is called an
)( pn complex random matrix given as
npnn
p
p
XXX
XXX
XXX
21
22221
11211
X . (2.57)
45
The likelihood function of ),(~ HθX pMVCN in (2.52) can be written as
. )()(exp)det(
)()(exp)det(
)()(exp)det(
.)()(exp)det(
)()(
1
1*
1*1
1
1*
1
1
1
1*1
1
n
i
ii
nnp
nn
p
p
n
i
ii
p
n
i
ifL
θxHθxH
θxHθxH
θxHθxH
θxHθxH
Hθ,;xHθ,X;
(2.58)
The term
n
j
jj
1
1* )()( θxHθx can be simplified using identity in Mardia et al. (1979)
(p. 97, 456) therefore yielding.
*1
1
*1
1
1* ))(( tr ))((tr )()( θxθxHxxxxHθxHθx
n
n
i
ii
n
i
ii . (2.59)
Let
n
i
iin
1
*))(( θxθxM and substitute (2.59) in (2.58) gives
. ))(( tr tr exp)det(
))(( tr )(tr exp)det()(
*11
*11
θxθxHMHH
θxθxHMHHHθ,X;
nn
nnL
nnp
nnp
(2.60)
Therefore, the log likelihood function of )(xXf can be written as
.))(( tr tr )det(loglog
))(( tr tr exp)det(log )(
*11
*11
θxθxHMHH
θxθxHMHHHθ,X;
nnnnp
nnl nnp
(2.61)
The maximum likelihood estimate (m.l.e) of the mean θ and variance matrix H
for the MVCN distribution is that value of the parameter which maximizes the
likelihood of the given observation. This value may be obtained by differentiation and
since the log likelihood )( Hθ,X;l is at a maximum when )( Hθ,X;L at a maximum, the
equation 0
θ
l and 0
H
l is solved for θ and H respectively.
Consider in (2.61). Using matrix differentiation theorems,
)(1θxH
θ
nl . (2.62)
)( Hθ,X;l
46
Solving 0
θ
l , 0)ˆ(1 θxHn . Therefore, the m.l.e of θ is xθ ˆ .
Define new parameter 1 HV . Equation (2.61) becomes
*))(( tr tr loglog )( θxθxVVMVHθ,X; nnnnpl . (2.63)
To calculate V
l , the right terms of (2.63) can be written as (Mardia et al, 1979, p. 104
and 481)
,)(22
)))(((2))((2
)))((())((222
))(())((2 )(2 )(2 )(
**
**
**
UU
θxθxMHθxθxMH
θxθxMHθxθxMH
θxθxθxθxMMHHHθ,X;
Diagn
Diagn
Diagn
DiagnDiagnDiagnl
(2.64)
where *))(( θxθxMHU . Solving 0
V
l,
.)ˆ)(ˆ(ˆ
0)ˆ)(ˆ(ˆ
0
*θxθxMH
θxθxMH
U
(2.65)
Since xθ ˆ , therefore the m.l.e of H is
MH ˆ . (2.66)
Analogous to the real multivariate normal distribution, the unbiased estimator for θ and
H are xθ ˆ and
n
i
iin
1
*))((1
1ˆ xxxxMH respectively.
2.9 Missing Values Analysis
Missing values is a common problem in data analysis. Deleting a particular
sample when one of the variables is unavailable seems to be the most common way to
handle missing data. A few deletion procedures were proposed such as the list wise and
pair wise deletion. However, this procedure is unfavorable particularly when the sample
size is small and a single sample deletion is crucial as it may affect the biased estimates,
statistical model and consequently the power of statistics (Barzi & Woodward, 2004).
47
Simple imputation approach seems to be the alternative whereby the missing
values of a particular variable will be imputed using the mean or mode of the respective
observable variable. Another sophisticated imputation approach is by generating
estimates of the missing values using a predicted probability distribution or regression
model from the knowledge of the observable data.
The following sub-section reviews some of the model-based imputation methods
for the analysis of missing values.
2.9.1 Data Augmentation (DA) Algorithm
2.9.1.1 Definition
The data augmentation algorithm was first proposed by Taner and Wong (1987).
The theory behind this method was derived from Gibbs sampling; a well known form of
Markov chain Monte Carlo which creates pseudorandom draws from a particular
probability distribution.
Suppose z is a random vector with two sub-vectors; ),( vuz , and its joint
distribution )(zP is not easily simulated or intractable., however, their conditional
distributions )|()|( vugvuP and )|()|( uvhuvP are. Let
,,,,,,,
,,,
)()()(
2
)(
2
)(
1
)(
1
)()(
2
)(
1
)(
00
0
t
n
t
n
tttt
t
n
ttt
vuvuvu
zzzZ
(2.67)
be a sample of size 0n simulated from a distribution that approximates the target
distribution )(zP at iteration t . The data augmentation algorithm updates this sample in
two steps.
Step 1: Generate the sub-vector
)1()1(2
)1(1
)1(
0,,, t
nttt uuuU , (2.68)
independently for 0,,2,1 ni from )|(~ )()1( ti
ti vugu .
Step 2: Generate the other sub-vector
48
)1()1(2
)1(1
)1(
0,,, t
nttt vvvV , (2.69)
independent and identically (i.i.d) sampled from the average of the conditionals
:)|( )1( tiuvh
0
1
)1(
0
)1( )|(1
)|(
n
i
ti
t uvhn
Uvh . (2.70)
Consequently completes the new sample
)1()1()1(2
)1(2
)1(1
)1(1
)1(
00,,,,,, t
nt
nttttt vuvuvuZ . (2.71)
Using functional analysis, the distribution of )(tZ will converge to )(zP as t
(Tanner & Wong, 1987).
The advantage of using the data augmentation is that the convergence does not
require a large value of 0n . With 10 n , data augmentation reduces to a special case of
the Gibbs sampler
,,,,|~
,,,,|~
,,,,|~
)1(
1
)1(
3
)1(
2
)()1(
)()(
3
)1(
2
)(
2
)1(
2
)()(
3
)(
2
)(
1
)1(
1
000
0
0
t
n
ttt
n
t
n
t
n
tttt
t
n
tttt
ZZZZPZ
ZZZZPZ
ZZZZPZ
(2.72)
with the random vector ),( vuz partitioned into two sub-vectors u and v .
2.9.1.2 Application of DA in the Missing Values Problem.
In many incomplete-data problems, the observed-data posterior distribution
)|( obsYP are sometimes difficult to determine and therefore the data from )|( obsYP
cannot be easily simulated. However, if obsY is augmented by an assumed value m isY , the
resulting complete-data posterior ),|( misobs YYP may be used to find its true posterior
when the data augmentation algorithm is adopted as follows:
49
Step 1 (Imputation step): Given a current guess of a parameter )(t , draw
independent 0n values of m isY
)1()1(2
)1(1
)1(
0,,, t
misnt
mist
mist
miss yyyY , (2.73)
generated from the conditional predictive distribution of m isY
),|(~ )()1( tobsmis
tmis YYPY . (2.74)
Step 2 (Posterior step): Draw new values of
)1()1(2
)1(1
)1(
0,,, t
nttt , (2.75)
sampled from the conditional distribution of obsY and )1( tmisY
),|(~ )1()1( tmisobs
t YYP . (2.76)
Repeating steps (1) – (2) from the starting value )0( for a value of t that
is relatively large yields a stochastic sequence
,2,1:, )()( tY tmiss
t , (2.77)
which possesses )|,( obsmis YYP as its stationary distribution (Schafer, 2010).
Consequently, the sub-sequences ,2,1:)( tt and ,2,1:)( tY tmiss have
)|( obsYP and )|( obsmiss YYP as their respective stationary distributions. A
stopping criterion can be introduced to determine the sufficient number of t .
The process of repeating steps (1) – (2) will continue until )(t and
)1( t
converge to a certain pre-specified tolerance. An extension for the multivariate
observation can be done using the same idea as above.
2.9.2 Expectation Maximization (EM) Algorithm
2.9.2.1 Definition
The EM algorithm is an iterative method for approximating the maximum
likelihood estimates (m.l.e) of parameters. It uses the idea of interdependence between
50
missing data m isY and parameters, say, . Since m isY holds relevant information for
estimating and may help in finding likely values of m isY , therefore is it also
possible to estimate in the presence of observable data obsY .
The scheme to estimate using this idea is as follows:
1. Replace the m isY (denoted as )(t
misY ) using arbitrary initial estimate of
(denoted as )(t ), where ,1t .
2. will be re-estimated (denoted as )1( t ) based on obsY and
)(tY
3. Step 1 is repeated iteratively until )(t converge to a specified
tolerance
Dempster et al. (1977) generalized the above idea for broader class of problems such as
in applications for cluster and truncated data, finite mixture models, factor analysis,
variance component estimation and missing value problems. The generalized EM
involves 2 steps as follows:
1. the Expectation or E-step, which require the estimation of )(t for the
complete data, when obsY is given.
2. the Maximization or M-step, where )1( t is estimated by maximum
likelihood similar to as though the estimated complete data were the obsY .
The EM algorithm was shown to be a reliable method whereby it is consistently
converging to a stationary point of the observed-data log likelihood (Dempster et al.,
1977; Wu, 1983). The application of EM is commonly used in computer vision and
machine learning for data clustering and analyzing data with missing values (Barzi &
Woodward, 2004; Bishop & Nasrabadi, 2006).
51
2.9.2.2 EM for Regular Exponential Families in Missing Values Problem
If the complete-data probability model has the regular exponential family, the
complete-data log likelihood on n i.i.d (possibly multivariate) observations
nyyyY ,,, 21 may be written as
cngYTYl T )()()()|( , (2.78)
where
Ts )(),(),()( 21 , (2.79)
is the canonical form of the parameter ,
Tsss YTYTYTYT )(),(),()( , (2.80)
is a (1 x s) vector of complete-data sufficient statistics and c is a constant. The
sufficient statistics when (2.68) holds, exist in an additive form of
n
i
ijj yhYT
1
)()( , (2.81)
for some function jh and j=1,2,…s.
The E-step estimates the )(YT jby finding
)(,|)(
tobsj YYTE . (2.82)
In many of the models in the form of (2.68), the expectations )(,|)(
tobsj YYTE can be
obtained in closed form and E-step will be computationally straightforward.
An example of i.i.d observations nyyyY ,,, 21 from a univariate normal
distribution with mean and variance , so that , is the unknown parameter,
is presented to illustrate the E-step. Let the first 1n components of the data vector Y are
observed, and the remaining 10 nnn are missing at random. The sufficient statistics
for , are given by:
52
Tn
i
i
n
i
iT yyTTYT
1
2
1
21 ,,),()( . (2.83)
The E-step estimates the 1T and 2T by finding
0
111
1
1
1
1
,| nyyyEYTE
n
i
i
n
ni
i
n
i
iobs
, (2.84)
and
)( ,| 20
1
2
1
2
1
22
1
1
1
nyyyEYTE
n
i
i
n
ni
i
n
i
iobs , (2.85)
respectively (Schafer, 2010).
The M-step determines )1( t by replacing the estimates from E-step into the
complete-data m.l.e as the solution of
tYTE |,)( , (2.86)
where t is the realized value of the vector )(YT .
It is known that the m.l.e for parameter from univariate normal distribution are
n
iiyny
1
1 and
n
i i
n
ii
n
ii ynynyyn
1
2
1
21
1
221 )( . The M-step
of i.i.d observations from univariate normal distributions may be obtained by inserting
the expected sufficient statistics in (2.74) and (2.75) into the expressions for the
complete-data m.l.e and yields
)(0
1
1)1( 1 tn
ii
tnyn , (2.87)
2
)(
10
222)(0
)(
10
21)1( 11
tn
ii
ttn
ii
tnynnnyn , (2.88)
Imputation of missing values using EM usually uses )1( t . Since the calculation
of )1( t does not depend on
)1( t , therefore it may be possibly extended to the case
of multivariate normal.
53
It can be shown that the observed-data likelihood )|( obsYL for normal
distribution is just a complete-data likelihood based on 1
,,, 21 nobs yyyY and the
observed-data m.l.e for and are
1
1
11
n
iiobs yny and
1
1
2211 )(
n
iobsi yyn
respectively. Although the observed-data m.l.e exist in closed form, one can also
compute them using the EM algorithm. Nevertheless, EM is very useful when dealing
with complicated functions whereby obtaining their maximum likelihood estimates of
parameters in statistical models appear to be difficult.
54
CHAPTER 3: EXPLORATORY DATA ANALYSIS OF SHAPE FEATURE AND
INVESTIGATION OF CATEGORIES OF THE DENTAL ARCH SHAPE
3.1 Introduction
Categories of dental arch shape may exist due to ethnicity, hereditary and
environmental factors. This may necessitate different considerations in dental treatment
planning. To categorize dental arches, a novel shape descriptor indicating the variation
of tooth position was derived from digital images of maxillary dental casts. Initial tests
on control samples of 47 dental casts showed that, firstly, angular measures can be
linearly approximated, and secondly, the maxillary arch is symmetrical. Then, clustering
methods, principal component analysis and cluster validity index were carried out to
show possible groups of dental arch shapes.
3.2 Data Collection
One hundred and twenty-two Malaysian dentate subjects (45 males and 77
females) of 74 Malay, 36 Chinese and 12 Indian ethnic groups aged between 19 and 48
years were randomly selected from University of Malaya students and dental personnel
attached to the Faculty of Dentistry University of Malaya and patients receiving dental
treatment in the out patients clinic at the University Malaya Medical Centre.
Impressions of the patients’ dental arches were made to obtain their dental casts.
Obtaining a reasonably large sample size is time consuming as subject inclusion
criteria were imposed. Forty seven dental casts were collected from March 1st until
November 30th 2010 with the assistance of a dentist from the Department of Restorative
Dentistry, Faculty of Dentistry, University of Malaya. These casts were used as control
samples to investigate symmetry of the dental arches and possible clusters or categories
55
of arch shape as well as to establish the existence of multivariate normal shape model.
Another set of test sample consisting of 75 dental casts was collected from October 1st
2011 until March 30th 2012 for verification studies needed in several steps of this study.
The study was approved by the Faculty of Dentistry, University of Malaya Medical
Ethics Committee (MEC no: DF PD1106/0089(L), dated 20 December 2011).
3.2.1 Selection of Samples
Certain inclusion criteria were set to select control samples for the study. The subjects
must have:
i) Well aligned maxillary and mandibular teeth with minimal attrition
ii) Angle's Class I dental relationship
iii) No history of orthodontic treatment, anterior restoration or fixed dental
prosthesis in the maxilla or mandible.
iv) No facial asymmetry.
The inclusion criteria (i) was not strictly imposed for the remaining 75 subjects.
3.2.2 Dental Impression and Stone Cast Making
An impression of the dental and oral structures in the mouth of each patient was
made using irreversible hydrocolloid (Duplast fast set alginate impression material;
Dentsply Dental Co Ltd, Tianjin, China) (Figure 3.1). Impressions were then cast using
type III dental stone (Moldano; Heraeus Kulzer GmbH, Hanau, Germany) (Figure 3.2).
Then, each cast was numbered and labelled to identify the subjects to whom the casts
belonged to.
56
Figure 3.1: Dental impression made in an impression tray.
Source: http://www.firstbtob.com.
Figure 3.2: Dental plaster was poured onto the impression to obtain the stone dental
cast. Source: www.howtomakevampireteeth.com/.
3.2.3 Cast Preparation
To avoid image variation that may resulted from differences in camera to cast
distance, standardization of each cast was made. The occlusal plane of each cast was
made parallel to the base of the cast by ensuring that the incisal edges, canine tips and
mesiopalatal cusps of the first molars touched the surface of the table in the most stable
position. Then, a geometric compass was positioned on a flat surface of a table with one
of its arm resting on and parallel to the flat surface. Adhesive tape was used to stabilize
the compass on the flat surface.
A line was then inscribed on the base of the cast using a pencil attached to the
compass so that it is parallel to the occlusal plane. The base of the cast was then
57
trimmed following this line to obtain an occlusal plane which is parallel to the
horizontal (Figure 3.3). This was done to standardize the height of each cast so that
image variation that may be resulted from differences in camera to cast distance can be
avoided.
Figure 3.3: Method used to make the base of the stone cast parallel to the occlusal
plane
3.2.4 Image Acquisition, Shape Alignment and Calibration of Measurements
Correspondences of the dental arch shapes were ensured by identifying
landmarks on the digital images. Anatomical landmarks are defined as prominent points
that match between organisms in some biologically-meaningful way (Dryden & Mardia,
1998). The incisive papilla and the hamular notches were identified as the common
anatomical landmarks as they have been shown to be relatively stable, even after the
loss of teeth (Grave & Becker, 1987; Nikola et al., 2005). They were marked on each
cast using a 0.5 mm pencil lead with a cross sign (x) (Figure 3.4).
Digital images of the casts were then captured by a high resolution digital
camera (Nikon D70s; Nikon Corp, Tokyo, Japan). The camera to object distance was
fixed at approximately 50 centimeters to ensure distortion-free images. Two metal
rulers fixed perpendicular to each other and positioned on a plane parallel to the
occlusal plane were used as frames for calibration of the measurements (Figure 3.4).
These images were then saved as JPEG files.
58
Figure 3.4: Two metal rulers positioned on a plane parallel to the occlusal plane
enabled measurements to be calibrated. Shape alignment is achieved by the creation of
the Cartesian plane defined from anatomical landmarks which were marked with
crosses.
Digital images of these casts were then imported to a program developed in
MATLAB software (version R2009b, The MathWorks Inc., USA). In order to align the
arch shapes, the line joining the two hamular notches was used geometrically as the
Cartesian x-axis, while the line perpendicular to the x-axis passing through the incisive
papilla was defined as the y-axis. The point where both axes meet was defined as the
origin of the coordinates.
Shape alignment of the dental arch shape is achieved by the creation of these
Cartesian coordinate axes on the digital images (Figure 3.4). In this way, the dental
arches were comparable to one another.
Hamular notches
Incisive papilla
59
Figure 3.5: Selected points on the teeth used to represent the dental arch shape.
3.3 Shape feature of the Dental Arch
The midpoint of the anterior teeth and distobuccal cusp of the first molars were
selected (Figure 3.5). The j-th point may be represented by the length of a line joining
the cusp tips of a tooth to the origin (jl ) and an angle (
jw ) with respect to the
horizontal axis. The arch shape for a given dental cast was described using the shape
feature vector
),,,,,,,,,,,,,,,( 4433221144332211RRRRRRRRLLLLLLLL lwlwlwlwlwlwlwlwq , (3.1)
where for example ),( 11LL lw and ),( 11
RR lw represent the point on the left central incisor and
right central incisor respectively, in reference to the Cartesian origin established from
the digital image of the dental cast.
Let (𝑂𝑥, 𝑂𝑦) be the coordinate of the origin and ),( 11LL yx be the coordinate of the
first landmark on the left side of the arch. The angle Lw1 and distance Ll1 of this
landmark from the origin (in pixels coordinate) was obtained by
x
y
Ox
Oy
1
11tan and
21
21 )()( yx OyOx respectively (Figure 3.6). Then, a standard image processing
technique was applied whereby a calibration method converts all measurements in terms
of pixel number into millimeters (mm). The same procedure was used to find the
60
distances and angles of the other teeth. To simplify the above tasks for measuring q , a
program was developed in MATLAB software.
Measurements were repeated three times by the same investigator and a pilot
study using 47 control samples for test-retest reliability using Pearson’s correlation
shows that the measurements are reliable as the measurements taken in different times
show high correlation, r (Table 3.1). Further, the hypothesis testing on the population
correlation coefficient 1:0 H demonstrates consistency of the data (Murphy &
Davidshofer, 1991). The data were assumed to follow normal distribution since the
variation of teeth position and human arch shape may be regarded as natural
phenomena. In the later sections, this matter will be established.
Table 3.1: The sample correlation coefficient r, and the test statistic for 1;0 H to test
the reliability of measurements taken at three different times. The lower and upper
critical values are -2.0141 and 2.0141 respectively when α =0.05 was used.
Times of
measurements
Variable
First and second
times
First and third
times
Second and third
times
r Test
statistic r
Test
statistic r
Test
statistic Rw1 0.9715 -0.8065 0.9644 -0.9037 0.9884 -0.5115
Rl1 0.9687 -0.8454 0.9423 -1.1558 0.9582 -0.9801 Lw1 0.9709 -0.8145 0.9653 -0.8918 0.9883 -0.5135
Ll1 0.9636 -0.9132 0.9328 -1.2512 0.9551 -1.0167 Rw2 0.9689 -0.8435 0.9617 -0.9369 0.9842 -0.5986
Rl2 0.9641 -0.9073 0.9323 -1.2552 0.9498 -1.0759
Lw2 0.9634 -0.9158 0.9583 -0.9790 0.9861 -0.5621
Ll2 0.9677 -0.8595 0.9353 -1.2268 0.9537 -1.0328
Rw3 0.9737 -0.7745 0.9589 -0.9722 0.9776 -0.7147
Rl3 0.9664 -0.8775 0.9307 -1.2711 0.9470 -1.1070
Lw3 0.9730 -0.7841 0.9602 -0.9560 0.9834 -0.6140
Ll3 0.9666 -0.8745 0.9305 -1.2730 0.9479 -1.0974
Rw4 0.9909 -0.4524 0.9526 -1.0455 0.9575 -0.9883
Rl4 0.9671 -0.8675 0.9446 -1.1318 0.9667 -0.8734
Lw4 0.9899 -0.4788 0.9632 -0.9178 0.9703 -0.8240
Ll4 0.9669 -0.8697 0.9577 -0.9865 0.9749 -0.7556
61
The average measurement was used as the final measure. The notations for the
points together with their summary statistics for jw and jl where j = 1, ..4 are tabulated
in Table 3.2 and Table 3.3, respectively.
Table 3.2: Mean, standard deviation (SD), minimum and maximum values of the
angular measurements (in degrees).
Variable Mean SD Min Max
Right central incisor ( Rw1) 85.51 2.14 80.68
89.39
Left central incisor ( Lw1) 85.66 2.07 81.28
89.79
Right lateral incisor ( Rw2) 77.26 2.32 72.33
82.47
Left lateral incisor ( Lw2) 77.31 2.16 73.05
82.27
Right canine ( Rw3) 69.62 2.48 64.90
75.27
Left canine ( Lw3) 69.62 2.47 63.86
75.60
Right distobuccal cusp of first
molar ( Rw4)
38.55
4.43
30.10 47.99
Left distobuccal cusp of first
molar ( Lw4)
38.63
4.54
30.20 47.21
Table 3.3: Mean, standard deviation (SD), minimum and maximum values of the length
measurements (in mm).
Variable Mean SD Min Max
Right central incisor ( Rl1 ) 56.20 3.33 49.20
62.66
Left central incisor ( Ll1 ) 56.27 3.22 50.02
63.03
Right lateral incisor ( Rl2) 54.03 3.03 48.70
60.42
Left lateral incisor ( Ll2) 54.06 3.14 48.54
60.30
Right canine ( Rl3) 50.56 2.88 44.94
56.82
Left canine ( Ll3) 50.79
2.87
45.09
56.41
Right distobuccal cusp of
first molar ( Rl4)
35.41
2.31
30.51
40.71
Left distobuccal cusp of first
molar ( Ll4)
35.78
2.84
31.66
43.04
62
Figure 3.6: Computation of angle and distance of the central incisor tooth from the
geometrical Cartesian origin.
3.4 Properties of the Shape Feature
Shape feature ),,,,,,,,,,,,,,,( 4433221144332211RRRRRRRRLLLLLLLL lwlwlwlwlwlwlwlwq was
proposed to represent the shape of human dental arches and teeth location. Obviously,
there would be a problem in analyzing these measures as the nature topology of jw and
jl are different. However, if the angles recorded in the range ]360 ,0(
degree were
analyzed using the linear metric, then the directions o f 360 are close to the
opposite end-points. This made them near neighbours in a circular metric but
maximally distant in linear metric (Figure 3.7). Further, if we apply the angles
using the conventional linear techniques, it can lead to paradoxes; for example, the
arithmetic mean (ordinary mean on the straight line) of the angles 1 and
359 is
180 whereas by geometrical (mean direction of circular data), the mean has to be
0 (Figure 3.8).
63
Figure 3.7: Differences of linear and circular measurement.
Figure 3.8: The arithmetic mean points the wrong way.
A distribution that is often used to describe the physical properties of circular
data is the von Mises distribution. As a continuous probability distribution, the von
Mises is analogous to the normal distribution for linear data and has some similar
characteristics with the normal distribution. Thus, the von Mises is also known as
the circular normal distribution.
