ANFIS: ADAPTIVE-NETWORK-BASED FUZZY INFERENCE SYSTEMS (J.S.R. Jang 1993,1995) bell x; a, b, c = 1 a

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1 ANFIS: ADAPTIVE-NETWORK-ASED FUZZ INFERENCE SSTEMS (J.S.R. Jang 993,995) Membership Functions triangular triangle( ; a, a b, c c) ma min = b a, c b, 0, trapezoidal trapezoid( ; a, b, a c, d d) ma min = b a,, d c, 0, Gaussian Generalised ell Sigmoidal MF.0 c gaussian( ; σ, c) = ep σ ( ) bell ; a, b, c = + ( ;,, ) sigmoid abc c a b = + ep [ a ( c) ] 0.5 slope=-b/ 0 c-a c c+a a Fig. Meaning of parameters in generalised bell function

2 Set Operations Containment or Subset, Union (disjunction), Intersection (conjunction), Complement (negation), Probabilistic AND, Probabilistic OR. Fuzzy If-Then Rules Fuzzy implication if is A then y is, where A and are linguistic values defined by fuzzy sets on universes of discourse and, respectively. is A - antecedent, y is - consequence or conclusion. Interpretation of the implication operator (fuzzy relation R). Material implication: Propositional calculus: R = A = A. R = A = A ( A ). Etended propositional calculus: R = A = ( A ). Generalisation of modus ponens: ~ y, = sup c * c y and 0 c { } ( ) ( ) ( ) R A

3 Fuzzy Reasoning - Approimate reasoning Compositional rule of inference Suppose that we have a curve y f( ) we can infer that y = b= f( ) a and b are real numbers (Fig..a)) a and b are intervals (Fig..b)) = and for a given =a y=f( y=f() y=b b =a a a) b) Fig.. Derivation of y=b a) a and b are points b) a and b are intervals a and b are fuzzy sets.

4 Algorithm Interval reasoning construct a cylindrical etension of a, find its intersection I with the interval valued curve, make a projection of I onto the y-ais what yields the interval b. Fuzzy reasoning A is a fuzzy set of and F is a fuzzy relation on (Fig. 3.a) and b)), construct a cylindrical etension c(a) with base of A, =, ( )( y) A( ) c A find the intersection of c(a) and F (Fig. 3.c)) [ ] ( ) ( y, ) = min ( )( y, ), ( y, ) = min [ A( y, ), F( y, )] c A F c A F make a projection of the intersection ca ( ) () () ( ) Fonto, [ ] [ A() F( )] y = ma min,, y =, y. A F This formula is refereed to as ma-min composition and is represented as = AoF where o denotes the composition operator.

5 Modus Ponens Classical logic premise (fact): is A, premise (rule): if is A then y is, consequence (conclusion): y is Fuzzy logic - generalised modus ponens premise (fact): is A, premise (rule): if is A then y is, consequence (conclusion): y is Fuzzy reasoning or ( ) = A or = A o A [ ] [ A () R( )] () () ( ) y = ma min,, y =, y A R Single rule with single antecedent { [ ]} () () () () ( ) y =, y y = w y. A R A A' ' Fig. 4. Fuzzy reasoning for a single rule and a single antecedent

6 Single rule with two antecedents premise (fact): is A and y is premise (rule): if is A and y is then z is C, consequence (conclusion): y is C The fuzzy rule in premise A C ( yz,, ) = ( ) ( yz,, ) = ( ) ( y) ( z) R A C A C The resulting C ( ) o( ) C = A A C { [, ]} () () () () () C z = y y y C z = w w C z 4 43 firing strength min A A' ' w C C' w Z Fig. 5. Fuzzy reasoning for a single rule and multiple antecedents

7 Multiple rules with multiple antecedents premise (fact): is A and y is premise (rule ): if is A and y is then z is C, premise 3 (rule ): if is A and y is then z is C, consequence (conclusion): y is C The resulting C { } ( ) o ( ) ( ) ( A ) o( R R) {( A ) or} {( A ) or} C = A A C A C = = = C C Sugeno fuzzy model min or product A w z p q y r = + + A w z = p + q y+ r weighted average y z = wz wz w + + w Fig. 6. The Sugeno fuzzy model

