A Fuzzy Set Tool in the Classi cation and Prediction Software System (CLAPSS)

Abstract

In the paper, we give the outline of a fuzzy set tool implemented in the Classi cation and Prediction Software System (CLAPSS). CLAPSS is being developed for solving di erent classi cation and prediction problems using, among others, some specialized approaches based mainly on fuzzy sets and rough sets which are not available in other machine learning software systems. Theoretical background as well as the module embedded in CLPASS, for fuzzi cation of attribute values in information/decision systems, are described. Moreover, possible further steps in the usage of CLAPSS (generation of fuzzy decision trees as well as fuzzy ow graphs) are mentioned.

Keywords

• Fuzzy sets
• Fuzzi cation
• Software system
• CLAPSS

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Most of the methods implemented in the CLAPSS system are applied for information/decision tables representing information/decision systems understood as Pawlak's knowledge representation systems (cf. [ 12 ]).

A decision system is a tuple DS = (U; C; D; fVaga2C[D; finf ; fdec), where U is the non-empty, nite set of objects, C is the non-empty, nite set of condition attributes, D is the non-empty, nite set of decision attributes, fVaga2C[D is the family of non-empty sets of condition and decision attribute values, finf : C U ! S Vc is the information function such that finf (c; u) 2 Vc for each c2C c 2 C and u 2 U , fdec : D

Fuzzi cation is the process that transforms the real value variables into linguistic variables whose domains contain linguistic values which can be described by fuzzy sets (their membership functions). Let IS = (U; A; fVaga2A; finf ) be an information system such that Va R for each a 2 A. For each attribute a 2 A, we can de ne a linguistic variable a. With each linguistic variable a, a set L a = fl1 a ; l2 a ; : : : ; lkaa g of linguistic values is associated. Each linguistic value li a , where i = 1; 2; : : : ; ka, is described by a membership function l a : R ! [0; 1]. In CLAPSS, the user has a broad set of membership functions i which can be used to make a fuzzi cation process. This set consists of: { a triangular shaped membership function, { a trapezoidal shaped membership function, { a Gaussian shaped membership function, { a generalized bell shaped membership function, { an S shaped membership function, { a shaped membership function, { a sigmoidal shaped membership function, { a fuzzy singleton membership function, { a sinusoidal shaped membership function, { a Z shaped membership function, { a pseudo-exponential shaped membership function, { an L-R shaped membership function, { a two Gaussian shaped membership function, { a D-sigmoidal shaped membership function, { a P-sigmoidal shaped membership function.

Let: { IS = (U; A; fVaga2A; finf ) be an information system, where card(U ) = n and card(A) = m, such that Va R for each a 2 A, { fL a ga2A be the family of sets of linguistic values associated with linguistic variables from the family f aga2A de ned for attributes from A, where L a = fl1 a ; l2 a ; : : : ; lkaa g for each a 2 A.

A fuzzi ed information system is a tuple F (IS) = (U F ; ; fV g 2 ; fiFnf ), where U F is the non-empty, nite set of objects such that each u 2 U F corresponds exactly to one u 2 U , = a1 [ a2 [ [ am is the non-empty, nite set of fuzzi ed attributes, fV g 2 is the family of sets of fuzzi ed attribute values, fiFnf : U F ! S V is the information function such that fiFnf ( l aj ; u ) 2 V 2 i for each li aj 2 and u 2 U F , fiFnf ( li aj ; u ) = li aj (finf (aj ; u)), where li aj is a membership function describing li aj . 2

CLAPSS

CLAPSS is our tool developed for solving di erent classi cation and prediction problems using, among others, some specialized approaches based mainly on fuzzy sets and rough sets. CLAPSS is equipped with the graphical user interface (see Figure 1). In general, our main idea is to implement in CLAPSS those specialized approaches which are not available in other machine learning software systems. Selected functionalities of CLAPSS were earlier described in [ 7 ], [ 8 ], and [ 11 ].

The general usage of methods, based on fuzzy sets, implemented in CLAPSS is shown in Figure 2. Information/decision systems (also those fuzzi ed) can be imported from/exported to other machine learning software systems, RSES [ 1 ], WEKA [ 4 ], ORANGE [ 2 ], as it was depicted in Figure 2.

For methods based on fuzzy sets, CLAPSS o ers, rst of all, a tool for fuzzication of attribute values in information/decision systems. The fuzzi cation process can be done in three ways: graphical, scripting, and external.

For graphical fuzzi cation, Membership Function Creator (MFC) has been developed (see Figure 3). MFC enables the user to: { determine linguistic values and membership functions (their shapes and parameters) associated with them, { manually modify membership functions created earlier (for example, characteristic points or slopes can be moved), { see calculated values of the fuzzi ed attribute (these values are automatically updated if some changes in membership functions are made).

After the fuzzi cation process of the selected attribute, a script (a special scripting language was designed for CLAPSS) is generated. The script, consisting of membership function de nitions for each attribute to be fuzzi ed, can also be created manually, i.e., in a scripting way (see an example below). ATTR[0]->fuzzification(lingvalues={low=(trapezoidal,0.0000,0.0000,1.0000,3.0000), medium=(triangular,1.0000,2.5000,4.0000),high=(trapezoidal,2.0000,4.0000,5.0000,5.0000)});

ATTR[ 1 ]->fuzzification(lingvalues={low=(trapezoidal,0.0000,0.0000,1.0000,3.0000), medium=(triangular,1.0000,2.5000,4.0000),high=(trapezoidal,2.0000,4.0000,5.0000,5.0000)}); ATTR[ 2 ]->fuzzification(lingvalues={low=(trapezoidal,0.0000,0.0000,1.0000,3.0000), medium=(triangular,1.0000,2.5000,4.0000),high=(trapezoidal,2.0000,4.0000,5.0000,5.0000)});

Fuzzi cation of attribute values in the external tool is also possible. Then, the user can import a fuzzi ed information/decision system into CLAPSS.

Further steps which can be performed in CLAPSS for fuzzi ed information/decision systems can be as follows: { Generation of fuzzy decision trees. Fuzzy decision trees are generated using the algorithm based on cumulative information estimations of initial data [ 5 ]. { Generation of fuzzy ow graphs [ 11 ]. Fuzzy ow graphs are generated using the fuzzy cardinality (power) of linguistic values (cf. [ 6 ]).

The visualization of fuzzy decision trees and fuzzy ow graphs is possible due to the option for exporting them to the DOT format [ 3 ].

The practical usage of CLAPSS was presented in case of analysis and classi cation of MMPI (Minnesota Multiphasic Personality Inventory) data. Fuzzied decision systems were used among others in determining the importance of ranges of MMPI scales [ 10 ] and classi cation of MMPI pro les using fuzzy decision trees [ 9 ]. 3

Conclusions

In the paper, we have brie y presented an important part of the CLAPSS system concerning implemented methods based on fuzzy sets. CLAPSS is constantly being developed. One of the main directions in further developing of CLAPSS, in the considered area, is to add other types of membership functions and to add graph algorithms for fuzzy ow graphs to extract rules, episodes, etc. Acknowledgments This work was partially supported by the Center for Innovation and Transfer of Natural Sciences and Engineering Knowledge at the University of Rzeszow.

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