dissimilarities and matching between symbolic objects prof. donato malerba department of...
TRANSCRIPT
DISSIMILARITIES AND MATCHING BETWEEN SYMBOLIC OBJECTS
Prof. Donato MalerbaDepartment of Informatics, University of Bari, [email protected]
ASSO SchoolAthens, GreeceOctober 6-8, 2003
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
2
COMPUTING DISSIMILARITIES: WHY?
• Several data analysis techniques are based on quantifying a dissimilarity (or similarity) measure between multivariate data. • Clustering• Discriminant analysis• Visualization-based approaches
• Symbolic objects are a kind of multivariate data.
• Ex.: [colour={red, black}][weight {60,70,80}][height []1.50,1.60]
• The dissimilarity measures presented here are among those investigated in the ASSO Project.
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
3
A case study
• Abalone features survey
• Abalones are members of a large class (Gastropoda) of molluscs having one-piece shells.
• 4177 cases of marine crustaceans described by the following attributes:
Attribute Name
Data Type Unit of meas.
Description
Sex Nominal M, F. I (inf ant) Length Continuous mm Longest shell measurement Diameter Continuous mm Perpendicular to length Height Continuous mm Measured with meat in shell Whole weight Continuous grams Weight of the whole abalone Shucked weight Continuous grams Weight of the meat Viscera weight Continuous grams Gut weight af ter bleeding Shell weight Continuous grams Weigh of the dried shell Rings I nteger Number of rings
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
4
The construction of SO
DB2SO: facility of the ASSO system to generate (Boolean or Probabilistic) symbolic objects from relational databases.
Input:
• a set of groups or classes C1, C2, …, CK
• a set of n individuals k each of which is described by p variables Y1, …, Yp and is assigned to one or more groups
Output:
•a set of K symbolic objects ei described by p variables Y1, …, Yp
Example: Nine symbolic objects, one for each interval of: Number of rings
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
5
TABLE OF BOOLEAN SYMBOLIC OBJECTS
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
6
COMPUTATION OF DISSIMILARITIES BETWEEN SYMBOLIC OBJECTS
Dissimilarity matrixDissimilarity matrix
SO 1 2 3 4 1 0.0000 2 0.2053 0.0000 3 12.8626 15.0793 0.0000 4 14.0338 15.0403 8.6463 0.0000 …
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
7
The MID property
the degree of dissimilarity between crustaceans computed on the independent attributes should be proportional to the dissimilarity in the dependent attribute (i.e., the difference in the number of rings). This property is called monotonic increasing dissimilarity (MID).
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
8
The MID property
Abalone - U_ 1
0
3
6
9
12
15
1-3 4-6 7-9 10-12
13-15
16-18
19-21
22-24
25-29
Abalone - U_ 2
0
0,5
1
1,5
2
2,5
1-3 4-6 7-9 10-12
13-15
16-18
19-21
22-24
25-29
Abalone - U_ 4
0
0,05
0,1
0,15
0,2
1-3 4-6 7-9 10-12
13-15
16-18
19-21
22-24
25-29
Abalone - SO_1
00,10,20,30,4
1-3 4-6 7-9 10-12
13-15
16-18
19-21
22-24
25-29
Abalone - SO_2
00,050,1
0,150,2
0,25
1-3 4-6 7-9 10-12
13-15
16-18
19-21
22-24
25-29
Abalone - SO_3
00,30,60,91,21,5
1-3 4-6 7-9 10-12
13-15
16-18
19-21
22-24
25-29
Abalone - SO_4
00,10,20,30,4
1-3 4-6 7-9 10-12
13-15
16-18
19-21
22-24
25-29
Abalone - SO_5
00,30,60,91,2
1-3 4-6 7-9 10-12
13-15
16-18
19-21
22-24
25-29
Abalone - U_3
00,30,60,91,21,5
1-3 4-6 7-9 10-12
13-15
16-18
19-21
22-24
25-29
Abalone - C_1
00,30,60,91,2
1-3 4-6 7-9 10-12
13-15
16-18
19-21
22-24
25-29
The degree of dissimilarity between crustaceans computed on the independent attributes should be proportional to the dissimilarity in the dependent attribute (i.e., the difference in the number of rings). This property is called monotonic increasing dissimilarity (MID).
