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### Fuzzy logic and fuzzy time series edited

1. 1. 1 Fuzzy Logic and Fuzzy Set Theory & Time Series Modeling Prof. Dr. S.M. Aqil Burney College of Computer Science and Information System (IoBM) aqil.burney@iobm.edu.pk Former Meritorious Professor and Chairman Dept. of Computer Science University of Karachi (1973-20110) burney@uok.edu.pk www.burney.net
2. 2. Reasoning • Inductive and Deductive Reasoning …… • AI and Logic see 2
3. 3. Vageness & Uncertainty • It is evident that as we are using ICT and other technologies and generating huge data we come across complexity which consists Vagueness and Uncertainty which could not be handled amicably without logic which , helps us to formulate & model and solve the problems. 3
4. 4. 4 Fuzzy Logic and Fuzzy Set Theory e.g. A={1,2,3,4}, so 1 belongs to A but 5 is not member of set A. • Fuzzy sets are a natural outgrowth and generalization of crisp sets. • It tells us besides “belongs to” and “not belongs to” way, other possibilities exist in the relation between an element and a set emerging in various practical processes. • A crisp set defines only two possibilities “belongs to=(1)” or “not belongs to=(0)”. Lotfi Zadeh
5. 5. Fuzzy Logic and Fuzzy Set Theory: (Cont…) • Unlike crisp sets with sharp boundaries as discussed above, a fuzzy set is a set with smooth (un-sharp) boundaries. • e.g. A set whose elements PARTIALLY belongs to that set. • e.g 40% belongs to and 60% not belongs to some particular set. • Binary sets contains only two values 0 and 1, Whereas fuzzy sets consider all the values in the interval [0,1]. • Classical/crisp sets are suitable for various applications and have proven to be important tool. But they do not reflect the nature of human concepts and thoughts, which tends to be abstract and imprecise. • Fuzzy logic is the study of imperfect, imprecise and ambiguous knowledge. This knowledge includes the linguistic chaos as well. 5
6. 6. 6 Some Examples of fuzzy data (Linguistic Chaos) •Today is very hot day. (Degree of hotness is not defined. Inexact value) • He is very intelligent (Here intelligence is a matter of degree (%) and differ from person to person) • If you work hard then you will get the success (Inference based on qualitative data)
7. 7. Fuzzy Logic : Degree of Relation • We know many things in life are degree rather than present or not present. • A green and red apple is not just green and red; there many levels of green and red shades, computer scientists ,technologist and engineers and to some extend statistician have accepted this theory. 7
8. 8. Example (Degree of Relation) • A pixel can have a bright ness level between 0 and 255. • 0 value = Black 255 value = White , • While every value between 0 and 255 gives a gray level. 8
9. 9. 9 Fuzzy Logic and Fuzzy Set Theory: (Cont…) • Fuzzy theory holds that many things in life are matter of degree. • Thus the association of each element in a universe of discourse is a matter of degree, which is a number between 0 and 1. e.g its 60% cold in this hall and 40% not cold. So degree of coldness is a fuzzy concept. • This is represented by where A= fuzzy set and X is the universe of discourse. • Relation of an element with its set A is partially true and partially false . • Law Contradiction(excluded middle) needs revisit. • Law Included Middle ( )xAµ
10. 10. 10 Fuzzy Logic and Fuzzy Set Theory: (Cont…) • Fuzzy sets are a natural outgrowth and generalization of crisp sets. • A fuzzy set can be defined in two ways.  Enumerating membership values of those elements in the set (completely or partially).(Discrete membershop function)  Defining membership function mathematically for the given universe of discourse. The universe of discourse may be discrete or continuous or may be mixture of the two types.
