on multiple-case diagnostics in linear subspace method · on multiple-case diagnostics in linear...
TRANSCRIPT
![Page 1: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/1.jpg)
On multiple-case diagnostics
in linear subspace method
Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA2
1Graduate School of Information Science and Technology,
Hokkaido University2Information Initiative Center, Hokkaido University
19th International Conference on Computational Statistics Paris – France, August 22-27 1/36
![Page 2: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/2.jpg)
Outline
1 Introduction
2 Sensitivity analysis in linear subspace
method
3 A multiple-case diagnostics with clustering
4 Numerical example
5 Concluding remarks
2/36
![Page 3: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/3.jpg)
Outline
1 Introduction
2 Sensitivity analysis in linear subspace
method
3 A multiple-case diagnostics with clustering
4 Numerical example
5 Concluding remarks
3/36
![Page 4: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/4.jpg)
Introduction4/36
Sensitivity analysis in pattern recognition
Single-case diagnostics in linear subspace
method (Hayashi et al., 2008)
Multiple-case diagnostics in linear subspace
method
We show the availability through a
simulation study.
![Page 5: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/5.jpg)
Outline
1 Introduction
2 Sensitivity analysis in linear subspace
method
3 A multiple-case diagnostics with clustering
4 Numerical example
5 Concluding remarks
5/36
![Page 6: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/6.jpg)
CLAss-Featuring Information Compression
(CLAFIC; Watanabe, 1967)
Training observations ( )KkniR k
pk
i ,,1;,,1 KK ==∈x
Class1
L LClass Classk K
*x
11
1 1,, nxx K
k
n
k
kxx ,,1 K
K
n
K
Kxx ,,1 K
Test observation
6/36
Linear subspace method
![Page 7: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/7.jpg)
[CLAFIC]Autocorrelation matrix of the training data in -th class :
∑=
Τ=
kn
i
k
i
k
i
k
kn
G1
1ˆ xx
0ˆˆˆ21 ≥≥≥≥ k
p
kk λλλ L
k
( )psk
s ,,1ˆ K=u
Eigenvalues :
Eigenvectors :
A projection matrix in -th class:k
( )ppP k
p
s
k
s
k
sk
k
≤≤=∑=
Τ1ˆˆˆ
1
uu
{ }** ˆmaxarg: xxT
kPk =
7/36
![Page 8: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/8.jpg)
CLAFIC
∗∗xx
T
2P
∗x
Class1
Class2
>
∗x Class1
8/36
∗∗xx
T
1P
Test observation
∗∗xx
T
2P
∗∗xx
T
1P
![Page 9: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/9.jpg)
[Discriminant score and Average]
Discriminant score for :kix
( ) ,1ˆˆk
k
ik
k
i
k
i niQz ≤≤= xxT
where .ˆˆ1
1ˆ
1
−
−= ∑
=
K
l
lkk PPKK
Q
.ˆ1ˆ
1
∑=
=kn
i
k
i
k
k zn
Z
Average discriminant score
9/36
![Page 10: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/10.jpg)
where is the number of the basis vectors in -th class.
