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_ Rough Sets

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Basic Concepts of Rough Sets _ Information/Decision Systems (Tables) _ Indiscernibility _ Set Approximation _ Reducts and Core _ Rough Membership _ Dependency of Attributes

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Information Systems/Tables _ IS is a pair (U, A) _ U is a non-empty finite set of objects. _ A is a non-empty finite set of attributes such that for every _ is called the value set of a. Age LEMS x １ 16-30 50 x2 16-30 0 x3 31-45 1-25 x4 31-45 1-25 x5 46-60 26-49 x6 16-30 26-49 x7 46-60 26-49

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Decision Systems/Tables _ DS: _ is the decision attribute (instead of one we can consider more decision attributes). _ The elements of A are called the condition attributes. Age LEMS Walk x １ 16-30 50 yes x2 16-30 0 no x3 31-45 1-25 no x4 31-45 1-25 yes x5 46-60 26-49 no x6 16-30 26-49 yes x7 46-60 26-49 no

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Indiscernibility _ The equivalence relation A binary relation which is reflexive (xRx for any object x), symmetric (if xRy then yRx), and transitive (if xRy and yRz then xRz). _ The equivalence class of an element consists of all objects such that xRy.

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Indiscernibility (2) _ Let IS = (U, A) be an information system, then with any there is an associated equivalence relation: where is called the B-indiscernibility relation. _ If then objects x and x’ are indiscernible from each other by attributes from B. _ The equivalence classes of the B-indiscernibility relation are denoted by

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An Example of Indiscernibility _ The non-empty subsets of the condition attributes are {Age}, {LEMS}, and {Age, LEMS}. _ IND({Age}) = {{x1,x2,x6}, {x3,x4}, {x5,x7}} _ IND({LEMS}) = {{x1}, {x2}, {x3,x4}, {x5,x6,x7}} _ IND({Age,LEMS}) = {{x1}, {x2}, {x3,x4}, {x5,x7}, {x6}}. Age LEMS Walk x １ 16-30 50 yes x2 16-30 0 no x3 31-45 1-25 no x4 31-45 1-25 yes x5 46-60 26-49 no x6 16-30 26-49 yes x7 46-60 26-49 no

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Observations _ An equivalence relation induces a partitioning of the universe. _ The partitions can be used to build new subsets of the universe. _ Subsets that are most often of interest have the same value of the decision attribute. It may happen, however, that a concept such as “Walk” cannot be defined in a crisp manner.

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Set Approximation _ Let T = (U, A) and let and We can approximate X using only the information contained in B by constructing the B-lower and B-upper approximations of X, denoted and respectively, where

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Set Approximation (2) _ B-boundary region of X, consists of those objects that we cannot decisively classify into X in B. _ B-outside region of X, consists of those objects that can be with certainty classified as not belonging to X. _ A set is said to be rough if its boundary region is non-empty, otherwise the set is crisp.

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An Example of Set Approximation _ Let W = {x | Walk(x) = yes}. _ The decision class, Walk, is rough since the boundary region is not empty. Age LEMS Walk x １ 16-30 50 yes x2 16-30 0 no x3 31-45 1-25 no x4 31-45 1-25 yes x5 46-60 26-49 no x6 16-30 26-49 yes x7 46-60 26-49 no

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An Example of Set Approximation (2) yes yes/no no {{x1},{x6}} {{x3,x4}} {{x2}, {x5,x7}} AWAW

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U set Ｘ U/R R : subset of attributes Lower & Upper Approximations

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Lower & Upper Approximations (2) Lower Approximation: Upper Approximation:

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Lower & Upper Approximations (3) X1 = {u | Flu(u) = yes} = {u2, u3, u6, u7} RX1 = {u2, u3} = {u2, u3, u6, u7, u8, u5} X2 = {u | Flu(u) = no} = {u1, u4, u5, u8} RX2 = {u1, u4} = {u1, u4, u5, u8, u7, u6} The indiscernibility classes defined by R = {Headache, Temp.} are {u1}, {u2}, {u3}, {u4}, {u5, u7}, {u6, u8}.

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Lower & Upper Approximations (4) R = {Headache, Temp.} U/R = { {u1}, {u2}, {u3}, {u4}, {u5, u7}, {u6, u8}} X1 = {u | Flu(u) = yes} = {u2,u3,u6,u7} X2 = {u | Flu(u) = no} = {u1,u4,u5,u8} RX1 = {u2, u3} = {u2, u3, u6, u7, u8, u5} RX2 = {u1, u4} = {u1, u4, u5, u8, u7, u6} u1 u4 u3 X1 X2 u5u7 u2 u6u8

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Properties of Approximations impliesand

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Properties of Approximations (2) where -X denotes U - X.