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
9
BOOLEAN SYMBOLIC OBJECTS (BSO’S)
A BSO is a conjunction of boolean elementary events: [Y1=A1] [Y2=A2] ... [Yp=Ap]
where each variable Yi takes values in Yi and Ai is a subset of Yi Let a and b be two BSO’s:a = [Y1=A1] [Y2=A2] ... [Yp=Ap]
b = [Y1=B1] [Y2=B2] ... [Yp=Bp]
where each variable Yj takes values in Yj and Aj and Bj are subsets of Yj. We are interested to compute the dissimilarity d(a,b).
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
10
CONSTRAINED BSO’S
Two types of dependencies between variables:
• Hierarchical dependence (mother-daughter): A variable Yi may be inapplicable if another variable
Yj takes its values in a subset Sj Yj. This
dependence is expressed as a rule:
if [Yj = Sj] then [Yi = NA]
• Logical dependence: This case occurs, if a subset
Sj Yj of a variable Yj is related to a subset Si Yi of
a variable Yi by a rule such as:
if [Yj = Sj] then [Yi = Si]
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
11
DISSIMILARITY AND SIMILARITY MEASURES
Dissimilarity MeasureDissimilarity Measure d: EER such that d*
a = d(a,a) d(a,b) = d(b,a) < a,bE
Similarity MeasureSimilarity Measures: EE R such that s*
a = s(a,a) s(a,b) = s(b,a) 0 a,bE
Generally: a E: d*
a = d* and s*a= s* and specifically, d* = 0 while s*= 1
Dissimilarity measures can be transformed Dissimilarity measures can be transformed into similarity measures (and viceversa):into similarity measures (and viceversa):
d=(s) ( s=-1(d) )where:(s) strictly decreasing function, and (1) = 0, (0) =
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
12
DISSIMILARITY AND SIMILARITY MEASURES: PROPERTIES
Some properties that a dissimilarity measure d on E may satisfy are:
1. d(a, b) = 0 c E: d(a, c) = d(b, c) (eveness)
2. d(a, b) = 0 a = b (definiteness)
3. d(a, b) d(a, c) + d(c, b) (triangle inequality)
4. d(a, b) max(d(a, c), d(c, b)) (ultrametric inequality )
5. d(a, b) + d(c, d) max(d(a, c) + d(b, d), d(a, d) +d(b, c)) (Buneman's inequality)
6. Let (E, +) be a group, then d(a, b) = d(a+c, b+c) (translation invariance )
A dissimilarity function that satisfies proprieties 2 and 3 is called metric.
A dissimilarity function that satisfies only property 3 is called pseudo metric or semi- distance.
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
13
DISSIMILARITY MEASURES BETWEEN BSO’S
Author(s) (Year) Notation from the SODAS Package • Gowda & Diday (1991) U_1• Ichino & Yaguchi (1994) U_2, U_3, U_4• De Carvalho (1994) SO_1, SO_2• De Carvalho (1996, 1998) SO_3, SO_4, SO_5, C_1
U: only for unconstrained BSO’sC: only for constrained BSO’sSO: for both constrained and unconstrained BSO’s
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
14
GOWDA & DIDAY’S DISSIMILARITY MEASURE
Gowda & Diday’s dissimilarity measures for two BSO’s a and b:
U_1
If Yj is a continuous variable:
D(Aj, Bj) = D(Aj, Bj) + Ds(Aj, Bj) + Dc(Aj, Bj)
while if Yj is a nominal variable:
D(Aj, Bj) = Ds(Aj, Bj) + Dc(Aj, Bj)
where the components are defined so that their values are normalized between 0 and 1:
• D(Aj, Bj) due to position,
• Ds(Aj, Bj) due to span,
• Dc(Aj, Bj) due to content
),(1
jj
p
jBAD
D(a, b) =
Aj Bj
D DsDc
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
15
GOWDA & DIDAY’S DISSIMILARITY MEASURE
Properties:Properties:D(a, b) = 0 a = b (definiteness property), No proof is reported for the triangle inequality
property
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
16
ICHINO & YAGUCHI’S DISSIMILARITY MEASURES
Ichino & Yaguchi’s dissimilarity measures are based on the Cartesian operators join and meet . For continuous variables:
Aj Bj
Aj Bj
while for nominal variables:
Aj Bj = Aj Bj
Aj Bj = Aj Bj
Given a pair of subsets (Aj, Bj) of Yj the componentwise
dissimilarity(Aj,Bj) is:
(Aj, Bj) =Aj Bj Aj Bj+ (2Aj BjAj Bj)
where 0 0.5 and Ajis defined depending on variable types.
