Chapter%208%20%20Fuzzy%20Associative%20Memories

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Chapter 8 Fuzzy Associative Memories

• Li Lin
• 2004-11-24

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CONTENTS

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

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Review

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

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Fuzzy systems
Output universe of discourse
Input universe of discourse

• Koskos fuzzy systems as between-cube
• mapping

Fig.1 A fuzzy system
The continuous fuzzy system behave as associative
memories, or fuzzy associative memories.
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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

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

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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.

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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

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FAMs as mapping
Fig.3 Three possible fuzzy subsets of
traffic-density and green light duration, space X
and Y.
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Fuzzy vector-matrix multiplication max-min
composition

• Max-min composition

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

• Example
• Suppose A(.3 .4 .8 1),
• Max-product composition

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Fuzzy Hebb FAMs

• Classical Hebbian learning law
• Correlation minimum coding
• Example

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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)
iff
(ii)
iff
(iii)
for any
(iv)
for any
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The bidirectional FAM theorem for
correlation-minimum encoding

• Proof

Then
So
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Correlation-product encoding

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

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Correlation-product encoding

• Correlation-product bidirectional FAM theorem
• if and A and B are nonnull fit
  vector
• then

(i)
iff
(ii)
iff
(iii)
for any
(iv)
for any
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FAM system architecture
FAM Rule 1
FAM Rule 2
Defuzzifier
B
A
FAM Rule m
FAM SYSTEM
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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

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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
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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
• ? the mode of the B distribution is not
  unique
• ? ignores the information in the waveform B

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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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Thank you!