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Fuzzy Inference Systems

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Review Fuzzy Models If then.

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FuzzificationDefuzzification Inferencing InputOutput Basic Configuration of a Fuzzy Logic System Target Error =Target -Output

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Types of Rules Mamdani Assilian Model R1: If x is A 1 and y is B 1 then z is C 1 R2: If x is A 2 and y is B 2 then z is C 2 A i, B i and C i, are fuzzy sets defined on the universes of x, y, z respectively Takagi-Sugeno Model R1: If x is A 1 and y is B 1 then z =f 1 (x,y) R1: If x is A 2 and y is B 2 then z =f 2 (x,y) For example: f i (x,y)=a i x+b i y+c i

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Types of Rules Mamdani Assilian Model Takagi-Sugeno Model

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Mamdani Fuzzy Models

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The Reasoning Scheme Both antecedent and consequent are fuzzy

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The Reasoning Scheme Both antecedent and consequent are fuzzy

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1: IF FeO is high & SiO 2 is low & Granite is prox & Fault is prox, THEN metal is highImplication (Max) 0 1 0 1 = 2: IF FeO is aver & SiO 2 is high & Granite is interm & Fault is prox, THEN metal is aver 30% 50% 70% 0 1 40% 55% 70% 0 km 10 km 20km 0 km 5 km 10km 0t 100t 1000t 3: IF FeO is low & SiO 2 is high & Granite is dist & Fault is dist, THEN metal is low FeO = 60%SiO 2 = 60% Granite = 5 km Fault = 1 km Metal = ? 0t 100t 1000t = =

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Defuzzifier Converts the fuzzy output of the inference engine to crisp using membership functions analogous to the ones used by the fuzzifier. Five commonly used defuzzifying methods: –Centroid of area (COA) –Bisector of area (BOA) –Mean of maximum (MOM) –Smallest of maximum (SOM) –Largest of maximum (LOM) Since consequent is fuzzy, it has to be defuzzified

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Defuzzifier

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Rule 1: Rule 2: Rule 3: Aggregate (Max) + + = Defuzzify (Find centroid) 125 tonnes metal Formula for centroid

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Sugeno Fuzzy Models Also known as TSK fuzzy model –Takagi, Sugeno & Kang, 1985

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If x is A and y is B then z = f(x, y) Fuzzy Rules of TSK Model Fuzzy Sets Crisp Function f(x, y) is very often a polynomial function w.r.t. x and y. The order of a Takagi-Sugeno type fuzzy inference system = the order of the polynomial used. While antecedent is fuzzy, consequent is crisp

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The Reasoning Scheme

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Examples R1: if X is small and Y is small then z = x +y +1 R2: if X is small and Y is large then z = y +3 R3: if X is large and Y is small then z = x +3 R4: if X is large and Y is large then z = x + y + 2

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TAKAGI-SUGENO SYSTEM 1.IF x is f 1x (x) AND y is f 1y (y) THEN z 1 = p 10 +p 11 x+p 12 y 2.IF x is f 2x (x) AND y is f 1y (y) THEN z 2 = p 20 +p 21 x+p 22 y 3.IF x is f 1x (x) AND y is f 2y (y) THEN z 3 = p 30 +p 31 x+p 32 y 4.IF x is f 2x (x) AND y is f 2y (y) THEN z 4 = p 40 +p 41 x+p 42 y The firing strength (= output of the IF part) of each rule is: s 1 = f 1x (x) AND f 1y (y) s 2 = f 2x (x) AND f 1y (y) s 3 = f 1x (x) AND f 2y (y) s 4 = f 2x (x) AND f 2y (y) Output of each rule (= firing strength x consequent function) : 1.o 1 = s 1 ∙ z 1 2.o 2 = s 2 ∙ z 2 3.o 3 = s 3 ∙ z 3 4.o 4 = s 4 ∙ z 4 Overall output of the fuzzy inference system is: o 1 + o 2 + o 3 + o 4 s 1 + s 2 + s 3 + s 4 z =

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Sugeno system 18 Rule1: IF FeO is high AND SiO 2 is low AND Granite is proximal AND Fault is proximal, THEN Gold =p 1 (FeO%)+q 1 (SiO 2 %) +r 1 (Distance2Granite)+s 1 (Distance2Fault)+t 1 Rule 2: IF FeO is average AND SiO 2 is high AND Granite is intermediate AND Fault is proximal, THEN Gold =p 2 (FeO%)+q 2 (SiO 2 %)+r 2 (Distance2Granite)+s 2 (Distance2Fault)+t 2 Rule 3: IF FeO is low AND SiO 2 is high AND Granite is distal AND Fault is distal, THEN Gold =p 3 (FeO%)+q 3 (SiO 2 %)+r 3 (Distance2Granite)+s 3 (Distance2Fault)+t 3

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Gold(R1) =p 1 (FeO%)+q 1 (SiO 2 %) + r 1 (Distance2Granite) +s 1 (Distance2Fault)+t 1 1: IF FeO is high X SiO 2 is low X Granite is prox X Fault is prox, THEN 0 1 0 1 2: IF FeO is aver X SiO 2 is high X Granite is interm X Fault is prox, THEN 30% 50% 70% 0 1 40% 55% 70% 0 km 10 km 20km 0 km 5 km 10km 3: IF FeO is low & SiO 2 is high & Granite is dist & Fault is dist, THEN FeO = 60%SiO 2 = 60% Granite = 5 km Fault = 1 km Metal = ? s1s1 Gold(R2) =p 2 (FeO%)+q 2 (SiO 2 %) + r 2 (Distance2Granite) +s 2 (Distance2Fault)+t 2 s2s2 Gold(R3) =p 3 (FeO%)+q 3 (SiO 2 %) + r 3 (Distance2Granite) +s 3 (Distance2Fault)+t 3 s3s3 Sugeno system