11. 11. 11 Fuzzy Logic and Fuzzy Set Theory: (Cont…) • A crisp set can be generalized to multiple categories and each category is assigned its relevant value called the membership value. • Larger values denote high degrees of set membership • For simplicity and completeness, a membership function (MF) maps every element of a universe of discourse X into real numbers [0,1]
12. 12. 12 Fuzzy Logic and Fuzzy Set Theory: (Cont…) • For notational purpose, membership function of a fuzzy set is given by ( ) ( ) ( ) ( ) nnAAAi i iA xxxxxxxxA /.../// 2211 µµµµ +++== ∑ where ix X∈ In case of continuous universe of discourse, ( )∫= x iiA xxA /µ
13. 13. 13 Fuzzy Logic and Fuzzy Set Theory: (Cont…) • Formally, a fuzzy set A along with its MF ( ) Xxx iA ∈;µ is defined as ( )( ){ }XxxxA ∈= A, µ Where ( ) Xxx iA ∈;µ can take any of the function that satisfies the conditions of a fuzzy membership function
14. 14. 14 Example: Discrete Universe of Discourse . One poAn insurer wants to classify the types of plan for late premium paymentsssible reason is the possible number of months of delay in premium payments (X). Let { }10,...,2,1=X be the set of available types of offers to the customers by the insurance company. Then the fuzzy set “ease of payment to the customer” may be described as follows: A={(1,0.2), (2,0.5), (3,0.6), (4,0.7), (5,0.8), (6,0.9)}
15. 15. 15 Example: Discrete Universe of Discourse 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Premium payements Membershipvalues Ease of Payment for a Customer
16. 16. 16 Properties of Fuzzy Sets The fuzzy logic provides us an intuitively pleasing method of representing one form of uncertainty. In designing fuzzy inference systems, some preliminary concepts need to be defined properly. Some definitions necessary for designing of fuzzy systems are given
17. 17. 17 Example: Discrete Universe of Discourse Given a fuzzy set A defined on X and any number [ ]0,1α ∈ , the α -cut, Aα and the strong α -cut, A+α are crisp sets given by ( ){ }αα ≥= xAxA | ( ){ }αα >=+ xAxA |
18. 18. 18 Example: Level Set Representation of Fuzzy Sets Providing Discrete Approximation to Continuous Membership Functions Age Fuzzy Set Young Middle Age Old A discrete approximation of A2 (Middle Age) is presented and the MF values of A2 are denoted by D2. ( ) ( )      ≥ <<− ≤ = 350 352015/35 201 1 x xx x xA ( ) ( ) ( )        ≤≤ <<− <<− ≥≤ = 45351 604515/60 352015/20 60200 2 x xx xx xorx xA ( ) ( )      ≥ <<− ≤ = 601 604515/45 450 3 x xx X xA
19. 19. 19 Discrete Approximation of Continuous Universe of Discourse 0 10 20 30 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1 1.2 YoungYoung Middle Aged Old
20. 20. 20 Age Factor and Related Risk Membership Function 0 10 20 30 40 50 60 70 80 90 0 0.2 0.4 0.6 0.8 1 1.2 X= Age MembershipGrades Young Middle Aged Old
21. 21. 21 Support ( ) of a Fuzzy Set ( ){ }0|)( >= xxASupport Aµ The support of a fuzzy set A denoted by supp(A), within a universe of discourse X is the crisp set that contains all the elements of X that have nonzero membership grades in A. A+0
22. 22. 22 Fuzzy convexity ( )( ) ( ) ( )1 2 1 21 min ,A x x A x A xµ λ λ  + − ≥   A set A in n R is called convex iff, for every pair of points is also in A Here the sets A1-A5 are convex and A6-A9 are non-convex sets.
23. 23. 23 Fuzzy symmetry A fuzzy set A is symmetric if its MF is symmetric around a certain point , x=c, namely, ( ) ( ) ,A Ac x c x x Xµ µ+ = − ∀ ∈
24. 24. 24 Some Membership Functions A membership function provides a gradual transition form regions completely outside a set to regions completely inside the set. Its usefulness depends critically on our capability to construct appropriate membership functions for various given concepts in various contexts. Even for similar contexts, the representation of a system using fuzzy logic may vary considerably.