Let be the circular random variable with circular normal distribution,
denoted by ),( 0 CN . The probability density function of is
)}cos(exp{)(2
1),;( 0
0
0
I
g , (3.2)
where )20( 00 is the mean direction, )0( is known as the
concentration parameter and )(0 I denotes the modified Bessel function of order
zero.
Linear:
Circular:
Circular:
64
Note that in the context of circular statistics, 0 does not give an average
value of n
n ...1 which is commonly used for linear data. However, the
interpretation of mean direction is similar to the arithmetic mean, and is sometimes
called as the “ preferred direction”.
The concentration parameter (also known as the variance for the angular
variable) measures the departure of observations in a circle. For large value of ,
the observations became very concentrated and are close around the mean direction,
0 . If approaches 0, the distribution tends to converge to a uniform
distribution (Fisher, 1993, p. 49).
3.4.1 Approximation of Circular Normal to Linear Normal Distribution
The restriction that jw
varies between 0 degree and 90 degrees is the first
suggestion it may be regarded as a linear variable. The work by Jammalamadaka and
Sengupta (2001) has shown that a circular random variable θ from a circular normal
distribution with mean direction 0 and concentration parameter , can be
approximated to a linear normal distribution with mean direction 0 and variance
1
when , and can be written as
)1
,(),(~ 00
NwCNw ,
(3.3)
Parameter 0 is estimated using the maximum likelihood using
,0,0 2)/(tan
0 )/(tan
0,0 )/(tan
ˆ
1
1
1
0
CSCS
CCS
CSCS
(3.4)
and the estimate of concentration parameter is approximated by
65
,85.0 34
1
85.00.53 1
43.039.14.0
53.0 6
52
ˆ
23
5
3
RRRR
RR
R
RR
RR
(3.5)
where
n
i iC1cos ,
n
i iS1sin and 22 )/()/( nSnCR is the mean
resultant length (Fisher, 1993, p. 88; Jammalamadaka & Sengupta, 2001, pp. 85-86).
A QQ plot for circular normal distribution which assesses the goodness-of-fit of
sample to the circular normal distribution is one of the graphical tools used to
verify the circular normal distribution. The plotting is done as follows:
1. The quantiles nqq ,,1 of the corresponding best fitted circular normal
distribution, )ˆ,ˆ( CN as estimated in equation (3.4) and (3.5), are
calculated.
2. Then, the sample quantiles
)ˆ(2
1sin iiz , (3.6)
are obtained and rearranged in ascending order.
3. Plot the pair wise of circular normal theoretical and sample quantiles as:
(sin(1
2
1q ),
1z ), . . . , (sin(nq
2
1),
nz ). (3.7)
Tests such as Watson U2 and Kuiper tests also provide another alternative to test
the goodness of fit of the data to the circular normal distribution. The Watsons U2 test
(Fisher, 1993; Mardia & Jupp, 2000) performs a goodness-of-fit test against a
specified distribution, either uniform or circular normal. The cumulative frequency
values niFz ii ,,1),ˆ(ˆ were rearranged into increasing order as
)()1( nzz . The Watson 2U test statistic calculates the mean square deviation
for the fitted distribution given as
66
n
znnizUn
i
i12
1
2
1)2/()12(
2
1
2
)(
2
, (3.8)
where
n
i i nzz1 )( / . If the deviation is too high (resulting in a high U and a low
probability) then the null hypothesis that the data fit the chosen distribution is
rejected. The critical values of this test can be found in Fisher (1993, p. 230).
Kuiper’s Test (Fisher, 1993; Mardia & Jupp, 2000) takes the alternative
approach of directly comparing the distribution of the data to the desired
distribution, either uniform or circular normal. The Kuiper’s statistic is based on the
largest vertical deviations above and below the diagonal line (representing the
desired distribution) given as
)/24.0155.0( 2/12/1 nnVV n , (3.9)
where DDVn ,
2,,
2 ,/)1(max , /max
)()1(
n
iii znizDzniD
,, and )()1( n are the re-arranged i in increasing order )()1( n . The critical
value for the statistic V was given in Fisher (1993). Greater deviation gives high value
of V and low probability, leads to rejection of the null hypothesis that the data fit
the distribution.
3.4.2 Results on Approximation of Circular Variables of the Shape Feature
In this study, the Oriana 2 software (Kovach Computing Services, 1994-2003)
was used to obtain the QQ circular normal plot, calculating the parameters of circular
normal distribution and performing the hypotheses testing for the circular data.
Evidence of the circular normal distribution is illustrated with the QQ circular
normal plot. Figure 3.9 suggests that RRRR wwww 4321 ,,, are following circular normal
distribution with sample quantiles for the entire plots lie on the straight line
obtained from quantiles of theoretical von Mises distribution. The corresponding
67
four landmarks on the left side of the arch gave similar results.
Further confirmation of circular normal distribution was shown from the Watson
U 2 and Kuiper tests. Results from Table 3.4 also showed that all the circular variables
follow the circular normal distribution with a large value of κ. The raw data plot
illustrated in Figure 3.10 clearly shown that RRRR wwww 4321 ,,, were concentrated
around the circular mean (indicated as the bold line) and indicates that the
concentration parameter κ may be large enough to be approximated to linear
variables.
Comparison of mean and variance for LRLRLRLR wwwwwwww 44332211 and ,,,,,, using
circular and linear statistic in Table 3.5 gives similar results indicating that they may be
regarded as linear variables.
(a) Rw1
(b) Rw2
(c) Rw3
(d) Rw4
Figure 3.9: QQ circular normal plot for RRRR wwww 4321 and ,,
68
(a) Rw1
(b) Rw2
(c) Rw3
(d) Rw4
Figure 3.10: Raw data plot for RRRR wwww 4321 and ,,
Table 3.4: The test statistic for the Watson U2 and Kuiper’s tests and estimated
concentration parameter κ. All the variables show non-significant results and indicate
normality.
Test Watson U2 Kuiper
Critical value
Variable
Lower 2.5% = -0.117
Upper 2.5% = 0.117
Lower 2.5% = -1.747
Upper 2.5% = 1.747 Rw1 0.065 1.141 729.812
Lw1 0.031 0.810 779.770
Rw2 0.056 1.184 621.983
Lw2 0.026 0.791 721.246
Rw3 0.067 1.324 546.268
Lw3 0.053 1.034 550.745
Rw4 0.035 0.815 171.063
Lw4 0.100 1.283 163.182
69
Table 3.5: Mean and variance of angular variables using circular and linear statistic.
Statistic Variable Linear statistic Circular statistic
Mean Rw1 85.5090 85.5090
Lw1 85.6593 85.6590
Rw2 77.2581 77.2580
Lw2 77.3090 77.3090
Rw3 69.6219 69.6220
Lw3 69.6192 69.6190
Rw4 38.5546 38.5540
Lw4 38.6323 38.6320
Variance Rw1 0.0803 0.0785
Lw1 0.0751 0.0735
Rw2 0.0942 0.0921
Lw2 0.0812 0.0794
Rw3 0.1073 0.1049
Lw3 0.1064 0.1040
Rw4 0.3431 0.3349
Lw4 0.3597 0.3511
3.5 Dental Arch Symmetry and Dimension Reduction
In the previous section, the angular components in shape feature q have been
shown to have similar properties with linear data. Further analysis of q may now be
incorporated using the analysis of linear statistic. The shape feature q using eight
landmarks is clearly a statistical multivariate problem with dimension 16p .
Dimension reduction will be carried out by investigating the symmetry of the dental
arch. The symmetry of the arch would in turn imply that the four selected teeth is a
reasonable way of representing arch shape.
By definition, symmetry of the dental arch means that identical teeth are facing
each other around the midline, and is studied by comparing the corresponding shape
feature on the left and the right sides of the arch. A summary of the work examining
dental arch symmetry in Appendix C shows ambiguous results. The dental arch is said
to be symmetrical when the shape feature representing the teeth position on one side of
the arch was a mirror image of the other-side (Ferrario et al., 1993; Shrestha &
Bhattarai, 2009; Sprowls et al., 2008; Tong et al., 2012). On the other hand, a certain
70
degree of morphological asymmetry was shown to be present, with smaller asymmetry
degree in Class I compared to Class II and Class III arches (Cassidy et al., 1998;
Scanavini et al., 2012). Further, depending on the measurements used to represent the
dental arches, some of the tests investigated arch symmetry based on size only. It is
therefore of interest to investigate the symmetry of the dental arch based on arch size
and shape simultaneously, using the shape feature q .
3.5.1 Test of Arch Symmetry
Let ),,,,,,,( 44332211
LLLLLLLLL lwlwlwlwv represent the four landmarks on the left
side of a given arch. Similarly let, ),,,,,,,( 44332211
RRRRRRRRR lwlwlwlwv be the
corresponding landmarks on the right side of the same arch. For purposes of notation,
let L
iv represent L
v for the i-th dental cast. Similarly, let R
iv be the corresponding
vector for the right side of the cast. Without loss of generality, if ),(11LL lw and ),(
11RR lw
represent the left central incisor and right central incisor respectively, symmetry of the
arch is implied if RL ll 11
and RL ww 11
. This implication is strengthening if similar
equalities are true for the other three teeth. In total, symmetry of the arch is regarded as
confirmed if 0xAT where
71
1100000000000000
0011000000000000
0000110000000000
0000001100000000
0000000011000000
0000000000110000
0000000000001100
0000000000000011
TA and
R
L
R
L
R
L
R
L
R
L
R
L
R
L
R
L
l
l
w
w
l
l
w
w
l
l
w
w
l
l
w
w
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
x .
If x can be shown to be multivariate normal such that ),(~ Σμx pN , then a test for
symmetry is equivalent to a test on the hypothesis 0μA T.
Since x is a 16 dimensional vector with sample size of 47, evidence of
multivariate normality was shown using the Kolmogorov-Smirnov test (Dudewicz &
Mishra, 1988) to investigate univariate normality for each element of x and the Chi-
square plot.
Let 47,,2,1, ii
T
i xAy where T
A is a )( qp matrix and denote the
normality of the x vectors as 47,,2,1),,(~ iNqi Σμx . Therefore, ) ,(~ yy Σμy pi N
where μAμTy and AΣAΣ y
T (Mardia et al., 1979).
Define nn
yyyy 21
1 and Tr
n
r
r yyyyS 1
y as the unbiased
estimators of yy and Σμ and )
1,(~ yy Σ0μyn
N p . Let
),(~ yy Σ0μyu pNn , (3.10)
and
yy ,1~ ΣSW nWp, (3.11)
72
follow the Wishart distribution.
The test statistic for the one sample Hotelling T2 test statistic under
0:0 μATH is given as
,)1(
)()())(1(
)()()1(
)1(
1
y
1
y
y
1
y
12
ySy
μAySμAy
μySμy
uWu
T
TTT
T
y
T
nn
nn
nnn
nT
(3.12)
such that )(,2
)(
)1(~ pnpF
pn
pnT
where p =8 and n = 47.
3.5.2 Results for Test of Symmetry
Kolmogorov-Smirnov tests (Dudewicz & Mishra, 1988) showed that each
element of TRLRLRLRLRLRLRLRL llwwllwwllwwllww ),,,,,,,,,,,,,,,( 4444333322221111x is univariate normal
(Table 3.6). Further evidence of multivariate normality of x is indicated in the Chi-
square plot (Figure 3.11).
Figure 3.11: Chi square plot for testing 16-variate normal distribution.
0 5 10 15 20 25 30 350
5
10
15
20
25
30
35
Chi-square quartiles (theoretical)
Squ
are
d d
ista
nce
(O
bse
rvati
on)
73
Table 3.6: Results of Kolmogorov-Smirnov (KS) goodness-of-fit test for testing
normality of residuals. Critical value of KS statistic used is 0.1743.
Variables KS test statistic
Lw1 0.0973
Rw1 0.067
Ll1 0.0691 Rl1 0.0722 Lw2 0.0826
Rw2 0.0639
Ll2 0.0650
Rl2 0.0869
Lw3 0.0928
Rw3 0.0849
Ll3 0.1028
Rl3 0.0635
Lw4 0.0729
Rw4 0.0922
Ll4 0.0790
Rl4 0.1373
Table 3.7 illustrates the result of the Hotelling T2 test. The test appears to accept
0H at 5% significant level which gives evidence of maxillary arch symmetry.
Therefore, using landmarks on one side of the arch only is a reasonable way of
representing arch shape.
Table 3.7: Test statistic for the Hotelling one-sample T2 test
Critical value T2statistics
Lower 2.5% = 2.4547
Upper 2.5% = 23.9492 8.3441
74
3.6 Categories of Dental Arch Shape
Studies on arch size and shape of people from different ethnic groups in
different populations have shown varying results. Significant variations in a specific
population such as in Italian Caucasian and Chilean, American blacks and whites and
among the Southern Chinese were found, while no differences could be found between
the Caucasian and Japanese, and Caucasian and non- Caucasians (Burris & Harris,
2000; Ferrario et al., 1999; Ling & Wong, 2009; Nojima et al., 2001; Radmer &
Johnson, 2009).
When investigating arch size and shape for disparities, some studies have at the
outset segregated the arches according to ethnic groups and then compared the average
values measured in each group using univariate two sample t-test or other appropriate
discriminant analysis tests (Ferrario et al., 1999; Ling & Wong, 2009; Nojima et al.,
2001). It was concluded that the arches were discriminated according to ethnic groups
when the statistical tests on individual variables show significant differences. Other
studies used the multivariate approach, whereby the arches were grouped using cluster
analysis and subsequently the ethnic and gender homogeneity for each cluster
established was investigated (Hao et al., 2000; Raberin et al., 1993). Though many
studies have made an attempt to find groups of dental arches and relate them to ethnicity
and arch shape, the uniqueness of the groups was not thoroughly verified (Hao et al.,
2000; Lee et al., 2011; Preti et al., 1986).
Malaysia is a multi-racial country with citizens comprising of three major ethnic
groups; the Malays, Chinese and Indians. In 2010, the Malays formed 50.1% of the
population, the Chinese 22.5% and the Indians 6.7%. The rest of the population in the
country is made up of other indigenous groups and non Malaysian citizens (Malaysia,
2012). It is advantageous to study differences in arch size and shape of the ethnic
75
population, although intermarriages do occur in Malaysia (4.6% of the population)
(Nagaraj, 2009).
This section explores the existence of groups of dental arches applied on
),,,,,,,( 44332211RRRRRRRRR
i lwlwlwlwv , i = 1,..,47 which in turn validates that the v -vectors
is a reliable shape feature for the dental arch.
3.6.1 Clustering Method
An indicator of arch shape variation is the existence of groups or clusters of the
v -vectors. Agglomerative hierarchical clustering (AHC) is the most commonly used
and efficient method in describing pattern similarities and differences in the data to
reveal characteristics given in homogenous groups or clusters. A measure of distance or
separation between the i-th dental cast and the j-th dental cast is as follows (Anderberg,
1973):
244
244
211
211 )l(l+)w(w++)l(l+)w(w=j)d(i, jijijiji . (3.13)
The whole set of distances may be expressed in the matrix:
.
0121
03,23,1
02,1
0i0,
)nd(n,)d(n,)d(n,
)d()d(
)d(
=j),D(
(3.14)
The first step in the AHC method was to merge two dental casts with the
smallest ),( jid value between them into the same group. The distance between this
new group and the remaining original dental casts was then redefined using one of the
76
following methods (Anderberg, 1973; Everitt et al., 2001; McKenzie & Goldman,
1998):
M1) Average linkage method; ),( jid = average distance between all pairs of
dental casts in cluster-i and cluster-j.
M2) Centroid linkage method; ),( jid = distance between centroid of cluster-
i and cluster-j.
M3) Complete linkage method;
),( jid = maximum distance between dental
casts in cluster-i and cluster-j.
M4) McQuitty’s linkage method; ),( jid = average distance from cluster-i to
other cluster (e.g., cluster-l) and cluster-j to cluster-l.
M5) Median linkage method; ),( jid = median distance between all pairs of
dental casts in cluster-i and cluster-j.
M6) Single linkage method; ),( jid = minimum distance between dental casts
in cluster-i and cluster-j.
M7) Ward’s linkage method; ),( jid = minimum error sum of squares of
cluster-i and cluster-j.
A summary of clusters formed from the control samples using methods M1 to
M7 with selected similarity values are shown in Table 3.8. The problem of misfit of
impression trays with patients’ mouths suggests that there should be at least 2 clusters
of dental arches. Further, previous studies discussing morphology of dental arches
reported about 3 to 5 different shapes based on geometrical representations (de la Cruz
et al., 1995; Raberin et al., 1993). It is seen that only with the complete linkage method
was there a realistic number of groups or clusters of dental arch shapes (2 to 6) for a
similarity level of up to 50%.
Using the linkage method, the set of distances after the merger would be found
in the matrix ),,1( jiD . The next step in the AHC method was to combine two dental
77
casts (or two groups of dental casts) with the smallest ),( jid obtained from the matrix
),,1( jiD . This process of combining two groups and merging their characteristics was
repeated until all dental casts were placed in one large group. The result of this
hierarchical cluster analysis is shown graphically in a dendrogram (Figure 3.12), where
all the samples are listed and the level of similarity showing how any two clusters were
joined are indicated. In Figure 3.12, the horizontal axis indicates the position of the
dental casts (relative to each other) whilst the height of the vertical axis is a measure of
the disparity among the casts. The similarity level at the m-th merger of clusters is
defined as (McKenzie & Goldman, 1998):
100% x 0,in max
,in min1
j)i,D(j)d(i,
j)i,D(mj)d(i,=S(m) . (3.15)
A large )(mS value suggests dental cast-i and dental cast-j (or two groups of dental
casts) have similar arch shape. The number of clusters may be chosen by selecting the
height of the vertical axis which represents the cut-off point.
The means of ),,,,,,,( 44332211
RRRRRRRRR
i lwlwlwlwv were denoted as
),,,,,,,()( 44332211
RRRRRRRR lwlwlwlwk v where kGk ,,1 and
kG is the number of group
for a given similarity value, may also be graphically represented by a box containing the
means (Figure 3.13).
78
Table 3.8: Number of clusters formed using the available linkage methods (M1 to M7)
at selected percentage of similarity levels.
Linkage
method
Similarity
Level (dij)
M1 M2 M3 M4 M5 M6 M7
1% 1 1 2 1 1 1 5
5% 1 1 2 1 1 1 5
10% 1 1 2 1 1 1 5
15% 1 1 2 1 1 1 5
20% 1 1 2 1 1 1 5
25% 1 1 3 1 1 1 5
30% 1 1 3 1 1 1 6
35% 1 1 3 1 1 1 8
40% 1 1 4 2 1 1 8
45% 2 1 5 2 1 1 8
50% 2 1 5 3 1 1 10
55% 2 1 6 4 1 1 10
60% 5 1 8 5 1 1 10
65% 7 1 9 7 2 1 11
79
Number of
clusters
Similarity
level
Dendrogram
2
1% - 20%
3
21% - 38%
4
39% - 44%
5
45% - 50%
Figure 3.12: Dendrograms showing number of clusters at different cut-off levels
obtained using complete linkage method.
4523371446313428423629413541644303826212027223233134011172518152431924743121039659871
0.00
33.33
66.67
100.00
Observations
Similarity
DendrogramComplete Linkage, Euclidean Distance
4523371446313428423629413541644303826212027223233134011172518152431924743121039659871
0.00
33.33
66.67
100.00
Observations
Similarity
DendrogramComplete Linkage, Euclidean Distance
4523371446313428423629413541644303826212027223233134011172518152431924743121039659871
0.00
33.33
66.67
100.00
Observations
Similarity
DendrogramComplete Linkage, Euclidean Distance
4523371446313428423629413541644303826212027223233134011172518152431924743121039659871
0.00
33.33
66.67
100.00
Observations
Similarity
DendrogramComplete Linkage, Euclidean Distance
80
2 clusters
(1 - 20%)
3 clusters
(21 - 38%)
4 clusters
(39 - 44%)
5 clusters
(45 - 50%)
Figure 3.13: The pyramid of cluster boxes where each cluster is represented by the
vector of means
83.16
57.30
74.85
54.91
67.27
51.49
36.82
36.43
87.30
59.51
79.43
57.13
72.21
53.30
43.76
36.97
85.55
53.55
77.07
51.61
69.15
48.35
36.11
33.91
84.75
54.80
76.33
52.71
68.53
49.39
36.35
34.75
87.30
59.51
79.43
57.13
72.21
53.30
43.76
36.97
83.16
57.30
74.85
54.91
67.27
51.49
36.82
36.43
84.66
51.76
75.84
49.74
67.55
46.32
32.69
32.70
87.30
59.51
79.43
57.13
72.21
53.30
43.76
36.97
86.05
54.57
77.78
52.68
70.07
49.50
38.06
34.60
83.16
57.30
74.85
54.91
67.27
51.49
36.82
36.43
84.66
51.76
75.84
49.74
67.55
46.32
32.69
32.70
84.59
61.83
76.92
59.75
70.21
55.87
42.50
38.80
86.05
54.57
77.78
52.68
70.07
49.50
38.06
34.60
88.38
58.58
80.44
56.08
73.00
52.27
44.26
36.24
81
3.6.2 Definitive Number of Clusters
The definitive number of clusters is determined when each group resulting from
a chosen cut-off point is well separated, rather than overlapping with another group
(Everitt et al., 2001). To assess the uniqueness of the formed clusters, a 2D graphical
method using the first two principal components of v vectors and the validity indices
were carried out.
3.6.2.1 Principal Component Analysis
Let Tk kkkk )()( )()( vvvvS be the sample covariance of clusters
),,1( gkGk resulting from a chosen cut-off point where g is the number of clusters and
)(kv are v vectors from kG . Let kn
S1
1
be an estimate of the cluster population, say
kΣ . The spectral decomposition of the estimate of kΣ is given by T
Q ΛQ where
)|||( 821 qqqQ and ),,,(diag 821 Λ are the matrices of eigenvectors and
eigenvalues respectively (Anderson, 2003).
Henceforth, each v vector in )(kv is represented by its first and second principal
components, namely i
Tvq1
and i
Tvq2
. The two principal components of all )(kv was then
plotted in a 2D scatter plot. The same procedure was repeated for different cut-off
points, where 3 and 4 clusters resulted.
The plots are shown in Figure 3.14(a) – 3.14(c) whereby the 2 principal
components explained about 75% of the variation of )(kv . Figure 3.14(a) and Figure
3.14(b) shows 2 and 3 distinct clusters respectively. In Figure 3.14(c), overlapping of
members is seen in two of the 4 clusters formed. They were not clearly separated and
may be considered as one cluster. Therefore, the sample studied strongly suggests the
existence of 3 distinct groups of dental arches.
82
(a)
(b)
(c)
Figure 3.14: Investigation of separation of clusters using the first two principal
components indicates the existence of three distinct groups of dental arches.
-120 -100 -80 -60 -40 -20 0 20 40-150
-100
-50
0
50
100
150
1st PC
2nd
PC
G1
G2
-150 -100 -50 0 50-60
-40
-20
0
20
40
60
80
100
120
1st PC
2nd
PC
G1
G2
G3
-150 -100 -50 0 50-200
-150
-100
-50
0
50
100
150
1st PC
2nd
PC
G1
G2
G3
G4
83
3.6.2.2 Dunn’s Validity Index
Apart from the principal components, the Dunn’s validity index was also used to
establish the definitive number of clusters formed. The Dunn’s validity index validates
the separation and compactness of the groups which resulted from a chosen cut-off
point and is defined as
,))((max
))(),((minmin
1
11
k
mkD
gk
gmgk v
vv (3.16)
where )(kv are v vectors from kG and where g is the number of clusters. Further,
),(max))((,
jiG
dkkji
vvvvv
and ),(max))(),((,
jiGG
dmkmjki
vvvvvv
are the intra-cluster
diameter and inter-cluster linkage respectively and ),( jid vv is the Euclidean distance
between iv and jv (Bolshakova & Azuaje, 2003). The optimal number of clusters is
suggested by the maximum value of D.
Table 3.9 gives the Dunn’s validity index for all possible number of clusters
which suggests 3 groups as an optimum number of clusters. The 3 clusters, denoted as
kG (k= 1, 2, 3), have group sizes 11, 22 and 14 respectively. Table 3.10 shows the
percentage of each cluster membership according to gender and ethnicity. There is a
mixture of gender and ethnicity which further suggests that this information may be
derived by well defined groups of arch shape using the shape feature v . The v -vector
has successfully captured the shape of the dental arch and may be regarded as a reliable
shape feature for the dental arch.
Table 3.9: Validation of number of clusters using Dunn’s index.
