Fuzzy Sets and Fuzzy Techniques. Joakim Lindblad. Outline. L1: Intro. L4: Constr. and uncertainty. L5: Features. Distances. L8: Set operations

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1 Fuzzy Fuzzy Lecture notes Fuzzy Lecture 15 Repetition Centre for Image Analysis Uppsala University , (1/144), (2/144) Fuzzy duction, motivation of fuzzy sets Topics of today Fuzzy duction, motivation L4: Constructing fuzzy sets, Uncertainty measures L5: Fuzzy thresholding, Fuzzy c-means clustering, Some features of spatial fuzzy sets on between fuzzy sets L8: Operations on fuzzy sets fuzzy arithmetics approximate reasoning, (3/144), (4/144)

2 Fuzzy What will we learn in this course? Fuzzy Fuzzy What is a fuzzy set? The basics of fuzzy sets How to define fuzzy sets How to perform on fuzzy sets How to extend crisp concepts to fuzzy ones How to extract information from fuzzy sets The very basics of fuzzy fuzzy reasoning We will look at some applications of fuzzy in Image processing Control systems Machine intelligence / expert systems Btw., what is a set? A set is a collection of its members.... to be an element... The notion of fuzzy sets is an extension of the most fundamental property of sets. Fuzzy sets allows a grading of to what extent an element of a set belongs to that specific set., (5/144), (6/144) Fuzzy Why Fuzzy? Fuzzy What is a fuzzy set? Precision is not truth. - Henri Matisse So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality. - Albert Einstein Fuzzy is not just another name for probability. The number 10 is not probably big!... number 2 is not probably not big. Uncertainty is a consequence of non-sharp boundaries between the notions/objects, not caused by lack of information. As complexity rises, precise statements lose meaning meaningful statements lose precision. - Lotfi Zadeh Statistical models deal with rom events outcomes; fuzzy models attempt to capture quantify nonrom imprecision., (7/144), (8/144)

3 Fuzzy What is a fuzzy set? Romness vs. Fuzziness Fuzzy What is a fuzzy set? Romness refers to an event that may or may not occur. Romness: frequency of car accidents. Using fuzzy techniques is Fuzziness refers to the boundary of a set that is not precise. Fuzziness: seriousness of a car accident. Prof. George J. Klir to avoid throwing away data early (by crisp decisions)., (9/144), (10/144) Fuzzy of fuzzy sets Fuzzy Fuzzy sets A fuzzy set of a reference set is a set of ordered pairs where µ F : X [0, 1]. F = { x, µ F (x) x X }, Where there is no risk for confusion, we use the same symbol for the fuzzy set, as for its membership function. Thus where F : X [0, 1]. F = { x, F (x) x X }, To define a fuzzy set To define a membership function, (11/144), (12/144)

4 Fuzzy Continuous (analog) fuzzy sets A : X [0, 1] Fuzzy sets Fuzzy Fuzzy sets of different types The membership function may be vague in itself. Fuzzy sets of type 2 Discrete fuzzy sets A : {x 1, x 2, x 3,..., x s } [0, 1] A : X F([0, 1]) Digital fuzzy sets If a discrete-universal membership function can take only a finite number n 2 of distinct values, then we call this fuzzy set a digital fuzzy set. 1 A : {x 1, x 2, x 3,..., x s } {0, n 1, 2 n 1, 3 n 2 n 1,..., n 1, 1}, (13/144), (14/144) Fuzzy Fuzzy sets of different levels Fuzzy Basic concepts terminology Also the domain of the membership function may be fuzzy. Fuzzy sets defined so that the elements of the universal set are themselves fuzzy sets are called level 2 fuzzy sets. A : F(X ) [0, 1] The support of a fuzzy set A in the universal set X is a crisp set that contains all the elements of X that have nonzero membership values in A, that is, supp(a) = {x X A(x) > 0} A crossover point of a fuzzy set is a point in X whose membership value to A is equal to 0.5. The height, h(a) of a fuzzy set A is the largest membership value attained by any point. If the height of a fuzzy set is equal to one, it is called a normal fuzzy set, otherwise it is subnormal., (15/144), (16/144)

5 Fuzzy Basic concepts terminology An α-cut of a fuzzy set A is a crisp set α A that contains all the elements in X that have membership value in A greater than or equal to α. α A = {x A(x) α} A strong α-cut of a fuzzy set A is a crisp set α+ A that contains all the elements in X that have membership value in A strictly greater than α. α+ A = {x A(x) > α} Fuzzy Basic concepts terminology The ordering of the values of α in [0, 1] is inversely preserved by set inclusion of the corresponding α-cuts as well as strong α-cuts. That is, for any fuzzy set A α 1 < α 2 it holds that α 2 A α 1 A. All α-cuts all strong α-cuts for two distinct families of nested crisp sets. The set of all levels α [0, 1] that represent distinct α-cuts of a given fuzzy set A is called a level set of A. Λ(A) = {α A(x) = α for some x X }. We observe that the strong α-cut 0+ A is equivalent to the support supp(a). The 1-cut 1 A is often called the core of A., (17/144), (18/144) Fuzzy Basic concepts terminology Fuzzy Basic concepts terminology A fuzzy set A defined on n is convex iff A(λx 1 + (1 λ)x 2 ) min (A(x 1 ), A(x 2 )), for all λ [0, 1], x 1, x 2 n all α [0, 1]. Or, equivalently, A is convex if only if all its α-cuts α A, for any α in the interval α (0, 1], are convex sets. Don t forget to read in the book here! Chapter 1.4 Chapter 2. Any property that is generalized from classical set theory into the domain of fuzzy set theory by requiring that it holds in all α-cuts in the classical sense is called a cutworthy property., (19/144), (20/144)

