Rough Set Theory Fundamental Assumption Approximation Space Information Systems Decision Tables (Data Tables)

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1 Rough Set Theory Fundamental Assumption Objects from the domain are perceived only through the values of attributes that can be evaluated on these objects. Objects with the same information are indiscernible. Classification of objects is based on the accessible information about them, not on objects themselves. Approximation Space An approximation space is a pair (U, R) where U is a nonempty finite set called the universe and R is an equivalence relation defined on U. Information Systems An information system is a pair S = (U, A), where U is a nonempty finite set called the universe and A is a nonempty finite set of attributes, i.e., a: U V a for a A, where V a is called the domain of a. Decision Tables (Data Tables) A decision table is a special case of information systems, S = (U, A {d}), where attributes in A are called condition attributes and d is a designated attribute called the decision attribute. 1

2 Example of Decision Tables. STUDENT CATEGORY MAJOR BIRTH_PLACE GPA 1 M.A. history Akron junior math Detroit junior literature Detroit M.S. physics Ottawa Ph.D. math Chicago sophomore chemistry Akron senior computing Cleveland Ph.D. biology Houston sophomore music Cleveland Ph.D. computing Akron M.S. statistics Louisville freshman literature Toronto 3.9 Table 1. A decision table of students represented by Category, Major, Birth_Place, and GPA. Generalized Decision Tables Conceptual Generalization (Hierarchy): 1. Category: graduate, undergraduate 2. Major: art and science 3. Birth_Place: states 4. Gpa: Excellent: gpa >= 3.5, Good: 3.5 > gpa >=3.0, Average: 3.0 > gpa >= 2.5 STUDENT CATEGORY MAJOR BIRTH_PLAC GPA E 1 graduate art Ohio excellent 2 undergraduate science Michigan excellent 3 undergraduate art Michigan average 4 graduate science Ontario excellent 5 graduate science Illinois good 6 undergraduate science Ohio average 7 undergraduate science Ohio excellent 8 graduate science Texas good 9 undergraduate art Ohio average 10 graduate science Ohio excellent 11 graduate science Kentucky good 12 undergraduate art Ontario excellent Table 2. A generalized decision table from Table 1. 2

3 Observations: If we select GPA as the decision attribute and Category, Major, and Birth_Place as condition attributes, then Student 6 and 7 are inconsistent. Conceptual Generalization: Denote implicit information in data tables Denote a user s view or interest, usually it is dynamic in nature Results of quantization on data Approximations of Sets. Let S = (U, R) be an approximation space and X be a subset of U. The lower approximation of X by R in S is defined as RX = { e U [e] X} and The upper approximation of X by R in S is defined as R X = { e U [e] X }, where [e] denotes the equivalence class containing e. A subset X of U is said to be R-definable in S if and only if RX = R X. The boundary set BN R (X) is defined as R X RX. Rough Sets. The pair (RX, R X) defines a rough set in S, which is a family of subsets of U with the same lower and upper approximations as RX and R X. 3

4 Dempster-Shafer Theory (Evidence Theory, Belief Function Theory) Frame of Discernment Θ: A finite nonempty set. Basic Probability Assignment (bpa) on Θ is any function m: 2 Θ [0, 1] such that (1) for all X Θ, 0 <= m(x) <= 1 (2) m( ) = 0 (3) m ( X ) = 1 X Θ Let X Θ. For a given bpa function m two functions are defined. Belief function Bel: 2 Θ [0, 1] such that Bel(X) = m ( Y ) Y X Plausibility function Pl: 2 Θ [0, 1] such that Pl(X) = m ( Y ) Y X φ 4

5 U A B C E F D δ A {1} {1, 2, 3} {2} {1, 3} {2} {1, 3} {3} {1, 2} {3} {1} {1, 2, 3} {1, 2} {2, 3} {2, 3} {1, 2} {2, 3} {3} {2} {2} {1, 2, 3} {2, 3} {1} {1, 2, 3} {1, 2} {1, 2, 3} {1, 2, 3} {1, 2, 3} {3} The column δ A is derived by using attributes A, B, C, E, and F as condition attributes and attribute D as the decision attribute. Let Θ = {1, 2, 3}. Then, the following is a bpa. X {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3} m(x) 3/28 1/7 1/7 1/7 1/14 1/7 1/4 5