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Four Basic Classes of Rough Sets _ X is roughly B-definable, iff and _ X is internally B-undefinable, iff and _ X is externally B-undefinable, iff and _ X is totally B-undefinable, iff and

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Accuracy of Approximation where |X| denotes the cardinality of Obviously If X is crisp with respect to B. If X is rough with respect to B.

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Issues in the Decision Table _ The same or indiscernible objects may be represented several times. _ Some of the attributes may be superfluous (redundant). That is, their removal cannot worsen the classification.

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Reducts _ Keep only those attributes that preserve the indiscernibility relation and, consequently, set approximation. _ There are usually several such subsets of attributes and those which are minimal are called reducts.

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Dispensable & Indispensable Attributes Let Attribute c is dispensable in T if, otherwise attribute c is indispensable in T. The C-positive region of D :

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Independent _ T = (U, C, D) is independent if all are indispensable in T.

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Reduct & Core _ The set of attributes is called a reduct of C, if T’ = (U, R, D) is independent and _ The set of all the condition attributes indispensable in T is denoted by CORE(C). where RED(C) is the set of all reducts of C.

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An Example of Reducts & Core Reduct1 = {Muscle-pain,Temp.} Reduct2 = {Headache, Temp.} CORE = {Headache,Temp} {MusclePain, Temp} = {Temp}

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Discernibility Matrix (relative to positive region) _ Let T = (U, C, D) be a decision table, with By a discernibility matrix of T, denoted M(T), we will mean matrix defined as: for i, j = 1,2,…,n such that or belongs to the C-positive region of D. _ is the set of all the condition attributes that classify objects ui and uj into different classes.

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Discernibility Matrix (relative to positive region) (2) _ The equation is similar but conjunction is taken over all non-empty entries of M(T) corresponding to the indices i, j such that or belongs to the C-positive region of D. _ denotes that this case does not need to be considered. Hence it is interpreted as logic truth. _ All disjuncts of minimal disjunctive form of this function define the reducts of T (relative to the positive region).

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Discernibility Function (relative to objects) _ For any where (1) is the disjunction of all variables a such that if (2) if (3) if Each logical product in the minimal disjunctive normal form (DNF) defines a reduct of instance

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Examples of Discernibility Matrix No a b c d u1 a0 b1 c1 y u2 a1 b1 c0 n u3 a0 b2 c1 n u4 a1 b1 c1 y C = {a, b, c} D = {d} In order to discern equivalence classes of the decision attribute d, to preserve conditions described by the discernibility matrix for this table u1 u2 u3 u2 u3 u4 a,c b c a,b Reduct = {b, c}

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Examples of Discernibility Matrix (2) u1 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u7 b,c,d b,c b b,d c,d a,b,c,d a,b,c a,b,c,d a,b,c,d a,b c,d c,d Core = {b} Reduct1 = {b,c} Reduct2 = {b,d}

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Rough Membership _ The rough membership function quantifies the degree of relative overlap between the set X and the equivalence class to which x belongs. _ The rough membership function can be interpreted as a frequency-based estimate of where u is the equivalence class of IND(B).

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Rough Membership (2) _ The formulae for the lower and upper approximations can be generalized to some arbitrary level of precision by means of the rough membership function _ Note: the lower and upper approximations as originally formulated are obtained as a special case with

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Dependency of Attributes _ Discovering dependencies between attributes is an important issue in KDD. _ Set of attribute D depends totally on a set of attributes C, denoted if all values of attributes from D are uniquely determined by values of attributes from C.

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Dependency of Attributes (2) _ Let D and C be subsets of A. We will say that D depends on C in a degree k denoted by if where called C-positive region of D.

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Dependency of Attributes (3) _ Obviously _ If k = 1 we say that D depends totally on C. _ If k < 1 we say that D depends partially (in a degree k) on C.

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A Rough Set Based KDD Process _ Discretization based on RS and Boolean Reasoning (RSBR). _ Attribute selection based RS with Heuristics (RSH). _ Rule discovery by GDT-RS.

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What Are Issues of Real World ? _ Very large data sets _ Mixed types of data (continuous valued, symbolic data) _ Uncertainty (noisy data) _ Incompleteness (missing, incomplete data) _ Data change

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Probability Logic Set Soft Techniques for KDD

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Deduction InductionAbduction GDT GrC RS&ILP RS TM A Hybrid Model

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GDT : Generalization Distribution Table RS : Rough Sets TM: Transition Matrix ILP : Inductive Logic Programming GrC : Granular Computing

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A Rough Set Based KDD Process _ Discretization based on RS and Boolean Reasoning (RSBR). _ Attribute selection based RS with Heuristics (RSH). _ Rule discovery by GDT-RS.