6 Fuzzy Stard fuzzy set Fuzzy De Morgan lattice/algebra Ā(x) = 1 A(x) fuzzy complement (A B)(x) = min[a(x), B(x)] fuzzy intersection (A B)(x) = max[a(x), B(x)] fuzzy union For stard fuzzy set, the law of contradiction A Ā the law of excluded middle, are violated. An equilibrium point of a fuzzy set is a point in X such that A(x) = Ā(x). (Same as crossover point for stard complement.) A Ā X, (21/144), (22/144) Fuzzy Basic concepts terminology Fuzzy Basic concepts terminology Set inclusion A B iff A(x) B(x) x X Equality A = B iff A(x) = B(x) x X Scalar cardinality A = A(x) x X Stard fuzzy intersection fuzzy union of two fuzzy sets are cutworthy strong cutworty. Due to associativity of min max, any finite intersection/union. However, caution with infinitely many intersections/unions. Decomposition theorems Each stard fuzzy set is uniquely represented by the family of all its α-cuts, or by the family of all its strong α-cuts., (23/144), (24/144)

7 Fuzzy Extension principle Any given function f : X Y induces two functions, Fuzzy Constructing fuzzy sets, Uncertainty measures f : F(X ) Y [f (A)](y) = sup A(x) x y=f (x) f 1 : F(Y ) X [f 1 (B)](x) = B(f (x)) Strong cutworthiness For any A F(X ), a function f : X Y, it holds that f (A) = f ( α+ A) α [0,1], (25/144), (26/144) Fuzzy Constructing fuzzy sets, Uncertainty measures Methods of construction Fuzzy Direct methods with one expert Direct methods indirect methods One expert multiple experts Define the complete membership function based on a justifiable mathematical formula Often based on mapping of directly measurable features of the elements of X Exemplifying it for some selected elements of X interpolate (/extrapolate) MF in some way. Expert of some kind, (27/144), (28/144)

8 Fuzzy Direct methods with multiple experts Fuzzy Indirect methods The opinions of several experts need to be aggregated. Example: Average (Probabilistic interpretation) It may be easier/more objective to ask simpler questions to the experts, than the membership directly. Example: Pairwise comparisons A(x) = 1 n n a i (x) i=1 Problem: Determine membership a i = A(x i ) Extracted information: Pairwise relative belongingness, matrix P with p ij a i a j, (29/144), (30/144) Fuzzy Uncertainty measures Fuzzy Nonspecificity of crisp sets Hartley function Hartley [1928] showed that a function Nonspecificity of crisp sets Nonspecificity of fuzzy sets Fuzziness of fuzzy sets U(A) = c log b A, where A is the cardinality of A, b > 1 c > 0 are constants, is the only sensible way to measure the amount of associated with a finite set of possible alternatives. b = 2 c = 1 measure in bits U(A) = log 2 A Relates to the nonspecificity inherent in each set. Larger sets correspond to less specific predictions., (31/144), (32/144)

9 Fuzzy Nonspecificity of fuzzy sets U- Fuzzy A measure of fuzziness is a function Fuzziness of fuzzy sets Generalized Hartley function U(A) = 1 h(a) h(a) 0 log 2 α A dα Weighted average of the Hartley function for all distinct α-cuts of the normalized counterpart of A. Fuzzy sets that are equal when normalized have the same nonspecificity. f : F(X ) For each fuzzy set A, f (A) expresses the degree to which the boundary of A is not sharp. The following three requirements are essential 1 f (A) = 0 iff A is a crisp set 2 f (A) attains its maximum iff A(x) = 0.5 for all x X 3 f (A) f (B) when set A is undoubtedly sharper than set B + a) A(x) B(x) when B(x) 0.5 b) A(x) B(x) when B(x) 0.5, (33/144), (34/144) Fuzzy Fuzziness of fuzzy sets Fuzzy Fuzziness of fuzzy sets One way to measure fuzziness of a set A is to measure the distance between A the nearest crisp set. Remaining is to choose the distance measure. Another way is to view the fuzziness of a set as the lack of distinction between the set its complement. The less a set differs from its complement, the fuzzier it is. Also this path (which is the one we will take) requires a distance measure. A simple intuitive distance measure is the Hamming distance. d(a, B) = A(x) B(x) The measure of fuzziness as the distance to the complement, then becomes f (A) = d(x, X ) d(a, Ā) = (1 A(x) (1 A(x)) ) = (1 2A(x) 1 ), (35/144), (36/144)

10 Fuzzy Information gain? Fuzzy L5: Fuzzy thresholding, Fuzzy c-means clustering, Some features of spatial fuzzy sets Fuzziness nonspecificity are distinct types of totally independent of each other. They are also totally different in their connections to information. When nonspecificity is reduced, we view this as a gain in information, regardless of any associated change in fuzziness. The opposite, however, is not true. A reduction of fuzziness is reasonable to consider as a gain of information only if the nonspecificity also decreases or remains the same., (37/144), (38/144) Fuzzy Thresholding Fuzzy Thresholding Thresholding fuzzy thresholding of fuzzy sets, based on different ways of measuring minimizing fuzziness. distributions assigned using Membership a) Pal Rosenfeld (1988) b) Huang Wang (1995) c) Fuzzy c-means (Bezdek 1981) algorithms., (39/144), (40/144)

11 Fuzzy Fuzzy c-means clustering Bezdek Fuzzy Fuzzy c-means clustering Chapter 13.2 Algorithm make initial guess for cluster means iteratively use the estimated means to assign samples to clusters update means until there are no changes in means a partition of the observed set is represented by a c n matrix U = [u ik ], where u ik corresponds to the membership value (anything between 0 1!) of the kth element (out of n), to the ith cluster (out of c) boundaries between subgroups are not crisp each element may belong to more than one cluster - its overall membership equals one objective function includes parameter ling degree of fuzziness, (41/144), (42/144) Fuzzy The fuzzification principle Given a function f : P(X ) Features of fuzzy sets Aggregating over α-cuts We can extends this function to f : F(X ) using one of the following equations f (A) =. 1 0, f ( α A) dα, (1) f (A) = sup [αf ( α A)] (2) α (0,1] Both these definitions provide consistency for the crisp case. Fuzzy Features of fuzzy sets The area of a fuzzy set A on X is area(a) = A(x) dx = X 1 0 area( α A) dα For a discrete fuzzy set, the area is equal to the cardinality of the set area(a) = A = A(x) X The perimeter of a fuzzy set A perim(a) = 1 0 perim( α A) dα, (43/144), (44/144)