6 Example: U a b c e f d Condition Attributes: {a, b, c, e, f} Decision Attribute: {d} Equivalence Classes: elementary sets [1] = {1, 10} [2] = {2, 11, 20, 23} [3] = {3, 5} [4] = {4, 6} [7] = {7, 9, 17} [8] = {8, 12} [13] = {13, 16} U a b c e f d [14] = {14, 21} [15] = {15, 24} [18] = {18, 19} [22] = {22} [25] = {25, 26, 27} [28] = {28} CLASS S (d) = {X 1, X 2, X 3 }, where X 1 = [1, 2, 4, 8, 10, 15, 22, 25] X 2 = [3, 5, 11, 12, 16, 18, 19, 21, 23, 24, 27], and X 3 = [6, 7, 9, 13, 14, 17, 20, 26, 28] AX 1 = [1] [22] = {1, 10, 22} AX 2 = [3] [18] = {3, 5, 18, 19} AX 3 = [7] [28] = {7, 9, 17, 28} A X 1. = AX 1 [2] [4] [8] [15] [25] = {1, 2, 4, 6, 8, 10, 11, 12, 15, 20, 22, 23, 24, 25, 26, 27} A X 2. = AX 2 [2] [8] [13] [14] [15] [25] = {2, 3, 5, 8, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 27} A X 3. = AX 3 [2] [4] [13] [14] [25] = {2, 4, 6, 7, 9, 11, 13, 14, 16, 17, 20, 21, 23, 25, 26, 27, 28} BN A (X 1 ) = A X 1. - AX 1 = {2, 4, 6, 8, 11, 12, 15, 20, 23, 24, 25, 26, 27} 6

7 Quality of Approximations. Quality of lower approximation: Quality of upper approximation: Example. γ γ A A AX ( X ) = U AX ( X ) = U γ A (X 1 ) = 28 3 γ A (X 2 ) = 7 1 γ A (X 3 ) = 7 1 γ A (X 1 ) = γ (X 2 ) = A γ (X 3 ) = A 28 Quality of approximations can be used to characterize dependencies of attributes. 7

8 Fuzzy Sets and Rough Sets Let U be a domain of objects. A fuzzy set X defined on U is characterized by a membership function µ X : µ X : U [0, 1] Let A and B be two fuzzy sets. µ A B = min(µ A, µ B ) µ A B = max(µ A, µ B ) Rough Membership Function Let S = (U, R) be an approximation space and X be a subset of U. Define µ X (e) = 1 if e in RX 1/2 if e in BN R (X) 0 if e in R X where X is the complement of X. Then, the rough membership function cannot be extended to the fuzzy union and intersection of sets. Remarks: In general, R(X Y) RX RY and R (X Y) R X R Y. The rough membership function will reduce to fuzzy set when R(X Y) = RX RY and R (X Y) = R X R Y 8

9 Rough Set Approach to Machine Learning LERS32: Inconsistent Example: 6 Inconsistent Example: 11 Inconsistent Example: 12 Inconsistent Example: 16 Inconsistent Example: 20 Inconsistent Example: 21 Inconsistent Example: 23 Inconsistent Example: 24 Inconsistent Example: 26 Inconsistent Example: 27 Certain Rules for Class # 1: (a = 0) (c = 1) (f = 0) ===> (class 1) (a = 0) (c = 1) (e = 0) ===> (class 1) Possible Rules for Class # 1: (c = 1) (b = 0) ===> (class 1) (c = 1) (f = 1) ===> (class 1) (a = 1) (e = 0) (f = 0) ===> (class 1) (b = 0) (a = 1) ===> (class 1) Certain Rules for Class # 2: (f = 0) (c = 0) ===> (class 2) Possible Rules for Class # 2: (f = 0) (c = 0) ===> (class 2) (b = 1) (e = 0) ===> (class 2) (f = 1) (b = 1) ===> (class 2) (c = 1) (b = 0) (e = 1) ===> (class 2) (c = 1) (b = 0) (a = 1) ===> (class 2) Certain Rules for Class # 3: (e = 1) (a = 0) (c = 0) ===> (class 3) (e = 1) (f = 0) ===> (class 3) Possible Rules for Class # 3: (e = 1) (c = 0) ===> (class 3) (a = 1) (f = 0) ===> (class 3) (a = 1) (e = 0) ===> (class 3) 9

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