8 Tsukamoto fuzzy model min or product A C w z Z A C w y z Z weighted average z = wz + wz w + w Fig. 7. The Tsukamoto fuzzy model Partition styles for fuzzy models Grid partition - often chosen in designing a fuzzy controller, problems when we have moderately large number of inputs. Tree partition - relives the problem of an eponential increase in the number of rules. Scatter partition. a) b) c) Fig. 6. Methods for partitioning: a) grid; b) tree; c) scatter.

9 ADAPTIVE NETWORKS ack-propagation neural network. Radial basis function network. Adaptive network Overall input-output behaviour is determined by the values of a collection of modifiable parameters. Each node is a process unit that performs a static node function on its incoming signals and generate a single node output. Each link specifies the direction of signal flow from one node to another. Usually a node function is a parametrized function with modifiable parameters; by changing this parameters, we are changing the node function. In most general case, an adaptive network is heterogeneous and each node may have a different node function. A node parameter set can be non-empty - adaptive node or empty - fied node input layer layer l (output laye adaptive node fied node Fig. 7. A feedforward network in layered representation.

10 Classification of adaptive networks feedforward - acyclic. recurrent - if there is a feedback link that forms a circular path in the network. Topological ordering representation of feedforward networks a special case of the layered representation, with one node per layer adaptive node fied node Fig. 8. A feedforward adaptive network in topological ordering representation. Constructing the network training data set - a number of desired input-output pairs for a target system learning rule or learning algorithm - a procedure to follow in order to adjust the parameters to improve the performance of the network error measure - discrepancy between the desired output and the network s output under the same input conditions.

11 Eamples of adaptive networks An adaptive network with a single linear node (, ;,, ) = f a a a = a + a + a f 3 3 Fig. 9. A linear single node adaptive network. An adaptive network with a single non-linear node - perceptron (, ;,, ) = f a a a = a + a + a ( ) = f = if if 3 < 0 f f 4 Fig. 0. A non-linear single node adaptive network. The sigmoid function 4 = f4( 3) =. + ep ( ) 3

12 HRID LEARNING RULE Hybrid learning rule combines the gradient method with the least-squares estimator. Assume that the adaptive network has only one output output = F( I, S) where I is the vector of input variables and S is the set of parameters. Assume that there eists a function H such that the composite function Ho S is linear in some of the elements of S, then these elements can be identified by the least-squares method such that H S = S S H( output) = H of( I, S), o S is linear in the elements of S Now given values of elements S, we can plug P training data into the above equation and obtain Aθ= where θ is an unknown vector whose elements are parameters in S. The above equation can be solved used the least squares method.

13 Off-line learning (batch learning) Each epoch is composed of a forward pass an a backward pass In the forward pass, after an input vector is presented, we calculate the node outputs in the network layer by layer until entries of the matrices A and are obtained. Then parameters of S are identified by the pseudoinverse approach. Net we can compute the error measure for each training data entry. In the backward pass, the error signals propagate from the output end toward the input end. Then the parameters in S are updated by a gradient method. On-line learning (pattern learning) If parameters are updated after each data presentation, we have an on-line learning or pattern learning scheme. The gradient descent should be based on the energy function for a particular pattern. Different ways of combining GD and LSE. One pass of LSE only; Nonlinear parameters are fied while linear parameters are identified by one-time application of LSE.. GD only; All parameters are updated by GD iteratively. 3. One pass of LSE followed by GD; LSE is employed only once at the very beginning to obtain the initial values of linear parameters and then GD takes over to update all parameters iteratively. 4. GD and LSE - hybrid learning rule. 5. Sequential (approimate) LSE only; The outputs of adaptive network are linearized with respect to its parameter, and then etended Kalman filter algorithm is employed to update all parameters.