Aj Bj
Aj Bj
Aj Bj
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
17
ICHINO & YAGUCHI’S DISSIMILARITY MEASURES
(Aj,Bj) are aggregated by an aggregation function such as the
generalised Minkowski’s distance of order q:U_2
Drawback: dependence on the chosen units of measurements.Solution: normalization of the componentwise dissimilarity:U_3
The weighted formulation guarantees that dq(a,b)[0,1].
U_4
qp
j
qjjq BAbad
1),(),(
j
jjjjq
p
j
qjjq
Y
BABABAbad
),(),( ,),(),(
1
qp
j
qjjjq BAcbad
1),(),(
The above measures are The above measures are metricsmetrics
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
18
DE CARVALHO’S DISSIMILARITY MEASURES
A straightforward extension of similarity measures for classical data matrices with nominal variables.
where (Vj) is either the cardinality of the set Vj (if Yj is a nominal variable) or the length of the interval Vj (if Yj is a continuous variable).
Agreement Disagreement TotalAgreement =(Aj
Bj) =(Ajc(Bj)) (Aj)
Disagreement =(c(Aj)Bj) =(c(Aj)c(Bj)) (c(Aj))Total (Bj) (c(Bj)) (Yi)
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
19
DE CARVALHO’S DISSIMILARITY MEASURES
Five different similarity measures si, i = 1, ..., 5, are defined:
The corresponding dissimilarities are di = 1 si. The di are aggregated by an aggregation function AF such as the generalised Minkowski metric, thus obtaining:SO_1 5i1 ),(),(
1
qp
j
qjjij
ia BAdwbad
si Comparison Function Range Property s1 / (++) [0,1] metric s2 2/ (2++) [0,1] semi metric s3 / (+2+2) [0,1] metric s4 ½[/ (+)+/ (+)] [0,1] semi metric S5 / [(+)(+)]½ [0,1] semi metric
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
20
DE CARVALHO’S EXTENSION OF ICHINO & YAGUCHI’S DISSIMILARITY MEASURE
A different componentwise dissimilarity measure:
where is defined as in Ichino & Yaguchi’s dissimilarity measure. The aggregation function AF suggested by De Carvalho is:SO_2
jj
jjjj BA
BABA
,
,
qp
j
q
jjq BAp
bad
1
),(1
),(
This measure is a metric. This measure is a metric.
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
21
THE DESCRIPTION-POTENTIAL APPROACH
All dissimilarity measures considered so far are defined by two functions: a comparison function (componentwise measure) and an aggregation function.A different approach is based on the concept of description potential (a) of a symbolic object a.
where (Vj) is either the cardinality of the set Vj (if Yj is a
nominal variable) or the length of the interval Vj (if Yj is a
continuous variable).
p
jjAa
1)()(
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
22
THE DESCRIPTION-POTENTIAL APPROACH
SO_3
SO_4
SO_5
The triangular inequality does not hold for SO_3 and The triangular inequality does not hold for SO_3 and SO_4, which are equivalent. SO_5 is a metric. SO_4, which are equivalent. SO_5 is a metric.