Number of clusters, g Dunn’s Index, D
2 Clusters 1.2561
3 Clusters 1.2977 (max)
4 Clusters 1.1064
5 Clusters 1.0365
6 Clusters 1.1163
84
Table 3.10: Ethnic and gender homogeneity for each cluster.
Cluster
Total
membership
(percentage)
Ethnic
Percentage
of
ethnicity
Gender Percentage
of gender
1G 11 (23.40%)
Malay (n = 6) 54.54% Male (n = 2) 18.18%
Chinese (n = 5) 45.45% Female (n = 9) 81.81%
2G 22 (46.81%)
Malay(n = 18) 81.81% Male (n = 5) 22.72%
Chinese (n = 4) 18.18% Female (n = 17) 77.27%
3G 14 (29.79%)
Malay (n = 10) 71.42% Male (n = 7) 50%
Chinese (n = 4) 28.57% Female (n = 7) 50%
3.6.3 Results on Categories of Dental Arch Shape
Three categories of arch shape (1G ,
2G and 3G ) have been established from
meticulous handling of the clustering methods. Their respective means are
)1(v (83.16, 57.30, 74.85,54.91,67.27,51.49,36.82,36.43) ,
)91.33,11.36,35.48,15.69,61.51,07.77,55.53,55.85()2( v ,
)97.36,76.43,30.53,21.72,13.57,43.79,51.59,30.87()3( v .
Interpolation of the shape feature using means of the 4 teeth used in this study with the
mirror image of the other half of the arch gives the general shape of the dental arch. The
shapes of the arch for kG (k= 1, 2, 3), illustrated in a 2-dimensional is shown in Figure
3.15. The shape of 2G and 1G are almost similar, however 2G is smaller than 1G and 3G
is medium sized. 3G also differs from 1G and 2G as it has more taper in the frontal arch.
Figure 3.16 further shows separation of the three categories of shape when each shape
feature was considered separately. For the purpose of comparison of arch dimension
with other studies, the mean anterior and posterior widths and lengths for the arches in
each of the shape category are shown in Table 3.11.
85
Figure 3.15: Mean shapes of kG (k= 1, 2, 3).
Figure 3.16: Boxplot of each shape feature for kG (k= 1, 2, 3).
-40 -30 -20 -10 0 10 20 30 400
10
20
30
40
50
60
70
width (mm)
leng
th (
mm
)
G1
G2
G3
theta1 length1 theta2 length2 theta3 length3 theta4 length4
30
40
50
60
70
80
90
Valu
es
G1
G2
G3
86
Table 3.11: The anterior and posterior width and length for each of the shape category.
Shape
category
Length (mm) Width (mm)
Anterior Posterior Total Anterior Posterior
1G 9.5 25.5 57.0 40.0 58.0
2G 8.5 27.5 59.0 48.0 63.0
3G 9.0 27.0 65.0 36.5 60.5
3.7 Discussion
A novel shape feature ),,,,,,,( 44332211RRRRRRRRR
i lwlwlwlwv of the dental arch was
derived from digital images. Precautions were taken when digitizing the dental casts,
whereby the base of the cast were standardized, distortion of the images were avoided
or minimized, calibration of the measurements were made each time the shape feature
was measured and the shape features were aligned to enable comparison between arch
shape. The shape features were accurately measured, as shown by the small standard
deviation values of the measurements (Table 3.2 and Table 3.3).
Nakatsuka et al. (2001) also used a combination of linear and angular variables
as shape feature which were obtained from different landmarks. However, unlike this
study, the properties of shape feature were not carefully investigated, particularly the
treatment of the angular variables. This study on the other hand addressed this issue and
found that the angular variables may be regarded as a linear measured vector therefore
enabling investigation of arch symmetry using the usual linear statistic (Table 3.5).
The symmetry of the v -vectors was also investigated. Most studies which
investigated arch symmetry have commonly used univariate tests such as two sample t
test and paired t test for normally distributed data, and Wilcoxon sum rank and signed
rank tests for non-normal data (Cassidy et al., 1998; Maurice & Kula, 1998; Shrestha &
Bhattarai, 2009; Šlaj et al., 2003; Tong et al., 2012). However, it should be noted that
the use of two sample t test may not be appropriate since variables depicting shape
features on the left and right sides of the arch are clearly dependent. Further, the use of
87
several univariate tests on each variable individually makes the overall error rate greater
than error rate per test (also known as the Bonferroni inequality). In this study, all the
variables indicating the shape feature were tested simultaneously using the paired
Hotelling T2 together with thorough check on its assumptions to ensure that the decision
made about the null hypothesis was reliable (Table 3.6 and Figure 3.11). Investigation
of arch symmetry in turn allowed reduction of dimension of the shape feature.
Clustering methods using the proposed shape feature yields 3 categories of
dental arch shape (Figure 3.15). Although the mean of the shape feature for each shape
category were illustrated, this study also reported on the arch width and length to enable
comparison with other studies (Table 3.11). The first shape category can be described as
being widest on the posterior and anterior widths with medium arch length. The second
shape category has medium posterior and anterior widths and the shortest arch length.
On the other hand, the third shape category has narrowest posterior and anterior widths
with the longest arch length.
Differences in clustering analyses may contribute to the different results
obtained. Lee et al. (2011) and Raberin et al. (1993) used the partitional clustering
method that requires the user to specify the number of clusters, which was set from a
subjective viewpoint. Nakatsuka et al. (2011) on the other hand employed similar
hierarchical clustering methods used in this study. However the choice of Ward linkage
as distance measure for merging the clusters was not properly justified.
The strength of this study in investigating categories of the dental arch shape
was on the proper statistical analysis in each step of the clustering analysis technique
used. Further verification of the groups established also ensured that the groups are
deemed meaningful (Figure 3.14 and Table 3.9). This chapter considers the categories
of arch shape without the knowledge on how the variation of shape in each category
88
deviates from its mean shape. Three shape models will be established in the following
chapter to describe the variation in each of the shape category.
89
CHAPTER 4. SHAPE MODELS OF THE DENTAL ARCH: AS A GUIDE IN
DETERMINING APPROPRIATE IMPRESSION TRAYS FOR ORAL
DIAGNOSIS AND TREATMENT PLANNING
4.1 Introduction
A clustering method applied on ),,,,,,,( 44332211
RRRRRRRRR
i lwlwlwlwv , i = 1,..,47 in
Section 3.7 has established 3 categories of the v -vectors denoted as )3,2,1)(( kkv .
Three shape models can be defined if the mean and covariance of )(kv can be derived
(Hufnagel, 2011). The probability distribution of v-vectors, if exists, may in turn allow
more accurate inferences to be made.
The shape models were derived in the following procedure. Firstly, the
performance of several multivariate normality tests for small sample size and high
dimensional data were investigated before each shape category is tested for multivariate
normality. The existence of 3 meaningful shape models were tested to investigate if the
three MVN models possess significant differences in shape. To ensure that the shape
models are valid, verification study was carried out using 75 test samples. The
knowledge about these shape models were then used in the design of 3 impression trays.
A modified COVRATIO statistics which incorporates the problem of small
sample size and minimal model assumptions was proposed as a discrimination method
of shape and compared to the linear discrimination method. Using this knowledge, a
guide for determining appropriate impression tray for a patient (without and with a
missing tooth) was proposed.
90
4.2 Shape Models of Maxillary Dental Arch
Since there was evidence of normality distribution for the shape feature (section
3.5.2), three shape models can be obtained if each of the shape categories )3,2,1( kGk
can be shown to have a multivariate normal distribution and possess significant
differences in shape. However, the sample size of each category kG )3,2,1( k was
small and possesses relatively high dimension. Some multivariate normal tests may
have failed to reject normality. The following subsection investigated the performance
of several multivariate normality tests in the case of small sample size and high
dimension, followed by the test of normality for the three shape categories.
4.2.1 A Simulation Study on Performance of Multivariate Normality Tests for
Small Sample Size
In real life situations, particularly in shape analysis, smaller sample sizes relative
to numbers of landmarks selected are likely to occur. Therefore, the assumption of
multivariate normality and the validity of the MVN tests are most critical. The sample
size for category 1G ,
2G and 3G were 11, 22 and 14 respectively. Before performing
multivariate normality tests on )3,2,1)(( kkv , the effect of sample size as small as
10n and higher dimension up to 8p which mimic the sample size and dimension
of the v-vectors from kG was investigated. Five multivariate normal (MVN) tests
which possess high power from the literature were considered (see section 2.6.1 to
2.6.4):
1. Mardia’s skewness (MS)
2. Mardia’s kurtosis (MK)
3. Doornik and Hansen (DH)
4. Henze-Zirkler (HZ)
91
5. Royston
The null hypothesis of the above tests was, 0H : Data set follow a MVN
distribution, ),( ΣμpN , and the performance of the tests was evaluated according to:
1. Type I error rates against the MVN distribution
2. Power of test against non-MVN distribution
4.2.1.1 Type I Error Rates
To estimate the Type I error rates for MS, MK, DH, HZ and Royston tests, the
following steps were taken:
Step 1. A random sample from ),(~ Σμx pN with n = 10 and p = 2,4,8 was generated
using R random number generator function. The number of parameters to be
fixed is )2( pp . To reduce the number of parameters to be fixed, let μ be a p
zeros vector and A be the eigenvector of Σ . Therefore,
),(~ D0Ax pN , (4.1)
where
pd
d
00
0
0
001
AAΣD is the canonical form of Σ and ),,( 1 pdd d
is the corresponding eigenvalues (Mardia et al., 1979, p. 62 & p. 214). Now, the
number of parameters is reduced to p . The parameter ),,( 1 pdd d is fixed such
that the covariance
pppp
p
p
21
22221
11211
Σ demonstrates various possible data
dispersion as follows:
92
Set A: pIΣ
Set B: The diagonal and off-diagonal elements of Σ are 8 and 0
respectively.
Set C: The diagonal and off-diagonal elements of Σ are 8 and 3
respectively.
Set D: The diagonal and off-diagonal elements of Σ are 8 and 5
respectively.
Step 2. Obtain the p-value from the test statistic of MS, MK, DH, HZ and Royston
tests using the generated data as described in step 1 (Korkmaz et al., 2015; Aho,
2015).
Step 3. The above process of getting the p-value was repeated for s = 10000 times.
Step 4. The proportion of 10000 samples for which the test rejects the null hypothesis
0H at 05.0 significance level was calculated. This is equivalent to
calculating the probability of making Type I error:
trueis |reject 00 HHP . (4.2)
The corresponding error rate was then obtained.
Step 5. Similar steps as above were repeated to calculate the Type I error rates for
sample size n = 20 and 50.
Examples of generated data using the above set of parameters for p = 2 are
illustrated in Figure 4.1 and the above steps were also repeated for p = 4 and 8.
93
Example of generated MVN data using
parameter set A. The data are expected to
distributed around 0 with covariance
10
01Σ .
Example of generated MVN data using
parameter set B. The data are expected to
distributed around 0 with covariance
80
08Σ .
Example of generated MVN data using
parameter set C. The data are expected to
distributed around 0 with covariance
83
38.
Example of generated MVN data using
parameter set D. The data are expected to
distributed around 0 with covariance
.85
58
Figure 4.1: Examples of different sets of generated MVN data for p = 2.
x y
z
94
Table 4.1: Empirical Type I error rate (in percentage) against the MVN distribution
with dimension p = 2 for different sets of parameters.
n MVN test Set A Set B Set C Set D
10 Mardia’s skewness (MS) 0.35 0.35 0.51 0.38
Mardia’s kurtosis (MK) 0.00 0.00 0.00 0.00
Doornik & Hansen (DH) 4.32 4.18 4.15 4.51
Henze-Zirkler (HZ) 3.40 3.32 3.66 3.61
Royston 5.79 5.32 5.80 5.67
20 Mardia’s skewness (MS) 2.69 2.47 2.69 2.53
Mardia’s kurtosis (MK) 0.65 0.40 0.46 0.46
Doornik & Hansen (DH) 4.63 4.54 4.66 4.70
Henze-Zirkler (HZ) 4.49 4.26 4.20 4.51
Royston 6.30 6.20 6.26 6.20
50 Mardia’s skewness (MS) 4.20 4.27 4.58 4.22
Mardia’s kurtosis (MK) 1.91 1.83 1.88 1.91
Doornik & Hansen (DH) 4.99 5.20 5.09 4.67
Henze-Zirkler (HZ) 4.80 4.64 4.95 4.86
Royston 6.50 6.40 6.46 6.40
Table 4.2: Empirical Type I error rate (in percentage) against the MVN distribution
with dimension p = 4 for different sets of parameters.
n MVN test Set A Set B Set C Set D
10 Mardia’s skewness (MS) 0.01 0.00 0.00 0.01
Mardia’s kurtosis (MK) 0.00 0.00 0.00 0.00
Doornik & Hansen (DH) 4.60 4.45 4.60 4.84
Henze-Zirkler (HZ) 3.17 2.88 2.96 3.43
Royston 5.23 5.14 5.98 5.49
20 Mardia’s skewness (MS) 1.53 1.51 1.52 1.61
Mardia’s kurtosis (MK) 0.23 0.13 0.18 0.24
Doornik & Hansen (DH) 4.99 4.64 4.73 4.80
Henze-Zirkler (HZ) 4.60 4.50 4.53 4.40
Royston 7.18 6.63 7.39 7.35
50 Mardia’s skewness (MS) 3.99 3.78 4.03 4.04
Mardia’s kurtosis (MK) 2.08 1.89 2.00 2.03
Doornik & Hansen (DH) 5.20 5.27 5.35 5.26
Henze-Zirkler (HZ) 5.28 5.28 5.03 5.05
Royston 7.11 6.90 7.15 7.25
95
Table 4.3: Empirical Type I error rate (in percentage) against the MVN distribution
with dimension p = 8 for different sets of parameters.
n MVN test Set A Set B Set C Set D
10 Mardia’s skewness (MS) 0.00 0.00 0.00 0.00
Mardia’s kurtosis (MK) 0.34 0.42 0.41 0.43
Doornik & Hansen (DH) 5.20 4.90 4.70 4.98
Henze-Zirkler (HZ) 6.39 6.72 6.55 6.62
Royston 4.96 5.12 6.26 6.36
20 Mardia’s skewness (MS) 0.11 0.09 0.08 0.06
Mardia’s kurtosis (MK) 6.97 6.23 6.74 6.31
Doornik & Hansen (DH) 4.74 4.22 4.36 4.55
Henze-Zirkler (HZ) 6.02 5.85 5.85 5.95
Royston 7.92 7.80 8.95 8.90
50 Mardia’s skewness (MS) 2.50 2.64 2.73 2.44
Mardia’s kurtosis (MK) 5.39 5.37 5.63 5.49
Doornik & Hansen (DH) 4.94 4.74 4.94 4.54
Henze-Zirkler (HZ) 5.06 5.01 5.58 5.32
Royston 7.38 7.80 9.00 9.07
Table 4.1, Table 4.2 and Table 4.3 show the empirical Type I error rates against
the generated multivariate normal data. Since the 0H is true, each test should reject
the0H at about the nominal rate of 5%. MS and MK tests are far below from reaching
5% when sample size is small. With increasing n, MS test appear to show reliable
performance, however it breaks down as p gets higher. MK test on the other hand is
reaching 5% as the p and n gets higher. DH test appears to be the best performer when
different sets of parameters were considered, whereby it consistently possess close to
5% Type I error rate, followed by HZ test with higher variation compared to DH. The
estimates for the Royston were also relatively good in most of the cases, with a slight
higher error from the nominal rate. In a particular case where n is small and p is high
(Table 4.3), DH test perform best followed by Royston and HZ tests.
4.2.1.2 Power of Test
To estimate the power of tests between MS, MK, DH, HZ and Royston tests, the
following procedure was carried out:
Step 1. Alternative distributions which reflect non-normality of n = 10 were generated
96
using R random number generator functions. The distributions considered are:
(a) Multivariate t distribution with 10 degrees of freedom (d.f) and covariance
pppp
p
p
21
22221
11211
Σ , where the diagonal and off-diagonal elements of Σ
are 8 and 3 respectively, for p = 2, 4, 8, and 16 (Genz and Azzalini, 2015).
This distribution represents very mild departure from MVN distribution.
Figure 4.2 shows an example of generated data from multivariate t
distribution.
(b) Uniform distribution ),( baU with minimum value a and maximum value b
were set as 36 and 87 respectively. The properties of ),( baU are symmetric
and having lower kurtosis than the MVN distribution. Figure 4.3 shows an
example of generated data from multivariate uniform distribution.
(c) Multivariate lognormal distribution with parameters 0μ and pIΣ . The
property of this distribution is highly skewed and represents a drastic
departure from MVN distribution (Boogaart et al., 2015). Figure 4.4 shows
an example of generated data from multivariate lognormal distribution.
Step 2. Obtain the p-value from the test statistic of the MS, MK, DH, HZ
and Royston tests using the generated data as described in step 1 (Korkmaz et
al., 2015; Aho, 2015).
Step 3. The above process of getting the p-value was repeated for s = 10000 times.
Step 4. The proportion of 10000 samples for which the test rejects the null hypothesis
0H at 05.0 significance level was calculated. This is equivalent to calculating the
probability of not making a Type II error:
false is |reject 1 00 HHP . (4.3)
97
The corresponding error rate in percent was then obtained as the power of the
test.
Step 5. Similar steps as above was repeated to calculate the power of tests for
n = 20 and 50.
Figure 4.2: Example of generated multivariate t distribution for d.f = 10 and
83
38Σ .
Figure 4.3: Example of generated )87,36(U data for p = 2.
98
Figure 4.4: Example of a multivariate lognormal distribution generated for 0μ and
2IΣ .
Figure 4.5 shows the power estimates for the MVN test against multivariate t
distribution. For small sample size of n = 10 the rate of rejection for all tests was very
low and generally below 20% due to mild differences between multivariate t and normal
distribution. As n and p increases, Royston and MS tests perform best with power of test
around 70%. This result supports the findings of Farrell et al. (2007) with an added
knowledge on how the tests behave when an extremely small sample size with higher
dimension data set is considered.
99
(a) n = 10
(a) n = 20
(c) n = 50
Figure 4.5: Empirical power for MS, MK, DH, HZ and Royston test statistic against
multivariate t distribution.
2 3 4 5 6 7 8
Dimension, p
0
20
40
60
80
100
Pow
er
of
test
MS
MK
DH
HZ
Royston
Power of test against MV t distribution with n=10
0 2 4 6 8 10 12 14 16
Dimension, p
0
20
40
60
80
100
Pow
er
of
test
MS
MK
DH
HZ
Royston
Power of test against MV t distribution with n=20
0 2 4 6 8 10 12 14 16
Dimension, p
0
20
40
60
80
100
Pow
er
of
test
MS
MK
DH
HZ
Royston
Power of test against MV t distribution with n = 50
100
Results for the empirical power for the 5 MVN tests against multivariate
uniform distribution are given in Figure 4.6. As one would expect, MS had virtually no
power against symmetric alternatives whereby the power of the test is consistently 0%
for all cases. Other tests show low power of rejecting multivariate uniform distribution
when n is small. The Royston test produced the best result especially as n and p
increases. At n = 50, the power of Royston, DH and MK tests were nearly 100%, except
for HZ. Our results for the Royston and DH tests differ noticeably from those in Farrell
et al. (2007). Specifically, when n = 50, a decrease of DH and Royston’s power can be
evident in Farrell et al. (2007). This may be due to the differences of parameter used to
generate the multivariate uniform data set, which was not given in Farrell et al. (2007).
(a) n = 10
Figure 4.6: Empirical power for MS, MK, DH, HZ and Royston test statistic against
multivariate uniform distribution.
2 3 4 5 6 7 8
Dimension, p
0
20
40
60
80
100
Pow
er
of
test
MS
MK
DH
HZ
Royston
Power of test against MV uniform distribution with n=10
101
(b) n = 20
(c) n = 50
Figure 4.6, continued.
Figure 4.7 illustrates the results of the power of MVN tests against the
lognormal distribution. The multivariate lognormal is a heavily skewed distribution that
is a drastic departure from normality. Here, the power of the tests of MVN was expected
to be very high, close to 100%. For small sample size, Royston’s had the power of
nearly 100%, while the power of other tests dipped as low as 0% when p increases.
When n = 10 and 20, the MK and MS tests decreased as p increases. This may
be due to higher dimension data relative to the sample size which resulted in singularity
of the covariance matrices. This reason is supported as MK and MS showed consistent
rate of rejection which are close to 100% when the sample size greatly increases to n =
0 2 4 6 8 10 12 14 16
Dimension, p
0
20
40
60
80
100
Pow
er
of
test
MS
MK
DH
HZ
Royston
Power of test against MV uniform distribution with n=20
0 2 4 6 8 10 12 14 16
Dimension, p
0
20
40
60
80
100
Pow
er
of
test
MS
MK
DH
HZ
Royston
Power of test against MV uniform distribution with n=50
102
50. Results in this study are similar to Mecklin and Mundfrom (2003) and Farrell et al.
(2007), where as n increased to 50, the power of all procedures was at least 99.9%.
(a) n = 10
(b) n = 20
Figure 4.7: Empirical power for MS, MK, DH, HZ and Royston test statistic against
lognormal distribution.
2 3 4 5 6 7 8
Dimension, p
0
20
40
60
80
100
Pow
er
of
test
MS
MK
DH
HZ
Royston
Power of test against MV lognormal distribution with n=10
0 2 4 6 8 10 12 14 16
Dimension, p
0
20
40
60
80
100
Pow
er
of
test
MS
MK
DH
HZ
Royston
Power of test against MV lognormal distribution with n=20
103
(c) n = 50
Figure 4.7, continued.
4.2.1.3 Summary of Simulation Results for Testing Multivariate Normality
The above simulation study considers 5 MVN distribution tests which were
recommended by many studies. Results for n = 50 in this study were mostly in
agreement with studies by Farrell et al. (2007) although the error and power rates may
have slight differences. However, results for n = 10 and 20 cannot be compared as, to
the best of our knowledge, no studies have considered these cases in the literature.
This study showed that for small sample size, DH is the best performer when
only Type I error is considered. HZ and Royston tests on the other hand perform well in
the sense that their Type I error rates are consistently small in all considered cases.
Further, their power of estimates against severe departure of non-normality is
considerably high, ranging from 70% to 100%. It is therefore suggested that multiple
tests should be carried out when testing for multivariate normal distribution, especially
when dealing with data from small sample size and higher dimension. Some tests may
not be working well due to non-singularity of the sample covariance matrix.
0 2 4 6 8 10 12 14 16
Dimension, p
0
20
40
60
80
100
Pow
er
of
test
MS
MK
DH
HZ
Royston
Power of test against MV lognormal distribution with n=50
104
4.2.2 Multivariate Normal Distribution of Categories of Shape
The MVN contour and univariate Kolmogorov-Smirnov (KS) tests were initially
carried out to investigate the multivariate normality of each shape category. The 50%
probability ellipsoid appears to capture half of the observations (Table 4.4). The 60%
ellipsoid and 70% ellipsoid show an increasing number of observations being captured.
This result may be considered as weak evidence of multivariate normality in view of the
small sample sizes. On the other hand, each of the univariate variables was shown to be
normally distributed using the KS test (Table 4.5). These results strongly suggest the
existence of multivariate normality of )(kv for each shape category )3,2,1( kGk.
Since DH, HZ and Royston tests managed to perform well in the simulation
study for the case of small n and high p (Table 4.3 and Figure 4.7), these tests were
employed to further confirm the multivariate normality for each shape category. Table
4.6, Table 4.7 and Table 4.8 show evidence of multivariate normality for each kG .
Table 4.4: The proportion of observations satisfying )())(())(( 21 pii kk vvSvv in
each shape category kG . Arbitrary values of 0.7 and 0.6 ,5.0 were used giving
3441.7)5.0(28 , 8.3505)6.0(2
8 and 9.5245)7.0(28 .
Shape category
1G 1G
1G
5.0 11
5= 0.4545
22
11=0.5000
14
6= 0.4285
6.0 11
6= 0.5455
22
12=0.5455
14
7= 0.5000
7.0 11
11= 1.0000
22
15=0.6818
14
9= 0.6429
105
Table 4.5: The T values for KS test with sample size .14,22,11 321 nnn
Shape category 1G 2G
3G
Critical value at 5%
sig. level
Variables
0.3912 0.2809 0.3489
1w 0.1800 0.1392 0.1787
1l 0.0946 0.1070 0.1425
2w 0.1457 0.0926 0.2213
2l 0.2298 0.1178 0.1534
3w 0.1642 0.1360 0.1927
3l
0.1927 0.1663 0.1400
4w 0.2293 0.1259 0.1709
4l 0.0980 0.1380 0.1683
Table 4.6: Test statistic to investigate multivariate normality using Doornik and Hansen
(DH) test. Lower and upper 2.5% critical values are 6.9076 and 28.8453 respectively.
