fuzzy min-max neural networks
TRANSCRIPT
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GENERAL FUZZY MIN-MANEURAL NETWORK
In the name of God
Presented by: Habib Alizadeh
Adviser: Dr. Farokhi
December 14
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FUZZY SETS & NEUROFUZZY SYSTEM
Fuzzysets have been proposed by Zadeh. Compared to the c
sets, fuzzy sets and their operations are more compatible with
worldsystems and are highly efficient in pattern recognitiona
machine learning problems.
Fuzzy logic usually is combined with a learning instrument.
Neurofuzzysystems are created by combining fuzzy logic andneural networks. Computational efficiency of neural networks
capability of fuzzy logic to present complex class boundaries m
these networks a perfect tool for pattern recognition.
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FUZZY MIN-MAX NEURAL NETWORK
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GFMM
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FMM & GFMM
The fuzzy min-max (FMM) clustering and classification neural
networks, with their representation of classesas hyper boxes
dimensionalpattern space and their conceptually simple but p
learning process, provided a natural basis for our developmen
The proposed generalized fuzzy min-max (GFMM) neural netw
incorporates significant modificationsthat improve the effectiv
the original algorithms.
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FMM & GFMM
An important development of the GFMM algorithm relate
interpretation of the membership values, both during the trai
the operationof the GFMM neural network.
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THE ORIGINAL FMM ALGORITHMS
The FMM neural networks are built using hyperbox fuzzy sets
A hyperbox defines a region of the n-dimensional pattern spac
all patterns contained within the hyperbox have full cluster/cla
membership.
A hyperbox is completely defined by its min point and its max
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FMMNN: 1-CLASSIFICATION (HYPERBOMEMBERSHIP FUNCTION)
The example of membership function bjpresented in FMM
Classification NN for the hyperbox defined by min point V=[0.2 0.2] and max point W= [0.3 0.4]: Sensitivity parameter =
4
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FMMNN: 2-CLUSTRING (HYPERBOXMEMBERSHIP FUNCTION)
The example of membership function bj
presented in FMM clustering NN for the
hyperbox defined by min point V = [0.2 0.2]
and max point W = [0.3 0.4]: Sensitivity
parameter = 4
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FMMNNLEARNING
The fuzzy min-max neural network learning algorithm is a fou
process consisting of:
1. Initialization
2. Expansion
3. Overlap Test4. Contraction
The last three steps repeated for each training input pattern.
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GFMM ALGORITHM
A- Basic Def in i t ions
1)Input:The input is specified as the ordered pair: ,
Where =
is the hth input pattern in a form of lower,
upper,
, limits vectors contained within the n-dimensionalun.
And 0,1, , is the index of one of the + 1classes, wher
means that the input vector is unlabeled.
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GFMM ALGORITHM (MEMBERSHIPFUNCTION)
2) Fuzzy Hyperbox Membership Function:
Where is the min point for thejth hyperbox
is the max point for thejth hyperbox, and
membership function for thejth hyperbox is
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GFMM ALGORITHM (MEMBERSHIPFUNCTION)
where
fis a two parameter ramp threshold function and is
sensitivity parameters regulating how fast the membevalues decrease.
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GFMM ALGORITHM (MEMBERSHIPFUNCTION)
One-dimensional (1-D) membership function
where
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GFMM ALGORITHM (MEMBERSHIPFUNCTION)
The 1-D illustration of membership value finding for an input in fo
of lower and upper bounds.
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GFMM ALGORITHM (MEMBERSHIPFUNCTION)
Two-dimensional (2-D) membership function
The hyperbox is defined by min point V= [0.2 0.2] and max point W= [0.3 0
Sensitivity parameter = 4
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GFMM ALGORITHM (LEARNING)
B - GFMM Learning Algorithm:
1) Initialization: The hyperbox is adjusted for the first
using the input pattern =
the min and max
of this hyperbox would be Vj=
and Wj=
.
2) Hyperbox Expansion: When the hth input patternX
presented, the hyperbox Bjwith the highest degree of
membership and allowing expansion (if needed) is fou
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GFMM ALGORITHM(EXPANSION)
The expansion criterion, consists of the following two parts:
and
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GFMM ALGORITHM (EXPANSION)
with the adjustBjoperation defined as:
The parameter is a user-defined value that impbound on the maximum size of a hyperbox and its value signi
affects the effectiveness of the training algorithm.
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GFMM ALGORITHM
3) Hyperbox Overlap Test:Assuming that hyperbox Bjwas ex
in the previous step, test for overlapping with Bkif
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GFMM ALGORITHM(OVERLAP TEST)
The four cases are being considered(where initially = 1).
If overlap for the th dimension has
been detected (one of the above four
cases is valid) and ,
then ,
.
If overlap for the ithdimension has not
been detected, set signifying
that the contraction step is not
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GFMM ALGORITHM(CONTRACTION)
4) Hyperbox Contraction: If
then only the th dimensions of
the two hyperboxes are
adjusted.
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AN EXAMPLE ILLUSTRATING THELEARNING ALGORITHM
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GFMM ALGORITHM ()
5) An Adaptive Maximum Size of the Hyperbox:
In the original FMM NNs the user defined parameter .
To find the best value of this parameter the network has to b
trained for several different s and verified by checking the num
misclassifications.
A large value of can cause too many misclassifications.
When is small, many unnecessary hyperboxes may be created.
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GFMM ALGORITHM ()
The training is completed when:
a) after presentation of all training patterns there have been no
misclassifications for the training data;
b) or the minimum user-specified value of the parameter has been
where is the coefficient responsible for the speed of decrease of
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GFMM ALGORITHM ()
The result of NN training for the 42 input pattern data set (thre
classes).
was constant during training.
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TOPOLOGY OF THE NETWORK
THE EXAMPLE OF
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THE EXAMPLE OFCLUSTERING/CLASSIFICATION OFLABELED AND UNLABELED FUZZY INP
PATTERNS The data set consists of 26patterns from which 15 are
labeled as belonging to one of
four classes and the remaining
11 are unlabeled.
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FMM & GFMM(3 REAL DATA SET
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COMPARISON OF THE PERFORMANCE OF TGFMMWITH SEVERAL OTHER TRADITIONALCLASSIFIERS
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Thanks from your attentio