25. 25. 25 Triangular Membership Function ( )          ≤ ≤≤ − − ≤≤ − − ≤ = xc cxb bc xc bxa ab ax ax cbaxtriangle 0 0 ,,; 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 trim f(x, [3 4 5]); trim f(x, [2 4 7]); trim f(x, [1 4 9]); Triangular M em bership Function 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 trim f(x, [2 3 5]); trim f(x, [3 4 7]); trim f(x, [4 5 9]); 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 trim f(x, [3 4 5]); trim f(x, [2 4 7]); trim f(x, [1 4 9]); Triangular M em bership Function 0.6 0.8 1 Different shapes of Triangular function for varying parameters
26. 26. 26 Trapezoidal Membership Function ( )            ≤ ≤≤ − − ≤≤ ≤≤ − − ≤ = xc dxc cd xd cxb bxa ab ax ax dcbaxtrapezoid 0 1 0 ,,,; 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 trapmf(x, [2 3 7 9]); trapmf(x, [3 4 6 8]); trapmf(x, [4 5 5 7]); trapmf(x, [5 6 4 6]); Trapezoidal Membership Function
27. 27. 27 A Gaussian membership function is specified by two parameters. Gaussian Membership Function ( )               −− = 2 2 2 2 1 exp,; σ σ cx cxgaussian 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 gaussm f(x, [0.5 5]); gaussm f(x, [1 5]); gaussm f(x, [2 5]); gaussm f(x, [3 5]); G aussian C urve M em bership Function 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 gaussm f(x, [0.5 5]); gaussm f(x, [1 5]); gaussm f(x, [2 5]); gaussm f(x, [3 5]); 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 gaussm f(x, [1 2]); gaussm f(x, [1 4]); gaussm f(x, [1 6]); gaussm f(x, [1 8]); Varying c and fixed variance Fixed c and varying variance
28. 28. 28 Sigmoidal Membership Function ( ) ( )[ ] , exp1 1 ,; cxa caxsig −−+ = Depending on the sign of a, a sigmoidal function is inherently open right or open left and thus is appropriate for representing concepts such as “very risky” or “very negative”. Sigmoidal functions of this type are used in training of neural networks
29. 29. 29 Sigmoidal Membership Function (Cont…) 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 sigmf(x, [-1 5]); sigmf(x, [-3 5]); sigmf(x, [4 5]); sigmf(x, [8 5]); Sigmoid Curve Mmbership Function 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 sigmf(x, [5 2]); sigmf(x, [5 4]); sigmf(x, [5 6]); sigmf(x, [5 8]); 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 sigmf(x, [-1 5]); sigmf(x, [-3 5]); sigmf(x, [4 5]); sigmf(x, [8 5]); Sigmoid Curve Mmbership Function 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 sigmf(x, [5 2]); sigmf(x, [5 4]); sigmf(x, [5 6]); sigmf(x, [5 8]); Varying a and fixed c Fixed a and varying c
30. 30. 30 Fuzzy Set Theoretic Operations • Corresponding to the crisp set operations, we can define fuzzy set operations for fuzzy sets. • The set operations intersection and union correspond to logic operations, conjunction (AND) and disjunction (OR), respectively. • Normally, for fuzzy intersection, we use ‘min’ or ‘AND’ operator and for fuzzy union, we usually apply ‘max’ or ‘OR’ operators.
31. 31. 31 Fuzzy Intersection (conjunction) The intersection of two fuzzy sets A and B is specified in general by a function [ ] [ ] [ ]1,01,01,0: →×T If membership values of A and B are ( )xAµ and ( )xBµ then fuzzy conjunction is given by ( ) ( ) ( ) ( ), ,A B x T A x B x x X ∩ = ∀ ∈  where T represents a binary operation for the fuzzy intersection.
32. 32. 32 T-norm This class of fuzzy intersection operators are usually referred to as T-norm (triangular norm) operators. T-norm operator satisfies the fuzzy arithmetic axioms like boundary condition, monotonicity, commutativity, associativity, continuity and etc. Examples of frequently used fuzzy intersection. ( ) ( )babaT ,min, = ( ) abbaT =, Standard intersection: Algebraic product:
33. 33. 33 Fuzzy Union (Disjunction) Like fuzzy intersection, fuzzy union operator is defined by a function [ ] [ ] [ ]1,01,01,0: →×S ( ) ( ) ( ) ( ), ,A B x S A x B x x X ∪ = ∀ ∈  where S represents a binary operation for the fuzzy union. Properties that a function S satisfies to be intuitively acceptable as a fuzzy union are exactly the same as properties of t-conorm. Some frequently used t-conorm operation ( ) ( )babaS ,max, = ( ) abbabaS −+=, Standard disjunction: Algebraic product:
34. 34. 34 Membership Functions of Two Dimensions In multivariate studies, we need to define membership functions of higher dimensions. For example, let X and Y be two fuzzy numbers and R be the 2-D fuzzy set on Z X Y= × Cylindrical extension is a natural way to extent one-dimensional MFs to two-dimensional MFs define below.