means without -th observation in -th class
[Sample influence function (SIF)]
( )( ) ( ) ( ) ( ){ },ˆˆ1ˆ; )( kjgkgk
g
j QQnQ vechvechvechxSIF −⋅−−=
( ))(ˆjgkQvech ( )kQvech j g
[Empirical influence function (EIF)]
( )( )( ) ( ) ( )
( ) ( ) ( ),
ˆˆˆˆˆˆˆˆˆ1
1
ˆˆˆˆˆˆˆˆˆ
ˆ;
1 1
1
1 1
1
≠
+−
−−
=
+−
=
∑ ∑
∑ ∑
= +=
−
= +=
−
kgGK
kgG
Qg
g
g
g
p
s
g
s
g
t
g
t
g
s
g
t
jg
g
g
s
p
pt
g
t
g
s
p
s
g
s
g
t
g
t
g
s
g
t
jg
g
g
s
p
pt
g
t
g
s
k
g
j
TTT
TTT
uuuuuuvech
uuuuuuvech
vechxEIF
λλ
λλ
gp g
10/36
![Page 11: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/11.jpg)
Case of
( ) ( )∑=
=kn
i
k
i
jg
k
k
i
k
jg
k Qn
Z1
,ˆ1ˆ xxT
( ) ( ) ( ).ˆˆ1ˆ)( kjgkg
jg
k QQnQ −⋅−−=
( )( )k
g
j Q; vechxSIF
Case of
( )∑=
≠=kn
i
k
i
jg
k
k
i
k
jg
k kgQn
Z1
,ˆ1ˆ xxT
( ) ( ).ˆˆˆˆˆˆˆˆˆ1
1ˆ
1 1
1
∑ ∑= +=
−+−
−−=
g
g
p
s
g
s
g
t
g
t
g
s
g
t
jg
g
g
s
p
pt
g
t
g
s
jg
k GK
QTTT
uuuuuuλλ
( )( )k
g
j Q; vechxEIF
11/36
![Page 12: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/12.jpg)
Single-case diagnostics
Influence of single observation for kZ
Index
12/36
( )jg
kZ
![Page 13: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/13.jpg)
Multiple-case diagnostics
Influence of multiple observations for with( ) ( )gg
jig
k njniZ ,,1;,,1ˆ ,KK ==
kZ
Index
Index( )jigkZ,ˆ
13/36
i
j
![Page 14: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/14.jpg)
Multiple-case diagnostics
( )Tw 1,,1,10 K=g
( )( )gkQL 0ˆ wvech
( )kQvech
hww tgg += 0
( )( )gk gQL wvechw
ˆ
Weights of unperturbed observations
in -th class
Maximum likelihood estimator
Maximum log likelihood
Weights of perturbed case in -th class
Maximum log likelihood in -th
perturbed class
g
14/36
1=h
g
g
![Page 15: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/15.jpg)
Based on Cook (1986) and Tanaka (1994),
( ) ( ) ( ) ( ) ( ).0
2
100 32
2
2
ttdt
Ddt
dt
dDDtD Ο+++=
( ) ( ) .][ 22
0
tL
ConsttDgg
gg⋅
∂∂
∂−⋅≅
=
hww
h
ww
T
T
.][
0
2
hww
h
ww
T
T
h
gg
gg
LC
=∂∂
∂−=
15/36
( ) ( )( ) ( )( )]ˆˆ[2 00
g
k
g
k
ggQLQLD wvechwvechw
w−=
![Page 16: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/16.jpg)
( )( ) ( )
( ).
ˆˆ 2
hw
vech
vechvechw
vechh
w
T
T
wT
h
∂
∂
∂∂
∂−
∂
∂=
g
k
kk
g
k gg Q
LQC
( )( ) ,][]ˆ][[ 1 hEIFvechacovEIFh TT
h gkkgk QC −∧
≅
[ ] ( )( ){ }k
g
rgk Q;vechxEIFEIF =
and are evaluated
at and , respectively.
( ) ]ˆ[ g
gkQ wvech
T
w
∂∂ ( ) ( ) ][ 2 Tvechvech kk QQL ∂∂∂−
gg
0ww = ( ) ( )kkQQ ˆvechvech =
By and ,( )( ) ( )0
ˆˆ;
=∂
∂⋅=
gr
g
g
r
k
gk
g
r
QnQ
w
w
w
vechvechxEIF ( )( )
( ) ( )
1
2
ˆ
−
∂∂∂
−=T
vechvechvechacov
kk
k
LEQ
we estimate ashC
where and .( )( )( ) ( )
∂∂
∂−=
∧
T
vechvechvechacov
kk
k
LQ
ˆˆˆ
2
16/36
![Page 17: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/17.jpg)
( ) ( ) ( ) ( ) ,ˆ1ˆˆ1ˆ1
1,1,1
1 1,1,1
T
vechvechvechvech
−
−
−= ∑∑ ∑
=−−
= =−−
N
jjNkiNk
N
i
N
jjNkiNk Q
NQQ
NQ
N
N
( ) ( )NrQrNk ,,1ˆ,1
K=−vech ( )k
rg
k QQ ˆˆ −
We solve the following eigenvalue problem. 17/36
( )( ) .0][ˆ][
1
=
−
−∧
hIEIFvechEIFT
acov λgkkgk Q
( )( )kQvechacov∧
Here, we estimate with jackknife method.
where we put as .
