Decision tables and decision spaces

Transcription

1 Abstract Decision tables and decision spaces Z. Pawlak 1 Abstract. In this paper an Euclidean space, called a decision space is associated with ever decision table. This can be viewed as a generalization of the indiscernibilit matri, basic tool to find reducts in information sstems. Besides, the decision space gives a clear insight in the decision structure imposed b the decision table. 1. Introduction This paper concerns granular structure of decision tables imposed b the indiscernibilit relation on data. With ever decision rule in a decision table three coefficients are associated: the strength, the certaint and the erage factors of the rule. It is shown that these coefficients satisf Baes theorem and the total probabilit theorem. This enables us to use Baes theorem to diser patterns in data in a different wa than that offered b standard Baesian inference technique emploed in statistical reasoning. Besides, it is shown that the decision rules define a relation between condition and decision granules, which can be represented b a flow graph. The certaint and erage factors determine flow of information in the graph, which shows clearl the relationship between condition and decision granules determined b the decision table. The decision structure of a decision table can be represented in a decision space, which is Euclidean space, in which dimensions of the space are determined b values of the decision attribute, points in the space are condition granules and coordinates of the points are strengths of the corresponding rules. Distance in the decision space between condition granules allows to determine how distant are decision makers in view of their decisions. This idea can be viewed as a generalization of the indiscernibilit matri, basic tool to find reducts in information sstems. Besides, the decision space gives a clear insight in the decision structure imposed b the decision table. 2. Information sstems and decision tables In this section we define basic concept of rough set theor, information sstem. An information sstem is a data table, whose columns are labeled b attributes, rows are labeled b objects of interest and entries of the table are attribute values. Formall, an information sstem is a pair S = (U,A, where U and A, are non-empt finite sets called the universe, and the set of attributes, respectivel such that a:u V a, where V a, is the set of all values of a called the domain of a. An subset B of A determines a binar relation I(B on U, which will be called an indiscernibilit relation, and defined as follows: (, I(B if and onl if a( = a( for ever a A, where a( denotes the value of attribute a for element. Obviousl I(B is an equivalence relation. The famil of all equivalence classes of I(B, i.e., a partition determined b B, will be denoted b U/I(B, or simpl b U/B; an equivalence class of I(B, i.e., block of the partition U/B, containing will be denoted b B(. If (, belongs to I(B we will sa that and are B-indiscernible (indiscernible with respect to B. Equivalence classes of the relation I(B (or blocks of the partition U/B are referred to as B-elementar sets or B-granules. If we distinguish in the information sstem two disjoint classes of attributes, called condition and decision attributes, respectivel, then the sstem will be called a decision table and will be denoted b S = (U, C, D, where C and D are disjoint sets of condition and decision attributes, respectivel and C D = A. C( and D( will be referred to as the condition class and the decision class induced b, respectivel. 1 Institute of Theoretical and Applied Informatics, Polish Academ of Sciences, ul. Bałtcka 5, Gliwice, Poland 1

2 Thus the decision table describes decisions (actions, results etc. determined, when some conditions are satisfied. In other words each row of the decision table specifies a decision rule which determines decisions in terms of conditions. An eample of a simple decision table is shown below. decision rule age se profession disease 1 old male es no 2 med. female no es 3 med. male es no 4 old male es es 5 oung male no no 6 med. female no no Table 1. Decision table In the table age, se and profession are condition attributes, whereas disease is the decision attribute. The table contains data concerning relationship between age, se, profession and certain vocational disease. 3. Decision rules In what follows we will describe decision rules more eactl. Let S = (U, C, D be a decision table. Ever U determines a sequence c 1 (,, c n (, d 1 (,, d m ( where {c 1,, c n } = C and {d 1,, d m } = D. The sequence will be called a decision rule induced b (in S and denoted b c 1 (,, c n ( d 1 (,, d m ( or in short C D. The number supp (C,D = C( D( will be called a support of the decision rule C D and the number D supp σ D =, U will be referred to as the strength of the decision rule C D, where X denotes the cardinalit of X. In Table 2 below a modified version of Table 1 is shown. decision rule age se profession disease support strength 1 old male es no med. female no es med. male es no old male es es oung male no no med. female no no Table 2. Support and strength This decision table can be understood as an abbreviation of a bigger decision table containing 1100 rows. Support of the decision rule means the number of identical decision rules in the original decision table. With ever decision rule C D we associate a certaint factor of the decision rule, denoted cer (C, D and defined as follows: 2

