# Introduction of Fuzzy Inference Systems Introduction of Fuzzy Inference Systems By Kuentai Chen Fuzzy Inference Systems Base on Fuzzy set theory Fuzzy If-Then rules Fuzzy Reasoning Fuzzy Inference Systems Also named

Fuzzy-rule-based system Fuzzy Expert system Fuzzy model Fuzzy associative memory Fuzzy logic controller Fuzzy system Fuzzy inference The process of formulating the mapping fr om a given input to an output using fuzzy l ogic. The mapping then provides a basis from w hich decisions can be made, or patterns di scerned. Fuzzy Logic Toolbox uses Mamdani-type a nd Sugeno-type: Vary in the way outputs a re determined. Applications Automatic control

Data classification Decision analysis Expert systems Computer vision Mamdani's fuzzy inference met hod Proposed in 1975 by Ebrahim Mamdani control a steam engine and boiler combination synthesizing a set of linguistic control rules obtained from experienced human operators. Based on Lotfi Zadeh's 1973 paper Fuzzy Logic Toolbox uses a modified version Fuzzy IF-THEN rules Mamdani style If pressure is high then volume is small high small

Sugeno style If speed is medium then resistance = 5*speed medium resistance = 5*speed Fuzzy inference system (FIS) If speed is low then resistance = 2 If speed is medium then resistance = 4*speed If speed is high then resistance = 8*speed MFs low medium high .8 .3 .1 2

Rule 1: w1 = .3; r1 = 2 Rule 2: w2 = .8; r2 = 4*2 Rule 3: w3 = .1; r3 = 8*2 Speed Resistance = (wi*ri) wi*ri) / = 7.12 wi First-order Sugeno FIS Rule base If X is A1 and Y is B1 then Z = p1*x + q1*y + r1 If X is A2 and Y is B2 then Z = p2*x + q2*y + r2 Fuzzy reasoning A1 B1 X

A2 x=3 w1 Y B2 X z1 = p1*x+q1*y+r1 y=2 w2 Y z2 = p2*x+q2*y+r2

z= w1*z1+w2*z2 w1+w2 Fuzzy modeling Unknown target system y xn Fuzzy Inference System y* ... x1 Given desired i/o pairs (wi*ri) training data set) of the form

(wi*ri) x1, ..., xn; y), construct a FIS to match the i/o pairs Two steps in fuzzy modeling structure identification --- input selection, MF numbers parameter identification --- optimal parameters Data Clustering Cluster analysis is a technique for grouping data and finding structures in data. The most common application of clustering methods is to partition a data set into clusters or classes, where similar data are assigned to the same cluster whereas dissimilar data should belong to different clusters. In real applications there is very often no sharp boundary between clusters so that fuzzy clustering is often better suited for the data. Membership degrees between zero and one are used in fuzzy clustering instead of crisp assignments of the data to clusters. Fuzzy clustering can be applied as an unsupervised learning strategy in order to group data Another area of application of fuzzy cluster analysis is image analysis and recognition. Segmentation and the detection of special geometrical shapes like circles and ellipses can be achieved by so-called shell clustering algorithms. Types of Fuzzy Cluster Algorithms

Classical Fuzzy Algorithms (cummulus like clusters) The fuzzy c-means algorithm The Gustafson-Kessel algorithm The Gath-Geva algorithm Linear and Ellipsodial (lines) The fuzzy c-varieties algorithm The adaptive clustering algorithm Shell (circles,ellipses, parabolas) Fuzzy c-shells algorithm Fuzzy c-spherical algorithm Adaptive fuzzy c-shells algorithm Fuzzy c-mean cluster analysis The Fuzzy c-mean algorithm (FCM) recognizes spherical clouds of points in p-dimensional space . Having a finite set of objects and the number of cluster centers c to be calculated, the assignment of the n objects to the c clusters is represented by the proximity matrix . With and , expressing the fuzzy proximity or affiliation of object to cluster center .

The fuzzy c-mean algorithm consists of the following steps: 1. Fix the number c of cluster centers to be calculated and a threshold for the stop condition in step 4. Initialize the proximity matrix . 2. Update the c cluster centers according to the actual proximity matrix . 3. Update to according to the actual cluster centers . 4. Stop the algorithm if is fulfilled, else go on with step 2. ANFIS Fuzzy reasoning B1 A1 A2 B2 w1

w2 z1 = p1*x+q1*y+r1 z= z2 = p2*x+q2*y+r2 w1*z1+w2*z2 w1+w2 y x ANFIS (wi*ri) Adaptive Neuro-Fuzzy Inference System) x y

A1 A2 B1 B2 w1 w1*z1 wi*zi w2*z2 w2 wi

z Four-rule ANFIS Input space partitioning y A2 A1 B2 x B2 B1 B1 y

A1 A2 ANFIS (wi*ri) Adaptive Neuro-Fuzzy Inference System) x y A1 A2 B1 B2

w1 w1*z1 wi*zi w4 w4*z4 wi z x