Category DH Test statistic
1G 27.2909
2G 10.1494
3G 23.5899
Table 4.7: Test statistic to investigate multivariate normality using Henze-Zirkler (HZ)
test.
Category Critical Value HZ Test statistic
1G Lower 2.5% = 0. 8186
Upper 2.5% = 0. 9614 0.9357
2G Lower 2.5% = 0.8499
Upper 2.5% = 0.9795 0.8870
3G Lower 2.5% = 0.8300
Upper 2.5% = 0.9683 0.9181
106
Table 4.8: Test statistic to investigate multivariate normality using Royston’s test.
Category Critical Value Royston’s Test
statistic
1G Lower 2.5% = 1.0944
Upper 2.5% = 13.9102 3.4711
2G Lower 2.5% = 0.7789
Upper 2.5% = 12.6017 3.1804
3G Lower 2.5% = 0.2140
Upper 2.5% = 9.3334 3.6874
Therefore, the three fitted categories of shape models are given as
3,2,1 ),(~)(: kkMVNk kk Svv , (4.4)
where the means are:
36.43), 36.82, 51.49, 67.27, 54.91, 74.85, 57.30, 83.16,()1( v
33.91), 36.11, 48.35, 69.15, 51.61, 77.07, 53.55, 85.55,()2( v
36.97) 43.76, 53.30, 72.21, 57.13, 79.43, 59.51, 87.30,()3( v , and the shape variability
can be explained by the estimated covariance matrices
47.4.......
40.398.7......
12.110.195.1.....
85.189.387.057.2....
20.127.188.188.004.2...
19.021.257.078.147.024.2..
52.059.005.258.021.238.092.2.
15.037.159.037.139.005.233.001.2
1S
,
33.3.......
98.243.9......
23.239.412.4.....
51.091.363.101.3....
84.100.478.365.173.3...
15.090.220.153.227.134.2..
26.137.375.370.184.353.162.4.
21.083.158.006.278.095.104.178.1
2
S
and
.
73.2.......
52.133.6......
03.204.011.4.....
37.195.101.264.2....
99.123.089.320.206.4...
64.108.251.296.274.254.3..
68.157.085.304.297.354.283.4.
72.152.191.279.210.350.399.270.3
3
S
107
Using the above results, the variation of a particular tooth may be illustrated as
Figure 4.8. The mean shape and variation for 1G ,
2G and 3G are shown in Figure 4.9,
Figure 4.10 and Figure 4.11 respectively.
Figure 4.8: Mean location of a particular tooth and its variations.
Figure 4.9: Mean shape and variation of
1G .
-40 -30 -20 -10 0 10 20 30 400
10
20
30
40
50
60
70
width (mm)
leng
th (
mm
)
where,
𝐴 = (��𝑗 , ��𝑗)
𝐵 = (��𝑗 + 𝜎𝑤𝑗 , 𝑙�� − 𝜎𝑙𝑗
)
𝐶 = (��𝑗 + 𝜎𝑤𝑗 , 𝑙�� + 𝜎𝑙𝑗
)
𝐷 = (��𝑗 − 𝜎𝑤𝑗 , 𝑙��+𝜎𝑗)
𝐸 = ( ��𝑗 − 𝜎𝑤𝑗 , 𝑙�� − 𝜎𝑗)
E
C
D B
𝑙��
��𝑗
A
Width (mm)
(mm
Length (mm)
(mm
108
Figure 4.10: Mean shape and variation of
2G .
Figure 4.11: Mean shape and variation of
3G .
-40 -30 -20 -10 0 10 20 30 400
10
20
30
40
50
60
70
width (mm)
leng
th (
mm
)
-40 -30 -20 -10 0 10 20 30 400
10
20
30
40
50
60
70
width (mm)
leng
th (
mm
)
109
4.2.3 Test of Separation between the Shape Models
The existence of 3 meaningful MVN shape models can be obtained if these
models are unique and possess significant differences in shape. The Hotelling two
sample T2 test (Johnson & Wichern, 1992) was used to compare mean vectors from two
populations. However since the sample size of each category kG (k=1, 2, 3) is
relatively small, further assumptions have to be met, that is, both populations are MVN
with equal covariance matrix (Johnson & Wichern, 1992). The equality of covariance
matrices was imposed to reduce the number of parameters in the model and
consequently helped in reducing the error of the T2 test. Since the normality assumption
was investigated earlier, the latter was further investigated.
4.2.3.1 Test for Equality of Covariance Matrices
This section investigated the equality of covariance matrices using the Box test
(Stevens, 2001). Let g be the number of shape category, N be the total sample size, kn
and 1 kk nr be the sample size and degrees of freedom of the k-th shape category,
respectively. Let the null hypothesis for the three MVN covariance matrices be
3210 : H . (4.5)
The Box’s M test statistic is defined as
g
k
kkpooled rgNM1
||ln||ln)( SS , (4.6)
where,
g
k
kkpooled
gN
n
1
)1( SS is the sample covariance matrix for the pooled data,
1
))()())(()((
k
T
kn
kkkk vvvvS is the corresponding covariance matrix for category-k and
p is the dimension of the v-vector.
110
When the sample size 20kn , MF is approximately distributed as an F
distribution with r and 0r degrees of freedom, where rrrCF 0/1 ,
2
)1)(1(
gppr ,
2
0
0
2
CC
rr
,
g
k
k gNrg
ppC
1
22
0 )/(1/1)1(6
)2)(1(,
g
k
k gNrgp
ppC
1
22
2
)/(1/1)1)(1(6
132. The test statistic was found to be 1.004
which is between the lower and upper 2.5% critical value, namely
3580.1004.16982.0 , indicating the covariance matrices are equal.
Therefore, the three categories of fitted shape models may be now written as
3,2,1 ),(~)(: kkMVNk pooledk Svv , (4.7)
where the common shape variability is
41.3.......
75.119.8......
92.136.263.3.....
26.133.338.080.2....
74.113.238.334.045.3...
37.050.203.049.209.067.2..
21.191.139.334.051.306.030.4.
64.063.144.012.245.043.231.040.2
pooledS . (4.8)
4.2.3.2 Investigating Separation of Mean Shape using the Hotelling T2 test
Let the null hypothesis for two mean vectors be 210 : μμ H . The Hotelling two
sample test statistic with 21 is
)()'( 21121
21
212 vvSvv
poolednn
nnT , (4.9)
such that )1(,
21
212
211
)2(~ pnnpF
pnn
pnnT
. Table 4.9 clearly shows rejection of
0H and
indicates meaningful existence of 3 MVN models with significant differences in shape.
111
Table 4.9: Hotelling 2T test for comparing two multivariate means.
Mean vector Critical value HotellingT2 statistics
)2()1( vv Lower 2.5% =2.6179
Upper 2.5% =28.7177 75.0389
)3()1( vv
Lower 2.5% =2.8213
Upper 2.5% =35.9355
113.8008
)3()2( vv
Lower 2.5% =2.5714
Upper 2.5% =27.2745
124.0771
4.2.4 Verification of the Arch Shape Models
The 3 MVN shape models were established using a relatively small sample of 47
casts. To ensure that the shape models are valid, verification study were carried out after
obtaining another 75 test samples.
The shape features v from 47 control casts and 75 test samples were pooled
together and denoted as iv where .122,,1i Using these v -vectors, 3 categories of
shape (denoted as 3,2,1 ,122 lGl ) together with its mean (denoted as 3,2,1 ,)( 122 kkv )
were obtained from AHC with complete linkage and the multivariate normal
distribution and equality of the covariance matrices of 3,2,1 ,122 lGl were investigated.
Then, the 3,2,1 ,)( 122 llv were compared to 3,2,1 ),( kkv from the 47 control casts.
If the hypothesis testing under lklkH for )()(: 122
0 vv fails and reject the 0H
lk for , these would imply the validity of the 3 shape models established using the 47
control sample.
The 3 shape categories 3,2,1 ,122 kGk , with group sizes 26, 40 and 56
respectively, followed the MVN distribution when tested using DH, HZ and Royston
normality tests (Table 4.10, Table 4.11 and Table 4.12). The test statistic for Box test
(Stevens, 2001) was found to be 1.1499 which is between the lower and upper 2.5%
critical value, namely 0.7001< 1.1499<1.353, indicating the covariance matrices
112
122
3
122
2
122
1 ΣΣΣ are equal. Table 4.13 shows the p-values of Hotelling two sample 2T
test when comparing the means of shape using 47 and 122 samples. This result verifies
the validity of the 3 shape models established using the 47 control sample.
Table 4.10: Doornik and Hansen (DH) test to investigate multivariate normality of the
shape categories using 122 casts. Lower and upper 2.5% critical values are 6.9077 and
28.8454 respectively.
Category DH Test statistic 122
1G 14.9283
122
2G 15.5736
122
3G 13.1403
Table 4.11: Henze-Zirkler (HZ) test to investigate multivariate normality of the shape
categories using 122 casts.
Category Critical Value HZ Test statistic
122
1G Lower 2.5% = 0.8570
Upper 2.5% = 0.9830 0.9138
122
2G Lower 2.5% = 0.8740
Upper 2.5% = 0.9910 0.9819
122
3G Lower 2.5% = 0.8850
Upper 2.5% = 0.9960 0.9505
Table 4.12: Royston’s test to investigate multivariate normality of the shape categories
using 122 casts.
Category Critical Value Royston’s Test
statistic
122
1G Lower 2.5% = 1.1698
Upper 2.5% = 14.1979 1.4984
122
2G Lower 2.5% = 0.9001
Upper 2.5% = 13.1270 8.5142
122
3G Lower 2.5% = 0.7636
Upper 2.5% = 12.5331 6.088
113
Table 4.13: p-value of the Hotelling two sample 2T test for comparing 3 clusters using
47 casts and 122 casts.
Mean vectors for k-th cluster )1(v )2(v )3(v
122)1(v 0.9046 0 0
122)2(v 0 0.4137 0
122)3(v 0 0 0.6956
4.3 Application of Shape Models to Impression Tray Design
4.3.1 Fabricating Three Impression Trays
This section presents the application of the shape models of the dental arch
defined by MVN distributions in designing three impression trays for the population
studied. Existing stock trays were generally U-shaped and the dimensions include space
for an adequate thickness of the impression material all around the dental arch (Figure
4.12(a)). To provide impression trays which would be suitable for the population
studied, three trays with mean dimensions derived from each of the three shape models
were produced. Each tray was fabricated based on the knowledge of the corresponding
mean shape models together with the maximum deviation obtained from common
covariance matrix to indicate the variability of the arch shape.
In the k-th category of arch shape, select the i-th dental cast and let d(i), i=1,.., kn
denote the distance between the two hamular notches. Further, denote the average of
these distances by kd . On the Cartesian x-coordinate, the points 0,5.0 kd and
0,5.0 kd represent the position of the two hamular notches. Using the vertical line
passing through the origin as the y-axis, the four selected teeth can be represented by
),( 11 lw , ),( 22 lw , ),( 33 lw and ),( 44 lw where for example, 1w represent the average of 1w
for all casts in a given category. The remaining four points on the other side of the arch
were similarly derived due to the symmetry of the arch (see Section 3.5). Straight lines
114
joining the eight teeth positions and the two hamular notches give good approximation
to the arch shape.
An additional 5 mm was added to the outside perimeter on both sides of each
shape, thereby increasing the transverse distance of the simulated casts by 10 mm. The
5 mm increase in width on either side of the arch was added taking into account a
maximum standard deviation of 3 mm, and an allowance of 2 mm for the impression
material to be used in the stock tray (Hatrick et al., 2003; McCabe & Walls, 1998)
(Figure 4.12(b)). Light cure material (Kemdent Works, Wiltshire, UK) was then used to
make three impression trays for each shape model and were labelled as C1, C2 and C3
(Figure 4.12(c)).
(a) (b) (c)
Figure 4.12: (a) An example of a stock impression tray with space around the tray to
allow for variation of arch sizes and adequate thickness of alginate impression material.
(b) Mean shape v was indicated by solid line. The broken lines indicate 5mm added to
the perimeter of the arch shape. (c) The light cure acrylic resin tray.
4.3.2 Verification of the Fabricated Impression Trays
The fabricated trays C1, C2 and C3 should be verified by inspecting whether or
not they were constructed according to the designed measurements. To confirm this, the
membership of the 47 control casts which belong to each of the 3 shape models were
tracked from the dendrogram (see Figure 3.12 in section (3.6.1)). All casts which made
up the shape model )1(v should fit tray C1. Similarly, casts which belong from shape
models )2(v and )3(v should fit tray C2 and C3 respectively.
A particular cast is said to fit the corresponding tray adequately if the amount of
-4 -3 -2 -1 0 1 2 3 40
1
2
3
4
5
6
7
5mm
115
impression material between the inside edge of the tray to the buccal surfaces of the
teeth is at least 2 mm (Hatrick et al., 2003; McCabe & Walls, 1998). Plasticine was used
as a simple method (instead of using impression material), to examine the space for the
material impression in the fabricated tray. This was done by first loading a tray with
plasticine and a cast was then placed vertically into the tray. When placing the cast
vertically into the loaded plasticine tray, the distance of the tray to the labial surface of
the central incisors on the cast was ensured to be about 3 mm. Once properly seated,
the cast was removed gently from the plasticine, to ensure minimal distortion of the
plasticine. Finally, a minimum of 2 mm amount of the plasticine between the inside
edge of the tray to the buccal surfaces of the teeth was ensured by a dentist to confirm
that the tray is suitable for the cast.
Table 4.14 shows that the minimum amount of plasticine in the impression tray
for all casts using its corresponding fabricated tray was more than 2 mm, indicating all
47 casts fit their corresponding trays. This result shows that the trays were constructed
according to its corresponding shape model and therefore verifies the fabricated trays.
Table 4.14: Amount of plasticine thickness in the impression tray and number of cast
which fit the fabricated trays C1, C2 and C3 when impression of 47 control samples
was taken.
Tray
Material
thickness(mm)
C1 C2 C3
<2 0 0 0
2-4 4 14 9
4-9 7 8 5
>9 0 0 0
Number (percentage) of
casts which fit the tray 11 (23.4%) 22 (46.8%) 14 (29.8%)
Further, the ability of the fabricated impression trays to fit the Malaysian
population was carried out by investigating the fit of the trays on 40 dental casts which
were randomly chosen from the 75 test casts (the remaining 35 test casts was used for
116
another verification study in the later part of this chapter). The impression of each test
cast was taken using the same procedure as above, and the thickness of the plasticine in
the impression tray was examined. In a situation when the condition was not satisfied,
the procedure was repeated using the other 2 trays. The trays were considered suitable
when the cast could go into a tray, regardless of whether the teeth were in good
alignment or not (Figure 4.13). The result illustrated in Table 4.15 show that all 40 test
casts could fit in at least one of the fabricated trays.
Table 4.15: Amount of plasticine thickness in the impression tray and number of cast
which fit the fabricated trays C1, C2 and C3 when impression of 40 test samples was
taken.
Tray
Material
thickness(mm)
C1 C2 C3
<2 0 0 0
2-4 1 7 6
4-9 2 12 10
>9 0 0 2
Number (percentage) of
casts which fit the tray 3 (7.50%) 19 (47.50%) 18 (45.0%)
Figure 4.13: Two of the mal-aligned arches in the 40 samples used for verification of
the three fabricated trays.
117
4.4 Discrimination of the Dental Arch Shape
Three impression trays were fabricated using the three MVN shape models
)(kv in equation (4.7). An immediate question is how to discriminate which tray is
suitable for a particular patients’ arch shape. This section aims at developing a
discrimination procedure for the shape of the maxillary dental arch using the established
statistical shape models.
By definition, discrimination is an act of assigning object to categories or classes
(Costa & Cesar, 2009). The PCA may provide information for discrimination as it
explains the variability of the data that quantifies shape differences. However the main
limitation of the use of PCA is that it does not consider class separability since there is
no mention of class label of the shape. Therefore, assigning new objects to a particular
class may be difficult to accomplish.
The nearest neighbours method is non-parametric and the simplest
discrimination approach. It basically identifies the sample in a set of N samples that is
closest to a new object and takes its class. However, the performance of this method is
generally inferior to the likelihood function method as it does not supply information
about the probability density function of the class (Costa & Cesar, 2009).
Since the probability distribution of the shape has been identified as MVN with
equal covariance matrices, the discriminant function, in particular the linear
discrimination function (LDF) can be used (Uetani et al., 2015). Let )47,1( iiv be the
i-th control cast and )(vkf be the probability density function of the k-th population.
The LDF is given as:
118
jkpooledpooled
jkjk
jk
jk
j
kkj
jkjkjk
jjkkjk
jjjkkk
jjvp
kkp
ff
f
fh
SSvvSvvvSvv
vSvvSvvSvvSv
vSvvSvvSvvSv
vvSvS
vvSvvS
vv
v
vv
)).()(())'()((2
1))'()((
)()'(2
1)()'(
2
1)'()'(
)()'(2
1)'()()'(
2
1)'(
))(())'((2
1||log
2
1)2log(
2
))(())'((2
1||log
2
1)2log(
2
)(log)(log
)(
)(log)(
11
1111
1111
1
1
(4.10)
where jk .The discriminant rule for the 2 populations is as follows (Johnson &
Wichern, 1992; Mardia et al., 1979)
if 0)( vkjh , then kv ,
if 0)( vkjh , then jv , and (4.11)
The Gaussian and equality of covariance matrices assumptions for the
discriminant function are particularly restrictive and sometimes difficult to satisfy. If
violated, it may reduce the performance of discriminant analysis. The following section
therefore proposes COVRATIO as an alternative discrimination method which uses the
determinant ratio of two covariance matrices that are not necessarily from Gaussian
distribution. Subsequently, comparison between the LDF and )( iCOVRATIO was carried
out via simulation study.
4.4.1 Proposed )( iCOVRATIO as Discrimination Method
The COVRATIO procedure dates back to Belsley et al. (1980). They proposed a
numerical statistic to identify the presence of influential observation in linear regression
models. This numerical statistic is based on the determinantal ratio given as:
119
COV
COV iCOVRATIO i
, (4.12)
where COV is the determinant of covariance matrix for full data set and ( )iCOV is
that for the reduced data set by excluding the thi observation. If the ratio is close to 1,
then there is no significant difference between them. In other words, the thi observation
is consistent with the other observations. Alternatively, if the value of ( ) 1iCOVRATIO
is close to or larger than a derived cut off point, then it indicates that the thi observation
is a candidate of an outlier. This procedure has been extended and employed for other
models such as the simple linear functional relationship model and circular regression
model (Uddameri et al., 2014; Ghapor et al., 2014; Ibrahim et al., 2013; Abuzaid et al.,
2008).
In this study, the idea of COVRATIO statistics for outlier detection was extended
for the purpose of discrimination. Let 1Σ and 2Σ be the pp covariance matrix for
population 1 and 2 respectively, where each of the population has a known
probability distribution. The determinant of 1Σ and 2Σ , denoted as 1Σ and
2Σ gives
the descriptive measures of multivariate variability (Peña and Rodrıguez, 2003). Next,
instead of using reduced observation as formulated by Belsley et al. (1980), define the
ratio of covariance determinant for 1 as:
Σ
Σ
1
)(1
)(1
i
iCOVRATIO
, (4.13)
where Σ)(1 i
is the determinant of covariance matrix )(1 iΣ which is estimated by adding
new thi observation in 1 .
120
The same thi observation will be added in 2 and its
)(2 iCOVRATIO is obtained
using
Σ
Σ
2
)(2
)(2
i
iCOVRATIO
. (4.14)
Smaller value of )( iCOVRATIO (which is close to 1) indicates no variation when new
observation was included in a particular population. In other words, the thi observation
is consistent with the other observations and therefore belongs to the population.
Therefore, the discrimination rule using )( iCOVRATIO for 2 populations is given
as:
(4.15)
The above result can be generalized for g populations, given as:
(4.16)
4.4.2 Simulation Study for Comparing Performance of LDF and )( iCOVRATIO
The performance of LDF and )( iCOVRATIO was examined via simulation study.
Random samples from two MVN populations were generated and re-assigned to their
corresponding population using LDF and )( iCOVRATIO
. The misclassification
Allocate new thi observation in 1 if
)(2)(1 ii COVRATIOCOVRATIO ,
Else, allocate to 2
Allocate new thi observation in
j if )()( ihij COVRATIOCOVRATIO ,
where ghj ,,2,1, and hj .
121
probabilities using these discrimination methods were then compared. This procedure
was done as follows:
Step 1: Two sets of random samples with sample size n from two unequal p-variate
normal distributions; 1111 ,~: ΣμX pN and 2222 ,~: ΣμX pN were generated
using R random number generator.
Step 2: Generate another two random samples A and B which were drawn from 1 and
2 respectively.
Step 3: Calculate the LDF in equation (4.10) using random sample A and B , denoted
as )(12 Ah and )(12 Bh respectively.
Step 4: Calculate )()(1 AiCOVRATIO , )()(1 BiCOVRATIO , )()(2 AiCOVRATIO and
)()(2 BiCOVRATIO using equation (4.13) and (4.14), by adding random sample A and
B in both sets of random samples generated from 1 and 2 in step 1.
Step 5: Discrimination rule for LDF and )( iCOVRATIO (in equations (4.11) and (4.15))
were used to determine the membership of A and B .
Step 6: The above steps were repeated for s = 10000 times.
Step 7: The proportion of A and B which does not assign to their corresponding true
population when discriminated using LDF and )( iCOVRATIO may be regarded as the
misclassification probability.
Table 4.16 shows the performance of LDF and )( iCOVRATIO when different values
of n, p, μ and Σ were considered. In general, the misclassification probability for LDF
and )( iCOVRATIO reduces when a particular value of
2μ was chosen to indicate larger
separation between 2 and
1 . The dispersion of the data in a particular population also
122
dictates the misclassification probability, whereby larger value of Σ may result in higher
chances of a sample to be assigned to an incorrect population. Increasing value of n
gives slightly higher misclassification probability. On the other hand, increasing value
of p gives smaller misclassification probability. However at 8p , the LDF breaks
down and shows extremely high misclassification probability.
Overall, the )( iCOVRATIO method gives smaller probability misclassification as
compared to the LDF in all cases, indicating better performance of discrimination. The
proposed )( iCOVRATIO is theoretically preferable, because unlike the LDF, it does not
require Gaussian and equality of covariance matrices assumptions, which may be
difficult to attain. It also uses the raw data instead of descriptive statistics when
discriminating therefore retaining the information about the population. Moreover, due
to its simplicity, the i)(COVRATIO can be employed when discriminating more than 2
groups. The drawback of )( iCOVRATIO
method is that it is computationally more
expensive compared to LDF, however with technology advancement in available
statistical software, such computation is straightforward and incredibly fast to compute.
123
Table 4.16: Misclassification probability for LDF and )( iCOVRATIO when different
sample size, dimension, mean vector and covariance matrices were considered.
Parameters 1 2
LDF )( iCOVRATIO
LDF )( iCOVRATIO
10
01
, 0
0
,2,20
21
1
21
ΣΣ
μ
pnn
2
22μ 0.0741 0.0622 0.0672 0.0577
3
32μ 0.0141 0.0049 0.0149 0.0046
4
42μ 0.0015 0.0001 0.0013 0
10
01
4
4,
0
0
,2,20
1
21
21
Σ
μμ
pnn
5.10
05.12Σ 0.0022 0.0005 0.0093 0.0005
20
022Σ 0.0026 0.0012 0.0203 0.0021
5.20
05.22Σ 0.0031 0.0022
0.0301
0.0070
30
032Σ 0.0031 0.0031 0.0460 0.0114
10
01
2
2,
0
0 ,2
21
21
ΣΣ
μμp
1021 nn 0.0640 0.0530 0.0634 0.0504
2021 nn 0.0741 0.0622 0.0672 0.0577
5021 nn 0.0820 0.0663 0.0799 0.0636
100
0
10
001
,
2
2
,
0
0
,20
21
21
21
ΣΣ
μμ
nn
2p 0.0741 0.0622 0.0672 0.0577
4p 0.0177 0.0087 0.0163 0.0104
6p 0.0040 0.0018 0.0050 0.0015
8p 0.4000 0 0.9000 0
124
4.4.3 Results for Discrimination of Dental Arch Shape
The membership of the 47 control samples which belong to each of the 3 shape
models were tracked from the dendrogram (see Figure 3.12 in section (3.6.1)). The LDF
and )( iCOVRATIO were employed to re-assign these casts into one of 3 populations of
the shape models. Their misclassification probability was calculated to ensure that these
methods are deemed good for shape discrimination.