12 Fuzzy Features of fuzzy sets Fuzzy Inter-relations Geometric moments: The moment m p,q (A) of a fuzzy set A defined on X m p,q (A) = A(x, y) x p y q dxdy. for integers p, q 0. Remark: The area of a set is the m 0,0 moment. Remark: The centroid (centre of gravity) of a set is ( m1,0 (A) (x c, y c ) = m 0,0 (A), m ) 0,1(A) m 0,0 (A) X 2, is All the definitions listed above reduce to the corresponding customary definitions for crisp sets. However, some inter-relations which these notions satisfy in the crisp case, do not hold for the generalized (fuzzified) definitions. For example: The isoperimetric inequality, 4πarea(µ) perim 2 (µ),, (45/144), (46/144) Fuzzy Estimation of features Fuzzy Estimation of features As is well know, features of a continuous spatial shape S, can be estimated from features of its digitization D(S). The precision of such estimates is in general limited by the spatial resolution of the digital representation. For object represented by digital spatial fuzzy sets, where the membership of a point indicates to what extent the pixel/voxel is covered by the imaged continuous (crisp) object, significant improvements in precision of feature estimates can be obtained. Especially so, for small objects/limited resolution. Significant improvement in the precision of feature estimates can be achieved using a fuzzy approach. Exploiting fuzzy can provide an alternative to increasing the spatial resolution of the image., (47/144), (48/144)

13 Fuzzy Fuzzy features values Fuzzy on between fuzzy sets Set to set distances Why give crisp values of features? Fuzzy feature values still in its infancy. (Point to set distances) Point to point distances A mix of notions The objects that the distance is measured between (start stop) - crisp or fuzzy, point or set The space where a path between start stop is embedded (spatial cost function) - Unconstrained (Euclidean) - Constrained (geodesic/cost function) Output: Crisp (number) or fuzzy, (49/144), (50/144) Fuzzy on between fuzzy sets Fuzzy Set to set distances Definition (Metric) A metric is a positive function d : X 1 d(x, x) = 0 (reflexivity) 2 d(x, y) = 0 x = y (separability) 3 d(x, y) = d(y, x) (symmetry) such that 4 d(x, z) d(x, y) + d(y, z) (triangular inequality) between fuzzy sets a) Membership focused (vertical) b) Spatially focused (horizontal) c) Mix of spatial membership (tolerance) d) Feature distances (low or high dimensional representations) e) Morphoal (mixed focus), (51/144), (52/144)

14 Fuzzy Membership focused L p norm Fuzzy Membership focused L p norm The functional approach The most common: Based on the family of Minkowski distances ( 1/p d p (A, B) = µ A (x) µ B (x) dx) p, p 1, X d EssSup (A, B) = lim d p(a, B) p d (A, B) = sup µ A (x) µ B (x). x X Discrete version: ( n ) 1/p d p (A, B) = µ A (x i ) µ B (x i ) p, p 1, i=1 d (A, B) = max i=1...n ( µ A(x i ) µ B (x i ) ). d p for p 1 are all metrics in the discrete case. Normalized variants, divide with X or u + v or similar., (53/144), (54/144) Fuzzy Membership focused Set approach Fuzzy Membership focused Set approach Tversky 1977, et al. An important aspect of Tversky s model is that similarity depends not only on the proportion of features common to the two objects but also on their unique features. Based on this several other assumptions, Tversky derived the following relationship: S(a, b) = θf (A B) αf (A B) βf (B A) Figure: Representation of two objects that each contains its own unique features also contains common features. Here, S is an interval scale of similarity, f is an interval scale that reflects the salience of the various features, θ, α β are parameters that provide for differences in focus on the different components., (55/144), (56/144)

15 Fuzzy Spatially focused Fuzzy Spatially focused Hausdorff Crisp: Nearest point Mean distance Hausdorff Three (four) approaches: fuzzy distance weighting with membership morphoal integration of alpha-cuts, (57/144) d H (A, B) = max{sup inf d(x, y), sup inf d(x, y)} = inf{r x A y B y A x B + A D r (B) B D r (A)} where D r (A) is the dilation of the set A by a ball of radius r D r (A) = {y X x A : d(x, y) r} The Hausdorff distance between A B is the smallest amount that A must be exped to contain B vice versa. Is a metric on the set of nonempty compact sets. Remark: Usually extended with: d H (A, ) = d H (, ) = 0, (58/144) Fuzzy Spatially focused Hausdorff Fuzzy Feature distances Pattern recognition approach Ralescu Ralescu (1984) d H1 (A, B) = 1 0 d H ( α A, α B) dα, d H (A, B) = sup d H ( α A, α B), α>0 where d H is the Hausdorff distance between two crisp sets, A serious problem is that the distance between two fuzzy sets A B is infinite if height(a) height(b). No good solution to that problem is found. Use of a feature representation of the observed sets as an intermediate step in the distance calculations. The distance between sets A B is then given in terms of the distance between their feature vectors. Often global shape features are used (think shape matching, image retrieval)., (59/144), (60/144)

16 Fuzzy Point to point distances Fuzzy Cost function shoveling snow distance between points in a fuzzy set Defining the cost of traveling along a path Similar to grey weighted distances (Levi & Montanari 70) put in a fuzzy framework (Saha 02). Define the distance along a path π i between points x y in the fuzzy set A d A (π i (x, y)) = A(t) dt s π The distance between points x y in A is the distance along the shortest path d A (x, y) = inf d A(π) π Π(x,y), (61/144), (62/144) Fuzzy Cost function variations Fuzzy Constrained distance Membership as another dimension integrate the arc-length Bloch 1995, Toivanen 1996: ( ) da(t) 2 d A (π) = 1 + dt dt s π Problem: Scale of membership relative to spatial distance Geodesic distance shortest path within the set; not allowed to go out of the set a path that descends the least in terms of membership, (63/144), (64/144)