)]()()(2[)()(),(1 bababababad
)(
)]()()(2[)()(),(2 Ea
bababababad
)()]()()(2[)()(
),(2 bababababa
bad
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
23
DESCRIPTION POTENTIAL FOR CONSTRAINED BSO’S
Given a BSO a and a logical dependence expressed by the rule:
if [Yj = Sj] then [Yi = Si]
the incoherent restriction a’ of a is defined as:
a’= [Y1=A1] ... [Yj-1=Aj-1] [Yj=Aj Sj] ... [Yi-1=Ai-1] [Yi=Ai (Yi\Si)] ... [Yp=Ap]
Then the description potential of a is:
A similar extension exists for hierarchical dependencies.
p
jj aAa
1)()()(
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
24
DISSIMILARITY MEASURES FOR CONSTRAINED BSO’S
•The extended definition of description potential can be applied to the computation of the distances SO_3, SO_4 and SO_5.•De Carvalho proposed an extension of ’, so that SO_2 can also be applied to constrained BSO.•He also proposed an extension of , , , and in order to take into account of constraints. Therefore, SO_1 can also be applied to constrained BSO.Finally, C_1 is defined as follows:where:
If all BSO’s are coherent, then the dissimilarity measures If all BSO’s are coherent, then the dissimilarity measures do not change.do not change.
q p
j
p
j
qjji
q
j
BAd
bad
1
1
)(
,
),(
otherwise1
if0)(
NAYj j
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
31
MATCHING
• Matching is the process of comparing two or more structures to discover their similarities or differences.• Similarity judgements in the matching process are directional: They have a• referent, a, a prototype or the description of a class of objects• subject, b, a variant of the prototype or an instance of a class of objects. • Matching two structures is a common problem to many domains, like symbolic classification, pattern recognition, data mining and expert systems.
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
32
MATCHING BSO’S
•Generally, a BSO represents a class description and plays the role of the referent in the matching process. a: [color = {black, white}] [height =[170, 200]]describes a set of individuals either black or white, whose height is in the interval [170,200]. Such a set of individuals is called extension of the BSO. The extension is a subset of the universe of individuals Given two BSO’s a and b, the matching operators define whether b is the description of an individual in the extension of a.• In the ASSO software two matching operators for BSO’s have been defined.
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
33
CANONICAL MATCHING OPERATOR
• The result of the canonical matching operator is either 0 (false) or 1 (true). • If E denotes the space of BSO’s described by a set of p variables Yi taking values in the corresponding domains Yi, then the matching operator is a function:
Match: E × E {0, 1}such that for any two BSO’s a, b E:
a = [Y1=A1] [Y2=A2] ... [Yp=Ap]
b = [Y1=B1] [Y2=B2] ... [Yp=Bp]it happens that:
• Match(a,b) = 1 if BiAi for each i=1, 2, , p,• Match(a,b) = 0 otherwise.
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
34
CANONICAL MATCHING OPERATOR
Examples:
District1 = [profession={farmer, driver}] [age=[24,34]]
Indiv1 = [profession=farmer] [age=28]
Indiv2 = [profession=salesman] [age=[27,28]]
Match(District1,Indiv1) = 1 Match(District1,Indiv2) = 0
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
35
CANONICAL MATCHING OPERATOR
• The canonical matching function satisfies two out of three properties of a similarity measure:
a, b E: Match(a, b) 0 a, b E: Match(a, a) Match(a, b)
while it does not satisfy the commutativity or simmetry property:
a, b E: Match(a, b) = Match(b, a)because of the different role played by a and b.
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
36
FLEXIBLE MATCHING OPERATOR
• The requirement BiAi for each i=1, 2, , p, might be too strict for real-world problems, because of the presence of noise in the description of the individuals of the universe. • Example:District1 = [profession={farmer, driver}] [age=[24,34]]
Indiv3 = [profession=farmer] [age=23]
Match(District1,Indiv3) = 0• It is necessary to rely on a flexible definition of matching operator, which returns a number in [0,1] corresponding to the degree of match between two BSO’s, that is
flexible-matching: E × E [0,1]
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
37
FLEXIBLE MATCHING OPERATOR
For any two BSO’s a and b, i) flexible-matching(a,b)=1 if Match(a,b)=true, ii) flexible-matching(a,b)[0,1) otherwise.
The result of the flexible matching can be interpreted as the probability of a matching b provided that a change is made in b.
Let Ea = {b' E | Match(a,b')=1} and P(b | b') be the conditional probability of observing b given that the original observation was b'. Then
that is flexible-matching(a,b) equals the maximum conditional probability over the space of BSO’s canonically matched by a.
