Application of the Dominance-based Rough Set Approach to Case-based Reasoning

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Application of the Dominance-based Rough Set

Approach to Case-based Reasoning

Marcin Szeląg

Institute of Computing Science, Poznań University of Technology,

60-965 Poznań, Poland

12.5.2010

1



Outline

1 Introduction

2 CBR-DRSA Methodology

Basic Notions and DefinitionsDecision RulesCertain and Possible Fuzzy Classification

3 Conclusions

2



Introduction

3



General Motivations

Case-based reasoning (CBR) is a natural way in which peoplesolve problems. It is a process of solving new problems basedon the solutions of similar problems from the past.

When calculating similarity of two objects, main difficulty

consists in aggregation of different criteria/attributes; usuallysuch aggregation is performed arbitrary , using weights oraggregation operators like sum, average or distance metrics.

Therefore, there is a need for multi-criteria/multi-attributemodelling method that allows to include domain knowledge,

can handle possible inconsistencies in data, and avoids anyaggregation operators.

4



Information and Decision Table

Information table is defined by a set of objects .

Objects are described by a set of criteria and regular attributes.

If the set of criteria and regular attributes is divided into twodisjoint subsets of conditions and decisions , then informationtable is called a decision table.

Criterion is an attribute with values ordered according to a scale of preference introduced (by a decision maker) as a part of domainknowledge.

We distinguish two types of criteria:

ordinal, with values expressed on ordinal scale,

cardinal, with values expressed on interval or ratio scale.

5



Multi-attribute Fuzzy Classification Problem Statement

There is given a set of objects described in terms of condition

attributes from set , and a set of decision classes , .Each decision class , is a fuzzy set withmembership function . For given object ,value reflects the credibility that object belongs todecision class .

There is given a marginal similarity function for each conditionattribute , , and objects from set are marked as reference objects (“known cases”).

The task is to build a similarity-based model that is capable of fuzzy classification, i.e., which can assign appropriate membershipvalue to each decision class , for new (test) objects.

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Motivations for Application of DRSA

The problem can be effectively solved using Dominance-based

Rough Set Approach (DRSA), which:can handle inconsistences in data (preprocessing), resulting,e.g., from imprecise of incomplete information,

takes into account domain knowledge:

domains of attributes, i.e., sets of values that an attribute may

take while being meaningful for user’s perception,

division of attributes into condition and decision attributes,

monotonicity constraints between attributes, addressed by the

dominance principle,

works with heterogenous attributes – nominal, ordinal andcardinal (no need of discretization),

enables to infer decision rule model from decision table(disaggregation-aggregation paradigm).

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f l d l



Motivations for Using Decision Rule Model

Advantages of decision rules:

comprehensible form of knowledge representation,

can represent any function (more general than utility functions

or binary relations),resistant to irrelevant attributes,

do not require aggregation operators,

support “backtracing”,

can explain past decisions and predict future decisions.

8



CBR-DRSA Methodology

9

M h d l f S l i F Cl ifi i P bl



Methodology for Solving Fuzzy Classification Problems

Example for “fuzzy” IRIS problem – part of initial data:

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CBR DRSA



CBR-DRSA

The first step consists in creation of similarity tables, one for eachreference object . At this stage chosen marginalsimilarity functions are used to calculate marginal similarities w.r.t.chosen reference objects .

Different marginal similarity functions can be used, depending onthe domain of attribute , . The minimalrequirement that each such function must

satisfy is that , where is a reference object .

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CBR DRSA M i l Si il it F ti



CBR-DRSA – Marginal Similarity Functions

Numeric attribute with values on interval or ratio scale –similarity is defined using a mathematical function, e.g.:

. . .

Attribute with nominal values – similarity is defined using atable , e.g.:

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CBR DRSA Si il it T bl



CBR-DRSA – Similarity Table

Example for “fuzzy” IRIS problem – part of the similarity tablecreated for reference object no. 36 (5.1,3.4,1.5,0.2|1.0,0.5,0.4),with

, for :

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CBR DRSA cuts of Decision Classes



CBR-DRSA – -cuts of Decision Classes

We consider each decision class (fuzzy set) separately fromthe other classes. For each such class, we identify possibleupward/downward -cuts in the following way:

,

.

For an upward -cut it is required that , while

for a downward -cut , it is required that ,

where denotes a set of membership values to observed indata.

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CBR DRSA Dominance Relation for Pairs of Objects



CBR-DRSA – Dominance Relation for Pairs of Objects

The dominance relation between pairs of objects and ,

w.r.t. set of condition attributes is defined as:

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CBR DRSA Dominance Cones



CBR-DRSA – Dominance Cones

Given and , let:

-positive dominance cone ,

-negative dominance cone .

In the pair , is considered to be a reference object, while

is called a limit object, because it conditions the membership of in

and .