17 Fuzzy Constrained distance Fuzzy Constrained distance Connectedness, Rosenfeld 1979 Strength of a path the strength of its weakest link Strength of a link between two points defined by the membership function. The of two points x y in A the strength of the strongest path between x y c A (x, y) = sup π Π(x,y) inf A(t) t π Bloch Maître 1995: d(x, y) = inf π Π ca (x,y) π ds c A (x, y) where c A (x, y) is the strength of of points x y, Π ca (x, y) is the set of path contained within the α-cut c A A. Does not provide triangle inequality., (65/144), (66/144) Fuzzy L8: Operations on fuzzy sets Fuzzy slide Ā(x) = 1 A(x) fuzzy complement (A B)(x) = min[a(x), B(x)] fuzzy intersection (A B)(x) = max[a(x), B(x)] fuzzy union, (67/144), (68/144)

18 Fuzzy Properties of the stard Fuzzy Aggregation operators They are generalizations of the corresponding (uniquely defined!) classical set. They satisfy the cutworthy strong cutworthy properties. They are the only ones that do. The stard fuzzy intersection of two sets contains (is bigger than) all other fuzzy intersections of those sets. The stard fuzzy union of two sets is contained in (is smaller than) all other fuzzy unions of those sets. Aggregation operators are used to combine several fuzzy sets in order to produce a single fuzzy set. Associative aggregation (general) fuzzy intersections - t-norms (general) fuzzy unions - t-conorms Non-associative aggregation averaging - idempotent aggregation, (68/144), (69/144) Fuzzy Fuzzy complements Axiomatic requirements Fuzzy Generators Increasing generators Ax c1. c(0) = 1 c(1) = 0. boundary condition Ax c2. For all a, b [0, 1], if a b, then c(a) c(b). monotonicity c1 c2 are called axiomatic skeleton for fuzzy complements Ax c3. c is a continuous function. Ax c4. c is involutive, i.e., c(c(a)) = a, for each a [0, 1]. Increasing generator is a strictly increasing continuous function g : [0, 1] R, such that g(0) = 0. A pseudo-inverse of increasing generator g is defined as 0 for a (, 0) g ( 1) = g 1 (a) for a [0, g(1)] 1 for a (g(1), ) An example: g(a) = a p, p > 0 0 for a (, 0) g ( 1) (a) = a 1 p for a [0, 1] 1 for a (1, ), (70/144), (71/144)

19 Fuzzy Theorem Generating fuzzy complements (First Characterization Theorem of Fuzzy Complements.) Let c be a function from [0, 1] to [0, 1]. Then c is a (involutive) fuzzy complement iff there exists an increasing generator g such that, for all a [0, 1] c(a) = g 1 (g(1) g(a)). Fuzzy For all a, b, d [0, 1], Ax i1. i(a, 1) = a. boundary condition Ax i2. b d implies i(a, b) i(a, d). Ax i3. i(a, b) = i(b, a). commutativity Ax i4. i(a, i(b, d)) = i(i(a, b), d). Fuzzy intersections Axiomatic requirements monotonicity associativity Theorem (Second Characterization Theorem of Fuzzy Complements.) Let c be a function from [0, 1] to [0, 1]. Then c is a (involutive) fuzzy complement iff there exists an decreasing generator f such that, for all a [0, 1] Axioms i1 - i4 are called axiomatic skeleton for fuzzy intersections. If the sets are crisp, i becomes the classical (crisp) intersection. c(a) = f 1 (f (0) f (a))., (72/144), (73/144) Fuzzy Fuzzy intersections Additional (optional) requirements Fuzzy Fuzzy intersections Examples of t-norms frequently used For all a, b, d [0, 1], Ax i5. i is a continuous function. Ax i6. i(a, a) a. subidempotency continuity Ax i7. a 1 < a 2 b 1 < b 2 implies i(a 1, b 1) < i(a 2, b 2). strict monotonicity Note: The stard fuzzy intersection, i(a, b) = min[a, b], is the only idempotent t-norm., (74/144) Drastic intersection 8 < a if b = 1 i(a, b) = b if a = 1 : 0 otherwise Bounded difference i(a, b) = max[0, a + b 1] Algebraic product i(a, b) = ab Stard intersection i(a, b) = min[a, b] i min (a, b) max(0, a + b 1) ab min(a, b). For all a, b [0, 1],, (75/144) i min (a, b) i(a, b) min[a, b].

20 Fuzzy Fuzzy intersections How to generate t-norms Fuzzy Fuzzy unions Axiomatic requirements For all a, b, d [0, 1], Theorem (Characterization Theorem of t-norms) Let i be a binary operation on the unit interval. Then, i is an Archimedean t-norm iff there exists a decreasing generator f such that i(a, b) = f ( 1) (f (a) + f (b)), for a, b [0, 1]. Ax u1. u(a, 0) = a. boundary condition Ax u2. b d implies u(a, b) u(a, d). monotonicity Ax u3. u(a, b) = u(b, a). commutativity Ax u4. u(a, u(b, d)) = u(u(a, b), d). associativity Axioms u1 - u4 are called axiomatic skeleton for fuzzy unions. They differ from the axiomatic skeleton of fuzzy intersections only in boundary condition. For crisp sets, u behaves like a classical (crisp) union., (76/144), (77/144) Fuzzy Fuzzy unions Additional (optional) requirements Fuzzy Combinations of set De Morgan laws duality of fuzzy For all a, b, d [0, 1], De Morgan laws in classical set theory: Ax u5. u is a continuous function. continuity Ax u6. u(a, a) a. superidempotency Ax u7. a 1 < a 2 b 1 < b 2 implies u(a 1, b 1) < u(a 2, b 2). strict monotonicity A B = Ā B A B = Ā B. The union intersection operation are dual with respect to the complement. Note: Requirements u5 - u7 are analogous to Axioms i5 - i7. The stard fuzzy union, u(a, b) = max[a, b], is the only idempotent t-conorm. De Morgan laws for fuzzy sets: c(i(a, B)) = u(c(a), c(b)) c(u(a, B)) = i(c(a), c(b)) for a t-norm i, a t-conorm u, fuzzy complement c. Notation: i, u, c denotes a dual triple., (78/144), (79/144)