)'|('max = ),matching(-flexible bbP
aEbdefba
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
38
FLEXIBLE MATCHING: AN APPLICATION
• Credit card applications (Quinlan)• Fifteen variables whose names and values
have been changed to meaningless symbols to protect the confidentiality of the data.
+• class variable: positive in case of approval
of credit facilities, negative otherwise.• Training set: 490 cases• 6 rules generated by Quinlan’s system C4.5
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
39
FLEXIBLE MATCHING: AN APPLICATION
• Such rules can be easily represented by means of Boolean symbolic objects.
• Both matching operators can be considered in order to test the validity of the induced rules.
Rule Class Conditions41 - [Y3 > 1.54] [ Y9 = f ] [ Y4 {u, y}]
[Y6{c,d, cc, i, j, k, m, r, q, w, e, aa, ff}]
43 - [ Y4 {u, y}] [ Y8 <= 1.71 ] [ Y9 = f ]6 - [ Y3 <= 0.835] [ Y6 {c,d,i,k,m,q,w,e,aa }]
[ Y7 {v,bb}] [Y14 > 102] [Y15 <= 500]30 + [ Y9 = t ]34 + [Y3 <= 0.125 ] [Y14 > 221 ]46 + [Y4 {l} ]
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
40
A new dissimilarity measure
Flexible matching is asymmetric. However it is possible to “symmetrize” it New dissimilarity measure SO_6
It is computed asd(a,b) =
= 1-(flexible_matching(a,b)
+flexible_matching(b,a))/2
Dissimilarity measure Parameters Constraints Default U_1 (Gowda & Diday) none U_2 (Ichino & Yaguchi) Gamma
Order of power [0 .. 0.5] 1 .. 10
0.5 2
U_3 (Normalized Ichino & Yaguchi)
Gamma Order of power
[0 .. 0.5] 1 .. 10
0.5 2
U_4 (Weighted Normalized Ichino & Yaguchi)
Gamma Order of power List of weights, one per var.
[0 .. 0.5] 1 .. 10 Sum(weights) = 1.0
0.5 2 Equal weights
C_1 (Normalized De Carvalho)
Comparison function Order of power
D1, D2, D3, D4, D5 1 .. 10
D1 2
SO_1 (De Carvalho) Comparison function Order of power List of weights, one per var.
D1, D2, D3, D4, D5 1 .. 10 Sum(weights) = 1.0
D1 2 Equal weights
SO_2 (De Carvalho) Gamma Order of power
[0 .. 0.5] 1 .. 10
0.5 2
SO_3 (De Carvalho) Gamma Order of power
[0 .. 0.5] 1 .. 10
0.5 2
SO_4 (Normalized De Carvalho)
Gamma Order of power
[0 .. 0.5] 1 .. 10
0.5 2
SO_5 (Normalized De Carvalho)
Gamma Order of power
[0 .. 0.5] 1 .. 10
0.5 2
SO_6 (Symmetrized Flexible Matching)
none
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
42
Probabilistic symbolic objects (PSO’s) involve modal (probabilistic) variables.
Each cell represents the set of weighted values that the variable can take for a symbolic object, where a probabilistic weighting system is adopted.
In case of PSO, it isn’t possible to use dissimilarity measures for BSO because they don’t take the probabilities into consideration and so this determines a notable information loss.Therefore, new dissimilarity measures for PSO are needed.
PROBABILISTIC SYMBOLIC OBJECT (PSO’S)
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
43
Defining dissimilarity measures for probabilistic symbolic objects
Steps:1. Define coefficients measuring the divergence
between two probability distributions
Kullback-Leibler divergence Chi-square divergence Hellinger K-divergence Variation distance
(*) from them two dissimilarity measures, namely the Renyi’s and Chernoff’s coefficients, are obtained
non-symmetric coefficients
symmetric coefficient
similarity coefficient (*)
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
44
Defining dissimilarity measures for probabilistic symbolic objects
Steps:2. Symmetrize the non symmetric
coefficientsm(P,Q)= m(Q,P) + m(P,Q)
3. Aggregate the contribution of all variables to compute the dissimilarity between two symbolic objects
PSO Dissimilarity measures
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
45
Mixture SO
Some SO’s can be described by both non-modal and modal variables
They are neither BSO’s nor PSO’sWhat dissimilarity measure, then? In ASSO it has been proposed to combine
the result of two dissimilarity measure, one for modal and the other for non-modal.