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Contextual Approximations of cuts of Decision Classes



Contextual Approximations of -cuts of Decision Classes

Definitions of contextual -lower approximations:

,

.

Definitions of contextual -upper approximations:

,

.

Definitions of contextual -boundaries: , .

17

Approximations of -cuts of Decision Classes



Approximations of -cuts of Decision Classes

Definitions of -lower approximations:

,

.

Definitions of -upper approximations:

,

.

Definitions of -boundaries:

, .

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CBR-DRSA – Decision Rules



CBR DRSA Decision Rules

Decision rules are induced in order to identify similarity-basedpatterns in data.

All pairs of objects , where belongs to lower approximationof some or , are basis for induction of certain rules.

All pairs of objects , where belongs to upper approximationof some or , are basis for induction of possible rules.

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Exemplary certain decision rule, for upward union ,generated for decision class setosa:

If - -

then belongs to class setosa to degree at least 0.8.

Exemplary possible decision rule, for downward union ,generated for decision class versicolor :

If -

then could belong to class versicolor to degree at most 0.5.

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There is a problem which decision rules resulting fromlower/upper approximations of -cuts , should beused for classification.

The minimal set of certain/possible rules is non-unique . Thechoice of such a set is arbitrary and non-trivial .

On the other hand, (explicit) generation of a set of allcertain/possible rules is computationally hard .

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Proposed approach:

Certain knowledge minimal set of certain rules (MCR),generated from lower approximations of and ,independently for each decision class .

Possible knowledge minimal set of possible rules (MPR),generated from upper approximations of and ,independently for each decision class .

Only minimal certain/possible rules are taken into account.

Decision rules are generated by VC-DomLEM [1,2] algorithm.

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CBR-DRSA – Fuzzy Classification of New Objects



C S u y C ass cat o o e Objects

Fuzzy classification of a new (test) object is performedindependently for each decision class .

For each such class two suggestions can be calculated:

certain suggestion, resulting from application of MCR,

possible suggestion, resulting from application of MPR.

Each suggestion is obtained using a VC-DRSA classifier [5].

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Summary and Conclusions



y

DRSA is a flexible modelling method that allows to includedomain knowledge and can handle possible inconsistencies in

data by calculating lower and upper approximations of sets.

DRSA allows to work with heterogenous attributes – nominal,ordinal and cardinal (no need of discretization).

DRSA can be applied to multi-attribute similarity-based fuzzy

classification problems, that employ pairwise comparisons expressed in similarity table .

Rule model has many advantages, e.g., comprehensibility,generality, lack of aggregation operators, predictive power,resistance to irrelevant attributes.

Definitions of rough approximations and syntax of decisionrules in CBR-DRSA are based on ordinal properties of similarity relations only – unlike in other methods (e.g., k-NN),no aggregation operators are used.

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References



1 J. Błaszczyński, R. Słowiński, M. Szeląg, Sequential Covering Rule Induction Algorithm forVariable Consistency Rough Set Approaches. Submitted to Information Sciences in 2009.

2 J. Błaszczyński, R. Słowiński, M. Szeląg, VC-DomLEM: Rule induction algorithm for variableconsistency rough set approaches. Research Report RA-07/09, Poznań University of Technology,2009.

3 S. Greco, B. Matarazzo, R. Słowiński, Granular Computing for Reasoning About Ordered Data:the Dominance-Based Rough Set Approach. Chapter 15 [in]: W. Pedrycz, A. Skowron, V.Kreinovich (eds.), Handbook of Granular Computing. John Wiley & Sons, Chichester, 2008, pp.347–373.

4 S. Greco, B. Matarazzo, R. Słowiński, Case-based reasoning using gradual rules induced fromdominance-based rough approximations. [In]: G. Wang, T. Li, J. W. Grzymała-Busse, D. Miao,A. Skowron, Y. Yao (eds.), Rough Sets and Knowledge Technology (RSKT 2008). Lecture Notesin Artificial Intelligence, vol. 5009, Springer-Verlag, Berlin, 2008, pp. 268–275.

5 J. Błaszczyński, S. Greco, R. Słowiński, Multi-criteria classification – A new scheme forapplication of dominance-based decision rules. European Journal of Operational Research,181(3), 2007, pp. 1030–1044.

6 S. Greco, B. Matarazzo, R. Słowiński, Dominance-based Rough Set Approach to Case-Based

Reasoning. [In]: V. Torra, Y. Narukawa, A. Valls, J. Domingo-Ferrer (eds.), Modelling Decisionsfor Artificial Intelligence. Lecture Notes in Artificial Intelligence, vol. 3885, Springer-Verlag,Berlin Heidelberg, 2006, pp. 7–18.

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Application of the Dominance-based Rough Set Approach to Case-based Reasoning

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