3 cer ( D( C( D ( C( C σ, = =, π where C( and C( π ( C( =. U The certaint factor ma be interpreted as a conditional probabilit that belongs to D( given belongs to C(, smbolicall π (D C, i.e., cer (C, D = π (D C. If cer (C, D= 1, then C D will be called a certain decision rule; if 0 < cer (C, D < 1 the decision rule will be referred to as an uncertain decision rule. Besides, we will also use a erage factor of the decision rule, denoted (C, D defined as ( D( D( D ( D( C σ, = =, π where D( and D( π ( D( =. U Similarl D π =. If C D is a decision rule then D C will be called an inverse decision rule. The inverse decision rules can be used to give eplanations (reasons for decisions. 4. Probabilistic properties of decision tables Decision tables have important probabilistic properties which are discussed net. Let C D be a decision rule, then the following properties are valid: C ( D D = 1 cer (1 ( D = 1 ( ( = cer D ( ( C( = π D, π σ, ( ( = C ( D ( ( D( = π C, π σ, cer D ( D( D π ( D( C ( ( C( π σ,, = = π C ( cer D π ( C( σ C, D C, D = cer C, D π C π D ( = D ( ( ( ( ( ( ( That is, an decision table, satisfies (1 - (6. Observe that (3 and (4 refer to the well known total probabilit theorem, whereas (5 and (6 refer to Baes' theorem. Thus in order to compute the certaint and erage factors of decision rules according to formula (5 and (6 it is enough to know the strength (support of all decision rules onl. The strength of decision rules can be computed from data or can be a subjective assessment. Certaint and erage factors for the decision table presented in Table 2 are shown in Table 3. (2 (3 (4 (5 (6 3

4 decision rule strength certaint erage Table 3. Certaint and erage factors Let us observe that according to formulas (5 and (6 the certaint and erage factors can be computed emploing onl the strength of decision rules. In Table 2 decision rules 3 and 5 are certain, whereas the remaining decision rules are uncertain. This means that medium age male having the profession and oung male not having the profession are certainl health. Old males having the profession are most probabl ill (probabilit = 0.69 and medium age females not having the profession are most probabl health (probabilit = The inverse decision rules sa that health persons are most probabl medium age males having the profession (probabilit = 0.43 and ill persons are most probabl old males having the profession (probabilit = Decision tables and flow graphs With ever decision table we associate a flow graph, i.e., a directed acclic graph defined as follows: to ever decision rule C D we assign a directed branch connecting the input node C( and the output node D(. Strength of the decision rule represents a throughflow of the corresponding branch. The throughflow of the graph is governed b formulas (1,...,(6. Σσ = 0.59 A 1,4 cer = 0.69 = 0.87 σ = 0.41 Σσ = 0.15 B 2,6 cer = 0.31 cer = ,2 es Σσ = 0.47 σ = 0.09 Σσ = 0.23 C 3 cer = 0.60 = 0.17 Σσ = 0.03 D 5 cer = ,3,5,6 no Σσ = 0.53 Fig. 1. Flow graph Flow graph associated with decision table presented in Table 2 is shown in Fig. 1. The application of flow graphs to represent decision tables gives a ver clear insight into the decision process. Classification of objects in this representation boils down to finding the maimal output flow in the flow graph, whereas eplanation of decisions is connected with the maimal input flow associated with the given decision (see also [2] and [3]. 4

5 6. Decision Space With ever decision table having one n-valued decision attribute we can associate n-dimensional Euclidean space, where values of the decision attribute determine n ais of the space and condition attribute values (equivalence classes determine point of the space. Strengths of decision rules are to be understood as coordinates of corresponding points. Distance δ (, between point and in an n-dimensional decision space is defined as δ n (, = ( i i where = ( 1,..., n and = ( 1,..., n are vectors of strengths of corresponding decision rules. Decision space for Table 1 is shown in Figure 2 no i= *C(0.00, 0.23 * A (0.41, 0.18 D(0.00,0.03* *B(0.09, Fig. 2. Decision space es Distances between granules A, B, C and D are shown in Table 4. A B C D A B C D Table 4. Distance matri It follows from the above eample that group of patients B, C and D are close and form a cluster which is distant from group A. 7. Conclusions Decision spaces associated with decision tables can be viewed as a generalization of discernibilit matrices. Besides, the enable us to get deeper insight in decision processes determined b decision tables. References 1. Skowron, A., Rauszer, C.: The discernibilit matrices and functions in information sstems, in: Słowiński (ed., Intelligent Decision Support, - Handbook of Applications and Advances in Rough Set Theor, Kluwer Academic Publishers, Dordrech, 1992, Ford, L.R., Fulkerson, D. R.: Flows in Networks. Princeton Universit Press, Princeton, New Jerse Pawlak, Z.: New look on Baes' theorem - the rough set outlook. Proceeding of International Workshop on Rough Set Theor and Granular Computing (RSTGC-2001, Matsue, Shimane, 5

6 Japan, Ma 20-22, S. Hirano, M. Inuiguchi and S. Tsumoto (eds., Bull. of Int. Rough Set Societ vol. 5 no. 1/2, 2001,