35. 35. 35 Cylindrical Extension If A is a fuzzy set in X, then its cylindrical extension in is a fuzzy set defined by YX × ( ) ( ) ( )∫× = YX A yxxAc ,µ The concept of cylindrical extension extends the dimensions of a given MFs.
36. 36. 36 Fuzzy Projection • Projects a fuzzy relation to a subset of selected dimensions. • Often used to extract marginal possibility distribution of a few selected variables from a given fuzzy relation. • It decreases the dimension of a given (multidimensional) MF. • Let R be a two-dimensional fuzzy set on .Then the projection of R onto X and Y are defined as YX × ( ) yyxR R Y x Y ,max µ∫= ( ) xyxR R X y X ,max µ∫=
37. 37. 37 Extension Principle • Provides a general procedure for extending crisp domains of mathematical expressions to fuzzy domains. • Plays fundamental role in extending any point-to-point operations to fuzzy operations.
38. 38. 38 Extension Principle For One-to-one Fuzzy Relations Assume X and Y are two crisp sets and let be a mapping from X into Y, ,YXf →: f ( ),x X f x y Y∈ = ∈ Assume A is a fuzzy subset of X, using extension principle, we can define as a fuzzy subset of Y such that,( )Af ( ) ( ) ( ) ( ) ( ) ( ) nnAAAiiA Xx B yxyxyxyxAfy /.../// 2211 µµµµµ +++=∪== ∈ ( ) .,...,2,1, nixfy ii ==where
39. 39. 39 Extension Principle for Many-to-one Fuzzy Relations If is a many-to-one relation, then we may have more than one possibility at each value of x. that is , Therefore, f(x)y = ( ) ( ) Yyxfxf ∈== * 21 2121 ,, xxXxx =∈ ( ) ( ) ( )xy A yfx B µµ 1 max− = =
40. 40. 40 Generalized Extension Principle iA Suppose that f is a mapping from an n-dimensional Cartesian space nXXX ××× ...21 to a single universe of discourse Y such that ( ) ,,...,, 21 yxxxf n = and for each , 1,2,...,iX i n= a fuzzy membership is defined. Thus total n fuzzy membership functions are defined. Then by the extension principle, we can define ( ) ( ) ( )[ ] ( ) ( )    = ≠ = − − = − φ φµ µ yfif yfifx y iAi yfxxx B i n 1 1 ,...,, ,0 minmax 1 21
41. 41. 41 Where to Apply In fuzzy neural networks, fuzzy projection and extension principle are used to generalize backpropagation algorithm to fuzzy backpropagation learning algorithm.
42. 42. 42 2. Fuzzy Relations Topics to be discussed in this section • Binary Relations • Linguistic Hedges • Fuzzy If-Then Rules • Fuzzy Reasoning
43. 43. 43 Fuzzy Relations • Generalizes the notion of crisp relation into one that allows partial membership. • Degree of association can be represented by membership grades in a fuzzy relation in the same way as the degrees of set membership are represented in the fuzzy sets. • A relation defined between two objects is represented by a binary relation. Similarly, we can form tern-ary, quartern-ary, quin-ary or n-ary relation between three, four, five or n objects, respectively.
44. 44. 44 Binary Crisp Relationship Mathematically speaking, if x and y be two variables from two domains X and Y respectively, then the binary relation between x and y, R(x,y) is a subset of Cartesian space of X and Y. define the relationX<y, as below ( ){ }RyxyxyxR ∈<= ,,|, ( ) YXyxR ×⊆,Where
45. 45. 45 Crisp n-Dimensional Relations ( )1 2 1 2, ,..., ...n nR X X X X X X⊆ × × × In general, for n-dimensional arguments taken from the domains ,then nxxx ,...,, 21 1 2, ,..., nX X X We see that a relation is again a set and thus follows the same rules as the domain of Cartesian product of nXXX ,...,, 21
46. 46. 46 Binary Fuzzy Relations If x and y are two fuzzy variables with domains X and Y then the binary fuzzy relation R is ( ) ( ) ( ){ }YXyxyxyxR R ×∈= ,|,,, µ [ ]1,0: →×YXRwhere
47. 47. 47 Fuzzy Relations • The binary fuzzy relation can be extended for n-arguments. • This allows the characteristic function of a crisp relation to allow tuples to have degree of membership within the relation. • The membership grades indicate the strength of the relation present between the elements of the tuple. • A fuzzy relation is a fuzzy set defined on the Cartesian product space of crisp sets where the tuples may have varying degrees of membership within the relation. ( )nxxx ,...,, 21
48. 48. 48 Fuzzy Composition Rule • Fuzzy relationship in different product spaces can be combined through a composition operation. • Different composition operations have been suggested for fuzzy relations • The best known is the max-min composition operation by Zadeh (1965).
