Graphical Technique of Inference

Graphical Technique of Inference Using max-product (or correlation product) implication technique, aggregated output for r rules would be:

Graphical Technique of Inference Case 3: input(i) and input(j) are fuzzy variables

Graphical Technique of Inference Case 4: input(I) and input(j) are fuzzy, inference using correlation product

Graphical Technique of Inference Example: Rule 1: if x1 is and x2 is , then y is Rule 2: if x1 is or x2 is , then y is input(i) = 0.35 input(j) = 55

Fuzzy Nonlinear Simulation Virtually all physical processes in the real world are nonlinear. Nonlinear System Input Output X Y Input vector and output vector in Rn space in Rm space

Approximate Reasoning or Interpolative Reasoning The space of possible conditions or inputs, a collection of fuzzy subsets, for k = 1,2,… The space of possible outputs p = 1,2,… 3. The space of possible mapping relations, fuzzy relations q = 1,2,…

Fuzzy Relation Equations We may use different ways to find a. look up table b. linguistic rule of the form IF THEN If the fuzzy system is described by a system of conjunctive rules, we could decompose the rules into a single aggregated fuzzy relational equation for each input, x, as follows:

Fuzzy Relation Equations

Fuzzy Relation Equations Equivalently R: fuzzy system transfer for a single input x. If a system has n non-interactive fuzzy inputs xi and a single output y If the fuzzy system is described by a system of disjunctive rules:

Partitioning How to partition the input and output spaces (universes of discourse) into fuzzy sets? 1. prototype categorization 2. degree of similarity 3. degree similarity as distance Case 1: derive a class of membership functions for each variable. Case 2: create partitions that are fuzzy singletons (fuzzy sets with only one element having a nonzero membership)

Partitioning

Partitioning

Nonlinear Simulation using Fuzzy Rule-Based System : If x is , then y is Rules can be connected by “AND” or “OR” or “ELSE” IF : x = xi THEN : y = yi It is a simple lookup table for the system description 2. Inputs are crisp sets, Outputs are singletons This is also a lookup table.

Nonlinear Simulation using Fuzzy Rule-Based System This model may also involve Spline functions to represent the output instead of crisp singletons.

Nonlinear Simulation using Fuzzy Rule-Based System 3. Input conditions are crisp sets and output is fuzzy set or fuzzy relation The output can be defuzzied.

Nonlinear Simulation using Fuzzy Rule-Based System 4. Input: fuzzy Output: singleton or functions. If fi is linear Quasi-linear fuzzy model (QLFM) Quasi-nonlinear fuzzy model (QNFM)

Nonlinear Simulation using Fuzzy Rule-Based System

Nonlinear Simulation using Fuzzy Rule-Based System

Fuzzy Associative Memories (FAMs) A fuzzy system with n non-interactive inputs and a single output. Each input universe of discourse, x1, x2, …, xn is partitioned into k fuzzy partitions The total # of possible rules governing this system is given by: l = kn or l = (k+1)n Actual number r << 1. r: actual # of rules If x1 is partitioned into k1 partitions x2 is partitioned into k2 partitions : . xn is partitioned into kn partitions l = k1  k2  …  kn

Fuzzy Associative Memories (FAMs) Example: for n = 2 A1 A2 A3 A4 A5 A6 A7 B1 C1 C4 C3 B2 C2 B3 B4 B5 A  A1  A7 B  B1  B5 Output: C  C1  C4

Fuzzy Associative Memories (FAMs) Example: Non-linear membership function: y = 10 sin x

Fuzzy Associative Memories (FAMs) Few simple rules for y = 10 sin x IF x1 is Z or P B, THEN y is z. IF x1 is PS, THEN y is PB. IF x1 is z or N B, THEN y is z IF x1 is NS, THEN y is NB FAM for the four simple rules x1 N B N S z P S P B y

Fuzzy Associative Memories (FAMs) Graphical Inference Method showing membership propagation and defuzzification:

Fuzzy Associative Memories (FAMs)

Fuzzy Associative Memories (FAMs) Defuzzified results for simulation of y = 10 sin x1 select value with maximum absolute value in each column. x1 -135 -45 45 135 y -7 7

Fuzzy Associative Memories (FAMs) More rules would result in a close fit to the function. Comparing with results using extension principle: Let x1 = Z or PB x1 = PS x1 = Z or NB x1 = NS Let B = {-10,-8,-6,-4,-2,0,2,4,6,8,10}

Fuzzy Associative Memories (FAMs) To determine the mapping, we look at the inverse of y = f(x1) i.e. x1 = f-1(y) in the table y x1 -10 -90 -8 -126.9 -53.1 -6 -143.1 -36.9 -4 -156.4 -23.6 -2 -168.5 -11.5 -180 180 2 11.5 168.5 4 23.6 156.4 6 36.9 143.1 8 53.1 126.9

Fuzzy Associative Memories (FAMs) For rule1, x1 = Z or PB Graphical approach can give solutions very close to those using extension principle

Fuzzy Synthetic Evaluation Fuzzy Decision Making Fuzzy Synthetic Evaluation An evaluation of an object, especially ill-defined one, is often vague and ambiguous. First, finding , for a given situation , solving

Fuzzy Ordering Given two fuzzy numbers I and J

Fuzzy Ordering It can be extended to the more general case of many fuzzy sets

Fuzzy Ordering

Fuzzy Ordering Then the ordering is: Sometimes the transitivity in ordering does not hold. We use relativity to rank. fy(x): membership function of x with respect to y fx(y): membership function of y with respect to x The relationship function is:

Fuzzy Ordering This function is a measurement of membership value of choosing x over y. If set A contains more variables A = {x1,x2,…,xn} A’ = {x1,x2,…,xi-1,xi+1,…,xn} Note: here, A’ is not complement. f(xi | A’) = min{f(xi | x1),f(xi | x2),…,f(xi | xi-1),f(xi | xi+1),…,f(xi | xn)} Note: f(xi|xi) = 1 then f(xi|A’) = f(xi|A) We can form a matrix C to rank many fuzzy sets. To determine overall ranking, find the smallest value in each row.