Table 4.17 shows the misclassification probabilities when the samples were re-
assigned to *
2
*
1 , or *
3 using the LDF and )( iCOVRATIO
. A relatively low
misclassification probabilities for LDF were obtained, however )( iCOVRATIO
outperformed the LDF method by giving zero misclassification probabilities. These
results demonstrate the ability of the )( iCOVRATIO to discriminate the dental arch shape
correctly and can be regarded as a better discrimination method.
Table 4.17: Misclassification probability when 47 dental casts were re-assigned into
either one of the population of the shape model using the LDF and )( iCOVRATIO .
Discrimination
True
population
*1 *
2 *3
LDF )( iCOVRATIO LDF )( iCOVRATIO
LDF )( iCOVRATIO
1 - - 0.18 0 0.18 0
2 0.09 0 - - 0.09 0
3 0.21 0 0.07 0 - -
125
4.5 Investigating Dental Arch Shape with Missing Tooth: A Simulation Study on
Performance of Data Augmentation and Expectation Maximization
In usual clinical practice, a dentist will choose an impression tray that to his
observation may best fit the patient’s arch. If the resultant cast is not adequate for
restoration to be constructed upon it, a custom impression tray will have to be
constructed from this cast. The construction of custom tray is time consuming, incur
more expenses and suitable for only one particular patient (Patil et al., 2008).
Shape discrimination using )( iCOVRATIO may provide a good guide (see Section
4.4.3) to assign a suitable tray for a particular patient; however, this is only possible if
the teeth used as shape descriptor v are available. Since missing tooth occurs for a
variety of reasons, assigning a suitable tray to a particular shape of dental arch may be
problematic if a tooth is lost.
This section considers the data augmentation (DA) and expectation
maximization (EM) to impute any missing values. The performance of the imputation
method was evaluated if a particular method has lower misclassification probability
when discriminated into its true shape category. The effect of sample size n, number of
missing value 0n and specific missing tooth was investigated through the following
steps:
Step 1. A random sample of n = 10 which mimics one of the three shape models was
generated from
3,2,1 ),(~)(: kkMVNk pooledk Svv (4.17)
where the parameters were given in section (4.2.2). Consider k = 1 and let the
data denoted as
126
44332211
2424232322222121
1414131312121111
,,,,,,,
,,,,,,,
,,,,,,,
nnnnnnnn lwlwlwlw
lwlwlwlw
lwlwlwlw
, (4.18)
Step 2. Missing values of size 20 n were introduced into the generated data at the j-th
tooth ( 4,3,2,1j ). Since one tooth was defined as a pair wise variable,
therefore the missing values for the j-th tooth and true population 1 are denoted
as follow
44332211
3434333332323131
2424232322222121
1414131312121111
,,,,,,,
,,,,,,,
,,,,,,,
,,,,,,,
nnnnnnnn
NANA
NANA
lwlwlwlw
lwlwlwlw
lwlwlwlw
lwlwlwlw
, (4.19)
Step 3. The estimated values from DA (section 2.9.1) and EM (section 2.9.2) were
imputed to the missing values.
Step 4. The sample with imputed values from DA and EM were then assigned to the
population using the )( iCOVRATIO in section (4.4.1).
Step 5. If any samples with imputed DA or EM were not correctly classified to their
true population, the number of misclassification for the first simulation, s is
counted as 11 misn , else 01 misn . The initial value was arbitrarily taken as 30
with 50 iterations and the stopping criterion for EM and DA was set as 0.1.
Step 6. The above process of counting ),,1( sinmis
i was repeated for s = 10000 and
the misclassification probability using EM and DA was calculated as
s
n
icationmisclassifP
s
i
misi
1)( . (4.20)
Step 7. The above steps were repeated to calculate )( icationmisclassifP as a performance
127
indicator of the imputation method for pair wise cases of 6,4,20 n and
50,20,10n respectively.
Table 4.18: The misclassification probability using DA and EM for missing values
imputation when the number of missing values 20 n .
n
j-th tooth
10 20 50
DA EM DA EM DA EM
j=1 (central incisor tooth) 0.4809 0.2495 0.4965 0.2321 0.4917 0.2167
j=2 (lateral incisor tooth) 0.5099 0.4159 0.4979 0.4045 0.4961 0.4090
j=3 (canine tooth) 0.6010 0.4765 0.5804 0.4627 0.5821 0.4717
j=4 (1st molar tooth) 0.3219 0.2313 0.3210 0.2231 0.3227 0.2177
Table 4.19: The misclassification probability using DA and EM for missing values
imputation when the number of missing values 40 n .
n
j-th tooh
10 20 50
DA EM DA EM DA EM
j=1 (central incisor tooth) 0.7026 0.4456 0.7293 0.4047 0.7372 0.3971
j=2 (lateral incisor tooth) 0.7419 0.6499 0.7535 0.6468 0.7556 0.6343
j=3 (canine tooth) 0.8176 0.7130 0.8259 0.7118 0.8317 0.7152
j=4 (1st molar tooth) 0.5354 0.4130 0.5345 0.3879 0.5452 0.3990
Table 4.20: The misclassification probability using DA and EM for missing values
imputation when the number of missing values 60 n .
n
j-th tooh
10 20 50
DA EM DA EM DA EM
j=1 (central incisor tooth) 0.7985 0.5893 0.8480 0.5306 0.8646 0.5246
j=2 (lateral incisor tooth) 0.8575 0.7882 0.8686 0.7839 0.8703 0.7802
j=3 (canine tooth) 0.8977 0.8324 0.9234 0.8426 0.9243 0.8467
j=4 (1st molar tooth) 0.6465 0.5358 0.6842 0.5310 0.6953 0.5330
Table 4.18 to Table 4.20 illustrate the results. The misclassification probability
increases when the number of missing values increase, with EM performs best in all
cases, particularly for larger n. This simulation study also shows that the
misclassification probabilities are highest when canine tooth was missing, followed by
lateral incisor, central incisor and molar teeth. This indicates the hierarchy of the teeth
and their weight in determining the dental arch shape.
128
4.6 Application of Shape Discrimination
4.6.1 A Proposed Guide for Determining Appropriate Impression Tray for
Patients (Without and With Missing Tooth).
This section proposed a guide for determining appropriate fabricated impression
tray for clinical use. The method is as follows:
1) An intraoral camera connected to a PC is inserted into a patient’s mouth. A
small ruler is placed near the teeth while capturing the dental arch (Figure
4.14).
2) The image is directly transferred to the PC.
3) A program developed in MATLAB software is used to import and also to
calibrate the image, obtain the shape descriptor of the patient’s dental arch,
and employ the )( iCOVRATIO to assign a shape model or tray which best fits
the patient’s arch shape (sections 3.3 and 4.4.1).
4) If a missing tooth is detected, the estimation of the missing values using EM
will be imputed into the shape descriptor (section 4.5). However, there may
be 0.2177 to 0.601 probabilities (depending on which tooth is missing - see
Table 4.18) that the tray assigned does not fit the patient, and minor
modifications to the tray may be required.
129
Figure 4.14: A small ruler was placed near the teeth (for image calibration) while
capturing the 2D image of the patient’s dental arch by using an intraoral camera.
4.6.2 Verification of the Proposed Guide for Determining Appropriate Impression
Tray
The proposed method for determining appropriate impression tray was verified
to ensure that the guide is satisfactory for use on patients. The v-vector shape feature of
the remaining 35 test samples were obtained and discriminated using )( iCOVRATIO to
assign each test cast to any one of the 3 fabricated trays (or shape models). Then, the
impression of each test cast was made using its corresponding tray and the thickness of
the plasticine (see section 4.3.2) in the impression was used to indicate the fit of the tray
to cast. Table 4.21 shows the percentage of the cast that fit the assigned tray.
130
Table 4.21: Plasticine thickness in the impression and number of casts (n=35) which fit
the fabricated trays C1, C2 and C3.
Tray
Space(mm)
C1 C2 C3
<2 0 0 3
2-4 5 16 4
4-9 1 4 1
>9 0 0 1
Number (percentage) of
casts which fit the tray 6 (17.14%) 20 (57.14%) 6 (17.14%)
* Three casts (8.58%) could not go into any of the assigned tray.
4.8 Discussion
Three shape models resulting from the clustering method were used to construct
three impression trays. In the verification study, 91.42% of an independent random
sample of 35 casts could fit into at least one of the three impression trays (Table 4.21).
The remainder 8.58% were regarded as outliers, being too long or too wide for them to
go into any of the trays. Although only four points were used to define the arch shape,
the ability of the impression trays to capture 91.42% of the test dental casts showed that
as far as impression tray design is concerned, the MVN shape model is adequate. This
is inspite of some of the test casts had teeth which were not well aligned in the dental
arch.
A simulation study on missing tooth shows that the selected 4 teeth used to
describe the arch shape may be arguable, and that all 18 teeth should represent the arch
shape of the maxillary arch for a more accurate arch shape representation. If all teeth
were used, the MVN shape model will clearly fail because of dimensionality problems.
Other techniques should be introduced for example using Fourier descriptor to represent
the maxillary arch shape. However, a more complex probability shape model needs to
be defined. Nevertheless, the v-vector as shape descriptor is capable of capturing the
general shape of the dental arch and assigning arch shape to its correct population.
131
The value of the study in this chapter was: (1) the establishment of 3 categories
of arch shape with mean and variation for each shape category; (2) verification that the
shape models and trays produces can accommodate adequately randomly selected casts.
(3) Proposal of a guide to select suitable impression tray for a particular patient. In the
next chapter, a more precise arch shape representation of the dental arches is considered
for rehabilitating the edentulous patients.
132
CHAPTER 5. SHAPE MODELS OF THE DENTAL ARCH: AS A GUIDE IN
ESTIMATING NATURAL TEETH POSITIONS ON COMPLETE DENTURES
FOR THE EDENTULOUS
5.1 Introduction
The knowledge of teeth positions on the dental arch and the categories of arch
shape are important factors in restoring aesthetics and function of the edentulous
patient. Limitations to the shape feature ),,,( 4,43,312,21,1 lwlwlwlwv proposed in the
previous chapter may be that only 4 teeth were considered and were regarded to be
symmetrical. It may be argued that 4 selected teeth on the standardized digital images
of the dental casts could be considered as insufficient with respect to representing shape.
However, increasing the number of teeth would create problems with dimensions and
proof of existence of the multivariate normal distribution is extremely difficult.
This chapter investigates the ability of Fourier descriptors (FD) to represent all
maxillary teeth as alternative shape models. To avoid the curse of dimensionality, a
relatively small number of FD terms are used to enable the formulation of a new shape
model. Using these FDs, the 3 categories of shape established from v were verified, and
tested for multivariate complex normality as its shape model. A hypothesis testing for
two sample means from MVCN based on the Hotelling 2T test was derived and
employed to confirm the existence of shape models that possess significant differences
in shape.
133
A shape discrimination procedure for the maxillary dental arch using the
established statistical shape models for MVCN was developed using the modified
COVRATIO statistics. Then, three anatomical landmarks which remain in edentulous
arch were linked to the 3 categories of MVCN shape models. A guide for estimation of
natural teeth positions on complete dentures for the edentulous patient was then
proposed. Test casts were used to verify the proposed guide whereby the estimated teeth
positions using the MVCN shape model will be compared to the original teeth positions.
5.2 Shape model of the dental arch using Fourier Descriptor (FD)
5.2.1 The Ability of Fourier Descriptor (FD) in Representing Dental Arch Shape
The same control samples of 47 standardized digital images of dental casts were
used. Twenty one points representing the dental arch shape were rearranged in an anti-
clockwise sequence starting from the origin and denoted as
)]1(),1([)],...,1(),1([)],0(),0([ NyNxyxyx (Figure 5.1). Each coordinate pair can be treated
as a complex number so that )()()( kjykxks , where k = 0, 1,…, N-1 and N=21.
The discrete Fourier transform of )(ks is
2
exp)(1
1
0
N
k
uN
kjuks
Na
, (5.1)
1,...,1,0for Nu (Oppenheim et al., 1983). The complex coefficients ua are known as
the Fourier descriptors (FD) of the boundary. The set of FD for each dental cast was
denoted as ],,,[ 110iN
iii aaa A , 47,,1i where
iu
iu
iu jdca , (5.2)
134
The inverse Fourier transform of these coefficients restores )(ks where
1,,1,0 Nk , as given by
1
0
2exp )(
N
u
uN
kjuaks
. (5.3)
A pilot study was carried out to investigate the ability of FD terms to
approximate the shape boundary. The results are illustrated in Figure 5.2. It can be seen
that the boundaries of the arch shape are closer to the original as the number of selected
FD terms increase. An approximation of boundary using all 21 FD terms gives exact
original boundary of the dental arch shape.
Figure 5.1: An example of a particular dental cast which shows the hamular notches
(HN) and incisive papilla (IP) that were used to establish the Cartesian axes. Twenty
one selected points as illustrated in the diagram were used as the shape boundary.
135
(a) Plot of the original and
approximation of boundaries using
2 FD terms
(b) Plot of the original and
approximation of boundaries using
3 FD terms
(c) Plot of the original and
approximation of boundaries using
4 FD terms
(d) Plot of the original and
approximation of boundaries using
5 FD terms
Figure 5.2: The ability of 2, 3, 4, 5, 6, 19, 20 and all 21 FD terms in representing the
dental arch shape.
-40 -30 -20 -10 0 10 20 30 40-10
0
10
20
30
40
50
60
70
width(mm)
leng
th(m
m)
Approx boundary
Original boundary
-40 -30 -20 -10 0 10 20 30 40-10
0
10
20
30
40
50
60
70
width(mm)
leng
th(m
m)
Approx boundary
Original boundary
-40 -30 -20 -10 0 10 20 30 40-10
0
10
20
30
40
50
60
70
width(mm)
leng
th(m
m)
Approx boundary
Original boundary
-40 -30 -20 -10 0 10 20 30 40-10
0
10
20
30
40
50
60
70
width(mm)
leng
th(m
m)
Approx boundary
Original boundary
136
(e) Plot of the original and
approximation of boundaries using
6 FD terms
(f) Plot of the original and
approximation of boundaries using
19 FD terms
(g) Plot of the original and
approximation of boundaries using
20 FD terms
(h) Plot of the original and
approximation of boundaries using
all 21 FD terms
Figure 5.2, continued.
5.2.2 Selecting Number of FD Terms as Shape Feature
Each of the amplitude of ua given by
22uuu dca , (5.4)
indicates the size implicating the synthesized signal and may be regarded as
quantification of contribution or weight in representing the boundary of the arch shape
-40 -30 -20 -10 0 10 20 30 40-10
0
10
20
30
40
50
60
70
width(mm)
leng
th(m
m)
Approx boundary
Original boundary
-40 -30 -20 -10 0 10 20 30 40-10
0
10
20
30
40
50
60
70
width(mm)
leng
th(m
m)
Approx boundary
Original boundary
-40 -30 -20 -10 0 10 20 30 40-10
0
10
20
30
40
50
60
70
width(mm)
leng
th(m
m)
Approx boundary
Original boundary
-40 -30 -20 -10 0 10 20 30 40-10
0
10
20
30
40
50
60
70
width(mm)
leng
th(m
m)
Approx boundary
Original boundary
137
(Mikami et al., 2010). Larger amplitudes account for global shape, and small ones carry
fine detail of a boundary (Gonzalez & Woods, 2002). An example of magnitude plot of
ua , were plotted in Figure 5.3. The first 6 and the last 2 terms show higher contribution
in representing the boundary of the arch shape.
(a) Sample N022
(b) Sample N006
Figure 5.3: The magnitude plot for 2 casts as an example. The first 6 and the last 2
terms shows higher contribution in representing the boundary of the arch shape.
The q largest ua values were selected from the magnitude plot and its
corresponding ua were re-substituted in (5.3), whilst the remaining terms were
ignored. The 21 points 21,,1, rqrx obtained by using q of
ua terms where Nq ,
have to be compared with the corresponding 21 points from the original boundary,
21,,1, rorx when all ua terms were considered. A measure of similarity between
these two shapes will show the appropriateness of the choice of q as the shape feature of
the dental arch.
0 5 10 15 20 250
100
200
300
400
500
600
700
u
Am
plitu
de
, |a
u|
0 5 10 15 20 250
100
200
300
400
500
600
700
u
Am
plitu
de
, |a
u|
138
Since the images are aligned, the Procrustes distance
PD
N
r
qr
or
qr
or
1
)()'( xxxx , (5.5)
can be used as a similarity measure (Zhang & Lu, 2001; Stegmann and Gomez, 2002).
The value of q that minimizes PD will be regarded as the optimal choice of q.
The ability of the 8 FD terms to represent the 21 points of the teeth position is
illustrated in Figure 5.4. The Procrustes distance decreases rapidly when q changes from
2 to 8 and gradually levels off for increasing q. The 8-FD terms used were
i
q
ii aaa ˆ , ,ˆ ,ˆˆ21 A , (5.6)
where iu
iu
iu djca ˆˆˆ and qu ,,1 , may be used as an approximation of shape boundary.
The 8 selected FD are from the first 6 and the last 2 terms (Figure 5.3).
Figure 5.4: The Procrustes distances between the original 21 points and their estimated
positions using q-FD terms gradually levels off at q=8.
Figure 5.5 illustrates 2 cases of the original 21 points of the dental arch overlaid
with the 21 points derived from the 8-FD terms.
0 5 10 15 20 250
5
10
15
20
25
30
35
q-FD terms
Pro
cru
ste
s d
ista
nce
(m
m)
Cast number N006
Cast number N022
q=8
139
(a) Sample N022
(b) Sample N006
Figure 5.5: Plot of the original arch shape oix and the shape boundary q
ix approximated
using q = 8 FD terms.
5.2.3 Categories of Arch Shape Using FD Shape feature
Three categories of arch shape (1G ,
2G and 3G ) established in section 3.6.3 may
be used to define categories of shape when 8 FD was used to represent the shape feature
of the dental arch. Let each of the membership in the corresponding category
represented as ,ˆ , ,ˆ ,ˆˆ21
i
q
ii
i aaa A where 11,,1i for the first category, 33,,12 i
for the second category, and 47,,34 i for the third category with )3,2,1(ˆ ggA as
the respective means.
Table 5.1 shows descriptive statistics for the total casts and casts segregated
according to each shape category. Smaller values of standard deviation can be seen in
each category as compared to the total casts. This result supports the existence of three
categories of shape. The mean shapes for the 3 shape categories using FD,
iii aaa 821 ˆ , ,ˆ ,ˆˆ A as shape feature are illustrated in Figure 5.6. Shape feature using FD
provides additional information of all locations of teeth and the hamular notches instead
of only 8 teeth locations provided when the shape of the dental arch was represented by
),,,,,,,( 44332211RRRRRRRRR
i lwlwlwlwv .
-40 -30 -20 -10 0 10 20 30 40-10
0
10
20
30
40
50
60
70
width(mm)
leng
th(m
m)
Approx boundary
Original boundary
-40 -30 -20 -10 0 10 20 30 40-10
0
10
20
30
40
50
60
70
width(mm)
leng
th(m
m)
Approx boundary
Original boundary
140
Table 5.1: Mean, standard deviation (SD), and range of the FD terms.
FD
terms 1a 2a 3a 4a 5a 6a 7a 8a
Total
casts Mean
-0.23
+61.69i
-0.32
-58.69i
0.02
-7.59i
0.02
-9.03i
0.03
-4.50i
0.05
-2.56i
0.01
+6.24i
0.02
+4.94i
SD 6.47 2.76 1.42 1.03 0.73 0.52 1.03 1.73
Min -9.75
+51.34i
-2.61
-63.84i
-0.86
-10.48i
-1.21
-11.37i
-0.33
-5.84i
-0.45
-3.51i
-0.83
+4.24i
-1.41
+0.91i
Max 9.25
+72.16i
2.96
-53.80i
0.64
-4.33i
0.71
-6.91i
0.82
-2.78i
0.80
-1.30i
1.23+
8.15i
1.26+
8.41i
Shape
category
1
Mean 2.98
+63.57i
-1.65
-59.41i
-0.14
-7.72i
-0.15
-9.43i
0.06
-4.55i
0.17
-2.67i
-0.01
+5.89i
0.11
+4.58i
SD 5.22 2.20 1.83 0.88 0.63 0.53 0.89 1.62
Min -2.07
+56.67i
-2.61
-63.5i
-0.86
-10.48i
-1.21
-10.77i
-0.28
-5.47i
0.01
-3.23i
-0.45
+4.72i
-0.99
+3.12i
Max 9.25
+68.68i
-0.16
-56.01i
0.31
-4.33i
0.26
-8.07i
0.33
-3.49i
0.34
-1.30i
0.46
+7.63i
0.98
+8.41i
Shape
category
2
Mean -0.65
+57.6i
-0.48
-57.6i
0.040
-7.22i
0.04
-8.35i
0.00
-4.27i
0.08
-2.45i
0.06
+6.74i
0.10
+5.57i
SD 4.83 2.40 1.30 0.63 0.80 0.47 0.98 1.26
Min -9.75
+51.34i
-2.23
-63.84i
-0.56
-9.59i
-0.34
-9.70i
-0.33
-5.71i
-0.43
-3.25i
-0.83
+5.45i
-1.41
+3.71i
Max 4.76
+64.03i
2.27
-53.8i
0.61
-5.02i
0.57
-6.91i
0.46
-2.78i
0.80
-1.57i
1.23
+8.15i
1.26
+8.03i
Shape
category
3
Mean -2.08
+66.58i
0.98
-59.83i
0.12
-8.07i
0.12
-9.77i
0.050
-4.81i
-0.10
-2.65i
-0.030
+5.73i
-0.17
+4.24i
SD 4.50 2.58 1.16 0.98 0.57 0.55 0.87 2.15
Min -7.22
+61.11i
-2.51
-62.7i
-0.31
-9.66i
-0.34
-11.37i
-0.26
-5.84i
-0.45
-3.51i
-0.53
+4.24i
-0.87
+0.91i
Max 4.08
+72.16i
2.96
-56.55i
0.64
-5.97i
0.71
-8.78i
0.82
-4.07i
0.27
-1.95i
0.58
+7.21i
0.40
+7.23i
141
Figure 5.6: Mean shapes of 21 ˆ,ˆ AA and 3A using 8 FD as shape feature.
5.2.4 Probability Distribution of Shape Categories using FD
For completeness, variation of each of the shape category must be stated and this
is done by seeking the probability distribution of the iq
iii aaa ˆ , ,ˆ ,ˆˆ
21 A . The existence
of mean and variation of the dental arch shape will consequently provide the statistical
shape model. From (5.2), iA be can be written as
iii jdcA ˆˆˆ , (5.7)
where )ˆ,,ˆ,ˆ(ˆ21
iq
iiTi ccc c , )ˆ,,ˆ,ˆ(ˆ
21iq
iiTi ddd d and 11,,1i for the first shape category,
34,,12 i for the second category, and 49,,35i for the third category. Now, for the
convenience of notation, let )(ˆ gA (g = 1, 2, 3) represent the k-the shape category. The
-40 -30 -20 -10 0 10 20 30 40
0
10
20
30
40
50
60
70
width (mm)
leng
th (
mm
)
A(1)
A(2)
A(3)
142
complex random variable in (5.7) is a univariate complex normal random variable if the
distribution of each ˆ
ˆ,,
ˆ
ˆ,
ˆ
ˆ
2
2
1
1
q
q
d
c
d
c
d
c is a bivariate normal distribution (see Definition
1 in section 2.7.1). Further, dc ˆˆ j will then have the multivariate complex normal
distribution (see Definition 4 in section 2.7.1).
Table 5.2: Test statistic to investigate bivariate normality of each ˆ
ˆ,,
ˆ
ˆ,
ˆ
ˆ
2
2
1
1
q
q
d
c
d
c
d
c
using HZ test in the 3 shape categories. LCV and UCV are the abbreviation for lower
and upper critical value respectively.
FD term
Shape category 1
2.5% LCV = 0.1403
2.5% UCV = 0.7352
Shape category 2
2.5% LCV = 0.1786
2.5% UCV = 0.8568
Shape category 3
2.5% LCV = 0.1531
2.5% UCV = 0.7784
Tdc 11ˆˆ 0.4072 0.5170 0.2277
Tdc22
ˆˆ 0.3208 0.5936 0.2922
Tdc 33ˆˆ 0.3692 0.2945 0.3634
Tdc 44ˆˆ 0.4929 0.7338 0.4376
Tdc 55ˆˆ 0.1847 0.3428 0.5068
Tdc 66ˆˆ 0.4918 0.3273 0.3212
Tdc 77ˆˆ 0.2401 0.5191 0.3006
Tdc 88ˆˆ 0.5462 0.2975 0.6890
143
Table 5.3: Test statistic to investigate bivariate normality of each ˆ
ˆ,,
ˆ
ˆ,
ˆ
ˆ
2
2
1
1
q
q
d
c
d
c
d
c
using DH test in the 3 shape categories. LCV and UCV are the abbreviation for lower
and upper critical value respectively.
