Fuzzy Min-Max Neural Networks

8/10/2019 Fuzzy Min-Max Neural Networks

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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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where

fis a two parameter ramp threshold function and is

sensitivity parameters regulating how fast the membevalues decrease.


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One-dimensional (1-D) membership function

where


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The 1-D illustration of membership value finding for an input in fo

of lower and upper bounds.


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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

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Fuzzy Min-Max Neural Networks