Transcript Chapter 8 Fuzzy Associative Memories
Chapter 8
Fuzzy Associative Memories
Li Lin 2004-11-24
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 universe of discourse
I n
Fig.1 A fuzzy system
I p
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
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
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 1 Light Medium Heavy 1 Short Medium Long 0 50 100
Traffic density
150
x n
200 0 10 20 30
Green light duration
Fig.3 Three possible fuzzy subsets of traffic-density and green light duration, space X and Y.
y n
40
Fuzzy vector-matrix multiplication: max-min composition Max-min composition
A
M
B
Where,
A
(
a
1 ,...
a n
),
B
(
b
1 ,...
b p
) , M is a fuzzy
b j I
max 1
i
n
min(
a i
,
m i
,
j
)
Fuzzy vector-matrix multiplication: max-min composition Example Suppose A=(.3 .4 .8 1),
B
A
M
.
8 .
4
M
.
5 .
2 .
7 .
8 0 .
8 .
6 .
1 .
2 .
.
.
7 6 5 .
3 Max-product composition
b j
max 1
i
n a i m ij
Fuzzy Hebb FAMs
Classical Hebbian learning law:
ij
m ij
S i
(
x i
)
S j
(
y j
) Correlation minimum coding:
m ij
min(
a i
,
b j
) Example
M
A T
B
a
1
a n
B
B
A T
.
3
M
A
B
.
4 .
8 .
8 1 .
4 .
5 .
3 .
.
.
4 8 8 .
3 .
4 .
4 .
4 .
.
.
.
3 4 5 5
b
1
A T
b m
The bidirectional FAM theorem for correlation-minimum encoding The height and normality of fuzzy set A
H
(
A
) max 1
i
n a i
fuzzy set A is normal, if H(A)=1 Correlation-minimum bidirectional theorem (i) (ii) (iii) (iv)
A A B
B
M M M M T T
A B
B A
iff iff
H H
( (
A
)
B
)
H H
( for any for any
A
B
(
B
)
A
)
The bidirectional FAM theorem for correlation-minimum encoding Proof
A
A T
max 1
i
n a i
A
max 1
i
n a i
H
(
A
) Then So
A
M
A
(
A T
M
) (
A
A T
)
B
H
(
A
)
B
H
(
A
)
B H
(
A
)
B
B iff H
(
A
)
H
(
B
)
Correlation-product encoding Correlation-product encoding provides an alternative fuzzy Hebbian encoding scheme
M M
A T B A T
B
.
3 .
4 .
8 1
and
.
8 .
4
m ij
.
5
a i b
.
12 .
16 .
32 .
4 .
.
15 2 .
.
4 5
Correlation-product encoding Correlation-product bidirectional FAM theorem then (i) (ii) (iii) (iv)
A A B
B
M M M M T T
A B
B A
iff
H
(
A
) iff
H
for any for any (
B
)
A
B
1 1
FAM system architecture A ( (
A
1 ,
B
1 )
A
2 ,
B
2 ) FAM Rule 1
B
1 1 FAM Rule 2
B
2 2 (
A m
,
B m
) FAM Rule m
B
m
m
FAM SYSTEM B
y j
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:
M
max 1
k
m M k or M
M k k
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
M k B
k A
M k
A
(
A T k
B k
)
B k
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.
Recalled outputs and “ defuzzification ” The recalled output B equals a weighted sum of the individual recalled vectors B m
k B
'
k B k
k 1 How to defuzzify?
1. maximum-membership defuzzification
m B
(
y
max ) max 1
j
p m B
(
y j
) 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 B p j 1 y j
m B
(
y j
)
j p
1
m B
(
y j
) The fuzzy centroid is unique and uses all the information in the output distribution B