21 Fuzzy Theorem Dual triples - Six theorems (1) Fuzzy Dual triples - Six theorems (2) The triples min, max, c i min, u max, c are dual with respect to any fuzzy complement c. Theorem Theorem Given a t-norm i an involutive fuzzy complement c, the binary operation u on [0, 1], defined for all a, b [0, 1] by u(a, b) = c(i(c(a), c(b))) is a t-conorm such that i, u, c is a dual triple. Given an involutive fuzzy complement c an increasing generator g of c, the t-norm the t-conorm generated by g are dual with respect to c. Theorem Let i, u, c be a dual triple generated by an increasing generator g of the involutive fuzzy complement c. Then the fuzzy i, u, c satisfy the law of excluded middle, the law of contradiction. Theorem Given a t-conorm u an involutive fuzzy complement c, the binary operation i on [0, 1], defined for all a, b [0, 1] by i(a, b) = c(u(c(a), c(b))) Theorem Let i, u, c be a dual triple that satisfies the law of excluded middle the law of contradiction. Then i, u, c does not satisfy the distributive laws. is a t-norm such that i, u, c is a dual triple., (80/144), (81/144) Fuzzy Aggregation Definition Fuzzy Axiomatic requirements Aggregations on fuzzy sets are by which several fuzzy sets are combined in a desirable way to produce a single fuzzy set. Definition Aggregation operation on n fuzzy sets (n 2) is a function h : [0, 1] n [0, 1]. Applied to fuzzy sets A 1, A 2,..., A n, function h produces an aggregate fuzzy set A, by operating on membership grades to these sets for each x X : A(x) = h(a 1(x), A 2(x),..., A n(x)). Ax h1 h(0, 0,..., 0) = 0 h(1, 1,..., 1) = 1. boundary conditions Ax h2 For any (a 1, a 2,..., a n) (b 1, b 2,..., b n), such that a i, b i [0, 1] a i b i for i = 1,..., n, h(a 1, a 2,..., a n) h(b 1, b 2,..., b n). h is monotonic increasing in all its arguments. Ax h3 h is continuous. Ax h4 h is a symmetric function in all its arguments; for any permutation p on {1, 2,..., n} h(a 1, a 2,..., a n) = h(a p(1), a p(2),..., a p(n) ). Ax h5 h is an idempotent function; for all a [0, 1] h(a, a,..., a) = a., (82/144), (83/144)

22 Fuzzy Averaging Fuzzy Generalized means: Averaging If an aggregation operator h is monotonic idempotent (Ax h2 Ax h5), then for all (a 1, a 2,..., a n ) [0, 1] n min(a 1, a 2,..., a n ) h(a 1, a 2,..., a n ) max(a 1, a 2,..., a n ). All aggregation operators between the stard fuzzy intersection the stard fuzzy union are idempotent. The only idempotent aggregation operators are those between stard fuzzy intersection stard fuzzy union. Idempotent aggregation operators are called averaging. ( a α h α (a 1, a 2,..., a n ) = 1 + a2 α + + aα n n for α R, α 0, for α < 0 a i 0. Geometric mean: For α 0, lim hα(a1, a2,..., an) = α 0 (a1 a2 an) n 1 ; Harmonic mean: For α = 1, n h 1(a 1, a 2,..., a n) = ; 1 a a a n Arithmetic mean: For α = 1, h 1(a 1, a 2,..., a n) = 1 (a1 + a an). n ) 1 α,, (84/144) Functions h α satisfy axioms Ax h1 - Ax h5., (85/144) Fuzzy Averaging Do we need more than stard? Fuzzy An Application: Fuzzy morphologies Morphoal Intersection: No positive compensation (trade-off) between the memberships of the fuzzy sets observed. Union: Full compensation of lower degrees of membership by the maximal membership. In reality of decision making, rarely either happens. (non-verbal) merging connectives (language) connectives {, or,...,}. Aggregation called compensatory are needed to model fuzzy sets representing to, e.g., managerial decisions. Mathematical morphology is completely based on set theory. Fuzzification started in 1980s. Basic morphoal are dilation erosion. Many others can be derived from them. Dilation erosion are, in crisp case, dual with respect to the complementation: D(A) = c(e(ca)). In crisp case, dilation erosion fulfil a certain number of properties. Main construction principles: double integration over all combinations of α-cuts; fuzzification of set., (86/144), (87/144)

23 Fuzzy fuzzy arithmetics Fuzzy Interval An interval number, representing an uncertain real number A = [a 1, a 2 ] = {x a 1 x a 2, x For intervals A B, operator {+,,, /} we define A B = {a b a A, b B} Division, A/B, is not defined when 0 B. The result of an arithmetic operation on closed intervals is again a closed interval. }, (88/144), (89/144) Fuzzy Interval For closed intervals A = [a 1, a 2 ] B = [b 1, b 2 ], the four arithmetic are defined as follows (equivalent with definition on previous slide) Fuzzy Fuzzy fuzzy intervals A fuzzy number is a fuzzy set on A + B = [a 1 + b 1, a 2 + b 2 ] A B = A + B = [a 1 b 2, a 2 b 1 ] A B = [min(a 1 b 1, a 1 b 2, a 2 b 1, a 2 b 2 ), max(a 1 b 1, a 1 b 2, a 2 b 1, a 2 b 2 )], if 0 / [b 1, b 2 ] A/B = A B 1 = [a 1, a 2 ] [ 1 b 2, 1 b 1 ] = [min( a 1 b 1, a 1 b 2, a 2 b 1, a 2 b 2 ), max( a 1 b 1, a 1 b 2, a 2 b 1, a 2 b 2 )]. A : [0, 1] such that (i) A is normal (height(a) = 1) (ii) α A is a closed interval for all α (0, 1] (iii) The support of A, Supp(A) = 0+ A, is bounded, (90/144), (91/144)