FD term
Shape category 1
2.5% LCV = 0.4844
2.5% UCV = 11.1433
Shape category 2
2.5% LCV = 0.4844
2.5% UCV = 11.1433
Shape category 3
2.5% LCV = 0.4844
2.5% UCV = 11.1433
Tdc 11ˆˆ 2.9185 9.5717 1.8439
Tdc22
ˆˆ 3.6049 11.5220 3.7941
Tdc 33ˆˆ 1.8408 1.7985 1.9049
Tdc 44ˆˆ 7.4605 8.8383 6.4315
Tdc 55ˆˆ 4.8990 3.0334 4.3175
Tdc 66ˆˆ 6.0771 6.2587 0.1502
Tdc 77ˆˆ 5.0529 4.7464 0.4338
Tdc 88ˆˆ 8.0012 3.4041 7.6285
Table 5.4: Test statistic to investigate bivariate normality of each ˆ
ˆ,,
ˆ
ˆ,
ˆ
ˆ
2
2
1
1
q
q
d
c
d
c
d
c
using Royston’s test in the 3 shape categories. The critical values were denoted in the
bracket as (Lower 2.5% , Upper 2.5%).
FD term
Shape category 1
Shape category 2
Shape category 3
Critical value Test
statistic Critical value
Test
statistic Critical value
Test
statistic
Tdc 11ˆˆ
(0.0503 , 7.3702)
0.5543
(0.0458 , 7.2788)
1.3450
(0.0539 , 7.4424)
0.0331
Tdc22
ˆˆ (0.0506 , 7.3776)
1.8940 (0.0507 , 7.3786) 4.8374 (0.0526 , 7.4165) 2.2363
Tdc 33ˆˆ
(0.0506 , 7.3777)
0.3441 (0.0506 , 7.3777) 0.1963 (0.0508 , 7.3803) 0.5318
Tdc 44ˆˆ
(0.0519 , 7.4029)
5.9471 (0.0506 , 7.3770) 3.8766 (0.0506 , 7.3778) 4.5656
Tdc 55ˆˆ
(0.0506 , 7.3778)
1.0985 (0.0506 , 7.3778) 1.9266 (0.0427 , 7.2115) 5.0150
Tdc 66ˆˆ
(0.0533 , 7.4303)
5.8561 (0.0504 , 7.3738) 0.3820 (0.0499 , 7.3625) 1.3295
Tdc 77ˆˆ
(0.0472 , 7.3076)
0.4836 (0.0507 , 7.3788) 2.1720 (0.0512 , 7.3887) 0.0134
Tdc 88ˆˆ (0.0506 , 7.3778) 6.0304 (0.0506 , 7.3778) 0.5082 (0.0507 , 7.3786) 2.9674
144
Table 5.2 to Table 5.4 strongly suggest that each ˆ
ˆ,,
ˆ
ˆ,
ˆ
ˆ
2
2
1
1
q
q
d
c
d
c
d
c has a
bivariate normal when making use of HZ, DH and Royston’s tests for normality.
Except for DH test on
2
2
ˆ
ˆ
d
c in shape category 2, and Royston’s test on
1
1
ˆ
ˆ
d
c and
7
7
ˆ
ˆ
d
c in
shape category 3 which appear to reject the null hypothesis. Nevertheless, if 1%
significance level is considered, the test is likely to show bivariate normality. Therefore,
dc ˆˆ j will have the multivariate complex normal probability distribution
ggpCN Hθ ,8 where 3 ,2 ,1g are the corresponding shape category.
The three fitted shape models are given as
3,2,1 ,ˆ~)(ˆ:
gMVCNg g
gg MAA , (5.8)
where the means are given as
), 4.5777i0.1148 5.8928i,0.0124- 2.6657i,-0.1717 4.5459i,-0.0583
9.4307i,-0.1546- 7.7225i,-0.1445- 59.4081i,-1.6451- 63.5671i,2.9753(ˆ 1
A (5.9)
),5.5652i0.0980 6.7442i,0.0559 2.4500i,-0.0793 4.2743i,-0.0037
8.3538i,-0.0374 7.2191,-0.0425 57.5987i,-0.4808- 57.6496i,-0.6514(ˆ 2
A (5.10)
).i2385.40.1655- i,7306.50.0302- i,6505.2i,-0.10328052.40.0519
9.7729i,-0.1155 8.0748i,0.1246 i,8310.590.9793 i,5773.66-2.0787(ˆ 3
A (5.11)
and shape variability can be explained by the estimated Hermitian covariance matrices
whereby the element in the i-th row and j-th column is equal to the complex conjugate
of the respective element for all indices i and j, given as
2.61.......
0.17i + 0.680.79......
0.00i - 0.04-0.01i + 0.29 0.28.....
0.14i + 0.17-0.00i - 0.360.02i - 0.210.40....
0.10i + 0.080.11i - 0.310.04i + 0.270.12i - 0.280.76...
0.23i + 0.59- 0.24i - 0.670.12i - 0.720.15i - 0.730.36i - 1.083.36..
0.36i - 0.05-0.57i -0.200.09i - 0.030.37i - 0.510.08i + 0.680.35i - 0.764.85.
1.97i - 2.34-1.22i + 3.19- 0.51i + 1.36-1.34i + 1.83-0.27i + 2.77-0.68i + 5.67-0.92i + 4.79-27.32
1M
, (5.12)
145
1.57.......
0.18i + 0.61 0.95......
0.13i + 0.15- 0.09i + 0.12 0.21.....
0.04- 0.02i + 0.430.01i - 0.15 0.64....
0.17i + 0.06-0.00i - 0.210.10i - 0.16 0.04i - 0.280.39...
0.03i - 0.59- 0.17i + 0.310.01i - 0.320.03i + 0.730.15i + 0.491.67..
0.15i + 1.06-0.36i + 1.09-0.13i + 0.04-0.17i + 0.05- 0.11i - 0.080.03i - 0.495.78.
1.70i - 0.59-1.23i - 2.71-0.50i + 1.03-0.15i - 2.32-1.97-1.24i + 3.61-3.87i + 2.86-23.28
2M
, (5.13)
4.63.......
0.27i + 0.840.76......
0.19i + 0.24-0.13i + 0.02- 0.30.....
0.07i + 0.07- 0.06i + 0.090.02i - 0.19 0.32....
0.01i - 0.580.07i + 0.06-0.04i + 0.200.00i + 0.260.96...
0.10i + 1.69-0.13i + 0.17-0.12i + 0.360.12i + 0.370.07i + 0.22 1.34..
0.00i + 2.61- 0.08i + 0.94-0.45i + 0.33-0.39i + 0.31-0.66i + 0.14-0.85i + 1.14 6.64.
5.26i - 1.91-0.78i - 1.02- 0.01i + 0.12-0.18i - 0.38-1.22i - 1.60-1.08i + 1.36-3.75i + 5.04- 20.27
3M
. (5.14)
The mean shape and shape variability of the arch shape model )(ˆ gA from the
MVCN distribution can be illustrated. The inverse DFT of gg jg dcA ˆˆ)(ˆ restores
),()()( kjykxks ,20 , ,0 k and the Cartesian coordinate pair
,)20(),20(,,)0(),0( yxyx were plotted as the g-th mean shape.
The variance of )(ˆ gA that can be obtained from the diagonal elements of
Hermitian covariance matrix gM , was used to illustrate the the variation of each shape
model from its mean shape without considering the covariances. Since )(ˆ gA may also
be written as i
u
i
u
i
u djca ˆˆˆ , where 8,...,1u , and 11,,1i indicates the first shape
category, 34,,12i for the second category, and 49,,35i for the third category,
consequently its mean, variance and standard error may also be written as i
u
i
u
i
u djca ˆˆˆ
, )ˆvar()ˆvar()ˆvar( i
u
i
u
i
u dca (Andersen, et al., 1995, p. 7) and
)ˆvar()ˆvar()ˆ( iu
iu
iu dcase , (5.15)
respectively. From (5.15), the standard error for iuc and
iud may be difficult to obtain.
Therefore, we define the standard error for iuc and
iud as
146
2
)ˆ()ˆ()ˆ(
i
ui
u
i
u
asedsecse . (5.16)
The shape variation of the dental arch shape may be represented by looking at the
variation of each mean of the u-th FD term, ua , and finding the following points
)ˆ(ˆ)ˆ(ˆ i
u
i
u
i
u
i
u
i
u dsedjcsecQ , (5.17)
)ˆ(ˆ)ˆ(ˆ i
u
i
u
i
u
i
u
i
u dsedjcsecR , (5.18)
)ˆ(ˆ)ˆ(ˆ i
u
i
u
i
u
i
u
i
u dsedjcsecS , (5.19)
)ˆ(ˆ)ˆ(ˆ i
u
i
u
i
u
i
u
i
u dsedjcsecT . (5.20)
The inverse DFT of the )8,,1( and ,, uTSRQ i
u
i
u
i
u
i
u yields and ,, i
k
i
k
i
k SRQ
)20,,0( kT i
k . The variation of the k-th point may be illustrated in Figure 5.7. The
mean shape and shape variability of the arch shape model 3.2.1),(ˆ ggA are illustrated
in Figures 5.8, Figures 5.9 and Figures 5.10 respectively.
ka
Sk
Tk
Rk
Qk
Figure 5.7: Variation of the k-th point from its mean.
* *
* *
Width (mm)
(mm
Length (mm)
(mm
147
Figure 5.8: Mean shape and variation for shape model )1(A .
Figure 5.9: Mean shape and variation for shape model )2(A .
-40 -30 -20 -10 0 10 20 30 40
-10
0
10
20
30
40
50
60
width (mm)
leng
th (
mm
)
-40 -30 -20 -10 0 10 20 30 40
-10
0
10
20
30
40
50
60
width (mm)
leng
th (
mm
)
148
Figure 5.10: Mean shape and variation for shape model )3(A .
5.2.5 Test of Separation between MVCN Shape Models
Three distinct MVCN shape models exist if these models possess significant
differences in shape. A review of the literature shows that only one sample test
concerning the complex mean of MVCN distribution has been considered (Giri, 1965;
Khatri, 1965; Andersen et al., 1995). The aim of this section was to develop hypothesis
testing for two sample means from MVCN based on the Hotelling 2T test by using the
established relationship of MVCN to the multivariate real normal distribution. The
proposed test was employed for confirmation of the existence of the 3 MVCN shape
models.
-40 -30 -20 -10 0 10 20 30 40
-10
0
10
20
30
40
50
60
width (mm)
leng
th (
mm
)
149
5.2.5.1 The Proposed Hotelling T2 test for MVCN Distribution
Let 1 1 1~ ,pCNX θ H and 2 2 2~ ,pCNX θ H denotes the first and
second shape category respectively. Let 1 21gn
g g g ggn
X X X X and
*
1
1
1
gnr r
g g g g grgn
M X X X X be the unbiased estimators of gθ and 1,2g g H
respectively and 1 1 1 2
1 2
1 1
2pooled
n n
n n
M MM as an estimate of pooled Hermitian
covariance matrix H , where *
denotes the conjugate and transpose of .
A relationship with the multivariate real normal distribution from equation
(2.53) gives 1 2 1 1
1~ ,
2pN
X θ H and
}{
2
1],[~][ 2222 HθX pN respectively.
Consequently,
1 2 2 1 11 2
1 1~ ,
2 2pN
n n
X X θ θ H , (5.21)
1 2 1 2 21 2
1 1~ , ,
2 2pN
n n
X X θ θ 0 H (5.22)
and let
12
1 2 1 2 21 2
1 1~ , ,
2 2pN
n n
u X X θ θ 0 H (5.23)
where H denotes the pooled Hermitian covariance matrix. From equation (2.56),
1 1 2 2 1 21 1 ~ , 2 ,pn n CW n n M M H (5.24)
and can also be written as
1 1 2 2 2 1 22 1 1 ~ { }, 2 ,pn n W n n W M M H (5.25)
since equation (2.54) applies.
150
The test statistic for the Hotelling 2T test statistic under 1 2H 0:θ θ 0 can be
derived similar to the real case:
1
2 *
1 2
12 * 1
1 2 1 21 2
12
1 2 1 21 2
2
1 12
2 2
1 1
2 2
Tn n
n n
n n
pooled
Wu u
X X θ θ M
X X θ θ
1* 1
1 2 1 21 2
1 1 1
2 2 2n n
pooledX X M X X
* 11 21 2 1 2
1 2
.n n
n n
pooledX X M X X (5.26)
and the distribution of 2 T is given as
1 2
2 1 22 ,( 2) (2 1)
1 2
( 2)2~ .
( 2) (2 1)p n n p
n n pT F
n n p
(5.27)
5.2.5.2 Simulation Study on Performance of the Hotelling T2 for MVCN
Distribution
Simulation study were carried out using R software to investigate the performance of
the proposed test under the null hypothesis, 0H : 1 2θ θ and was evaluated according to:
1. Type I error rates against equal means
2. Power of test against unequal means
151
5.2.5.2.1 Type I Error Rates
To estimate the Type I error rates, the following steps were taken:
Step 1: Two random samples of p -variate complex normal distributions with size n
was drawn from 1 1 1~ ,pCNX θ H and 2 2 2~ ,pCNX θ H and generated by using R
package cmvnorm (Hankin, 2015), where 1 2θ θ and 1 2H H .
Step 2: Calculate the test statistic in equation (5.26) and find its corresponding p -
value using equation (5.27).
Step 3: The above steps of obtaining the p -value were repeated for s = 10000 times.
Step 4: The proportion of 10,000 samples for which the test rejects the 0H at
05.0 was calculated. This is equivalent to calculating the probability of making
Type I error, 0 0reject | is trueP H H . The corresponding error rate in percentage was
then obtained.
The examples of the generated complex normal data for 4p and 8p with
equal means are illustrated in Figure 5.11. Since the 0H is true, the test statistic should
reject the 0H at about the nominal rate of 5%. However, Table 5.5 shows high values of
Type I error rate, which gets higher as p increases. A correction factor imposed to the
test statistic 2T may be required to decrease the chance of the test getting rejected.
152
Table 5.5: Empirical Type I error rate against equal means for different sets of
parameters
Parameters
Without
Correction
Factor
With
Correction Factor
Type I Error 5 %
Type I
Error 5 %
Type I
Error 10 %
1 2
1 2
4,
(H ) (H )
1
,
1
0 0
1 0θ θ .
1
0
p
diag diag
i
i
i
i
20n 20.99 % 4.20 % 8.59 %
30n 20.14 % 4.45 % 8.97 %
50n 18.45 % 4.58 % 9.34 %
100n 18.10 % 4.80 % 9.65 %
250n 18.45 % 4.95 % 9.36 %
500n 17. 38 % 5.16 % 10.12%
1
2
30,
(H )
(H )
1
.
1
n
diag
diag
1 2
2,
θ θ
0 0
1 0
p
i
i
8.66 % 4.29 % 9.44 %
1 2
4,
θ θ
0 0
1 0
1
0
p
i
i
i
i
20.14 % 4.45 % 8.97 %
1 2
8,
θ θ
1 2
2
2
1 2
1 2
2
2
1 2
p
i
i
i
i
i
i
i
i
48.35 % 4.75 % 9.94 %
153
(a) 4p (b) 8p
Figure 5.11: Examples of the generated complex normal data for 4p and 8p .
Note that this test was derived using the complex and real normal relationships
which involves transformation of p complex dimension to 2 p (see equation (2.53)).
Therefore, the best guesstimate of the correction factor is two times what it should be
and the test statistic with correction factor is given as
*2 11 21 2 1 2
1 2
2 .pooled
n nT
n n
X X M X X (5.28)
Simulation study was repeated after correction factor was imposed on the test
statistic 2T . The Type I error rates for different sets of parameters are close to the
nominal rates when 5% and 10% significance level were considered, especially for a
large sample size.
5.2.5.2.1 Power of Test
To estimate the power of test, the following procedure was carried out:
Step 1: Two random samples of size n from two p -variate complex normal
distributions were generated similar to Step 1 and Step 2 in Section 4.1, with 1 2θ θ .
Step 2: Calculate the test statistic in equation (5.28) with correction factor imposed,
and find its corresponding p -value using equation (5.27).
Step 3: The above steps of obtaining the p -value were repeated for s = 10000 times.
-1 -0.5 0 0.5 1 1.5 2-1
-0.5
0
0.5
1
1.5
2
2.5
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
154
Step 4: The proportion of 10,000 samples for which the test rejects the null
hypothesis 0H at 0.05 significance level was calculated. This is equivalent to
calculating the probability of not making a Type II error,
false) is |reject (1 00 HHP .
The corresponding error rate in percentage was
then obtained as the power of the test.
The examples of the generated complex normal data for 4p with unequal
means are illustrated in Figure 5.12. Here, the test was expected to reject the 0H . Table
5.6 shows the power estimates when different sets of mean and covariance matrix were
considered, representing data with mild to complete separation. As expected, the power
of test gets higher as the differences between the means gets bigger and as the value of
covariance matrices gets smaller. With obvious separation of the means and smallest
values of covariance matrices, the power of the test is nearly 100%.
Figure 5.12: Examples of generated complex normal data
for Different Values of 1θ and
2θ .
-1 -0.5 0 0.5 1 1.5 2 2.5 3-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1 0 1 2 3 4 5-1
0
1
2
3
4
5
155
Table 5.6: Empirical power of test against unequal means for different sets of
parameters.
Parameters Power of
test
1
1 2
4, 30,
0 0
1 0θ ,
1
0
(H ) (H )
1
1
p n
i
i
i
i
diag diag
2
0.1 0.1
1.1 0.1θ
1.1 0.1
0.1 1.1
i
i
i
i
88.90 %
2
0.3 0.3
1.3 0.3θ
1.3 0.3
0.3 1.3
i
i
i
i
96.41 %
2
0.1 0.1
1.1 0.1θ
1.1 0.1
0.1 1.1
i
i
i
i
99.90 %
2
1 1
2 1θ
2 2
1 2
i
i
i
i
100 %
1
4, 30,
0 0
1 0θ ,
1
0
p n
i
i
i
i
2
0.3 0.3
1.3 0.3θ
1.3 0.3
0.3 1.3
i
i
i
i
1
2
(H )
(H )
0.5
0.5
diag
diag
99.54 %
1
2
(H )
(H )
1
1
diag
diag
96.41 %
1
2
(H )
(H )
1.5
1.5
diag
diag
89.88 %
5.2.5.3 Results on Test of Separation between MVCN Shape Models of Dental Arch
Let
g
gMVCNg MAA ,ˆ~)(ˆ8
denotes the g-th shape model. Estimate the
pooled Hermitian covariance matrix 2
)1()1(
21
2111
nn
nnpooled
MMM and use equations (5.28)
and (5.27) to test for mean separation between two MVCN means. The result is shown
in Table 5.7.
156
Only 1 2ˆ ˆA A and 2 3ˆ ˆA A appear to reject 0H at 5% significance level, which
indicates separation of shape category 2 from the other two shape categories. 1 3ˆ ˆA A on
the other hand appear to accept the 0H , however they are significantly separated when
10% significant level is considered. This result confirms the existence of the three
MVCN shape models.
Table 5.7: Hotelling T2 test for comparing two MVCN mean.
Mean vector Critical value HotellingT2 statistics
21 ˆˆ AA Lower 2.5% = 11.2264
Upper 2.5% = 85.6021 97.66
21 ˆˆ AA Lower 2.5% = 14.7208
Upper 2.5% = 187.5004 16.17
Lower 2.5% = 10.6120
Upper 2.5% = 74.1745 135.47
5.3 Discrimination of Shape for MVCN Model
This section developed shape discrimination procedure for the maxillary dental
arch using the established statistical shape models for MVCN. Since the probability
distribution of the shape has been identified as MVCN, the linear discrimination
function (LDF) for MVCN may be derived and used as discrimination method.
However, the Gaussian assumptions for the discriminant function are particularly
restrictive and sometimes difficult to satisfy if the sample size for the shape category is
small. If violated, it may reduce the performance of discriminant analysis. An
alternative discrimination method will be proposed which uses the determinant ratio of
two covariance matrices that are not necessarily from Gaussian distribution.
Subsequently, comparison between the LDF and )( iCOVRATIO will be carried out via
simulation study.
157
5.3.1 LDF for MVCN Distribution
Let x be the landmarks of the i-th object and the probability density function for
the g-th population ( g ) from the MVCN distribution be
)()(exp)det()( 1*1gggg
p
gf θxHθxHxX
, (5.29)
where gθ and gH are the mean and Hermitian covariance matrix respectively. The
LDF for MVCN is given as
, ,)()()()(
)()()()()det()det(
)()()det(log
)()()det(log
)( log)( log
)(
)(log)(
1*1*
1*1*
1*
1*
HHHθxHθxθxHθx
θxHθxθxHθxHH
θxHθxH
θxHθxH
xx
x
xx
kgkkgg
kkkgggkg
kkkk
gggg
kg
k
gMVCN
gk
p
p
ff
f
fh
(5.30)
where the covariance matrices are equal and kg . The unbiased estimator of
estimators of θ and H are given as
1ˆ 21 gn
ggggn
xxxθ and
2
1 2
)1(
gg
g
N
nHH
respectively (Goodman, 1963).
Let 1x . The )(1 xf from equation (5.29) is maximized because the
exponent is minimized and )()( 21 xx ff (Mardia et al., 1979). Consequently, it is
expected to obtain 0)(12 xMVCNh (equation (5.30)). On the contrary, if 2x , then it
is expected to have 0)(12 xMVCNh . The discrimination rule for two CN populations
1
and 2 can be therefore generalized as:
158
If 0)(12 xMVCNh , then 1x and,
if 0)(12 xMVCNh , then 2x . (5.31)
Similar to the real MVN case, the LDF for 2 populations in equation (5.31) can be
extended for 3 populations as follows (Johnson and Whicern 1992; Mardia et al. 1979):
if 0)(12 xMVCNh and 0)(13 xMVCNh , then 1x ,
if 0)(12 xMVCNh and 0)(23 xMVCNh , then 2x , (5.32)
if 0)(13 xMVCNh and 0)(23 xMVCNh , then 3x .
5.3.2 Proposed MVCN
iCOVRATIO )( for MVCN Distribution as Discrimination Method
The COVRATIO statistics for the MVCN model will be formulated using the
same idea proposed in section (4.4.1) for the purpose of discrimination. Let 1H and 2H
be the Hermitian matrix which corresponds to a real matrix being symmetry for
population 1 and 2 respectively, where each of the population has a known
probability distribution. Instead of using reduced observation as formulated by Belsley
et al. (1980), define the ratio of covariance determinant for 1 as:
H
H
1
)(1
)(1
iMVCNiCOVRATIO
, (5.33)
where H)(1 i
is the determinant of Hermitian covariance matrix )(1 iH which is
estimated by adding new thi observation in 1 and
1H is the determinant of 1H .
The same thi observation will be added in 2 and its
MVCNiCOVRATIO )(2 is obtained
using
159
H
H
2
)(2
)(2
iMVCNiCOVRATIO
. (5.34)
Smaller value of MVCN
iCOVRATIO )( (which is close to 1) indicates no variation when new
observation was included in a particular population. In other words, the thi observation
is consistent with the other observations and therefore belongs to the population.
Therefore, the discrimination rule using MVCN
iCOVRATIO )( for 2 MVCN
populations is given as:
(5.35)
The above result can be generalized for g populations, given as:
(5.36)
5.3.3 Simulation Study on Performance of LDF and MVCN
iCOVRATIO )(
A simulation study was designed to examine the performance of LDF and
)( iCOVRATIO for CN model. Random samples from two CN populations were
generated and re-assigned to their corresponding population using LDF and
i)(COVRATIO . The misclassification probabilities using these discrimination methods
were then compared. This procedure was carried out as follows:
Step 1: Two sets of complex random samples with sample size n from two unequal p-
variate MVCN distributions 1111 ,~: HθX pMVCN and
Allocate new thi observation in 1 if
MVCNi
MVCNi COVRATIOCOVRATIO )(2)(1 ,
Else, allocate to 2
Allocate new thi observation in
j if MVCN
ihMVCN
ij COVRATIOCOVRATIO )()( ,
where ghj ,,2,1, and hj .
