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**Graphical Technique of Inference**

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**Graphical Technique of Inference**

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) = 0.35 input(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 Rn space in Rm space

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**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,…

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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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**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:

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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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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” 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.

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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: fuzzy Output: singleton or functions. If fi 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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**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, 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

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**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

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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 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

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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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**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

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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 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}

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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 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

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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 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

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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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**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:

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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 = {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.

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Practical Considerations When t , the supp, then there will be a value of t when supp folds, it becomes multi-w-to-one-x mapping.

Practical Considerations When t , the supp, then there will be a value of t when supp folds, it becomes multi-w-to-one-x mapping.

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