24 Fuzzy Fuzzy fuzzy intervals Theorem (4.1) Let A F( ). Then, A is a fuzzy number iff there exists a closed interval [a, b] such that 1 for x [a, b] A(x) = l(x) for x (, a) r(x) for x (b, ) where l : (, a) [0, 1] is monotonic non-decreasing, continuous from the right, l(x) = 0 for x < ω 1 r : (b, ) [0, 1] is monotonic non-increasing, continuous from the left, r(x) = 0 for x > ω 2. Fuzzy Arithmetics on fuzzy Moving from interval, we can define arithmetics on fuzzy based on two principles: 1 Cutworthiness (thanks to inclusion monotonicity of intervals) α (A B) = α A α B in combination with A B = 2 or the extension principle α (0,1] α(a B) (A B)(z) = sup min [A(x), B(x)] z=x y, (92/144), (93/144) Fuzzy Arithmetics on fuzzy Fuzzy MIN MAX operators Theorem (4.2) Let {+,,, /}, let A, B denote continuous fuzzy. Then, the fuzzy set A B defined by the extension principle (prev. slide) is a continuous fuzzy number. Lemma (A B)(z) = sup min [A(x), B(x)] α (A B) = α A α B z=x y So the two definitions are equivalent for continuous fuzzy. (The proof is built on continuity.) MIN(A, B)(z) = MAX(A, B)(z) = sup min [A(x), B(x)], z=min(x,y) sup min [A(x), B(x)] z=max(x,y) Again, for continuous fuzzy, this is equivalent with a definition based on cutworthiness. α (MIN(A, B)) = MIN( α A, α B), α (MAX(A, B)) = MAX( α A, α B), α (0, 1]. Where, for intervals [a 1, a 2 ], [b 1, b 2 ] MIN([a 1, a 2 ], [b 1, b 2 ]) = [min(a 1, b 1 ), min(a 2, b 2 )], MAX([a 1, a 2 ], [b 1, b 2 ]) = [max(a 1, b 1 ), max(a 2, b 2 )]., (94/144), (95/144)

25 Fuzzy MIN MAX operators Fuzzy Arithmetics on fuzzy Not all fuzzy are comparable (only partial order). However, values of linguistic variables are often defined by fuzzy that are comparable. For example: Figure: Comparison of the operators MIN, min, MAX, max., Fuzzy Linguistic variables, (97/144) Fuzzy Linguistic variables When fuzzy are connected to linguistic concepts, such as very small, small, medium, so on, interpreted in a particular context, the resulting constructs are usually called linguistic variables. A linguistic variable is fully characterized by a quintuple hv, T, X, g, mi, in which v is the name of the variable, T is the set of linguistic terms of v that refer to the base variable whose values range over a universal set X, g is a syntactic rule (a grammar) for generating linguistic terms, m is a semantic rule that assigns to each linguistic term t T its meaning, m(t), which is a fuzzy set on X (i.e., m : T F(X )). (96/144) very small ¹ small ¹ medium ¹ large ¹ very large, (98/144) Figure: An example of a linguistic variable., (99/144)

26 Fuzzy A + X = B Let X = [x 1, x 2 ]. Interval equations Equation A + X = B Then [a 1 + x 1, a 2 + x 2 ] = [b 1, b 2 ] follows immediately. Clearly: x 1 = b 1 a 1 x 2 = b 2 a 2. Since X must be an interval, it is required that x 1 x 2. That is, the equation has a solution iff b 1 a 1 b 2 a 2. Then X = [b 1 a 1, b 2 a 2 ] is the solution. Fuzzy Fuzzy equations The solution to a fuzzy equation can be obtained by solving a set of interval equations, one for each nonzero α in the level set Λ(A) Λ(B). The equation A + X = B has a solution iff (i) α b 1 α a 1 α b 2 α a 2 for every α (0, 1], (ii) α β implies α b 1 α a 1 β b 1 β a 1 β b 2 β a 2 α b 2 α a 2. If a solution α X exists for every α (0, 1] (property (i)), property (ii) is satisfied, then the solution X is given by X = αx α (0,1], (100/144), (101/144) Fuzzy Fuzzy equations Equation A X = B Fuzzy Linear programming An example Similarly as A + X = B The equation A X = B has a solution iff (i) α b 1 / α a 1 α b 2 / α a 2 for every α (0, 1], (ii) α β implies α b 1 / α a 1 β b 1 / β a 1 β b 2 / β a 2 α b 2 / α a 2. If the solution exists, it has the form X = αx α (0,1] where α X = [ α b 1 / α a 1, α b 2 / α a 2 ]. Again, X = B/A is not a solution of the equation. Minimize z = x 1 2x 2 Subject to 3x 1 x 2 1 2x 1 + x x x 1 The feasible set, i.e., the set of vectors x that satisfy all constraints, is always a convex polygon (if bounded). Figure: An example of a classical linear programming problem., (102/144), (103/144)

27 Fuzzy Fuzzy linear programming Fuzzy approximate reasoning In many practical situations, it is not reasonable to require that the constraints or the objective function are specified in crisp precise terms. The most general case of fuzzy linear programming grows rather complex, is not discussed in the book. A realistic special case to provide the feeling: The situation where the right-h-side vector the constraint matrix are expressed by fuzzy triangular. Simple membership functions allows transformation of the problem, (104/144), (105/144) Fuzzy Classical : A brief overview Logic functions Logic function assigns a truth value to a combination of truth values of its variables: f : {true, false} n {true, false} 2 n choices of n arguments 2 2n functions of n variables., (106/144) Fuzzy Classical : A brief overview Logic functions of two variables v Function Adopted v name symbol ω Zero function 0 ω NOR function v 1 v 2 ω Inhibition v 1 > v 2 ω Negation v 2 ω Inhibition v 1 < v 2 ω Negation v 1 ω Exclusive OR v 1 v 2 ω NAND function v 1 v 2 ω Conjunction v 1 v 2 ω Equivalence v 1 v 2 ω Assertion v 1 ω Implication v 1 v 2 ω Assertion v 2 ω Implication v 1 v 2 ω Disjunction v 1 v 2, ω (107/144) One function 1