Combination can be either additive or multiplicative.
This possibility should be taken with great care!!!
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
46
REFERENCES• Esposito F., Malerba D., V. Tamma, H.-H. Bock. Classical
resemblance measures. Chapter 8.1• Esposito F., Malerba D., V. Tamma. Dissimilarity measures
for symbolic objects. Chapter 8.3• Esposito F., Malerba D., F.A. Lisi. Matching symbolic
objects. Chapter 8.4in H.-H. Bock, E. Diday (eds.): Analysis of Symbolic Data.
Exploratory methods for extracting statistical information from complex data. Springer Verlag, Heidelberg, 2000.
• D. Malerba, L. Sanarico, & V. Tamma (2000). A comparison of dissimilarity measures for Boolean symbolic data. In P. Brito, J. Costa, & D. Malerba (Eds.), Proc. of the ECML 2000 Workshop on “Dealing with Structured Data in Machine Learning and Statistics”, Barcelona.
• D. Malerba, F. Esposito, V. Gioviale, & V. Tamma. Comparing Dissimilarity Measures in Symbolic Data Analysis. Pre-Proceedings of EKT-NTTS, vol. 1, pp. 473-481.
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
47
REFERENCES• D. Malerba, F. Esposito, M. Monopoli (2002). Estrazione e
matching di oggetti simbolici da database relazionali. Atti del Decimo Convegno Nazionale su Sistemi Evoluti per Basi di Dati SEBD’2002, 265-272.
• D. Malerba, F. Esposito, & M. Monopoli (2002). Comparing dissimilarity measures for probabilistic symbolic objects. In A. Zanasi, C. A. Brebbia, N.F.F. Ebecken, P. Melli (Eds.) Data Mining III, Series Management Information Systems, Vol 6, 31-40, WIT Press, Southampton, UK.
• E. Diday, F. Esposito (2003). An Introduction to Symbolic Data Analysis and the Sodas Software, Intelligent Data Analysis, 7, 6, (in press).
• Other project reports
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
48
METHOD DISS
• Dissimilarity measures between both BSO’s and PSO’s.
• Input: Asso file of SO’s• Output for
dissimilarities: Report + Asso file with dissimilarity matrix
• Developer: Dipartimento di Informatica, University of Bari, Italy.
DI method
Report file
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
49
TWO USE CASE DIAGRAMS
Run the DISS method and generate a new ASSO file with a dissimilarity matrixUser
Create a new chaining with the new ASSO file
Create an ASSO chaining with the DISS method
Set up parameters of the DISS method
Run the DISS method and generate a report file
User
View report file
Create an ASSO chaining with the DISS method
Set up parameters of the DISS method
Run VDISS and visualize the dissimilarity measure, the bi-dimensional mapping & the graphical representation
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
50
PARAMETER SETUP
• The user can select a subset of variables Yi on which the dissimilarity measure or the matching operator has to computed .
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
51
PARAMETER SETUP• The user can select a number of parameters.
Dissimilarity measure
Name of the new ASSO file ?
combine
?
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
52
OUTPUT SODAS FILE
• The output ASSO file contains both the same input data and an additional dissimilarity matrix. The dissimilarity between the i-th and the j-th BSO is written in the cell (entry) (i, j) of the matrix.
• Only the lower part of the dissimilarity matrix is reported in the file, since dissimilarities are symmetric.
abalone output file
OUTPUT REPORT FILE
The report file is organized as follows:
Output report file
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
54
Output
Visualization of the dissimilarity table
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
55
OutputVisualization of a line graph of dissimilarities
Each line represents the dissimilarity between a given SO and the subsequent SOs in the file
The number of lines in each graph is equal to the number of SOs minus one
Fare clic per modificare lo stile del titolo dello schema
Fare clic per modificare gli stili del testo dello schema Secondo livello
Terzo livello• Quarto livello
– Quinto livello
56
Output
Visualization of a scatterplot of Sammon’s nonlinear mapping into a bidimensional space