49. 49. 49 Max-Min Fuzzy Composition Rule of Inference Let and be two fuzzy relation defined on and ,respectively. The max-min composition of and is a fuzzy set defined by 1R 2R YX × ZY × 1R 2R ( ) ( ) ( )( ){ }ZzYyXxzyyxzxRR RR y ∈∈∈= ,,|,,,minmax,, 2121 µµ or equivalently, ( ) ( ) ( )( )zyyxzx RR y RR ,,,minmax, 2121 µµµ = ( ) ( )[ ]zyyx RR y ,, 21 µµ ∧∨ The composition rule of inference is not uniquely defined. By choosing different fuzzy conjunction and disjunction operators, we can get different composition rules of inference.
50. 50. 50 Max-Product Fuzzy Composition Rule of Inference An alternate to max-min composition called max-product composition is used due to its higher mathematical tractability then max-min composition and can be defined as same as max-min composition: ( ) ( ) ( )( )zyyxzx RR y RR ,,,max, 2121 µµµ =
51. 51. 51 Properties of Fuzzy Relations The fuzzy composition rules follow several properties common to binary relations. If A, B and C are binary relations on and then ZYYX ×× , WZ × Associativity Distributive over union Weak distributivity over intersection Monotonicity ( ) ( )A B C A B C=    ( ) ( ) ( )CABACBA  ∪=∪ ( ) ( ) ( )CABACBA  ∩⊆∩ CABACB  ⊆⇒⊆
52. 52. 52 3. Fuzzy IF-Then Rules Topics to be discussed in this section • Linguistic variables • Linguistic Hedges • Fuzzy If-Then Rules • Fuzzy Reasoning
53. 53. 53 Fuzzy IF-Then Rules Fuzzy If-then rules or fuzzy inferencing is an extension of crisp propositional statements. They allow human knowledge and common sense representation using modes-ponen rule of inference and are able to make conclusions in the presence of uncertainty and chaos. To deal with variety of decisions on a single problem, give place to include fuzzy hedging operators that enables a fuzzy inference system to deal with extremities.
54. 54. 54 Linguistic variables The concept of fuzzy numbers plays a fundamental role in formulating quantitative fuzzy variables, i.e., the variables whose states are fuzzy numbers. When in addition, the fuzzy numbers represents linguistic concepts, such as very low, high, extreme and so on, as interpreted in a particular context, the resulting constructs are usually called linguistic variables. A linguistic variable is characterized by a quintuple ( )( ), , , ,x T x X G M x =is the name of variable; X is the universe of discourse. T(x) =is the linguistic term set of x; G =is a syntactic rule which generates the terms in T(x) and M =is the semantic rule which associates with each linguistic value A its meaning M(A), where A denotes a fuzzy set in X.
55. 55. 55 Linguistic Hedges A linguistic variable enables its values to be described qualitatively by a linguistic term and quantitatively by a corresponding membership function. The linguistic term is used to express concepts and knowledge in human communication, whereas membership function is useful for processing numerical input data. For example, very, more or less, fairly, or extremely are all hedges defined for linguistic variables.
56. 56. 56 Linguistic Hedges (Cont...) Let A be a linguistic value characterized by a fuzzy set with membership function , Then is interpreted as modified version of the original value expressed as ( ).Aµ k A ( )[ ]∫== X k A k xxA µ The linguistic hedges can either concentrate (increase) or dilate (decrease) the significance of a fuzzy set. ( ) 0,ACON ≥= kAk ( ) 10, <<= kAADIL k Using linguistic hedges, we can define composite linguistic terms in fuzzy reasoning.