28 Fuzzy Classical : A brief overview Logic primitives Fuzzy Classical : A brief overview Logic formulae We can express all the functions of n variables by using only a small number of simple functions. Such a set is a complete set of primitives. Examples: {negation, conjunction, disjunction}, {negation, implication}. Definition 1. If v is a variable, then v v are formulae; 2. If v 1 v 2 are formulae, then v 1 v 2 v 1 v 2 are also formulae; 3. Logic formulae are only those defined (obtained) by the two previous rules., (108/144), (109/144) Fuzzy Fuzzy propositions Fuzzy Linguistic hedges (modifiers) The range of truth values of fuzzy propositions is not only {0, 1}, but [0, 1]. The truth of a fuzzy proposition is a matter of degree. Classification of fuzzy propositions: Unconditional unqualified propositions The temperature is high. Unconditional qualified propositions The temperature is high is very true. Conditional unqualified propositions If the temperature is high, then it is hot. Linguistic hedges are linguistic terms by which other linguistic terms are modified. Tina is young is true. Tina is very young is true. Tina is young is very true. Tina is very young is very true. Fuzzy predicates fuzzy truth values can be modified. Crisp predicates cannot be modified. Examples of hedges: very, fairly, extremely. Conditional qualified propositions If the temperature is high, then it is hot is true., (110/144), (111/144)

29 Fuzzy Modifiers Fuzzy Fuzzy quantifiers Strong modifier reduces the truth value of a proposition. Weak modifier increases the truth value of a proposition (by weakening the proposition). One commonly used class of modifiers is h α (a) = a α, for α R + a [0, 1]. For α < 1, h α is a weak modifier. Example: H : fairly h(a) = a. For α > 1, h α is a strong modifier. Example: H : very h(a) = a 2. h 1 is the identity modifier. Absolute quantifiers: about 10 ; much more than 100,... Relative quantifiers: almost all ; about half,... Examples: p: There are about 3 high-fluent students in the group. q: Almost all students in the group are high-fluent., (112/144), (113/144) Fuzzy Fuzzy propositions Unconditional unqualified propositions Fuzzy Fuzzy propositions Unconditional qualified propositions The canonical form p : ν is F ν is a variable on some universal set V F is a fuzzy set on V that represents a fuzzy predicate (e.g., low, tall, young, expensive...) The degree of truth of p is T (p) = F (v), for v ν. T is a fuzzy set on V. Its membership function is derived form the membership function of a fuzzy predicate F. The role of a function T is to connect fuzzy sets fuzzy propositions. The canonical form p : ν is F is S (truth qualified proposition) where ν is a variable on some universal set V, F is a fuzzy set on V that represents a fuzzy predicate, S is a fuzzy truth qualifier. To calculate the degree of truth T (p) of the proposition p, we use: T (p) = S(F (v)) In case of unconditional unqualified propositions, the identity function is used., (114/144), (115/144)

30 Fuzzy Fuzzy propositions Conditional unqualified propositions Fuzzy Fuzzy propositions Conditional qualified propositions The canonical form p : If X is A, then Y is B, where X, Y are variables on X, Y respectively, A, B are fuzzy sets on X, Y respectively. Alternative form: X, Y is R where R(x, y) = J (A(x), B(x)) is a fuzzy set on X Y representing a suitable fuzzy implication. The canonical form p : If X is A, then Y is B is S where X, Y are variables on X, Y respectively, A, B are fuzzy sets on X, Y respectively, S is a truth qualifier., (116/144), (117/144) Fuzzy Fuzzy implications Definition(s) Fuzzy Fuzzy implications How to select fuzzy implication A fuzzy implication J of two fuzzy propositions p q is a function of the form J : [0, 1] [0, 1] [0, 1], which for any truth values a = T (p) b = T (q) defines the truth value J (a, b) of the conditional proposition if p, then q. Fuzzy implications as extensions of the classical implication: Crisp implication a b Fuzzy implication J (a, b) (S) ā b u(c(a), b) (R) max{x {0, 1} a x b} sup{x [0, 1] i(a, x) b} (QL) ā (a b) u(c(a), i(a, b)) (QL) (ā b) b u(i(c(a), c(b)), b) Look at Table 11.2, Table 11.3, Table 11.4 (pp ). One good choice: J s (a, b) = { 1 a b 0 a > b One frequently used implication: Lukasiewicz J a (a, b) = min[1, 1 a + b], (118/144), (119/144)

31 Fuzzy Inference rules Fuzzy Inference rules Example: Generalized modus ponens Fuzzy inference rules are basis for approximate reasoning. As an example, three classical inference rules (Modus ponens, Modus Tollens, Hypothetical syllogism) are generalized by using compositional rule of inference For a given fuzzy relation R on X Y, a given fuzzy set A on X, a fuzzy set B on Y can be derived for all y Y, so that B (y) = sup min[a (x), R(x, y)]. x X In matrix form, compositional rule of inference is B = A R Rule: If X is A, then Y is B Fact: X is A Conclusion: Y is B In this case, R(x, y) = J [A(x), B(y)] B (y) = sup min[a (x), R(x, y)]. x X, (120/144), (121/144) Fuzzy General schema is of the form: Multiconditional approximate reasoning Rule 1: If X is A 1, then Y is B 1 Rule 2: If X is A 2, then Y is B 2... Rule n: If X is A n, then Y is B n Fact: X is A Conclusion: Y is B A, A j are fuzzy sets on X, B, B j are fuzzy sets on Y, for all j., (122/144) Fuzzy Multiconditional approximate reasoning Method of interpolation Most common way to determine B is by using method of interpolation. Step 1. Calculate the degree of consistency between the given fact the antecedent of each rule. Use height of intersection of the associated sets: r j (A ) = h(a A j ) = sup min[a (x), A j (x)]. x X Step 2. Truncate each B j by the value r j (A ) determine B as the union of truncated sets: B (y) = sup min[r j (A ), B j (y)], for all y Y. j n Note that interpolation method is a special case of the composition rule of inference, with R(x, y) = sup min[a j(x), B j(y)] j n where then B (y) = sup x X min[a (x), R(x, y)] = (A R)(y)., (123/144)

