Chapter 8 Fuzzy Associative Memories Li Lin 2004-11-24.
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Transcript of Chapter 8 Fuzzy Associative Memories Li Lin 2004-11-24.
![Page 1: Chapter 8 Fuzzy Associative Memories Li Lin 2004-11-24.](https://reader034.fdocuments.us/reader034/viewer/2022050704/56649e995503460f94b9bdde/html5/thumbnails/1.jpg)
Chapter 8 Fuzzy Associative Memories
Li Lin2004-11-24
![Page 2: Chapter 8 Fuzzy Associative Memories Li Lin 2004-11-24.](https://reader034.fdocuments.us/reader034/viewer/2022050704/56649e995503460f94b9bdde/html5/thumbnails/2.jpg)
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
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.
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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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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:
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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 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
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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.
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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
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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
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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
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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
iniaAH
1max)(
BMA AMB T BMA AMB T
)()( BHAH )()( AHBH
AB
(i)
(ii)
(iii)
(iv)
iffifffor any
for any
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The bidirectional FAM theorem for correlation-minimum encoding
Proof)(maxmax
11AHaAaAA i
nii
ni
T
Then )( MAAMA T BAA T )(
BAH )(BAH )(
)()()( BHAHiffBBAH 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
jiijT bamandBAM
5.4.8.
4.32.64.
2.16.32.
15.12.24.
5.4.8.
1
8.
4.
3.
BAM T
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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
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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
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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
kkk
mkMMorMM
1max
kkTkk BBAAMA )(
kBkM
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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: 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
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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
p
jjB
jB
ym
ym
1
p
1jj
)(
)(y
B
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Thank you!