57. 57. 57 Fuzzy If-Then Rules A fuzzy if-then rule (or fuzzy implication) defines a relation between x and y. A fuzzy if-then rule be defined as a binary fuzzy relation R on the product space YX × If A and B are two fuzzy sets defined over X and Y and then the implication is given as YyXx ∈∈ , A B→ if x is A then y is B Here "x is A" is the antecedent or premise, while "y is B" is called the consequent or conclusion.
58. 58. 58 Fuzzy If-Then Rules (Cont…) Some examples of fuzzy if-then rules are: • if risk is high then premium is high • if interest rate is high then liquidity is low. • if rate of return is adequate then investment will increase
59. 59. 59 Fuzzy Reasoning/Fuzzy Expert System An inferential procedure that derives conclusions from a set of if- then rules and known facts. Using compositional rules of inference and generalized modes- ponens (GMP) rule, we can define three possible cases in fuzzy reasoning • Single rule with single antecedent. (SRSA) • Single rule with multiple antecedents. (SRMA) • Multiple rules and multiple antecedents. (MRMA)
60. 60. 60 4. Fuzzy Aggregation Operations
61. 61. 61 Fuzzy Aggregation/Averaging Operations • A class of types of defuzzification in fuzzy inference/expert systems. • Using fuzzy aggregation operations on fuzzy sets, we can obtain appropriate single fuzzy set. • In fuzzy inference engines, these operations allow us to combine multiple rules using single rule of inference.
62. 62. 62 Axioms of Fuzzy Aggregation Operations (Cont…) Formally, we define [ ] [ ]1,01,0: → n h nAAA ,...,, 21 are fuzzy sets defined on X. Thus the aggregated fuzzy set A defined over X will be ( ) ( ) ( ) ( )( )xAxAxAhxA n...,,, 21= for each Xx∈ Fuzzy Aggregation operations are necessary in defuzzification of ordered/unordered fuzzy knowledge.
63. 63. 63 Axioms of Fuzzy Aggregation Operations In order to qualify as an intuitive and meaningful aggregation function (h), it must satisfy at least following three requirements. Axiom 1: ( ) ( ) 11,...,1,100,...,0,0 == handh (Boundary condition) Axiom 2: ( )naaa ,...,, 21 and ( )nbbb ,...,, 21 such that, if ii ba ≤ ,then ( ) ( )nn bbbhaaah ,...,,...,, 2,121 ≤ (Monotonically increasing) Axiom 3: Fuzzy aggregation operation, h, is a continuous function.
64. 64. 64 Axioms of Fuzzy Aggregation Operations (Cont…) Other two additional axioms are:- h is symmetric function in all its arguments; i.e., Axiom 5: Axiom 4: ( ) ( ) ( ) ( )( )npppn aaahaaah ,...,,,...,, 2121 = For any permutation of p on N h is an idempotent function; that is, ( ) [ ]1,0,...,, ∈∀= aaaaah
65. 65. 65 Fuzzy Idempotency for Aggregation Operations Note that, if any aggregation operation satisfies axioms 2-5, then it also satisfy the inequality ( ) ( ) ( )nnn aaaaaahaaa ,...,,max,...,,,...,,min 212121 ≤≤ ( ) [ ]n naaa 1,0,...,, 21 ∈for all n-tuples
66. 66. 66 Generalized Fuzzy Aggregation Operation ** All aggregation operations between the standard fuzzy intersection and union are idempotent. These aggregation operations are usually called averaging operations. The generalized mean is defined as: ( ) αααα α 1 21 21 ... ,...,,       +++ = n aaa aaah n n
67. 67. 67 5. Fuzzy Time Series Pioneer Researchers and practitioners: 1. Zadeh (1975) 2. Yager R. R. (2005) 3. Kacpryzk (Germany) (Springer-Verlag) 4. Klir (2005) 5. Chen S. M. and Lee L.W. (2004, 2006,2007,2008) 6. Huang K. (2001, 2003, 2004, 2007, 2008) 7. Zimmermann (2002) 8. Oscar Castillo (2007) 9. Burney and Jilani (2006, 2007, 2008)
68. 68. 68 Each observation is assumed interval based fuzzy variable alongwith associated membership function. Less than one and half decade of history of fuzzy time series. Based on fuzzy relation and fuzzy inference rules, efficient modeling and forecasting of fuzzy time series is possible. Fuzzy Times Series
69. 69. 69 Fuzzy Times Series Time series analysis plays vital role in most of the actuarial related problems. Based on fuzzy relation, section 2.2.7 and fuzzy inference rules, section 2.2.8, efficient modeling and forecasting of fuzzy time series is possible. This field of fuzzy time series analysis is not very mature due to the time and space complexities in most of the actuarial related issue.
