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Graphical Technique of Inference

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

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

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

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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) = 0.35input(j) = 55

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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 R n space in R m space

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

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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:

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Fuzzy Relation Equations

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

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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)

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Partitioning

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Nonlinear Simulation using Fuzzy Rule-Based System : If x is, then y is Rules can be connected by “AND” or “OR” or “ELSE” 1.IF : x = x i THEN : y = y i It is a simple lookup table for the system description 2. Inputs are crisp sets, Outputs are singletons This is also a lookup table.

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

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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.

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

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Nonlinear Simulation using Fuzzy Rule-Based System

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Fuzzy Associative Memories (FAMs) A fuzzy system with n non-interactive inputs and a single output. Each input universe of discourse, x 1, x 2, …, x n is partitioned into k fuzzy partitions The total # of possible rules governing this system is given by: l = k n or l = (k+1) n Actual number r << 1.r: actual # of rules If x 1 is partitioned into k 1 partitions x 2 is partitioned into k 2 partitions :. x n is partitioned into k n partitions l = k 1 k 2 … k n

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Fuzzy Associative Memories (FAMs) Example: for n = 2 A1A2A3A4A5A6A7 B1C1C4 C3 B2C1C2 B3C4C1 C2 B4C3 C1 C2 B5C3C4 C1C3 A A1 A7 B B1 B5 Output: C C1 C4

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

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Fuzzy Associative Memories (FAMs) Few simple rules for y = 10 sin x 1.IF x1 is Z or P B, THEN y is z. 2.IF x1 is PS, THEN y is PB. 3.IF x1 is z or N B, THEN y is z 4.IF x1 is NS, THEN y is NB FAM for the four simple rules x1N BN SzP SP B yzN BzP Bz

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

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Fuzzy Associative Memories (FAMs)

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Defuzzified results for simulation of y = 10 sin x1 select value with maximum absolute value in each column. x1 -135 -45 45 135 y0000 -7007 7

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Fuzzy Associative Memories (FAMs) More rules would result in a close fit to the function. Comparing with results using extension principle: Let 1.x1 = Z or PB 2.x1 = PS 3.x1 = Z or NB 4.x1 = NS Let B = {-10,-8,-6,-4,-2,0,2,4,6,8,10}

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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 yx1 -10-90 -8-126.9-53.1 -6-143.1-36.9 -4-156.4-23.6 -2-168.5-11.5 0-180180 211.5168.5 423.6156.4 636.9143.1 853.1126.9

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

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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

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Fuzzy Ordering Given two fuzzy numbers I and J

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

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Fuzzy Ordering

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

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Fuzzy Ordering This function is a measurement of membership value of choosing x over y. If set A contains more variables A = {x 1,x 2,…,x n } A’ = {x 1,x 2,…,x i-1,x i+1,…,x n } Note: here, A’ is not complement. f(x i | A’) = min{f(x i | x 1 ),f(x i | x 2 ),…,f(x i | x i-1 ),f(x i | x i+1 ),…,f(x i | x n )} Note: f(x i |x i ) = 1 then f(x i |A’) = f(x i |A) We can form a matrix C to rank many fuzzy sets. To determine overall ranking, find the smallest value in each row.

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