DISSIMILARITIES AND MATCHING BETWEEN SYMBOLIC OBJECTS Prof. Donato Malerba Department of Informatics, University of Bari, Italy ASSO.

Presentazione sul tema: "DISSIMILARITIES AND MATCHING BETWEEN SYMBOLIC OBJECTS Prof. Donato Malerba Department of Informatics, University of Bari, Italy ASSO."— Transcript della presentazione:

1 DISSIMILARITIES AND MATCHING BETWEEN SYMBOLIC OBJECTS Prof. Donato Malerba Department of Informatics, University of Bari, Italy malerba@di.uniba.it ASSO School Porto, Portugal May 13-15, 2002

2 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo 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 SODAS Project.

3 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 3 COMPUTING DISSIMILARITIES: AN APPLICATION zAbalone features survey zTwenty-eight Boolean symbolic objects, one for each combination of: yNumber of rings (+1.5 gives the age in years ) Abalones are members of a large class (Gastropoda) of molluscs having one-piece shells.

4 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 4 FEATURES OF ABALONES zAttributes: each symbolic object is described by 8 variables representing different features of abalones NameData TypeMeas.Description SexNominalM, F, and I (infant) LengthContinuousmmLongest shell measurement DiameterContinuousmmperpendicular to length HeightContinuousmmwith meat in shell Whole weightContinuousGramswhole abalone Shucked weightContinuousGramsweight of meat Viscera weightContinuousGramsgut weight (after bleeding) Shell weightContinuousGramsafter being dried

5 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 5 TABLE OF BOOLEAN SYMBOLIC OBJECTS

6 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 6 COMPUTATION OF DISSIMILARITIES BETWEEN BOOLEAN SYMBOLIC OBJECTS Dissimilarity matrix

7 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 7 BOOLEAN SYMBOLIC OBJECTS (BSOS) A BSO is a conjunction of boolean elementary events: [Y 1 =A 1 ] [Y 2 =A 2 ]... [Y p =A p ] where each variable Y i takes values in Y i and A i is a subset of Y i Let a and b be two BSOs: a = [Y 1 =A 1 ] [Y 2 =A 2 ]... [Y p =A p ] b = [Y 1 =B 1 ] [Y 2 =B 2 ]... [Y p =B p ] where each variable Y j takes values in Y j and A j and B j are subsets of Y j. We are interested to compute the dissimilarity d(a,b).

8 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 8 CONSTRAINED BSOS Two types of dependencies between variables: Hierarchical dependence (mother-daughter): A variable Y i may be inapplicable if another variable Y j takes its values in a subset S j Y j. This dependence is expressed as a rule: if [Y j = S j ] then [Y i = NA] Logical dependence: This case occurs, if a subset S j Y j of a variable Y j is related to a subset S i Y i of a variable Y i by a rule such as: if [Y j = S j ] then [Y i = S i ]

9 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 9 DISSIMILARITY AND SIMILARITY MEASURES Dissimilarity Measure d: E E R such that d * a = d(a,a) d(a,b) = d(b,a) < a,b E Similarity Measure s: E E R such that s * a = s(a,a) s(a,b) = s(b,a) 0 a,b E Generally: a E: d * a = d * and s * a = s * and specifically, d * = 0 while s * = 1 Dissimilarity measures can be transformed into similarity measures (and viceversa): d= (s) ( s= -1 (d) ) where: (s) strictly decreasing function, and (1) = 0, (0) =

10 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 10 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.

11 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 11 DISSIMILARITY MEASURES BETWEEN BSOS 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 BSOs C: only for constrained BSOs SO: for both constrained and unconstrained BSOs

12 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 12 GOWDA & DIDAYS DISSIMILARITY MEASURE Gowda & Didays dissimilarity measures for two BSOs a and b: U_1 If Y j is a continuous variable: D(A j, B j ) = D (A j, B j ) + D s (A j, B j ) + D c (A j, B j ) while if Y j is a nominal variable: D(A j, B j ) = D s (A j, B j ) + D c (A j, B j ) where the components are defined so that their values are normalized between 0 and 1: D (A j, B j ) due to position, D s (A j, B j ) due to span, D c (A j, B j ) due to content D(a, b) = AjAj BjBj D DsDs DcDc

13 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 13 GOWDA & DIDAYS DISSIMILARITY MEASURE Properties: D(a, b) = 0 a = b (definiteness property), No proof is reported for the triangle inequality property

14 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 14 ICHINO & YAGUCHIS DISSIMILARITY MEASURES Ichino & Yaguchis dissimilarity measures are based on the Cartesian operators join and meet. For continuous variables: A j B j while for nominal variables: A j B j = A j B j Given a pair of subsets (A j, B j ) of Y j the componentwise dissimilarity (A j,B j ) is: (A j, B j ) = A j B j A j B j + (2 A j B j A j B j ) where 0 0.5 and A j is defined depending on variable types. AjAj BjBj A j B j

