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Table 1.4: Hiring: The reordered decision table. Example 1.4.7 The Hiring decision table in Tab. 1.4 is now used to illustrate the construc- tion of the corresponding decision-relative discernibility matrix and function. The rows are reordered for convenience putting the accepted objects in the top rows. The corresponding

Table 1 4: Hiring: The reordered decision table. Example 1.4.7 The Hiring decision table in Tab. 1.4 is now used to illustrate the construc- tion of the corresponding decision-relative discernibility matrix and function. The rows are reordered for convenience putting the accepted objects in the top rows. The corresponding

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Table 1.1: An example information system. Example 1.1.1 A very simple information system is shown in Tab. 1.1. There are seven cases or objects'and two attributes (Age and Lower Extremity Motor Score LEMS).
Table 1.1: An example information system. Example 1.1.1 A very simple information system is shown in Tab. 1.1. There are seven cases or objects'and two attributes (Age and Lower Extremity Motor Score LEMS).
Table 1.2: Walk: An example decision table The reader may again notice that cases 73 and x4 as well as x5 and ‘7 still have exactly the same values of conditions(’but the first pair has a different outcome (different value of the decision attribute) while the second pair also has the same outcome.
Table 1.2: Walk: An example decision table The reader may again notice that cases 73 and x4 as well as x5 and ‘7 still have exactly the same values of conditions(’but the first pair has a different outcome (different value of the decision attribute) while the second pair also has the same outcome.
Table 1.5: Hiring: The decision-relative discernibility matrix. discernibility matrix in Tab. 1.5 is symmetrical and the diagonal is emptyland so are all th entries for which the decisions are equal.
Table 1.5: Hiring: The decision-relative discernibility matrix. discernibility matrix in Tab. 1.5 is symmetrical and the diagonal is emptyland so are all th entries for which the decisions are equal.
Table 1.6: Discretization process. (a) The original decision system A. (b) P-discretization of Alwhere P = {(a,0.9), (a,1.5), (b,0.75), (b,1.5)} The sets of possible values of a and 6 are defined by: The sets of values of a and b on objects from U are given by
Table 1.6: Discretization process. (a) The original decision system A. (b) P-discretization of Alwhere P = {(a,0.9), (a,1.5), (b,0.75), (b,1.5)} The sets of possible values of a and 6 are defined by: The sets of values of a and b on objects from U are given by
Figure 1.6: A geometrical representation of data and cuts.
Figure 1.6: A geometrical representation of data and cuts.
Figure 1.7: The set of cuts (a, cz), (a, c2), (a, c3), (a,c4) on al'the set of propositional variables pilp$lp§l pt and the set of intervals corresponding to these variables in A (see Example 1.7.1) of ®4 defines uniquely the family of all minimal set of cuts discerning objects in A. Moreoverl any valuation satisfying this formula determines the set of cuts discerning all object pairs to be discerned.
Figure 1.7: The set of cuts (a, cz), (a, c2), (a, c3), (a,c4) on al'the set of propositional variables pilp$lp§l pt and the set of intervals corresponding to these variables in A (see Example 1.7.1) of ®4 defines uniquely the family of all minimal set of cuts discerning objects in A. Moreoverl any valuation satisfying this formula determines the set of cuts discerning all object pairs to be discerned.
Table 1.7: Decision system A* constructed from A
Table 1.7: Decision system A* constructed from A
Figure 1.9: Interpretation of the Test and Projection functions in two-dimensional space.
Figure 1.9: Interpretation of the Test and Projection functions in two-dimensional space.
Table 1.8: The decision table A?
Table 1.8: The decision table A?
Figure 1.10: Geometrical presentation of the best surface. we can define decision rules:
Figure 1.10: Geometrical presentation of the best surface. we can define decision rules:
Table 1.9: Data tables stored in the UC Irvine Repository entire original decision table). We first generate a decision algorithm (i.e. a set of decision rules or a decision tree) for the training table and next we check its quality on the testing table. We denote by N the number of objects of the testing table and by N; the number of objects in this table properly recognized by the decision algorithm. The classification quality (accuracy rate) of learning algorithm is defined by gq = 41 100%.
Table 1.9: Data tables stored in the UC Irvine Repository entire original decision table). We first generate a decision algorithm (i.e. a set of decision rules or a decision tree) for the training table and next we check its quality on the testing table. We denote by N the number of objects of the testing table and by N; the number of objects in this table properly recognized by the decision algorithm. The classification quality (accuracy rate) of learning algorithm is defined by gq = 41 100%.
Table 1.11: Results of experiments on inverted pendulum and autopilot data Table 1.10: Results of experiments on Machine Learning data. attributes with real values and one decision attribute representing the angle of the rudder. The problem is to predict this angle with error not exceeding 1° for cases with known values of 6 other attributes.
Table 1.11: Results of experiments on inverted pendulum and autopilot data Table 1.10: Results of experiments on Machine Learning data. attributes with real values and one decision attribute representing the angle of the rudder. The problem is to predict this angle with error not exceeding 1° for cases with known values of 6 other attributes.
