GEOG 5113 Special Topics in GIScience. Why is Classical set theory restricted? Contradiction & Excluded Middle. Fuzzy Set Theory in GIScience
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1 GEOG 5113 Special Topics in GIScience Fuzzy Set Theory in GIScience -Basic Properties and Concepts of Fuzzy Sets- Why is Classical set theory restricted? Boundaries of classical sets are required to be drawn precise (sharp) Set membership with complete certainty (In or Out) Coin types vs. set of tall people Continuous transition btw. Tall and not tall Contradiction & Excluded Middle These properties depend on the sharpness of set boundaries LoC: Proposition that affirms AND denies a fact at the same time is FALSE (so a location cannot be a member of forest and non-forest area at the same time) LEM: Any proposition must be true or false, but not both (a location must be a member of either a set or its complement) 1
2 Are we crisply thinking in every-day life? We end up imposing arbitrary (artificial) partitions to determine who s healthy and who s not healthy That s the reason why classical theory cannot handle situations that involve vagueness or other kinds of uncertainty Why is artificial intelligence +/- ruled out? So why should we? Allowing for imprecise boundaries of sets makes the two basic laws of classic set theory invalid A person would be to some degree tall and to some degree considered not tall Ufffh -- You can imagine the consequences in logic and set theory? However this provides us with additional ways to solve complex problems of imprecision and vagueness Membership functions Membership of an individual is a matter of degree to which it meets the operating concept of the fuzzy set ( tall ) The degree of membership expresses the degree of compatibility with the concept represented by the fuzzy set Here we use functions (re characteristic function in classic set theory) for assignment 2
3 Membership functions This function assigns to each element x of the universal set X a number A(x), in the closed unit interval [0,1] This number is the degree of membership of x in the fuzzy set A A: X [0,1] (X is assumed to be a classic set) A stands for the fuzzy set AND its membership function (we will see how to do this differently) Membership functions In contrast to classic set theory the numbers of memberships in fuzzy sets have quantitative meaning ( symbolic in classic set theory) If 0 and 1 have quantitative meaning they are a special type of fuzzy sets: A so-called crisp set Graphical Representation of Membership Functions Crisp set Teenager Fuzzy set Young people 3
4 The Experienced Undergrad Freshman - Sophomore - Junior - Senior The Middle-Aged Person Bell-Shape Transitional shape from 0 to 1 is often not clearly defined Empirical evidence is needed (often incomplete) Normally low sensitivity to the shape of A Tabular/ List Representation of Membership Functions Fuzzy sets can be presented as tables: Student (Label) Degree of Membership in A Kay (x1 ) 0.8 Roy (x2 ) 0.3 Tom (x3 ) 0.5 Or as Lists: A = 0.8/Kay + 0.3/Roy + 0.5/Tom (/ - corr. Btw. An element and its membership degree) (+ - element connector) General notation: A= A(x) / x " 4
5 Geometric Representations of Membership Functions n-dimensional unit cube (subset of Euclidean space with coordinates [0,1]) Each point in the cube is defined by n coordinates Fuzzy sets that can be defined on X with n elements can be represented as points in the n-dimensional unit cube 2D Unit Cubes n = 2: each point covered by the square is a fuzzy set; corner points are crisp sets <0,0> <0,1> <1,0> <1,1> 3D Unit Cubes n = 3: each point of the cube is a fuzzy set; corner points are crisp sets <0,0,0> <0,0,1> <1,0,0> <1,1,1> 5
6 Analytic Representations of Membership Functions If the universal set X is infinite (real numbers) it is impossible to list all members and their membership values Fuzzy numbers have the universal set of R About 6 represented by analytic form to describe the shape of this fuzzy number Analytic Representations of Membership Functions About 6 Triangular-, Trapezoid- and Bell-Shaped membership functions Shape and Context A membership function tries to capture the intended meaning of linguistic terms adequately, which is context-dependent 6
7 Constructing Fuzzy Sets An issue of Knowledge Acquisition Involves Experts and knowledge engineers to formalize operational forms of FS Ideal prototypes & Similarity functions Or: Exemplified on a subset of individuals (Compatibility to concept) Neural networks (learning from samples) Constructing Fuzzy Sets Several Experts - The diving survey Write down the Fuzzy set using a list Membership degree = Ratio of # of favorable answers to the # of referees Operations on Fuzzy Sets Basic operations in classical set theory (complement, intersection, union) are unique Their extensions in fuzzy set theory are not unique (broad class of operations for each of these) Distinct meanings of the linguistic terms NOT, AND and OR in different contexts Standard fuzzy operations: Special operations providing good approximations 7
8 Operations on Fuzzy Sets Standard Fuzzy Complement Standard Fuzzy Union Standard Fuzzy Intersection Standard Fuzzy Complement Standard Fuzzy Complement 2 the degree to which x does not belong to A Complements overlap Fundamental difference to classical set theory A (x) = 1" A(x) 8
9 Standard Fuzzy Union Law of excluded middle does not hold (for union and complement) A " A = X A(x) # {0,1} Violated for all x of X \ A " B(x) = max[ A(x),B(x)] Standard Fuzzy Intersection Law of contradiction does not hold (for union and complement) A " A = # Violated for e.g. A(x) = 0.6 A " A (x) = 0.4 A " B(x) = min[ A(x), B(x)] Does Involution hold? 9
10 Fundamental laws do not hold? Consequence of imprecise boundaries? Commutativity, Associativity, Idempotence Distributivity DeMorgan s Law hold for standard fuzzy operations Note that standard fuzzy operations do not fully utilize the expressive power of fuzzy set theory There is a broad variety of the meanings of the linguistic terms or, and and not Alpha cuts 10
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