70. 70. 70 Review of Fuzzy Time Series Song and Chissom (1993a; 1993b; 1994) presented the concept of fuzzy time series based on the concepts of fuzzy set theory to forecast the historical enrollments of the University of Alabama. Huarng (2001b) presented the definition of two kinds of intervals in the universe of discourse to forecast the TAIFEX. Chen (2002) presented a method for forecasting based on high-order fuzzy time series. Lee et. al. (2004) presented a method for temperature prediction based on two-factor high-order fuzzy time series. Melike and Konstsntin (2004) proposed forecasting method using first order fuzzy time series. Lee, Wang and Chen (2006) presented handling of forecasting problems using two-factor high order fuzzy time series for TAIFEX and daily temperature in Taipei, Taiwan.
71. 71. 71 Jilani T. A. and Aqil Burney S. M. (2008), A Refined Fuzzy Time Series Model for Enrollments Problem, Physica A, Elsevier Publishers. Jilani T. A. and Aqil Burney S. M. (2008), Multivariate Stochastic Fuzzy Forecasting Models, Expert Systems with Applications 37(2), Elsevier Publishers. Jilani T. A. and Aqil Burney S. M. (2007), M-Factor High Order Fuzzy Time Series Forecasting for Road Accident Data, In IEEE-IFSA 2007, World Congress, Cancun, Mexico, June 18-21, In Castillo, O.; Melin, P.; Montiel Ross, O.; Sepúlveda Cruz, R.; Pedrycz, W.; Kacprzyk, J. (Eds.), Design and Analysis of Intelligent Systems usingFuzzy Logic and Soft Computing vol. 41, Advances in Soft Computing, Berlin: Springer-Verlag. Jilani T. A., Aqil Burney S. M. and Ardil C.(2008), Multivariate High Order Fuzzy Time Series Forecasting for Car Road Accidents. International Journal of Computational Intelligence, Vol. 4, no. 1. pp. 7-16. Review of Fuzzy Time Series (Cont…)
72. 72. 72 Jilani T. A., Aqil Burney S. M. and Ardil C. (2007), Fuzzy metric approach for fuzzy time series forecasting based on frequency density based partitioning, International Conference on Machine Learning and Pattern Recognition. August 24-26 (2007), Berlin, Germany. Jilani T. A. and Aqil Burney S. M. (2007), A New Quantile Based Fuzzy Time Series Forecasting Model, submitted in Computers and Mathematics With Applications (CMWA), Elsevier Publishers. Jilani T. A. and Aqil Burney S. M. (2007), Fuzzy Time Series Forecasting Using Frequency Density Based Partitioning for Enrollments Problem, submitted in Expert Systems with Applications (ESWA), Elsevier Publishers. Review of Fuzzy Time Series (Cont…)
73. 73. 73 Two-Factor kth-Order Fuzzy Time Series Model We can extend the concept of single antecedent and single consequent (one-to-one) to many antecedents and single consequent (many-to-one). For example, in designing two-factor kth-order fuzzy time series model with X be the primary and Y be second fact. We assume that there are k antecedent and one consequent ( ) ( ) ( )( )kk YXYXYX ,,...,,,, 2211 1+kX ( ) ( ) ( ) ( )1122221111 ,,...,,,, ++ =→====== kkkkkk xXyYxXyYxXyYxXIf
74. 74. 74 Two- Factor Fuzzy inferencing We can extend the concept of single antecedent and single consequent (one-to-one) to many antecedents and single consequent (many-to-one). For example, in designing two-factor kth-order fuzzy time series model with X be the primary and Y be second factor. ( ) ( ) ( ) ( )1122221111 ,,...,,,, ++ =→====== kkkkkk xXyYxXyYxXyYxXIf We assume that there are k antecedent ( ) ( ) ( )( )kk YXYXYX ,,...,,,, 2211 and one consequent 1+kX
75. 75. 75 M-Factor and k-th Order Fuzzy Time Series Model ( ) ( ) ( ) ( ) 11 11 12 12 1 1 21 21 22 22 2 2 1 1 2 2 1, 1 1, 1 , ,..., , , ,..., ,..., , ,..., 1,2,..., , 1,2,..., k k k k m m m m mk mk m k m k X x X x X x X x X x X x X x X x X x X x for i m j k + + + + = = = = = = = = = = = = If then In the similar way, we can define m-factor (i=1,2,…,m) and kth order (k=1,2,…,k) fuzzy time series as