32 Fuzzy Multiconditional approximate reasoning Method of interpolation-example Fuzzy, (124/144) Fuzzy Defuzzification is a process that maps a fuzzy set to a crisp set. Defuzzification to a point Approaches: Centre of gravity (Set of real ) COG (A) = Defuzzification to a point. Defuzzification to a set. Generating a good representative of a fuzzy set. Recovering a crisp original set. (126/144) Pxmax xmin x A(x) P. xmax xmin A(x) Mean of maxima (Set of real ) MeOM(A) =, Examples of common methods (125/144), Fuzzy P x core(a) core(a) x. Centre of area (COA) COA(A) is the value that minimizes the expression ŕ ŕ COA(A) ŕ ŕ ŕ, ŕ X x=inf(x ) (127/144) sup(x ) A(x) X ŕ ŕ ŕ A(x)ŕŕ. ŕ x=coa(a)

33 Fuzzy Defuzzification to a set A.k.a. Averaging procedures Fuzzy Defuzzification to a set Average α-cuts Let a fuzzy set A be given by a membership function µ : R [0, 1]. α-cuts chosen at various levels α. Average α-cuts based on an integration of set-valued function, called Kudo-Aumann integration. Feature distance minimization find the crisp set at the minimal feature distance to the given fuzzy set. Sets F (w) are α cuts, A α of the fuzzy set A, for α [0, 1]; Selectors are ϕ(α) = inf A α ϕ(α) = sup A α. Then, the average α-cut of A is [ A µ = inf A α dα, [0,1] [0,1] sup A α dα ]., (128/144), (129/144) Fuzzy Fuzzy Definition An optimal defuzzification D(A) of a fuzzy set A on a reference set X, with respect to the distance d, is D(A) {C P(X ) d(a, C) = min [d(a, B)]}. (3) B P(X ) Use a feature distance containing both local global features. Find minimum using heuristic search methods, e.g., simulated annealing., (130/144), (131/144)

34 Fuzzy Conventional system Fuzzy Fuzzy Methodology first developed by Mamdani in 1975 used to a steam plant. Based on work by Zadeh (1973) on fuzzy algorithms for complex systems decision processes. PID Control (Proportional-Integral-Derivative) The PID ler is the workhorse of the process industries. In a manner analogous to conventional systems, inputs of a system are mapped to outputs using fuzzy rather than differential equations. Output = bias + K P ε + K I t 0 ε dt + K D dε dt, (132/144), (133/144) Fuzzy Motivation Can be used for systems that are difficult or impossible to model mathematically. Can also be applied to processes that are too complex or nonlinear to be led with traditional strategies. In fact, a detailed precise mathematical description is not always necessary for optimized operation of an engineering process. Human operators often are capable of managing complex situations of a plant without knowing anything about differential equations. Such a rule based system can be used to define a ler that emulates the heuristic rule-of-thumb strategies of an expert. Fuzzy Useful cases Motivation 1 The processes are too complex to analyze by conventional quantitative techniques. 2 The available sources of information are interpreted qualitatively, inexactly, or uncertainly. Advantage of Fuzzy Flexible Universal approximator Easy to underst Powerful yet simple Linguistic linguistic terms human knowledge Tolerant of imprecision / Robust more than 1 rules - an error of a rule is not fatal limited trust in input data Parallel or distributed, (134/144) multiple fuzzy rules - complex nonlinear system, (135/144)

35 Fuzzy Motivation Fuzzy Fuzzy Disadvantages More complex than PID More parameters to tune Un-mathematical (stability?) Four main components 1 The fuzzification interface : transforms input crisp values into fuzzy values 2 The knowledge base : contains a knowledge of the application domain the goals. 3 The decision-making : performs inference for fuzzy actions 4 The defuzzification interface : provides a crisp action out, (136/144), (137/144) Fuzzy Five steps... How to build a fuzzy ler in five easy steps... Fuzzy 1 Partition input output spaces: Select meaningful linguistic states for each variable express them as appropriate fuzzy sets. 2 Fuzzification of input: Introduce a fuzzification function for each input variable to express the associated measurement. 3 Formulate a set of inference rules: If ɛ = A dɛ dt = B, then C. 4 Design an inference engine: Use method of interpolation (Lecture 10). 5 Select a suitable defuzzification method (Lecture 11)., (138/144), (139/144)

36 Fuzzy Fuzzy Hanging-togetherness natural grouping of voxels constituting an object a human viewer readily sees in a display of the scene as a Gestalt in spite of intensity heterogeneity. Basic idea: Compute global hanging-togetherness from local hanging-togetherness. Strength of a path the strength of its weakest link A. Rosenfeld 1979 Strength of a link between two points defined by the membership function. J. K. Udupa S. Samarasekera 1996 Strength of a link between two points defined by affinity The of two points x y in A the strength of the strongest path between x y c A (x, y) = sup π Π(x,y) inf A(t) t π, (140/144), (141/144) Fuzzy Fuzzy spel adjacency is a reflexive symmetric fuzzy relation α in n assigns a value to a pair of spels (c, d) based on how close they are spatially. Fuzzy spel affinity is a reflexive symmetric fuzzy relation κ in n assigns a value to a pair of spels (c, d) based on how close they are spatially intensity-based-property-wise (local hanging-togetherness). µ κ (c, d) = h(µ α (c, d), µ(c), µ(d), c, d) The fuzzy κ- assigns a value to a pair of spels (c, d) that is the maximum of the strengths of assigned to all possible paths from c to d (global hanging-togetherness). Fuzzy Components of fuzzy affinity Fuzzy spel adjacency µ α (c, d) indicates the degree of spatial adjacency of spels The homogeneity-based component µ ψ (c, d) indicates the degree of local hanging-togetherness of spels due to their similarities of intensities The object-feature-based component µ ϕ (c, d) indicates the degree of local hanging-togetherness of spels with respect to some given object feature Example: µ κ = 1 2 µ α(µ ψ + µ ϕ ), (142/144), (143/144)

37 Fuzzy Computation A graph search problem Dynamic programming solution (think distance transform or level sets computation) Practical usage examples: Seed foreground (one or multiple seeds), threshold at some level of fuzzy. Seed different regions let them compete (relative fc, iterated relative fc)., (144/144)