( )( ) kk VQ /ˆ
JACKvechacov ≅∧
hC : Cook’s Curvature
![Page 18: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/18.jpg)
1 Get the eigenvectors with
2 Plot them in a lower
dimensional space
3 Detect observations that
have similar influence
4 Evaluate the values of
( )( ) .0][ˆ][
1
=
−
−∧
hIEIFvechEIFT
acov λgkkgk Q
18/36
( ) { }( )g
Jg
k nJZ ,,1ˆ K∈
![Page 19: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/19.jpg)
Outline
1 Introduction
2 Sensitivity analysis in linear subspace
method
3 A multiple-case diagnostics with clustering
4 Numerical example
5 Concluding remarks
19/36
![Page 20: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/20.jpg)
A multiple-case diagnostics with clustering
We can not always represent influence directions in
low dimensions.
We solve the problem with clustering.
Eigenvalues of :
A component of the eigenvectors : ( )gij nij ,,1, K=a
021 ≥≥≥≥gn
ξξξ L
( )( )
−
−∧
IEIFvechEIF T
acov λ][ˆ][
1
gkkgk Q
20/36
![Page 21: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/21.jpg)
Euclidean norm
( ) ( ) ( ).,,1,1 g
n
i ijijijijjj njjdg
K=−−= ∑ =aaaa
T
( )jjdD=Apply Ward’s method to
Reduce the number of combinations of deleting
observations based on the result
21/36
![Page 22: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/22.jpg)
Outline
1 Introduction
2 Sensitivity analysis in linear subspace
method
3 A multiple-case diagnostics with clustering
4 Numerical example
5 Concluding remarks
22/36
![Page 23: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/23.jpg)
Numerical example23/36
[Simulation]
Classes : Group1, Group2
Distribution : Multivariate normal distribution
Number of variables : 6
Number of observations : 10
We compared the result of the proposed method
with the result of deleting observations in all
possible combinations.
Threshold of linear subspace method : 0.997
![Page 24: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/24.jpg)
Group1 :
[ ]Tµ 352.12244.18030.1037.0241.127857.1051 −=
=∑
44.963101.179-5.694-0.045-13.106-245.875
101.179-250.45912.1270.076209.257525.731-
5.694-12.1272.1710.0019.090-21.393-
0.045-0.0760.0010.0000.208-0.384-
13.106-209.2579.090-0.208-1674.438789.092
245.875525.731-21.393-0.384-789.0928676.883
1
Data [,1] [,2] [,3] [,4] [,5] [,6]
[1,] 3.251 74.731 0.04 2.868 19.71 -15.56
[2,] -18.51 87.221 0.038 0.316 17.876 -13.239
[3,] 162.092 130.641 0.03 0.835 -2.196 -2.52
[4,] 120.625 175.7 0.033 3.239 39.097 -17.958
[5,] 70.893 104.692 0.064 -0.848 12.563 -10.372
[6,] 61.044 139.198 0.044 1.157 14.358 -10.993
[7,] 277.224 98.071 0.051 1.425 18.551 -15.139
[8,] 40.983 188.004 0.042 0.915 48.729 -23.67
[9,] 212.658 175.569 0.008 -1.482 -2.177 -1.233
[10,] 128.307 98.585 0.018 1.873 15.924 -12.834
24/36
![Page 25: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/25.jpg)
Group2 : [ ]Tµ 1.3141.3250.089-0.000131.57897.4922 =
=∑
101.04150.9006.4070.119311.49677.281
50.90092.73232.2380.025-171.388145.029
6.40732.23818.1230.00210.82717.550
0.1190.025-0.0020.0010.125-1.018-
311.496171.38810.8270.125-1934.00284.440-
77.281145.02917.5501.018-84.440-5736.272
2
Data [,1] [,2] [,3] [,4] [,5] [,6]
[1,] 90.704 96.139 -0.021 -0.622 -0.666 -3.766
[2,] 39.285 95.397 0.026 3.493 0.681 1.991
[3,] 31.19 243.481 -0.001 1.623 8.291 16.302
[4,] -18.781 113.201 0.011 2.88 7.173 -4.303
[5,] 193.847 161.998 0.024 0.688 15.552 22.948