15 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 15 ICHINO & YAGUCHIS DISSIMILARITY MEASURES (A j,B j ) are aggregated by an aggregation function such as the generalised Minkowskis 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 d q (a,b) [0,1]. U_4 The above measures are metrics

16 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 16 DE CARVALHOS DISSIMILARITY MEASURES A straightforward extension of similarity measures for classical data matrices with nominal variables. where (V j ) is either the cardinality of the set V j (if Y j is a nominal variable) or the length of the interval V j (if Y j is a continuous variable).

17 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 17 DE CARVALHOS DISSIMILARITY MEASURES Five different similarity measures s i, i = 1,..., 5, are defined: The corresponding dissimilarities are d i = 1 s i. The d i are aggregated by an aggregation function AF such as the generalised Minkowski metric, thus obtaining: SO_1

18 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 18 DE CARVALHOS EXTENSION OF ICHINO & YAGUCHIS DISSIMILARITY MEASURE A different componentwise dissimilarity measure: where is defined as in Ichino & Yaguchis dissimilarity measure. The aggregation function AF suggested by De Carvalho is: SO_2 This measure is a metric.

19 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 19 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 (V j ) is either the cardinality of the set V j (if Y j is a nominal variable) or the length of the interval V j (if Y j is a continuous variable).

20 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 20 THE DESCRIPTION-POTENTIAL APPROACH SO_3 SO_4 SO_5 The triangular inequality does not hold for SO_3 and SO_4, which are equivalent. SO_5 is a metric.

21 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 21 DESCRIPTION POTENTIAL FOR CONSTRAINED BSOS Given a BSO a and a logical dependence expressed by the rule: if [Y j = S j ] then [Y i = S i ] the incoherent restriction a of a is defined as: a= [Y 1 =A 1 ]... [Y j-1 =A j-1 ] [Y j =A j S j ]... [Y i-1 =A i-1 ] [Y i =A i ( Y i \S i )]... [Y p =A p ] Then the description potential of a is: A similar extension exists for hierarchical dependencies.

22 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 22 DISSIMILARITY MEASURES FOR CONSTRAINED BSOS 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 BSOs are coherent, then the dissimilarity measures do not change.

23 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 23 DISSIMILARITY MEASURES FOR CONSTRAINED BSOS 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: where:

24 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 24 DISSIMILARITY MEASURES FOR CONSTRAINED BSOS where = [Y 1 =A 1 ]... [Y j-1 =A j-1 ] [Y j =A j B j ] … [Y p =A p ] = [Y 1 =B 1 ]... [Y j-1 =B j-1 ] [Y j =A j B j ] … [Y p =B p ]

25 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 25 DISSIMILARITY MEASURES FOR CONSTRAINED BSOS where = [Y 1 =A 1 ]... [Y j-1 =A j-1 ] [Y j =A j c(B j )] … [Y p =A p ]

26 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 26 DISSIMILARITY MEASURES FOR CONSTRAINED BSOS where = [Y 1 =B 1 ]... [Y j-1 =B j-1 ] [Y j =c(A j ) B j ] … [Y p =B p ]

27 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 27 DISSIMILARITY MEASURES FOR CONSTRAINED BSOS De Carvalho proposed an extension of,, in order to take into account of constraints

28 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 28 DISSIMILARITY MEASURES FOR CONSTRAINED BSOS The previous extension of,, in order to take into account of constraints, can be used in SO_1. Finally, C_1 is defined as follows: where: If all BSOs are coherent, then the dissimilarity measures do not change.

29 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 29 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.

30 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 30 MATCHING BSOS 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 BSOs a and b, the matching operators define whether b is the description of an individual in the extension of a. In the SODAS software two matching operators for BSOs have been defined.

31 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 31 CANONICAL MATCHING OPERATOR The result of the canonical matching operator is either 0 (false) or 1 (true). If E denotes the space of BSOs described by a set of p variables Y i taking values in the corresponding domains Y i, then the matching operator is a function: Match: E × E {0, 1} such that for any two BSOs a, b E: a = [Y 1 =A 1 ] [Y 2 =A 2 ]... [Y p =A p ] b = [Y 1 =B 1 ] [Y 2 =B 2 ]... [Y p =B p ] it happens that: Match(a,b) = 1 if B i A i for each i=1, 2,, p, Match(a,b) = 0 otherwise.

32 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 32 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

33 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 33 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.

