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Chapter 8 Fuzzy Associative Memories
Li Lin2004-11-24
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
nI pIFig.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 transforma
tions FAM map fuzzy sets to fuzzy sets, units cube to units cube. Access the associative matrices in parallel a
nd store them separately Numerical point inputs permit this simplification binary input-out FAMs, or BIOFAMs
FAMs as mapping
200nx1 5 01 0 05 00
1L ig h t M ed iu m Heav y
Tra f f ic de n s ity
40ny3 02 01 00
1M ed iu mS h o r t L o n g
G re e n lig h t du ra t io n
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 “ ”
BMA
),...(),,...( 11 pn bbBaaA Where, , M is a fuzzy
n-by-p matrix (a point in )pnI
),min(max ,1
jiini
j mab
Fuzzy vector-matrix multiplication: max-min composition
ExampleSuppose A=(.3 .4 .8 1),
Max-product composition
3.2.0
5.1.8.
6.6.7.
7.8.2.
M
5.4.8. MAB
ijniij mab
1
max
Fuzzy Hebb FAMs Classical Hebbian learning law:
Correlation minimum coding:
Example
),min( jiij bam mTT
n
T bAbA
Ba
Ba
BAM
1
1
5.4.8.
5.4.8.
4.4.4.
3.3.3.
5.4.8.
1
8.
4.
3.
BAM
)()( jjiiijij ySxSmm
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
iniaAH
1max)(
BMA AMB T BMA AMB T
)()( BHAH )()( AHBH
AB
(i)
(ii)
(iii)
(iv)
iffifffor any
for any
The bidirectional FAM theorem for correlation-minimum encoding
Proof)(maxmax
11AHaAaAA i
nii
ni
T
Then )( MAAMA T BAA T )(
BAH )(BAH )(
)()()( BHAHiffBBAH So
Correlation-product encoding
Correlation-product encoding provides an alternative fuzzy Hebbian encoding scheme
Example
Correlation-product encoding preserves more information than correlation-minimum
jiijT bamandBAM
5.4.8.
4.32.64.
2.16.32.
15.12.24.
5.4.8.
1
8.
4.
3.
BAM T
Correlation-product encoding
Correlation-product bidirectional FAM theorem
if and A and B are nonnull fit vector then
BAM T
BMA AMB T BMA AMB T
1)( BH1)( AH
AB
(i)
(ii)
(iii)
(iv)
iffifffor any
for any
FAM system architecture
jy
FAM Rule m
FAM Rule 1
FAM SYSTEM
),( 11 BA
),( 22 BAFAM Rule 2
),( mm BA
1B
2B
mB
1
2
m
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
kkk
mkMMorMM
1max
kkTkk BBAAMA )(
kBkM
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
kBkkB'B
m
1k
)(max)(1
max jBpj
B ymym
Recalled outputs and “defuzzification”
2. Fuzzy centroid defuzzification
The fuzzy centroid is unique and uses all the information in the output distribution B
p
jjB
jB
ym
ym
1
p
1jj
)(
)(y
B
Thank you!