160
),(~: 2222 HθX pMVCN were generated using R package cmvnorm
(Hankin, 2015). The parameters for 1 when 4p were set as
i
i
i
i
0
1
01
00
1θ and
5.0
5.0
)( 1 Hdiag ,
where )( 1Hdiag denotes the diagonal matrix with its off-diagonal
elements are 0.
Step 2: Generate another two complex random samples A and B which were drawn
from 1 and
2 respectively.
Step 3: Calculate the LDF for MVCN in equation (5.30) using random sample A and
B , denoted as )(12 AMVCNh and )(12 BMVCNh respectively.
Step 4: Calculate )()(1 AMVCN
iCOVRATIO , )()(1 BMVCN
iCOVRATIO , )()(2 AMVCN
iCOVRATIO and
)()(2 BMVCN
iCOVRATIO using equation (5.33) and (5.44), by adding random sample A
and B in both sets of random samples generated from 1 and 2 in step 1.
Step 5: Discrimination rule for LDF (in equation (5.31)) and MVCN
iCOVRATIO )( (in
equations (5.35) and (5.36)) were used to determine the membership of A and B .
Step 6: The above steps were repeated for s = 10000 times.
Step 7: The proportion of A and B which does not assign to their corresponding true
population when discriminated using LDF and MVCN
iCOVRATIO )( may be regarded as
the misclassification probability.
161
Table 5.8: Misclassification probability of LDF and MVCN
iCOVRATIO )( for MVCN
model when different sample size, dimension, mean vector and covariance matrices
were considered.
Parameters
1 2
LDF MVCN
iCOVRATIO )(
LDF MVCN
iCOVRATIO )(
,30,4 np
)(
)(
2
1
H
H
diag
diag
i
i
i
i
8.18.0
8.08.1
8.08.1
8.08.0
2θ 0.0140 0.0092 0.0122 0.0110
i
i
i
i
21
22
12
11
2θ 0.0026 0.0011 0.0030
0.0013
i
i
i
i
2.22.1
2.22.2
2.12.2
2.12.1
2θ 0. 0005 0 0. 0004 0
,30,4 np
i
i
i
i
21
22
12
11
2θ ,
3.0
3.0
)( 2 Hdiag 0.0022 0.0003 0.0001 0
5.0
5.0
)( 2 Hdiag 0.0026 0.0011 0.0030
0.0013
7.0
7.0
)( 2 Hdiag 0.0034 0.0012 0.0083 0.0042
,4p
i
i
i
i
21
22
12
11
2θ ,
)(
)(
2
1
H
H
diag
diag
20n 0.0211 0.0123 0.0203 0.0125
30n 0.0026
0.0011 0.0030 0.0013
50n 0.0024 0.0009 0.0015 0.0008
162
Table 5.8: Continue.
,30n
)(
)(
2
1
H
H
diag
diag
2p ,
i
i
01
001θ ,
i
i
12
112θ
0.0222 0.0147
0.0224 0.0148
4p ,
i
i
i
i
21
22
12
11
2θ
0.0026 0.0011 0.0030
0.0013
8p ,
i
i
i
i
i
i
i
i
0
1
2
3
03
02
01
00
1θ ,
i
i
i
i
i
i
i
i
21
22
23
24
4
3
2
1
2θ
0 0
0.0001 0
163
The examples of the generated complex normal data for p = 4 and p = 8 with unequal
means are illustrated in Figure 5.13. Table 5.8 shows the performance of LDF and
MVCNiCOVRATIO )( for MVCN model when different values of n, p, μ and Σ were
considered. Overall, the misclassification probabilities for LDF and i)(COVRATIO in
all cases are less than 0.05. However, the misclassification probability reduces when a
particular value of 2θ was chosen to indicate larger separation between
2 and 1 .
The dispersion of the data in a particular population also dictates the misclassification
probability, whereby larger value of H may results in higher chances of a sample to be
assigned to an incorrect population. Increasing value of n and p gives smaller
misclassification probability.
The MVCN
iCOVRATIO )( method gives smaller probability of misclassification as
compared to LDF in all cases, indicating better performance of shape discrimination.
The )( iCOVRATIO is theoretically desirable when discriminating shape of the dental
arch, since differences of shapes may be barely noticeable and the discriminatory
information may not necessarily in the mean shape, but rather in shape variability.
Further, the )( iCOVRATIO can be employed when discriminating k groups of shape and
without any assumptions on the distribution of the shape.
Figure 5.13: Examples of the generated complex normal data for p = 4 and p = 8.
164
5.3.4 Results for Shape Discrimination of Dental Arch using FD
The membership of the 47 control samples which belong to each of the 3 shape
models of the dental arch using FD iq
iii aaa ˆ , ,ˆ ,ˆˆ
21 A as shape feature were tracked
from the dendrogram (see Figure 3.12 in section (3.6.1)). The LDF and MVCN
iCOVRATIO )(
were employed to re-assign these casts into one of 3 populations of the MVCN shape
models. Their misclassification probability was investigated to ensure that these
methods are deemed good for shape discrimination.
Table 5.9 shows the misclassification probabilities when the samples were re-
assigned to *
2
*
1 , or *
3 using LDF and MVCN
iCOVRATIO )( . The results for LDF show
relatively small misclassification probabilities except for the discrimination of 1 from
3 . However this is expected since the test of separation between these 2 shape models
was significantly separated when 10% significant level is considered (see Table (5.7)).
MVCNiCOVRATIO )( outperformed the LDF method by having zero misclassification
probabilities. These results demonstrate the ability of the MVCN
iCOVRATIO )( to discriminate
the dental arch shape correctly as a better discrimination method.
Table 5.9: Misclassification probability when the 47 dental casts were re-assigned into
either one of the population of the shape model using the LDF and MVCN
iCOVRATIO )( .
Discrimina-
tion
True
population
*1 *
2 *3
LDF MVCN
iCOVRATIO )( LDF MVCN
iCOVRATIO )( LDF MVCN
iCOVRATIO )(
1 - - 0.18 0 0.18 0
2 0.09 0 - - 0.09 0
3 0.21 0 0.07 0 - -
165
5.4 Linking Anatomical Landmarks to the MVCN Shape Models on Edentulous
Arch
The only landmarks that a dentist may obtain from an edentulous patient are the
incisive papilla and hamular notches (Figure 5.14). Investigation on these landmarks in
dentate arches which correspond to MVCN shape category models may provide
indicator for determining the shape category or shape model for the edentulous arches.
Consequently, setting up of artificial teeth for edentulous arch may be carried out
according to the assigned MVCN shape category model. This knowledge may provide a
guide for the inexperienced dentists and dental technicians when setting up teeth on
complete dentures.
Since each of the shape categories follows the MVCN, the corresponding shape
category using the anatomical landmarks as shape feature may have the same MVCN
distribution. Confirmation of group separation for the same 47 casts (using only the
incisive papilla and hamular notches, without taking into account the teeth) was carried
out and referred to as the landmark models. The casts were then assigned into either one
of 3 populations of the landmark models using the proposed MVCN
iCOVRATIO )( . Their
misclassification probabilities were also investigated for verification of the landmark
models.
Figure 5.14: Edentulous cast showing the incisive papilla (IP) and hamular notches
(HN).
IP
HN
166
5.4.1 Shape Model of Anatomical Landmarks
Let the set of FD of anatomical landmarks; the right and left hamular notches
and incisive papilla, for the i-th control cast be denoted as ),,( 321 iiii LLLL , where
47 ,...,1i . The membership of the casts were identified according to groups
established and the vector of the anatomical landmarks were denoted by
ggg LLLg 321 ,,)( L . Since each of the shape categories are following the MVCN, the
corresponding shape category when using the anatomical landmarks as shape feature
may have the same MVCN distribution. Therefore, let ) ,(~)( 3
landmark
g
gMVCNg MLL ,
where gL and
landmark
g M are the estimator of gθ and gH (g = 1,2,3). Confirmation of
group separation using the anatomical landmarks will be investigated using test derived
in (5.28).
Table 5.10 indicates separation of shape categories for )(gL where .3,2,1g
Table 5.11 on the other hand shows descriptive statistics for the total casts and casts
segregated according to each shape category. Smaller values of standard deviation can
be seen in each category as compared to the total casts. These further support the
existence of three categories of shape using anatomical landmarks.
Table 5.10: The T2 test for comparing two MVCN mean for anatomical landmarks.
Mean vector Critical value HotellingT2 statistics
21LL
Lower 2.5% = 1.4035
Upper 2.5% = 21.0661 26.7173
31LL
Lower 2.5% = 1.4738
Upper 2.5% = 24.6937 52.6035
32LL
Lower 2.5% = 1.3868
Upper 2.5% = 20.2874 94.3139
167
Table 5.11: Mean, standard deviation (SD) and range of the FD terms for the
anatomical landmarks.
FD
terms 1L 2L 3L
Total casts Mean -0.07 + 4.73i 7.45 - 4.33i -0.78 - 0.44i
SD 0.57 0.45 0.47
Min 0 + 4.02i 6.60 - 3.77i -0.23 - 0.07i
Max 0.33 + 5.44i 8.21 - 4.73i -1.52 - 0.71i
Shape
category 1 Mean 0.25 + 4.72i 7.46 - 4.29i -0.92 - 0.26i
SD 0.42 0.38 0.49
Min -0.04 + 4.47i 6.79 – 4.14i -0.26 - 0.12i
Max 0.85 + 5.03i 7.94 - 50i -1.55 - 0.43i
Shape
category 2 Mean -0.11 + 4.54i 7.27 - 4.25i -0.59 - 0.32i
SD 0.52 0.46 0.32
Min -0.12 + 4.03i 6.60 - 3.87i -0.23 - 0.071i
Max -0.82 + 5.13i 7.88 - 4.92i -0.99- 0.50i
Shape
category 3 Mean -0.25 + 5.10i 7.71 - 4.49i -0.98 - 0.78i
SD 0.51 0.32 0.39
Min -0.64 + 4.68i 7.44 – 4.10i -0.48 - 0.38i
Max 0.33 + 5.44i 8.21 - 4.73i -1.52 - 0.71i
The above results consequently allow the three fitted landmarks models be labelled as
3,2,1 ,~)(: 3 gMVCNg landmarkg
gg MLL , (5.37)
where the means are given as
)0.2408i - 0.8090- 4.2612i, - 7.4342 4.7012i, + 0.2515(1 L , (5.38)
)0.3231i - 0.5906- 4.2528i, - 7.2656 4.5401i, + -0.1192(2 L , (5.39)
)0.7243i - 0.9210- 4.4183i, - 7.5993 5.0657i, + -0.1738(3 L , (5.40)
168
and shape variability can be explained by the estimated Hermitian covariance matrices
given as
0.2463..
0.0259i + 0.0369-0.1387 .
0.0479i + 0.1028-0.0178i - 0.0217-0.1556
1
landmarkM , (5.41)
0.0993..
0.0190i + 0.0158 0.2156.
0.0211i + 0.0554-0.0102i + 0.0490-0.26526
2
landmarkM , (5.42)
0.1318..
0.0321i + 0.0363- 0.2333.
0.0607i + 0.0832- 0.0019i + 0.0063 0.2886
3
landmarkM . (5.43)
5.4.2 Verification of the Anatomical Landmark Models
A verification study was carried out to confirm the ability of the anatomical
landmarks in indicating the shape category. This is important to enable estimation of
teeth positions for the edentulous patients. The membership of iii
i aaa 821ˆ , ,ˆ ,ˆˆ A from
the 47 control samples which belong to each of the 3 shape models 3,2,1 , )(ˆ: ggg A ,
were tracked from the dendrogram (See Figure 3.12 in section (3.6.1)). The
MVCNiCOVRATIO )( was employed to re-assign these casts to one of 3 populations of )(ˆ gA
using the landmarks ),,,( 321 iiii LLLL where 47 ,...,1i . Their misclassification
probability was then investigated to verify that the landmarks can be used to
discriminate shape category of the dental arch.
Then, shape discrimination using shape feature teeth location, iii
i aaa 821ˆ , ,ˆ ,ˆˆ A
and anatomical landmarks, ),,( 321 iiii LLLL was carried out for comparison.
Discrimination using iii
i aaa 821ˆ , ,ˆ ,ˆˆ A was regarded as the true population. Similar
classification of population is expected using both shape features to further verify the
ability of the anatomical landmarks in indicating the shape category.
169
Table 5.12 and Table 5.13 show the misclassification probability of the 47
control casts and 40 test casts respectively, that were re-assigned to 21, or 3 . Low
misclassification probabilities were obtained in each of the true shape population.
These results support the ability of the ),,( 321 iiii LLLL in discriminating the categories
of arch shape correctly.
Table 5.12: Misclassification probability when 47 control casts was discriminated using
the anatomical landmarks ),,( 321 iiii LLLL of the dental arch.
Assigned using
)( iCOVRATIO
True population
*1 *
2 *3
1 0 0.0909 0
2 0.1818 0 0
3 0.0714 0.0714 0
Table 5.13: Misclassification probabilities when 40 test casts were discriminated using
anatomical landmarks ),,( 321 iiii LLLL and teeth location iii
i aaa 821ˆ , ,ˆ ,ˆˆ A .
Assigned using ),,( 321 iii LLL
True population:
using iii
i aaa 821ˆ , ,ˆ ,ˆˆ A
*1 *
2 *3
1 0 0.14 0
2 0 0 0.17
3 0 0.03 0
5.5 Application of Shape Discrimination for MVCN model
5.5.1 A Proposed Guide to Teeth Positioning on Complete Dentures
The conventional construction of complete dentures requires at least five clinical
appointments or visits. In the first visit, the patient is examined, primary impressions are
made and diagnosic casts are poured from the impressions. In the second visit, final
impressions are made using custom tray from which final casts are poured. In the third
visit, jaw relationship registrations are made for transferring necessary information from
the patient to construct the dentures. In the fourth visit, the trial denture is evaluated in
170
the patient’s mouth and on the articulator for aesthetics and occlusion. In the fifth visit,
the final dentures are inserted into the patient’s mouth amd delivered to the patient.
The schematic diagram below (Table 5.15) is a brief outline of the steps
performed by the dentist and technician in the construction of complete dentures
according to clinical visits.
Number
of clinical
visit
Dentist (clinic) Technician (laboratory)
1st visit - Examination
- Making primary impression
- Making primary cast
- Fabrication of custom tray from the
primary cast
Figure 5.15: Brief outline of steps in the construction of complete dentures.
171
2nd visit - Taking final impression using
custom
tray
- Making final cast
- Fabrication of denture baseplate
and
wax rim
3rd visit - Checking patient’s upper and
lower arch relationship
- Selection of artificial teeth
- Artificial teeth positioning
4th visit - Trial denture visit
- Modification on dentures
5th visit - Final denture and follow up
Figure 5.15, continued.
172
Previous sections have established 3 categories of shape models and they can be
linked using the anatomical landmarks. This section is aimed at using the knowledge to
provide a guide for developing the wax rim and also for teeth positioning on complete
dentures. The proposed guide is as follows:
Number
of clinical
visit
Dentist (clinic) Technician (laboratory)
1st visit - Examination
- Make primary impression
- A 2D image of the edentulous
patient’s arch is captured using an
intraoral camera with a small ruler
attached.
- Make primary cast
- Fabrication of custom tray from the
primary cast
- A program developed in MATLAB
software will be used to import and
calibrate the image, and obtain 3
anatomical landmarks from the
edentulous arch. Consequently, the
landmarks may be used to determine
which shape category the patient
belongs to (section 5.7.2). The
primary wax rim will be developed
using the estimated teeth position
from the corresponding MVCN
model (section 5.5.3).
Figure 5.16: Proposed guide for construction of complete dentures.
173
2nd visit - Make final impression using
custom tray
- Check patient’s upper and
lower arch relationship using
wax rim built on primary casts.
- Make final cast
- Fabrication of final wax rim and
recording jaw relationships using
primary wax rim.
- Selection of artificial teeth
- Artificial teeth positioning will be
carried out using the estimated
teeth positions obtained from the
corresponding MVCN shape
model (section 5.5.3).
3rd visit - Trial denture visit
- Modification on dentures
4th visit - Final denture and follow up
Figure 5.16, continued.
174
There are several advantages of the above proposed guide as compared to the
conventional steps. First, it may reduce one clinical visits. Secondly, shorter clinical
sessions may be needed in the third (trial denture) visit and thirdly, the development of
wax rim, teeth positioning and consequently the whole process of preparing the final
dentures is facilitated.
5.5.2 Verification of the Proposed Teeth Positioning Guide for the Edentulous
The proposed guide for teeth positioning for the edentulous patient was verified
by investigating another 35 selected dentate casts. The MVCN
iCOVRATIO )( in equation (5.33)
together with its discrimination rule established in equation (5.36) was employed using
the anatomical landmarks to determine the membership of shape category. Then, the
teeth position was estimated using the MVCN model corresponding to the assigned
shape category (equation (5.8)). The Procrustes distance (PD) in equation (5.5)
calculates the differences between the original and the estimated teeth position. Table
5.14 shows the membership percentage of the 35 casts in the three shape categories.
Table 5.15 on the other hand indicates that 80% of the 35 casts have 20 mm PD
or sum squared of difference between the 21 estimated and original teeth position. This
signifies that an average squared error for each tooth position is 0.95 mm. Figure 5.17,
Figure 5.18 and Figure 5.19 illustrate the smallest (PD = 9.7520 mm), average (PD =
17.4664 mm) and largest (PD = 26.1236 mm) Procrustes distances respectively, for
comparison of the estimated and original teeth positions.
175
Table 5.14: Group membership when the anatomical landmarks of the 35 test cast was
assigned using MVCN
iCOVRATIO )( .
Population *1 *
2 *3
Number
(Percentage) of
casts assigned
using
)( iCOVRATIO
6
(17.14%)
21
(60.00%)
8
(22.86%)
Table 5.15: The number and percentage of casts according to the Procrustes distance
(PD) intervals indicating sum squared of difference between the estimated and original
teeth position.
PD interval 10 mm )15,10[ mm )20,15[ mm )20,15[ mm 25 mm
Number
(Percentage)
of casts
2
(5.71%)
13
(37.14%)
13
(37.14%)
4
(11.43%)
3
(8.58%)
Figure 5.17: Smallest Procrustes distance PD = 9.7520 mm between the original and
estimated teeth position. Red dots indicate the estimated teeth position (sample N110).
176
Figure 5.18: An average Procrustes distance PD = 17.4664 mm, between the original
and estimated teeth position. Red dots indicate the estimated teeth position (sample
N130).
Figure 5.19: Largest Procrustes distance PD = 26.1236 mm, between the original and
estimated teeth position. Red dots indicate the estimated teeth position (sample N117).
177
5.6 Discussion
The 8 Fourier descriptors provide a more precise arch shape and teeth position
and was capable of closely representing the dental arch shape. The Fourier descriptor
was considered as it is a reversible linear transformation which retains all the
information in the original boundary of the object (Keyes & Winstanley, 1999). This
important feature makes the modelling of arch shape and teeth position using the FDs
easier and more precise. Further, the FDs used in this study were derived from the
origin established from anatomical landmarks which made them valid for use in the
edentulous patient, rather than using teeth which may be lost (Mikami et al., 2010;
Nakatsuka et al., 2011).
Test of separation of two mean vectors from MVCN was derived and the
anatomical landmarks were found capable in discriminating the shape category
established with relatively low misclassification probabilities. The use of shape models
3,2,1 , ,ˆ~)(ˆ
gMVCNg g
g SAA gives the precise locations of the 18 teeth on the
maxillary arch. The proposed guide for estimating teeth positioning on complete
dentures shows that the three categories of shape models may estimate the original teeth
position correctly in 80% of the Malaysian dental arches with an average error margin
of 0.95 mm for each tooth position. This will help the dentist to position the maxillary
artificial teeth as close as possible to the position originally occupied by the natural
teeth.
178
CHAPTER 6: CONCLUDING REMARKS
6.1 Conclusion
The primary goal of this thesis was to propose shape features and statistical
shape models to develop a novel discrimination procedure for the shape of the maxillary
dental arch. This work has been applied to two applications in dentistry, namely in
designing and selection of impression trays and in predicting arch shape and teeth
positions for the edentulous patients.
A review of the literature related to the shape analysis of the dental arch shows
that this study is the first to propose shape discrimination procedure and to demonstrate
its applications in dentistry. The first step in discriminating shape of the dental arch to
enable reconstruction of the arch shape even in edentulous situation was to propose a
shape feature derived from stable anatomical landmarks. Then, the properties of the
shape feature were investigated. The probability distribution of each shape category
which was clustered from the shape feature provides shape variation and was used as
shape model of the dental arch. A new hypothesis testing for two sample means from
multivariate complex normal based on Hotelling T2 test was also derived and employed
to test the distinction of two shape models.
A modified COVRATIO statistics, denoted as )( iCOVRATIO , was then proposed
as a novel shape discrimination method and compared to the linear discrimination
function. Simulation results showed that )( iCOVRATIO
as discrimination method
performs better than the LDF with lower misclassification probabilities in all considered
cases. The )( iCOVRATIO is theoretically desirable when discriminating shape of the
179
dental arch, since differences of the dental arch shapes are barely noticeable and the
discriminatory information may not necessarily be in the mean shape, but rather in
shape variability. Further, the )( iCOVRATIO can be employed when discriminating k
groups of shape and without any assumptions on the distribution of the shape. The
resulting statistical shape discrimination procedure enables the development of two
proposed guides in choosing suitable impression trays and predicting natural teeth
positions for the edentulous. Verification study shows that 91.42% of the test sample
studied indicates appropriate fitting when assigned using the proposed guide. As for the
latter guide, 80% of the studied arches may adequately estimate the original teeth
position, with an average error of 0.95 mm for each tooth position.
The presented statistical shape discrimination procedure therefore may be useful
in assisting inexperienced dentists and dental laboratory technicians to choose the most
appropriate impression tray for the Malaysian population and facilitate the estimation
natural teeth positions for the edentulous. Dental visits may be shortened, and
unnecessary cost of repeated dental procedures may be avoided.
The new contribution of the work can be summarized as follows:
1. Shape features indicating the location of the teeth have been proposed which
allows the reconstruction of the arch shape even when all teeth are lost.
2. Shape models of the dental arch has been proposed and provides not only
presents the mean shape, but also shape variation.
3. A new hypothesis testing for two sample means from multivariate complex
normal based on Hotelling T2 test was derived and employed to test the
distinction of two shape models.
4. A modified COVRATIO statistics, denoted as )( iCOVRATIO , was then proposed
as a novel shape discrimination method.
180
5. A guide in assisting inexperienced dentists and dental laboratory technicians to
choose the most appropriate stock tray for dentate or partially edentulous
patients has been proposed.
6. A guides in assisting inexperienced dentists and dental laboratory technicians to
estimate natural teeth position on the complete dentures for the edentulous
patients has been proposed.
6.2 Limitations of the study
The limitation of this study was the sampling method used to represent the
Malaysian population. Nevertheless, the results of this study may be generalized since
the sample was collected from one of the hospitals in the Klang Valley region in
Malaysia, which is the heart of commercial, business and industrial center of the
country and consists of the 3 major ethnic groups in the population.
The shape feature using angular measurements compared to Fourier descriptor
may be a limitation to the shape model. It may only work for semi-angular measures
and must be approximated to the linear measures. In addition, the shape model should
follow normal distribution. A non-normal distribution of shape model may be explored
in the future research.
It should be noted that the design of the trays, guide in assigning suitable
impression tray and guide in estimating teeth positions on complete dentures are limited
as only two dimensional shapes on the maxillary (upper) arch were considered. Minor
modifications to the impression trays, predicted arch shape or teeth positions may be
necessary to accommodate factors which were not investigated in this study and some
clinical work is needed to further scrutinize the statistical methods to be used in clinical
practices. It has to be pointed out that the palatal shape and depth of maxillary and
mandibular (lower) arches are also important considerations for making stock
181
impression tray and guiding the teeth positioning. This would be the subject of a future
study.
6.3 Direction for Future Research
6.3.1 Exploring other Applications of Shape Model and Shape Discrimination
Procedure in Dental Problems
Two applications were discussed in this study. More potential applications of the
discrimination procedure for the dental arch shape can be extended, namely in
fabricating the arch wire used in orthodontic and matching teeth position from the bite
mark in forensic dentistry. Lee et al. (2011) classified the dental arch forms with the
objective to provide a guide in designing pre-formed orthodontic arch wire forms.
Results from their study may only be used to determine the number of preformed wire
to be designed, since the knowledge of mean shape variability was not provided. The
shape model and discrimination procedure proposed in this study may determine the
elasticity of the arch wire and suitable pre-formed wire that would fit the patient’s arch.