76. 76. 76 • We can define relationship among present and future state of a time series with the help of fuzzy sets. Fuzzy Time Series kA , respectively, where UAA kj ∈, ,then kj AA → represented the Assume the fuzzified data of the thi ( )thi 1+ day areand jA and andjAfuzzy logical relationship between kA
77. 77. 77 Fuzzy Time Series • Let ( ) ( ),...2,1,0...,,tY =t be the universe of discourse and ( ) RtY ⊆ Assume that ( ) ,...2,1, =itfi is defined in the universe of discourse ( )tY and ( )tF is a collection of ( ) ( ),...2,1,0...,,tf i =i , then ( )tF is called a fuzzy time series of ( ) ,...2,1,tY =i • Using fuzzy relation, we define, ( ) ( ) ( )1,1 −−= ttRtFtF  , where ( ), 1R t t − is a fuzzy relation and “  ” is the max–min composition operator, then ( )tF is caused by ( )1−tF ( )tF ( )1−tF, where and are fuzzy sets.
78. 78. 78 ( )tF ( )tF ( ) ( ) ( )1 , 2 ,...,F t F t F t n− − − Let be a fuzzy time series. If is caused by then the fuzzy logical relationship is represented by ( ) ( ) ( ) ( )tFtFtFntF →−−− 1,2,..., is called the one-factor nth order fuzzy time series forecasting model. Univariate Vector Fuzzy Logic Inferencing New Forecasting Methods Based on M-Factors High-Order Fuzzy Time Series
79. 79. 79 Bivariate Vector Fuzzy Logic Inferencing New Forecasting Methods Based on M-Factors High-Order Fuzzy Time Series Let ( )tFbe a fuzzy time series. If( )tF is caused by then this fuzzy logical relationship is represented by ( ) ( )( ) ( ) ( )( ) ( ) ( )( )1 2 1 2 1 21 , 1 , 2 , 2 ,..., ,F t F t F t F t F t n F t n− − − − − − ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )tFtFtFtFtFntFntF →−−−−−− 1,1,2,2,...,, 212121 and is called the two-factors nth order fuzzy time series forecasting model, where ( )tF1 ( )tF2 are called the main factor and the Secondary factor FTS respectively.
80. 80. 80 • In the similar way, we can define m-factor nth-order fuzzy logical relationship as Mutivariate Vector Fuzzy Logic Inferencing ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )tFtFtFtF tFtFtFntFntFntF m mm →−−− −−−−−− 1,...,1,1 ,2,...,2,2,...,,...,, 21 2121 Here ( )1F t is called the main factor and ( ) ( ) ( )2 3, ,..., mF t F t F t secondary factor FTS. are called New Forecasting Methods Based on M-Factors High-Order Fuzzy Time Series
81. 81. 81 New Forecasting Methods Based on M-Factors High- Order Fuzzy Time Series Steps: Step 1) Define the universe of discourse, U of the main factor [ ]min 1 max 2,U D D D D= − − where minD and maxD are the minimum and the maximum values of the main 1D , 2D proper positive real numbers to divide the universe of discourse into n-equal length intervals 1 2,, ..., lu u u factor of the known historical data, respectively, and are two
82. 82. 82 Some Observations • Here we can implement any of the fuzzy membership function to define the FTS in above equations. • Comparative study by using different membership functions is also possible. However, we have used triangular membership function due to low computational cost. • Using fuzzy composition rules, we establish a fuzzy inference system for FTS forecasting with higher accuracy • Using fuzzy composition rules, we establish a fuzzy inference system for FTS forecasting with higher accuracy • The accuracy of forecast can be improved by considering higher number of factors and higher dependence on history.
83. 83. Some work in Actuarial Science , Health Management • E- Health Management etc.Management • Mobile Health Management(Internet of Things) • Temporal Database and Fuzzy Logic • Software Engineering (SCM) • Medical Image Analysis using Soft Computing Techniques 83
84. 84. THANK YOU 84