34 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 34 FLEXIBLE MATCHING OPERATOR The requirement B i A i 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 BSOs, that is flexible-matching: E × E [0,1]

35 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 35 FLEXIBLE MATCHING OPERATOR For any two BSOs 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 E a = {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 BSOs canonically matched by a.

36 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 36 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 Quinlans system C4.5

37 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 37 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.

38 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 38 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 8.4 in 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.

39 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 39 METHOD DI Both dissimilarity measures and matching operators between BSOs are available in the method DI (Distance matrix) of the SODAS software. Input: Sodas file of BSOs Output for dissimilarities: Report + Sodas file with distance matrix Output for matching operators: Report Developer: Dipartimento di Informatica, University of Bari, Italy. DI method Report file

40 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 40 TWO USE CASE DIAGRAMS Run the DI method and generate a new SODAS file with a dissimilarity matrix User Create a new chaining with the new SODAS file Create a SODAS chaining with the DI method Set up parameters of the DI method Run the DI method and generate a report file User View report file Create a SODAS chaining with the DI method Set up parameters of the DI method

41 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 41 PARAMETER SETUP The user can select a subset of variables Y i on which the dissimilarity measure or the matching operator has to computed.

42 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 42 PARAMETER SETUP The user can select a number of parameters. Name of the new SODAS file Dissimilarity measure or matching operator

43

44 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 44 OUTPUT SODAS FILE The output SODAS 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. TRIANGE_MATRIX = ( (0), (0.20531, 0), (12.8626, 15.0793, 0), (14.0338, 15.0403, 8.64626, 0), (14.1655, 15.1651, 11.4512, 6.90074, 0), …)

45 OUTPUT REPORT FILE The report file is organized as follows: Page 1 SODAS 07/26/01 Sodas The Statistical Package for Symbolic Data Analysis Version 1.0 - 05 January 2001 **************D I S T A N C E M E A S U R E S*********** Data Information:

46 OUTPUT REPORT FILE Input Sodas File: C:\ASSO_L~1\SPERIM~3\NUOVAS~1\ABALO.SDS 28 Boolean Symbolic Objects (BSOs) read. 8 Variables selected for each BSO: 1 -- 8 Selected Distance Function: U_1 Gowda & Diday Distance Matrix BSO 1 2 3 4 1 0 2 0.2053 0 3 12.86 15.08 0 4 14.03 15.04 8.646 0 5 14.17 15.17 11.45 6.901 ----------------------------------------------------------------------- -- Page 2 SODAS 07/26/01

47 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 47 PARAMETER SETUP Only one parameter has to be specified for matching BSOs. The Class BSO represents the BSO that will be used as referent (default: 2nd BSO). In the case of canonical matching the subject can be any BSO. In the case of flexible matching the subject must be a BSO representing an individual.

48 OUTPUT REPORT FILE The report file is organized as follows: Page 1 SODAS 12/31/99 Sodas The Statistical Package for Symbolic Data Analysis *********** C A N O N I C A L M A T C H I N G ********** Data Information 29 Boolean Symbolic Objects (BSOs) read. 1 : Boolean Symbolic Object (BSO) selected as class. Matching Vector BSO 1 1 Match 2 No Match... 29 No Match This procedure was completed at 17:57:33

49 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 49 FURTHER DEVELOPMENTS IN THE ASSO PROJECT Two modules, DISS and MATCH. Both modules will be upgraded to work with both BSOs and Problabilistic Symbolic Objects (PSOs). Formally, a PSO, which is made up of M probabilistic elementary events (PEE) a i, is defined as: a = = Each PEE a i is attached to an observed quantitative variables Y i which takes k i values c ij a with probability p ij a (which sum up to 1).

50 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 50 Probabilistic symbolic objects (PSOs) are represented by a logical conjunction of internal modal elementary events. Each elementary event is attached to an observed variable. It represents the set of weighted values that the variable can take, where a probabilistic weighting system is adopted. PROBABILISTIC SYMBOLIC OBJECT (PSOS)

51 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 51 PROBABILISTIC SYMBOLIC OBJECT (PSOS) Formally, a PSO, which is made up of M probabilistic elementary events (PEE) a i, is defined as: a = = Each PEE a i is attached to an observed quantitative variables Y i which takes k i values c ij a with probability p ij a (which sum up to 1).

52 Fare clic per modificare lo stile del titolo dello schema zFare clic per modificare gli stili del testo dello schema ySecondo livello xTerzo livello Quarto livello –Quinto livello 52 WHY ARE NEEDED NEW DISSIMILARITY MEASURES FOR PSO? In case of PSO, it isnt possible to use dissimilarity measures for BSO because they dont take the probabilities into consideration and so this determines a notable information loss. Therefore, new dissimilarity measures for PSO are needed.