This study may also help to provide evidence to exclude suspects in a judicial
procedure. The bite marks may be first discriminated from the three shape models to
determine which category it belongs to. Then, the teeth positions from all the suspects
are discriminated and those which belong to the same category as the bite marks can be
narrowed as the potential assailant.
6.3.2 Extension to Mandibular and 3D Shape of the Dental Arches
The current study focuses on the 2D digital images of the maxillary dental arch.
This research can be extended to the mandibular arches, whereby the same shape feature
can be derived from stable anatomical landmarks of the mandicular arches, which are
the retromolar pads and lingual frenum (Celebic et al., 1995; Roraff, 1977).
182
A study by Wu et al. (2012) has considered a 3D statistical model of gingival
contours from training dataset for reconstructing missing gingival contours in partially
edentulous patients. Recently, Elhabian & Farag (2014) formulated a statistical model
of teeth surface from training dataset using shape-from-shading approaches to estimate
the 3D teeth shape surface for clinical crowns to aid dentists dealing problems with
irregular teeth and dental pulp. Ideas from these works in obtaining the 3D images of
the dental arch may be incorporated this study to develop 3D shape discrimination
procedure. An extension of the study for 3D images which includes the depth of the
palate of the arches would be beneficial in more accurate design of the stock tray and
teeth positioning on complete denture.
6.3.3 Multivariate Complex Normal with Relation Matrix
The multivariate complex normal distribution considered in this study assumes
zero relation matrix. Piccinboho (1996) showed the importance of the relation matrix to
complete the description of the second-order statistics. Further development of
multivariate complex normal with relation matrix as a shape model may be explored.
6.3.4 Comparing Shape Discrimination using COVRATIO and Bayesian Methods
In real situations, particularly in shape analysis, smaller sample size relative to
numbers of landmarks or variables selected to represent the shape of the subject are
likely to occur. A Bayesian approach mentioned in Mardia et al. (1979) may be an
alternative to deal with this problem. It is an interest to compare the proposed
COVRATIO and the Bayesian methods for shape discrimination.
183
6.3.5 Regression Ideas for Shape Models and Discrimination
Let ),,,,,,,()( 44332211
kkkkkkkk lwlwlwlwk v be the k-th ( 2,1k ) shape category
where ),( k
j
k
j lw represent the j-th selected teeth. Two regression models of the j-th teeth
gives
ˆˆ and ˆˆ 22221111
jjjjjjjj lwlw .
Shape categories )1(v and )2(v are said to be different in shape if 21 ˆˆjj and
21 ˆˆjj for all j. It is an interest to develop an alternative shape model and
discrimination of the dental arch based on this idea and compare with shape model
presented in this study.
184
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LIST OF PUBLICATIONS AND PAPERS PRESENTED
1. N.A. Abdullah1, O.M. Rijal1, Y.Z. Zubairi2 and Z.M. Isa3 (2015). Hotelling two
sample T2 test for complex normal distribution. Accepted for publication in the
Pakistan Journal of Statistics, (ISI-Cited Publication).
2. Rijal, O. M., Abdullah, N. A., Isa, Z. M., Noor, N. M., & Tawfiq, O. F. (2013,
July). Shape model of the maxillary dental arch using Fourier descriptors with
an application in the rehabilitation for edentulous patient. In Engineering in
Medicine and Biology Society (EMBC), 2013 35th Annual International
Conference of the IEEE (pp. 209-212) (IS-Cited Publication).
3. Rijal, O. M., Abdullah, N. A., Isa, Z. M., Noor, N. M., & Tawfiq, O. F. (2012,
August). A probability distribution of shape for the dental maxillary arch using
digital images. In Engineering in Medicine and Biology Society (EMBC), 2012
Annual International Conference of the IEEE (pp. 5420-5423) (ISI-Cited
Publication).
4. Rijal, O. M., Abdullah, N. A., Isa, Z. M., Davaei, F. A., Noor, N. M., & Tawfiq,
O. F. (2011, August). A novel shape representation of the dental arch and its
applications in some dentistry problems. In Engineering in Medicine and
Biology Society, EMBC, 2011 Annual International Conference of the IEEE (pp.
5092-5095) (ISI-Cited Publication).
5. Isa, Z. M., Tawfiq, O. F., Abdullah, N. A., Noor, N. M., & Rijal, O. M. (2011).
Statistical clustering of maxillary dental arches. Scientific Research and Essays,
6(13), 2710-2719 (SCOPUS-Cited Publication).
6. N.A. Abdullah1, O.M. Rijal1, Y.Z. Zubairi2 , Z.M. Isa3 . COVRATIO Statistics as
discrimination method for complex normal distribution. Sent to Statistical
Methods in Medical Research on October 2015.
199
APPENDIX A: Ethics Approval
200
APPENDIX B: Examples of Research Data for ),,,,,,,( 44332211
RRRRRRRRR lwlwlwlwv , iii aaa 821ˆ , ,ˆ ,ˆˆ A and ),,( 321 LLLL .
Patient
No.
Gender Race Age Shape feature using ),,,,,,,( 44332211
RRRRRRRRR lwlwlwlwv . The selected tooth on the right side of
the arch is represented by the length of a line joining the cusp tips of a tooth to the origin (R
jl )
and an angle (R
jw ) with respect to the horizontal axis.
Rw1 Rl2
Rw2 Rl2
Rw3 Rl3
Rw4 Rl4
N001 Male Malay 21
83.312621 57.944650 74.485064 54.575058 65.099794 51.244767 30.104148 33.004298
N002 Female Malay 27
85.165569 49.202121 75.641731 48.695497 68.162934 44.940678 33.092770 33.263693
N093 Male Chinese 30
89.377146 59.761346 81.352490 57.074108 73.565726 52.713229 47.993159 37.722920
20
1
Patient
No.
Gender Race Age Shape feature using 8 FD terms, iii aaa 821
ˆ , ,ˆ ,ˆˆ A used to represent all teeth on the dental arch.
1a 2a 3a 4a 5a 6a 7a 8a
N001 Male Malay 21 3.156537 +
56.666303i
-2.612131 -
60.720536i
-0.588401 -
4.331291i
0.179727 -
8.073411i
0.103580 -
3.488072i
0.341993 -
1.295228i
0.103206 +
7.633484i
-0.530347 +
4.967460i
N002 Female Malay 27 -2.735498 +
64.028726i
-1.390996 -
59.425994i
0.142704 -
9.589793i
0.528870 -
8.970231i
0.351559 -
5.507708i
0.260832 -
2.708948i
0.605500 +
6.252824i
0.348833 +
6.299513i
N003 Male Chinese 30 -5.047854 +
68.200273i
2.648933 -
58.916754i
0.012492 -
8.904548i
0.076124 -
11.062200i
-0.151649 -
5.378759i
-0.390196 -
3.510903i
0.050893 +
5.386973i
-0.382475 +
4.431804i
Patient
No.
Gender Race Age Shape feature using 3 FD terms, ),,( 321 LLLL depicting 3 anatomical landmarks on the
dental arch.
1L 2L
3L
N001 Male Malay 21
-0.055406 + 4.026401i 6.681682 - 3.877263i -0.274842 - 0.178272i
N002 Female Malay 27
-0.117819 + 4.029429i 6.599693 - 3.864965i -0.360941 - 0.193751i
N003 Male Chinese 30
-0.252726 + 5.027066i 7.790054 - 4.481303i -0.755679 - 0.812290i
20
2
APPENDIX C: Summary of Works on Dental Arch Symmetry
Authors Purpose of
investigation
of arch
symmetry
Subject
(adult or
adolescent)
and results
of the arch
symmetry
Measurements used Size and
shape
demonst-
rated
Image
registration
demonstra-
ted
Method used to assess symmetry
of arch
Lundstrom
(1961)
To investigate
arch symmetry
in genetically
simiar
individuals.
Children aged
13 years old.
Asymmetry
arches.
The center of the arch is
defined from the palatal
raphae line and the
transverse distance
connecting on each side
of the teeth to the center
were obtained.
Size only.
Not using
image.
Distance of the left side of the teeth
to the raphae line was greater than
the right side.
Lavelle &
Plant
(1969)
To provide a
record of the
British
dentition.
Adults aged
18 years.
Symmetry.
Mesiodistal crown
diameters from 2nd molar
to 1st incisor on each side
of the dental arch.
Size only. No Unclear method of symmetry test.
Uses p-value to indicate
insignificance of the statistical test.
Ferrario et
al. (1993)
Patients with
normal arch
and
malocclusion
were evaluated
for the
symmetry.
Adult 20-27
years.
Symmetry,
but not in
maxillary
female arch.
Each patient’s dental arch
is represented by linear
distances between all
possible pairs of
landmarks on the left and
right arch (21 distances
for each side of arch).
Size only. No. Ratio of corresponding linear
distances on the left and right arch
for each patient and the average
were calculated. Then, T1=max
average ratio/ min average ratio
was obtained. Hypothesis testing is
tested on the null hypothesis H0: T1
= 1 (indicating that the size is
similar and therefore symmetry).
20
3
Araujo et
al. (1994)
To investigate
dental
symmetry in
individuals
with normal
dental
occlusions.
Adults
Mean age 22.4
years
Symmetry
dental arches.
(1) Difference between
mesiobuccal cusps of
right and left first molars
and median axis of the
defined grid (molar
transverse asymmetry
using the transferred
palatal raphae).
(2) Molar
anteroposterior
asymmetry using a plane
that passess by the
buccal surfaces of
incisors.
(3) Mandibular midline
deviation to the
transferred palatal
raphae.
Shape
only.
Not using
image.
Student’s t test and chi-square tests
were conducted to check for
statistically signifiacant
asymmetries between right and left
sides of the arch.
Cassidy et
al. (1998)
To investigate
the role
heredity plays
in determining
dental arch
size and shape.
Children aged
10-19 years
old.
Not symmetry.
Linear and angular
measurements on the left
and right arch
(See Fig 3A-D in this
paper).
Size only. No. Paired t-test testing for mu_d=0
Use mixed model ANOVA.
Maurice &
Kula (1998)
Describe the
degree of
asymmetry.
For
orthodontic
treatment and
to evaluate
arch symmetry
0 to 9 years.
Symmetry.
Assymmetry
only present at
mandibular
canine region.
Linear distances from
teeth to medial palatal
plane.
Size only. No. One tailed median signed test
mu(|d|)>2mm. (nonparametric-
using median instead of mean)
Paired t-test indicate significant
differences from 0
20
4
Šlaj et al.
(2003)
Analysze
changes of
arch symmetry
for developing
occlusal
relationship.
For
orthodontic
treatment.
3-13 years
Symmetry.
Assymmetry
only present at
mandibular
canine region.
Arch width and length
of left and right arch
measured from medial
palatal plane.
Size only. No. Two sample t test for independent
sample.
Schaefer et
al. (2006)
To investigate
is ethnic origin
influences
dental arch
symmetry.
Children 7-15
years
Symmetry. No
results/tables
were
displayed.
3D Digitized coordinate
(x,y,z)
Shape and
size.
Yes. In text: The mirror image of each
dental cast was produced using one
side of the arch. Then, the forms of
the original and the mirror were
superimposed using generalized LS
Procrustes.
Modified Hotelling 2T test were
used to test the symmetry of arch
simultaneously (Mardia et al.,
2000).
Sprowls et
al. (2008)
to determine
the
relationship
between dental
arch
asymmetry
and
asymmetry.
11.5 to 48.3
years.
Symmetry.
Distance between
permanent teeth and
MPR (medial palatal
raphae)
On the right and left
side of arch.
Shape
only.
No. Two way ANOVA for testing:
H0_d: tooth position depends on
the presence/absence of directional
asymmetry
Ho_e: tooth position depends on
whether the remaining fluctuation
asymmetry exceeds the within
observer measurement error or not.
Shrestha &
Bhattarai
Of interest to
the
Adult 17-32
years. Symmetry
Linear distance from
(1) midpoint of central
Size only. No. Two sample t-test testing for
independent sample
20
5
(2009) orthodontist
for functional
and aesthetic
value. Also
useful for
odontometric
and
anthropometric
purposes.
in both
maxillary and
mandibular
female and male
arches.
incisor to canine and
(2) midpoint of central
incisor to first molar
cusp teeth
on the left and right
arch.
- Left and right arch is
independent?
Scanavini
et al. (2012)
To verify the
degree of
asymmetry in
Brazilian
dental arches.
For
orthodontic
treatment.
12 to 21 years
subjects
Asymmetry in
the dental arches
was found in all
individuals.
Normal arches.
has smaller
asymmetry
degree
Liner and Angular
measurements of teeth
position in digital
images and protractor
(standardized) on the
left and right arch.
Shape
only.
Not using
digital
image.
ANOVA.
Test of Pearson’s correlation
coefficient, 0 for differences
between left and right side of the
arch.
Tong et al.
(2012)
to obtain the
angulations
and inclination
for all teeth in
near-normal
occlusion.
Adult subjects.
Almost perfect
symmetry
between the
right and left
side
measurements
allowed them to
combine the 2-
side data.
Used 76 CBCT scans
before treatment,
They measured the
mesiodistal and
faciolingual
angulations between
Long axis of upper-
lower teeth.
Shape
only
(tooth
position in
dental
arch).
Yes,
comparable
somehow to
present
study.
Paired t test for normally
distributed data.
And wilcoxon signed rank test for
non-normal data.
Use Bonferoni correction.
206
APPENDIX D:
MATLAB Program for Determine Appropriate Impression Tray for Dentate
Patients
%new cast image to assign to a suitable fabricated tray %select only 4 teeth on the right arch: central incisor, lateral
incisor, %canine and distobuccal of first molar teeth
%% retrieve image
cd c:/; % Change directory [f,d] = uigetfile('*'); % * means can select any file. d=directory
name. f=filename S=imread([d,f]); imshow(S) h = helpdlg('select the desired area and double click to crop the
picture'); uiwait (h); Y=imcrop(S); Y=rgb2gray(Y); figure; imshow(Y) h = helpdlg('select 2 points from the ruler to calibrate the distance
for 1cm'); uiwait (h); [x1,y1]=ginput(1); hold on; plot(x1,y1,'-c+') [x2,y2]=ginput(1); hold on; plot(x2,y2,'-c+')
D1 = (x2-x1)^2 + (y2-y1)^2; oneCm = sqrt(D1);
h = helpdlg('Click on 2 hamular notches'); uiwait (h); [x3,y3]=ginput(1); %Titik hamular notch pertama hold on; plot(x3,y3,'-c+') [x4,y4]=ginput(1); %Titik hamular notch kedua hold on; plot(x4,y4,'-c+') h = helpdlg('Click on the incisive papilla'); uiwait (h); [x5,y5]=ginput(1); %Titik incisive papilla hold on; plot(x5,y5,'-c+')
%------------------------------------- % Dapatkan persamaan garislurus %-------------------------------------
m1 = (y3-y4)/(x3-x4); m2 = -1/m1; c2 = y3 - m1*x3; %Equation for the hamular notch c1 = y5 - m2*x5; % Equation perpendicular to hamular notch line
207
%Dapatkan titik origin (u, w) (in pixels) antara 2 garisan u = (c2-c1)/(m2-m1); w = m2*u + c1; xHamular=[x3 u x4]; yHamular=[y3 w y4]; plot(xHamular,yHamular)
xIncisive=[x5 u]; yIncisive=[y5 w]; plot(xIncisive,yIncisive)
data=double([]); cnt=1; button = questdlg('Click at the point (cusp tips of the teeth) you
wish to measure from the origin. Next point?',... 'Continue Operation','Yes','No','No'); while strcmp(button,'Yes') [x,y]=ginput(1); hold on; plot(x,y,'-c+')
xx=x-u; yy=y-w;
if xx<0 xx=-xx; end
theta=abs(atan(yy/xx))*180/pi; lengthcm=sqrt(xx^2+yy^2)/oneCm; length=lengthcm*10;
data=[data;theta,length];
button = questdlg('Take the next point?',... 'Continue Operation','Yes','No','No'); cnt=cnt+1; if strcmp(button,'No') | cnt >= 20 break end end
save data
X=[data(1,1) data(1,2) data(2,1) data(2,2) data(3,1) data(3,2)
data(4,1) data(4,2)];
%mean and covariance matrix for C1 meanc1=[83.16 57.30 74.85 54.91 67.27 51.49 36.82
36.43];
Sc1=[2.012123598 0.337432009 2.052163747 0.399304037 1.374913377
0.592820423 1.372807356 -0.150785691 0.337432009 2.922243216 0.381430085 2.214038665 0.58930365
2.054974648 0.597471715 0.525553419 2.052163747 0.381430085 2.241715613 0.47811376 1.78768176
0.573042135 2.210770242 0.192481994 0.399304037 2.214038665 0.47811376 2.046860634 0.886332222
1.882684422 1.270886268 1.206468829
208
1.374913377 0.58930365 1.78768176 0.886332222 2.579873342
0.877251632 3.890586272 1.854193936 0.592820423 2.054974648 0.573042135 1.882684422 0.877251632
1.953917399 1.100585987 1.124200586 1.372807356 0.597471715 2.210770242 1.270886268 3.890586272
1.100585987 7.9871065 3.406908715 -0.150785691 0.525553419 0.192481994 1.206468829 1.854193936
1.124200586 3.406908715 4.472910114];
%mean and covariance matrix for C2 meanc2=[85.55 53.55 77.07 51.61 69.15 48.35 36.11
33.91];
Sc2=[1.787781732 1.043992557 1.952824521 0.781918286 2.065771236
0.58693496 1.834273356 -0.214270913 1.043992557 4.62791343 1.538818563 3.845541984 1.700776267 3.75215652
3.377773901 1.25664875 1.952824521 1.538818563 2.342034167 1.274220143 2.53759831
1.201756339 2.900485286 0.153779997 0.781918286 3.845541984 1.274220143 3.737919597 1.659892627
3.782797801 4.007685346 1.841344663 2.065771236 1.700776267 2.53759831 1.659892627 3.019286244
1.639383456 3.91572139 0.514891077 0.58693496 3.75215652 1.201756339 3.782797801 1.639383456
4.129723757 4.398579999 2.23415283 1.834273356 3.377773901 2.900485286 4.007685346 3.91572139
4.398579999 9.438333114 2.989053801 -0.214270913 1.25664875 0.153779997 1.841344663 0.514891077
2.23415283 2.989053801 3.33636598];
%mean and covariance matrix for C3 meanc3=[87.30 59.51 79.43 57.13 72.21 53.30 43.76
36.97];
Sc3=[3.705327047 -2.997999787 3.507847911 -3.104122741
2.792800093 -2.914275848 1.522582331 -1.72004541 -2.997999787 4.835122711 -2.544628522 3.971776525 -2.041828018
3.852192782 0.570477185 1.687803257 3.507847911 -2.544628522 3.545887763 -2.742954926 2.962438686 -
2.51232233 2.087531852 -1.648866261 -3.104122741 3.971776525 -2.742954926 4.06513582 -2.202586106
3.898300992 -0.238788688 1.999575682 2.792800093 -2.041828018 2.962438686 -2.202586106 2.64042465 -
2.013866607 1.95582961 -1.374264035 -2.914275848 3.852192782 -2.51232233 3.898300992 -2.013866607
4.118655357 0.049982414 2.031833986 1.522582331 0.570477185 2.087531852 -0.238788688 1.95582961
0.049982414 6.333348038 -1.524028874 -1.72004541 1.687803257 -1.648866261 1.999575682 -1.374264035
2.031833986 -1.524028874 2.736373079];
%% finding mahalanobis distance:
%pooled sigma Spooled=(11*Sc1 + 22*Sc2 + 14*Sc3)/(47-3); %3groups, n for c_j=11, 22
and 14
XMinusMuC1=double([]); XMinusMuC2=double([]); XMinusMuC3=double([]); Z2C1=double([]); Z2C2=double([]);
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Z2C3=double([]);
sizeX=size(X,1);
for i=1:sizeX %for C1 xMinusMuC1=X(i,:)-meanc1; XMinusMuC1=[XMinusMuC1; xMinusMuC1];
z2C1=XMinusMuC1(i,:)*inv(Spooled)*transpose(XMinusMuC1(i,:)); Z2C1=[Z2C1; z2C1];
%for C2 xMinusMuC2=X(i,:)-meanc2; XMinusMuC2=[XMinusMuC2; xMinusMuC2];
z2C2=XMinusMuC2(i,:)*inv(Spooled)*transpose(XMinusMuC2(i,:)); Z2C2=[Z2C2; z2C2];
%for C3 xMinusMuC3=X(i,:)-meanc3; XMinusMuC3=[XMinusMuC3; xMinusMuC3];
z2C3=XMinusMuC3(i,:)*inv(Spooled)*transpose(XMinusMuC3(i,:)); Z2C3=[Z2C3; z2C3]; end
mahalnob=[Z2C1 Z2C2 Z2C3];
if Z2C1<Z2C2 && Z2C1<Z2C3 disp ('Assign to tray 1') else if Z2C2<Z2C1 && Z2C2<Z2C3 disp('Assign to tray 2') else if Z2C3<Z2C1 && Z2C3<Z2C1 disp('Assign to tray 3') end end end
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APPENDIX E:
MATLAB program for Estimating Natural Teeth Positions on Complete Dentures
for the Edentulous
%select MVCN shape model 1, 2 or 3
%% reterieve image cd c:/; % Change directory [f,d] = uigetfile('*'); % * means can select any file. d=directory
name. f=filename S=imread([d,f]); imshow(S) h = helpdlg('select the desired area and double click to crop the
picture'); uiwait (h); Y=imcrop(S); Y=rgb2gray(Y); figure; imshow(Y) h = helpdlg('select 2 points from the ruler to calibrate the distance
for 1cm'); uiwait (h); [x1,y1]=ginput(1); [x2,y2]=ginput(1);
D1 = (x2-x1)^2 + (y2-y1)^2; oneCm = sqrt(D1);
h = helpdlg('Click on 2 hamular notches'); uiwait (h); [x3,y3]=ginput(1); %Titik hamular notch pertama %select on right side
first hold on; plot(x3,y3,'-c+') [x4,y4]=ginput(1); %Titik hamular notch kedua hold on; plot(x4,y4,'-c+') h = helpdlg('Click on the incisive papilla'); uiwait (h); [x5,y5]=ginput(1); %Titik incisive papilla hold on; plot(x5,y5,'-c+') hold on;
%------------------------------------- % Dapatkan persamaan garislurus %-------------------------------------
m1 = (y3-y4)/(x3-x4); m2 = -1/m1; c2 = y3 - m1*x3; %Equation for the hamular notch c1 = y5 - m2*x5; % Equation perpendicular to hamular notch line
%Dapatkan titik origin (u, w) (in pixels) antara 2 garisan
u = (c2-c1)/(m2-m1); w = m2*u + c1;
xHamular=[x3 u x4]; yHamular=[y3 w y4];
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plot(xHamular,yHamular) hold on;
HamularX=[x3-u x4-u]'/oneCm; yHamularX=[y3-w y4-w]'/oneCm; Hamular=[HamularX yHamularX];
xIncisive=[x5 u]; yIncisive=[y5 w]; plot(xIncisive,yIncisive) hold on;
IP=[x5-u w-y5]/oneCm; landmark=[Hamular;IP];
%% choose mean of assigned shape (AS) category in FD % meanc1 % AS=[2.975253561 63.56708953 % -1.645126476 -59.40807746 % -0.144520751 -7.722531286 % -0.154638083 -9.4306985 % 0.058291401 -4.545900269 % 0.171673126 -2.665689093 % -0.012414048 5.892822351 % 0.114782775 4.577652201];
% meanc2 % AS=[-0.651386156 57.64963381 % -0.480754733 -57.59873011 % 0.04254522 -7.219076249 % 0.037382931 -8.35381279 % 0.003718754 -4.274265505 % 0.079266153 -2.450033353 % 0.055917418 6.744189641 % 0.09803187 5.565180921];
%meanc3 AS=[-2.078679961 66.5773416 0.979342286 -59.83098877 0.12456784 -8.074783287 0.11547679 -9.77286547 0.0518563 -4.805206904 -0.103224426 -2.650462174 -0.030196412 5.730557796 -0.165540068 4.238463245];
as1=AS(:,1)+AS(:,2)*i; %change in complex number zero=zeros(13,1); as=[as1(1:6) ;zero;as1(7:8)]; %make it in 21 points xas=ifft(as); %inverse transform of the 8FD giving 21 coordinates in
complex form
X1=(xas(1:2)*oneCm)+(u+w*i); X2=transpose(((xas(3:20)*oneCm)+(u-w*i))'); X3=(xas(21)*oneCm)+(u+w*i); X=[X1;X2;X3];
scatter(real(X),imag(X),'MarkerEdgeColor','r') scatter(real(X),imag(X),'fill','r')