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Sugeno system: Output Gold(R1) =p 1 (FeO%)+q 1 (SiO 2 %) + r 1 (Distance2Granite) +s 1 (Distance2Fault)+t 1 s1s1 Gold(R2) =p 2 (FeO%)+q 2 (SiO 2 %) + r 2 (Distance2Granite) +s 2 (Distance2Fault)+t 2 s2s2 Gold(R3) =p 3 (FeO%)+q 3 (SiO 2 %) + r 3 (Distance2Granite) +s 3 (Distance2Fault)+t 3 s3s3 Firing strength Rule output

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A neural fuzzy system Implements FIS in the framework of NNs Fuzzification Nodes Antecedent Nodes Output Nodes xy

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Fuzzification Nodes Represents the term sets of the features. If we have two features x and y and two linguistic variables defined on both of it say BIG and SMALL. Then we have 4 fuzzification nodes. xy BIG SMALL We use Gaussian Membership functions for fuzzification --- They are differentiable, triangular and trapezoidal membership functions are NOT differentiable.

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Fuzzification Nodes (Contd.) and are two free parameters of the membership functions which needs to be determined How to determine and Two strategies: 1) Fixed and 2) Update and , through any tuning algorithm

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Consequent nodes p, q and k are three free parameters of the consequent polynomial function How to determine p, q, k Two strategies: 1) Fixed 2) Update through any tuning algorithm

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Fuzzification nodes xy BIG SMALL μ x1 μ x2 μ y1 μ y2 Antecedent nodes e.g. If x is Small & y is Small Consequent nodes w1w1 w2w2 w3w3 w4w4 e.g. z 4 = p 4 x + q 4 y + k 4 z1z1 z2z2 z3z3 z4z4 Output node O = (w 1 z 1 +w 2 z 2 +w 3 z 3 +w 4 z 4 )/ (w 1 +w 2 +w 3 +w 4 Target (t) Error = ½(t-o) 2

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ANFIS Architecture Squares: Adaptive nodes Circles: Fixed nodes

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ANFIS Architecture Layer 1 (Adaptive) Contains adaptive nodes, each with a Gaussian membership function: Number of nodes = number of variables x number of linguistic values In the previous example there are 4 nodes (2 variable x 2 linguistic values for each) Two parameters to be estimated per node: mean (centre) and standard deviation (spread) These are called premise parameters Number of premise parameters = 2 x number of nodes = 8 in the example

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ANFIS Architecture Layer 2 (Fixed) Contains fixed nodes, each with product operator (T-norm operator). Returns the firing strength of each If-Then Rule. The firing strength can be normalized. In ANFIS, each node returns a normalized firing strength – Fixed nodes – no parameter to be estimated.

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ANFIS Architecture Layer 3 (Adaptive) Each node contains an adaptive polynomial, and returns output for each fuzzy If-Then rule Number of nodes = number of If-Then Rules. The parameters ps are called consequent parameters.

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ANFIS Architecture Layer 4 (Fixed) Sums up the output of each node in the previous layer: A single node in this layer. No parameter to be estimated.

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ANFIS Training Linear in the consequent parameters P ki, if the premise parameters and, therefore, the firing strengths s k of the fuzzy if-then rules are fixed. ANFIS uses a hybrid learning procedure (Jang and Sun, 1995) for estimation of the premise and consequent parameters. The hybrid learning procedure estimates the consequent parameters (keeping the premise parameters fixed) in a forward pass and the premise parameters (keeping the consequent parameters fixed) in a backward pass.

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Squares: Adaptive nodes Circles: Fixed nodes The forward pass: Propagate information forward until Layer 3 Estimate the consequent parameters by the least square estimator. The backward pass: Propagate the error signals backwards and update the premise parameters by gradient descent. ANFIS Training

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ANFIS Training : Least Square Estimation 1.Data assembled in form of (x n ; y n ) 2.We assume that there is a linear relation between x and y: y = ax + b 3.Can be extended to n dimensions: y = a 1 x 1 + a 2 x 2 + a 3 x 3 + … + b The problem: Given the function f, ﬁnd values of coefﬁcients a i s such that the linear combination best fits the data

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ANFIS Training : Least Square Estimation Given data {(x 1 ; y 1 (x N ; y N )}, we may deﬁne the error associated to saying y = ax + b by: This is just N times the variance of data : {y1 - (ax1+b),…., y n - ( ax N +b)} The goal is to ﬁnd values of a and b that minimize the error. In other words minimize the partial derivative of the error wrt a and b:

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ANFIS Training : Least Square Estimation Which gives us: We may rewrite them as: The values of a and b which minimize the error satisfy the following matrix equation: Hence a and b are estimated using:

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ANFIS Training : Least Square Estimation For the following data find least square estimator SNoXYX2X2 XY 129418 2311933 34131652 461736102 582164168 61717 729418 81127121297 91433196462 TOTAL511474511157

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ANFIS Training : Least Square Estimation

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ANFIS Training : Gradient descent

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