Figure 1.11: The decision table and the discernibility matrix
Figure 1.11: The decision table and the discernibility matrix
Figure 1.12: Coloring of attribute value graphs and the reduced table. decision table with these attributes to search for minimal set of new attributes that discerns all pairs of objects from different decision classes. After extracting of these setsI for each attribute a; we construct graph T, = (Vy, Eq) where E, is defined as the set of all new attributes (propositional variables) found for the attribute a. Any vertex coloring of [', defines a partition of Vz. The colorability problem is solvable in polynomial time for k = 2’ but remains NP- complete for all k > 3. Butlsimilarly to discretizationlone can apply some efficient heuristic searching for optimal partition.
Figure 1.12: Coloring of attribute value graphs and the reduced table. decision table with these attributes to search for minimal set of new attributes that discerns all pairs of objects from different decision classes. After extracting of these setsI for each attribute a; we construct graph T, = (Vy, Eq) where E, is defined as the set of all new attributes (propositional variables) found for the attribute a. Any vertex coloring of [', defines a partition of Vz. The colorability problem is solvable in polynomial time for k = 2’ but remains NP- complete for all k > 3. Butlsimilarly to discretizationlone can apply some efficient heuristic searching for optimal partition.
Table 1.12: The quality comparison between decision tree methods. MD: MD-heuristics; MD- G: MD-heuristics with symbolic value partition
Table 1.12: The quality comparison between decision tree methods. MD: MD-heuristics; MD- G: MD-heuristics with symbolic value partition
Table 1.13: The quality of matched tolerance class strategy. Classifiers are defined by conjunc- tive form relation (Acc.: accuracy of classification; Rec.: Recognition percent).
Table 1.13: The quality of matched tolerance class strategy. Classifiers are defined by conjunc- tive form relation (Acc.: accuracy of classification; Rec.: Recognition percent).
Table 1.15: The quality of the ”closest” strategy. Classifiers are defined by conjunction and combination form relation. Table 1.14: The quality of ”matching” strategy. Classifiers are defined by conjunctive form relation.(Acc.: accuracy of classification; Rec.: Recognition percent)
Table 1.15: The quality of the ”closest” strategy. Classifiers are defined by conjunction and combination form relation. Table 1.14: The quality of ”matching” strategy. Classifiers are defined by conjunctive form relation.(Acc.: accuracy of classification; Rec.: Recognition percent)
Table 1.17: The simplified version of discernibility matrix M(A]|q) and association rules
Table 1.17: The simplified version of discernibility matrix M(A]|q) and association rules
Table 1.18: A comparison results for Monk’s problems
Table 1.18: A comparison results for Monk’s problems
Table 1.20: A comparison results for the StatLog’s data
Table 1.20: A comparison results for the StatLog’s data
Table 1.19: A comparison results for medical data
Table 1.19: A comparison results for medical data
Table 2.2: Initial rough inclusion for H Table 2.1: The information system H 2.1.1 Rough Inclusions from Information Systems Rough inclusions can be generated from a given information system A; for instancelfor a given partition P = {Aj,..., Ag} of the set A of attributes into non-empty sets Ai,...,A,Tand a given set W = {w1,..., wx} of weightsl'w; € [0,1] for i =1,2,...,4 and ye w; = 1 we let
Table 2.2: Initial rough inclusion for H Table 2.1: The information system H 2.1.1 Rough Inclusions from Information Systems Rough inclusions can be generated from a given information system A; for instancelfor a given partition P = {Aj,..., Ag} of the set A of attributes into non-empty sets Ai,...,A,Tand a given set W = {w1,..., wx} of weightsl'w; € [0,1] for i =1,2,...,4 and ye w; = 1 we let
Table 2.4: The information system HB Table 2.3: The information system B
Table 2.4: The information system HB Table 2.3: The information system B
the problem of uncertainty propagation i.e. we have to decide in what way is the measure [4 related to measures ju, Uc, -.-. Formally! we have to find an (approximation) to a function f satisfying the following property: In practicel it is unrealistic to expect the existence of a function f satisfying the above condition globally; thereforeI ‘we localize this notion. To this endl let us select a subset S4 CU, of objects which we will call standard objects; we will use the symbol xyz... to denote the object oa(x,y,...) € U4. Given a standard s = xyz..., we will call a function f an (approximation to) rough mereological connective relative to o,4 and s in case it satisfies the condition:
the problem of uncertainty propagation i.e. we have to decide in what way is the measure [4 related to measures ju, Uc, -.-. Formally! we have to find an (approximation) to a function f satisfying the following property: In practicel it is unrealistic to expect the existence of a function f satisfying the above condition globally; thereforeI ‘we localize this notion. To this endl let us select a subset S4 CU, of objects which we will call standard objects; we will use the symbol xyz... to denote the object oa(x,y,...) € U4. Given a standard s = xyz..., we will call a function f an (approximation to) rough mereological connective relative to o,4 and s in case it satisfies the condition:
We outline an algorithm which may be used to extract from information systems approxi- mations to uncertainty functions (rough mereological connectives).
We outline an algorithm which may be used to extract from information systems approxi- mations to uncertainty functions (rough mereological connectives).
Table 2.6: The conditions for f (second part) Table 2.5: The conditions for f (first part)
Table 2.6: The conditions for f (second part) Table 2.5: The conditions for f (first part)
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