1 / 38

Fuzzy Inference Systems

Fuzzy Inference Systems. Review Fuzzy Models. If <antecedence> then <consequence> . Basic Configuration of a Fuzzy Logic System. Inferencing. Fuzzification. Defuzzification. Input. Output. Target. Error =Target -Output. Types of Rules. Mamdani Assilian Model

alida
Download Presentation

Fuzzy Inference Systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Fuzzy Inference Systems

  2. Review Fuzzy Models If <antecedence> then <consequence>.

  3. Basic Configuration of a Fuzzy Logic System Inferencing Fuzzification Defuzzification Input Output Target Error =Target -Output

  4. Types of Rules MamdaniAssilian Model R1: If x is A1 and y is B1 then z is C1 R2: If x is A2 and y is B2 then z is C2 Ai , Bi and Ci, are fuzzy sets defined on the universes of x, y, z respectively Takagi-Sugeno Model R1: If x is A1 and y is B1 then z =f1(x,y) R1: If x is A2 and y is B2 then z =f2(x,y) For example: fi(x,y)=aix+biy+ci

  5. Types of Rules MamdaniAssilian Model Takagi-Sugeno Model

  6. MamdaniFuzzy Models

  7. The Reasoning Scheme Both antecedent and consequent are fuzzy

  8. The Reasoning Scheme Both antecedent and consequent are fuzzy

  9. 1: IFFeOis high & SiO2 is low & Granite isprox& Fault isprox, THEN metal is high Implication (Max) 1 = 1 1 0 2: IFFeOis aver & SiO2 is high & Granite isinterm& Fault isprox, THEN metal is aver = 30% 50% 70% 0 km 10 km 20km 0 0 40% 55% 70% 3: IFFeOis low & SiO2 is high & Granite is dist & Fault is dist, THEN metal is low = 0t 100t 1000t 0 km 5 km 10km 0t 100t 1000t FeO = 60% SiO2 = 60% Metal = ? Granite = 5 km Fault = 1 km

  10. Defuzzifier Since consequent is fuzzy, it has to be defuzzified • 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)

  11. Defuzzifier

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

  13. Sugeno Fuzzy Models • Also known as TSK fuzzy model • Takagi, Sugeno & Kang, 1985

  14. Crisp Function Fuzzy Sets Fuzzy Rules of TSK Model While antecedent is fuzzy, consequent is crisp If x is A and y is B then z = f(x, y) 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.

  15. The Reasoning Scheme

  16. 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

  17. TAKAGI-SUGENO SYSTEM • IF x is f1x(x) AND y is f1y(y) THEN z1 = p10+p11x+p12y • IF x is f2x(x) AND y is f1y(y) THEN z2 = p20+p21x+p22y • IF x is f1x(x) AND y is f2y(y) THEN z3 = p30+p31x+p32y • IF x is f2x(x) AND y is f2y(y) THEN z4 = p40+p41x+p42y • The firing strength (= output of the IF part) of each rule is: • s1 = f1x(x)AND f1y(y) • s2 = f2x(x) AND f1y(y) • s3 = f1x(x)AND f2y(y) • s4 = f2x(x) AND f2y(y) • Output of each rule (= firing strength x consequent function) : • o1 = s1 ∙ z1 • o2 = s2 ∙ z2 • o3 = s3 ∙ z3 • o4 = s4 ∙ z4 • Overall output of the fuzzy inference system is: • o1+ o2+ o3+ o4 • s1+ s2+ s3+ s4 z =

  18. Sugeno system Rule1: IFFeOis high ANDSiO2 is low ANDGranite is proximal AND Fault is proximal, THENGold =p1(FeO%)+q1(SiO2%) +r1(Distance2Granite)+s1(Distance2Fault)+t1 Rule 2: IFFeOis average ANDSiO2 is high ANDGranite is intermediate AND Fault is proximal, THEN Gold =p2(FeO%)+q2(SiO2%)+r2(Distance2Granite)+s2(Distance2Fault)+t2 Rule 3: IFFeOis low ANDSiO2 is high ANDGranite is distal AND Fault is distal, THEN Gold =p3(FeO%)+q3(SiO2%)+r3(Distance2Granite)+s3(Distance2Fault)+t3

  19. Sugeno system 1 1: IFFeOis high XSiO2is low XGranite isproxXFault isprox, THEN Gold(R1) =p1(FeO%)+q1(SiO2%) + r1(Distance2Granite) +s1(Distance2Fault)+t1 s1 1 1 0 2: IFFeOis aver XSiO2 is high XGranite isintermXFault isprox, THEN Gold(R2) =p2(FeO%)+q2(SiO2%) + r2(Distance2Granite) +s2(Distance2Fault)+t2 30% 50% 70% 0 km 10 km 20km 0 0 40% 55% 70% s2 3: IFFeOis low & SiO2 is high & Granite is dist & Fault is dist, THEN Gold(R3) =p3(FeO%)+q3(SiO2%) + r3(Distance2Granite) +s3(Distance2Fault)+t3 s3 0 km 5 km 10km FeO = 60% SiO2 = 60% Metal = ? Granite = 5 km Fault = 1 km

  20. Sugeno system: Output Firing strength Rule output s1 Gold(R1) =p1(FeO%)+q1(SiO2%) + r1(Distance2Granite) +s1(Distance2Fault)+t1 Gold(R2) =p2(FeO%)+q2(SiO2%) + r2(Distance2Granite) +s2(Distance2Fault)+t2 s2 Gold(R3) =p3(FeO%)+q3(SiO2%) + r3(Distance2Granite) +s3(Distance2Fault)+t3 s3

  21. A neural fuzzy system Implements FIS in the framework of NNs Output Nodes Antecedent Nodes Fuzzification Nodes x y

  22. 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. BIG SMALL BIG SMALL x y We use Gaussian Membership functions for fuzzification --- They are differentiable, triangular and trapezoidal membership functions are NOT differentiable.

  23. 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

  24. 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

  25. Target (t) Error = ½(t-o)2 Output node O = (w1z1+w2z2+w3z3+w4z4)/ (w1+w2+w3+w4 z1 z4 z2 z3 Consequent nodes e.g. z4 = p4x + q4y + k4 w3 w1 w4 w2 Antecedent nodes e.g. If x is Small & y is Small Fuzzificationnodes μx1 μx2 μy2 μy1 BIG SMALL BIG SMALL x y

  26. ANFIS Architecture Squares: Adaptive nodes Circles: Fixed nodes

  27. 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

  28. 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.

  29. 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.

  30. 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.

  31. ANFIS Training Linear in the consequent parameters Pki, if the premise parameters and, therefore, the firing strengths sk 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.

  32. ANFIS Training 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. Squares: Adaptive nodes Circles: Fixed nodes

  33. ANFIS Training : Least Square Estimation Data assembled in form of (xn; yn) We assume that there is a linear relation between x and y: y = ax + b Can be extended to n dimensions: y = a1x1 + a2x2 + a3x3 + … + b The problem: Given the function f, find values of coefficientsais such that the linear combination best fits the data

  34. ANFIS Training : Least Square Estimation Given data {(x1; y1 (xN ; yN)}, we may define the error associated to saying y = ax + b by: This is just N times the variance of data : {y1 - (ax1+b),…., yn- (axN +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:

  35. 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:

  36. ANFIS Training : Least Square Estimation For the following data find least square estimator

  37. ANFIS Training : Least Square Estimation

  38. ANFIS Training : Gradient descent

More Related