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Observations _ A real world data set always contains mixed types of data such as continuous valued, symbolic data, etc. _ When it comes to analyze attributes with real values, they must undergo a process called discretization, which divides the attribute’s value into intervals. _ There is a lack of the unified approach to discretization problems so far, and the choice of method depends heavily on data considered.

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Discretization based on RSBR _ In the discretization of a decision table T = where is an interval of real values, we search for a partition of for any _ Any partition of is defined by a sequence of the so-called cuts from _ Any family of partitions can be identified with a set of cuts.

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Discretization Based on RSBR (2) In the discretization process, we search for a set of cuts satisfying some natural conditions. U a b d x1 0.8 2 1 x2 1 0.5 0 x3 1.3 3 0 x4 1.4 1 1 x5 1.4 2 0 x6 1.6 3 1 x7 1.3 1 1 U a b d x1 0 2 1 x2 1 0 0 x3 1 2 0 x4 1 1 1 x5 1 2 0 x6 2 2 1 x7 1 1 1 PP P = {(a, 0.9), (a, 1.5), (b, 0.75), (b, 1.5)}

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A Geometrical Representation of Data 00.81 1.3 1.4 1.6 a b 3 2 1 0.5 x1 x2 x3 x4 x7 x5 x6

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A Geometrical Representation of Data and Cuts 00.81 1.3 1.4 1.6 a b 3 2 1 0.5 x1 x2 x3 x4 x5 x6 x7

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Discretization Based on RSBR (3) _ The sets of possible values of a and b are defined by _ The sets of values of a and b on objects from U are given by a(U) = {0.8, 1, 1.3, 1.4, 1.6}; b(U) = {0.5, 1, 2, 3}.

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Discretization Based on RSBR (4) _ The discretization process returns a partition of the value sets of condition attributes into intervals.

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A Discretization Process _ Step 1: define a set of Boolean variables, where corresponds to the interval [0.8, 1) of a corresponds to the interval [1, 1.3) of a corresponds to the interval [1.3, 1.4) of a corresponds to the interval [1.4, 1.6) of a corresponds to the interval [0.5, 1) of b corresponds to the interval [1, 2) of b corresponds to the interval [2, 3) of b

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The Set of Cuts on Attribute a 0.81.01.31.41.6 a

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A Discretization Process (2) _ Step 2: create a new decision table by using the set of Boolean variables defined in Step 1. Let be a decision table, be a propositional variable corresponding to the interval for any and

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A Sample Defined in Step 2 U* (x1,x2) (x1,x3) (x1,x5) (x4,x2) (x4,x3) (x4,x5) (x6,x2) (x6,x3) (x6,x5) (x7,x2) (x7,x3) (x7,x5) 1 0 0 0 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 1 1 1 1 1 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0

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The Discernibility Formula _ The discernibility formula means that in order to discern object x1 and x2, at least one of the following cuts must be set, a cut between a(0.8) and a(1) a cut between b(0.5) and b(1) a cut between b(1) and b(2).

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The Discernibility Formulae for All Different Pairs

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The Discernibility Formulae for All Different Pairs (2)

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A Discretization Process (3) _ Step 3: find the minimal subset of p that discerns all objects in different decision classes. The discernibility boolean propositional formula is defined as follows,

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The Discernibility Formula in CNF Form

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The Discernibility Formula in DNF Form _ We obtain four prime implicants, is the optimal result, because it is the minimal subset of P.

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The Minimal Set Cuts for the Sample DB 00.81 1.3 1.4 1.6 a b 3 2 1 0.5 x1 x2 x3 x4 x5 x6 x7

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A Result U a b d x1 0.8 2 1 x2 1 0.5 0 x3 1.3 3 0 x4 1.4 1 1 x5 1.4 2 0 x6 1.6 3 1 x7 1.3 1 1 U a b d x1 0 1 1 x2 0 0 0 x3 1 1 0 x4 1 0 1 x5 1 1 0 x6 2 1 1 x7 1 0 1 PP P = {(a, 1.2), (a, 1.5), (b, 1.5)}

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A Rough Set Based KDD Process _ Discretization based on RS and Boolean Reasoning (RSBR). _ Attribute selection based RS with Heuristics (RSH). _ Rule discovery by GDT-RS.

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Observations _ A database always contains a lot of attributes that are redundant and not necessary for rule discovery. _ If these redundant attributes are not removed, not only the time complexity of rule discovery increases, but also the quality of the discovered rules may be significantly depleted.

