Chapter 8 Fuzzy Associative Memories Li Lin

CONTENTS Review Fuzzy Systems as between-cube mapping Fuzzy and Neural Function Estimators Fuzzy Hebb FAMs Adaptive FAMs

Review In Chapter 2, we have mentioned BAM theorem Chapter 7 discussed fuzzy sets as points in the unit hypercube What is associative memories?

Fuzzy systems Koskos: fuzzy systems as between-cube mapping Fig.1 A fuzzy system Output universe of discourse Input universe of discourse The continuous fuzzy system behave as associative memories, or fuzzy associative memories.

Fuzzy and neural function estimators Fuzzy and neural systems estimates sampled function and behave as associative memories Similarities: 1. They are model-free estimator 2. Learn from samples 3. Numerical, unlike AI Differences: They differ in how to estimate the sampled function 1. During the system construction 2. The kind of samples used

Fig.2 Function f maps domains X to range Y 3. Application 4. How they represent and store those samples 5. How they associatively inference Differences:

Neural vs. fuzzy representation of structured knowledge Neural network problems: 1. computational burden of training 2. system inscrutability There is no natural inferential audit tail, like an computational black box. 3. sample generation

Neural vs. fuzzy representation of structured knowledge Fuzzy systems 1. directly encode the linguistic sample (HEAVY,LONGER) in a matrix 2. combine the numerical approaches with the symbolic one Fuzzy approach does not abandon neural-network, it limits them to unstructured parameter and state estimate, pattern recognition and cluster formation.

FAMs as mapping Fuzzy associative memories are transformations FAM map fuzzy sets to fuzzy sets, units cube to units cube. Access the associative matrices in parallel and store them separately Numerical point inputs permit this simplification binary input-out FAMs, or BIOFAMs

FAMs as mapping Fig.3 Three possible fuzzy subsets of traffic-density and green light duration, space X and Y.

Fuzzy vector-matrix multiplication: max-min composition Max-min composition “ ” Where,, M is a fuzzy n-by-p matrix (a point in )

Fuzzy vector-matrix multiplication: max-min composition Example Suppose A=( ), Max-product composition

Fuzzy Hebb FAMs Classical Hebbian learning law: Correlation minimum coding: Example

The bidirectional FAM theorem for correlation-minimum encoding The height and normality of fuzzy set A fuzzy set A is normal, if H(A)=1 Correlation-minimum bidirectional theorem (i) (ii) (iii) (iv) iff for any

The bidirectional FAM theorem for correlation-minimum encoding Proof Then So

Correlation-product encoding Correlation-product encoding provides an alternative fuzzy Hebbian encoding scheme Example Correlation-product encoding preserves more information than correlation-minimum

Correlation-product encoding Correlation-product bidirectional FAM theorem if and A and B are nonnull fit vector then (i) (ii) (iii) (iv) iff for any

FAM system architecture FAM Rule m FAM Rule 1 FAM SYSTEM FAM Rule 2 A B Defuzzifier

Superimposing FAM rules Suppose there are m FAM rules or associations The natural neural-network maximum or add the m associative matrices in a single matrix M: This superimposition scheme fails for fuzzy Hebbian encoding The fuzzy approach to the superimposition problem additively superimposes the m recalled vectors instead of the fuzzy Hebb matrices

Superimposing FAM rules Disadvantages: Separate storage of FAM associations consumes space Advantages: 1 provides an “ audit trail ” of the FAM inference procedure 2 avoids crosstalk 3 provides knowledge-base modularity 4 a fit-vector input A activates all the FAM rules in parallel but to different degrees. Back

Recalled outputs and “ defuzzification ” The recalled output B equals a weighted sum of the individual recalled vectors How to defuzzify? 1. maximum-membership defuzzification simple, but has two fundamental problems: ① the mode of the B distribution is not unique ② ignores the information in the waveform B

Recalled outputs and “ defuzzification ” 2. Fuzzy centroid defuzzification The fuzzy centroid is unique and uses all the information in the output distribution B

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