[6,] 122.154 110.92 0.024 4.865 7.591 -4.361
[7,] 174.834 111.309 0.007 -0.382 -7.5 -1.789
[8,] 48.081 122.082 0.024 -9.056 -15.391 -2.228
[9,] 87.225 122.684 -0.013 -5.637 -9.465 -4.696
[10,] 206.376 138.573 -0.082 1.261 6.988 -6.954
25/36
![Page 26: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/26.jpg)
26/36
{ }10,7 Large influential subset
Average scores in Group1
Average scores in Group2
All possible combinations
( ) { }( )1
1 ,,1;2,1ˆ nJkZ J
k K∈=
![Page 27: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/27.jpg)
27/36
( ) { }( )2
2 ,,1;2,1ˆ nJkZ J
k K∈=
{ }8,5 Large influential subset
Average scores in Group1
Average scores in Group2
All possible combinations
![Page 28: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/28.jpg)
28/36
8 clusters8 clusters
1023 → 255 1023 → 255( )128 −= ( )128 −=
Clusters of influence directions
Proposed method
![Page 29: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/29.jpg)
{ }10,7
Large influential subset
Average scores in Group1
Average scores in Group2
29/36Proposed method
![Page 30: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/30.jpg)
{ }8,5Large influential subset
Average scores in Group1
Average scores in Group2
30/36Proposed method
![Page 31: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/31.jpg)
Case 231/36
[Simulation]
Classes : Group1, Group2
Distribution : Multivariate normal distribution
Number of variables : 6
Number of observations : 100
Threshold of linear subspace method : 0.997
![Page 32: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/32.jpg)
32/36
[ ]Tµ 352.12244.18030.1037.0241.127857.1051 −=
=∑
44.963101.179-5.694-0.045-13.106-245.875
101.179-250.45912.1270.076209.257525.731-
5.694-12.1272.1710.0019.090-21.393-
0.045-0.0760.0010.0000.208-0.384-
13.106-209.2579.090-0.208-1674.438789.092
245.875525.731-21.393-0.384-789.0928676.883
1
[ ]Tµ 1.3141.3250.089-0.000131.57897.4922 =
=∑
101.04150.9006.4070.119311.49677.281
50.90092.73232.2380.025-171.388145.029
6.40732.23818.1230.00210.82717.550
0.1190.025-0.0020.0010.125-1.018-
311.496171.38810.8270.125-1934.00284.440-
77.281145.02917.5501.018-84.440-5736.272
2
Setting up 3 outliers that
have similar influence
direction
{ }94,93,42
Group1 Group2
![Page 33: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/33.jpg)
19 clusters
1.267651 1030
→ 524287
Our Multiple-case
diagnostics in Group1
{ }94,93,423 outliers
33/36
×
![Page 34: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/34.jpg)
Outline
1 Introduction
2 Sensitivity analysis in linear subspace
method
3 A multiple-case diagnostics with clustering
4 Numerical example
5 Concluding remarks
34/36
![Page 35: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/35.jpg)
Concluding remarks
We proposed a multiple-case diagnostics with
clustering in linear subspace method.
In simulation study, we could confirm the availability
of our method.
In the future, we would like to show the results of
some simulation studies and their Monte Carlo
simulations.
35/36
![Page 36: On multiple-case diagnostics in linear subspace method · On multiple-case diagnostics in linear subspace method Kuniyoshi HAYASHI1 Hiroyuki MINAMI2 Masahiro MIZUTA 2 1Graduate School](https://reader030.vdocument.in/reader030/viewer/2022040515/5e715b280ebbdf33c46855a8/html5/thumbnails/36.jpg)
Thank you for your attention!
36/36