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The Goal of Attribute Selection Finding an optimal subset of attributes in a database according to some criterion, so that a classifier with the highest possible accuracy can be induced by learning algorithm using information about data available only from the subset of attributes.

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Attribute Selection

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The Filter Approach _ Preprocessing _ The main strategies of attribute selection: –The minimal subset of attributes –Selection of the attributes with a higher rank _ Advantage –Fast _ Disadvantage –Ignoring the performance effects of the induction algorithm

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The Wrapper Approach _ Using the induction algorithm as a part of the search evaluation function _ Possible attribute subsets (N-number of attributes) _ The main search methods: –Exhaustive/Complete search –Heuristic search –Non-deterministic search _ Advantage –Taking into account the performance of the induction algorithm _ Disadvantage –The time complexity is high

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Basic Ideas: Attribute Selection using RSH _ Take the attributes in CORE as the initial subset. _ Select one attribute each time using the rule evaluation criterion in our rule discovery system, GDT-RS. _ Stop when the subset of selected attributes is a reduct.

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Why Heuristics ? _ The number of possible reducts can be where N is the number of attributes. Selecting the optimal reduct from all of possible reducts is time-complex and heuristics must be used.

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The Rule Selection Criteria in GDT-RS _ Selecting the rules that cover as many instances as possible. _ Selecting the rules that contain as little attributes as possible, if they cover the same number of instances. _ Selecting the rules with larger strengths, if they have same number of condition attributes and cover the same number of instances.

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Attribute Evaluation Criteria _ Selecting the attributes that cause the number of consistent instances to increase faster –To obtain the subset of attributes as small as possible _ Selecting an attribute that has smaller number of different values –To guarantee that the number of instances covered by rules is as large as possible.

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Main Features of RSH _ It can select a better subset of attributes quickly and effectively from a large DB. _ The selected attributes do not damage the performance of induction so much.

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An Example of Attribute Selection Condition Attributes: a: Va = {1, 2} b: Vb = {0, 1, 2} c: Vc = {0, 1, 2} d: Vd = {0, 1} Decision Attribute: e: Ve = {0, 1, 2}

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Searching for ＣＯＲＥ Removing attribute a Removing attribute a does not cause inconsistency. Hence, a is not used as CORE.

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Searching for ＣＯＲＥ (2) Removing attribute ｂ Removing attribute b cause inconsistency. Hence, b is used as CORE.

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Searching for ＣＯＲＥ (3) Removing attribute c does not cause inconsistency. Hence, c is not used as CORE.

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Searching for ＣＯＲＥ (4) Removing attribute d does not cause inconsistency. Hence, d is not used as CORE.

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Searching for ＣＯＲＥ (5) CORE(C)={b} Initial subset R = {b} Attribute b is the unique indispensable attribute.

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R={b} The instances containing b0 will not be considered. TT’

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Attribute Evaluation Criteria _ Selecting the attributes that cause the number of consistent instances to increase faster –To obtain the subset of attributes as small as possible _ Selecting the attribute that has smaller number of different values –To guarantee that the number of instances covered by a rule is as large as possible.

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Selecting Attribute from {a,c,d} 1. Selecting {a} R = {a,b} u3,u5,u6 u4 u7 U/{e} u3 u4 u7 U/{a,b} u5 u6

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Selecting Attribute from {a,c,d} (2) 2. Selecting {c} R = {b,c} u3,u5,u6 u4 u7 U/{e}

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Selecting Attribute from {a,c,d} (3) 3. Selecting {d} R = {b,d} u3,u5,u6 u4 u7 U/{e}

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Selecting Attribute from {a,c,d} (4) 3. Selecting {d} R = {b,d} Result: Subset of attributes ＝ {b, d} u3,u5,u6 u4 u7 U/{e} u3, u4 u7 U/{b,d} u5,u6

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A Heuristic Algorithm for Attribute Selection _ Let R be a set of the selected attributes, P be the set of unselected condition attributes, U be the set of all instances, X be the set of contradictory instances, and EXPECT be the threshold of accuracy. _ In the initial state, R = CORE(C), k = 0.

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A Heuristic Algorithm for Attribute Selection (2) _ Step 1. If k >= EXPECT, finish, otherwise calculate the dependency degree, k, _ Step 2. For each p in P, calculate where max_size denotes the cardinality of the maximal subset.

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A Heuristic Algorithm for Attribute Selection (3) _ Step 3. Choose the best attribute p with the largest and let _ Step 4. Remove all consistent instances u in from X. _ Step 5. Go back to Step 1.

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Experimental Results

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ECE 667 Synthesis and Verification of Digital Systems

ECE 667 Synthesis and Verification of Digital Systems

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