Introduction of Fuzzy Inference Systems By Kuentai Chen

Fuzzy Inference Systems Base on Base on –Fuzzy set theory –Fuzzy If-Then rules –Fuzzy Reasoning

Fuzzy Inference Systems Also named 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 from a given input to an output using fuzzy logic. The process of formulating the mapping from a given input to an output using fuzzy logic. The mapping then provides a basis from which decisions can be made, or patterns discerned. The mapping then provides a basis from which decisions can be made, or patterns discerned. Fuzzy Logic Toolbox uses Mamdani-type and Sugeno-type: Vary in the way outputs are determined. Fuzzy Logic Toolbox uses Mamdani-type and Sugeno-type: Vary in the way outputs are determined.

Applications Automatic control Automatic control Data classification Data classification Decision analysis Decision analysis Expert systems Expert systems Computer vision Computer vision

Mamdani's fuzzy inference method Proposed in 1975 by Ebrahim Mamdani 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 Based on Lotfi Zadeh's 1973 paper Fuzzy Logic Toolbox uses a modified version Fuzzy Logic Toolbox uses a modified version

Fuzzy IF-THEN rules high small medium resistance = 5*speed Mamdani style If pressure is high then volume is small Sugeno style If speed is medium then 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 Rule 1: w1 =.3; r1 = 2 Rule 2: w2 =.8; r2 = 4*2 Rule 3: w3 =.1; r3 = 8*2 Speed lowmedium high Resistance =  (wi*ri) /  wi = 7.12 MFs

First-order Sugeno FIS Rule base If X is A 1 and Y is B 1 then Z = p 1 *x + q 1 *y + r 1 If X is A 2 and Y is B 2 then Z = p 2 *x + q 2 *y + r 2 Fuzzy reasoning A1A1 B1B1 A2A2 B2B2 x=3 X X Y Y y=2 w1 w2 z 1 = p 1 *x+q 1 *y+r 1 z = z 2 = p 2 *x+q 2 *y+r 2 w 1 +w 2 w 1 *z 1 +w 2 *z 2 

Fuzzy modeling Given desired i/o pairs (training data set) of the form (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 Unknown target system Fuzzy Inference System y y* x1 xn...

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

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. Fuzzy c-mean cluster analysis 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.4 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 2.2

ANFIS Fuzzy reasoning A1A1 B1B1 A2A2 B2B2 w1 w2 z 1 = p 1 *x+q 1 *y+r 1 z 2 = p 2 *x+q 2 *y+r 2 z = w 1 +w 2 w 1 *z 1 +w 2 *z 2 x y ANFIS (Adaptive Neuro-Fuzzy Inference System) A1A1 A2A2 B1B1 B2B2      x y w1w1 w2w2 w 1 *z 1 w 2 *z 2  w i *z i wiwi z

Four-rule ANFIS ANFIS (Adaptive Neuro-Fuzzy Inference System) A1A1 A2A2 B1B1 B2B2    x y w1w1 w4w4 w 1 *z 1 w 4 *z 4  w i *z i wiwi z     Input space partitioning A1A1 B1B1 A2A2 B2B2 x yx y A1A1 A2A2 B1B1 B2B2