A Method for Membership Function Generation from Training Samples
Yvonne C. Lucero and Patricia A. Nava, Ph.D.
Department of Electrical and Computer Engineering
University of Texas at El Paso
El Paso, TX 79968
Voice: (915) 747-5994
Fax: (915) 747-7871
Email: ylucero@utep.edu
Email: pnava@ece.utep.edu
ABSTRACT:In the early 1980’s, Japan became
the first to implement fuzzy logic in various
commercial products. Since then, fuzzy systems
have been touted as the solution to complex problems because there is no need to develop a mathematical model for the problem to arrive at a solution. The solution consists of constructing rules
that apply when input values lie within certain
designer-defined categories. The major disadvantage of this approach lies in defining the categories, called membership functions, for each
input parameter – i.e. use of this method requires
some form of expert knowledge in order to define
these membership functions. The efficiency or
accuracy of the fuzzy system is then proportional
to the designer’s expertise in the application area.
In cases where expert knowledge is unavailable,
defining the membership functions is a major
hurdle. Several methods have been proposed for
automatic generation of membership functions.
This study describes the implementation of one
such method, and examines its application to the
identification of vowel sounds in spoken English.
1. INTRODUCTION
Fuzzy logic and fuzzy sets are effective tools
for modeling complex mathematical problems
with parameters that demonstrate uncertainty [1].
Many current commercial applications, such as
vacuum cleaners and cameras, benefit from using
fuzzy logic. Extensive research has led to the creation of fuzzy logic software tools used to model
and create solutions to many types of problems.
The main obstacle faced by most software tools is
the problem of effectively mapping specific, numerical data into linguistic variables and their
associated membership functions.
Linguistic variables are used for system input
and output, and are represented by words such as
“size,” “age,” and “temperature”. These words
are ambiguous or uncertain. A fuzzy set is creat-
ed to describe the linguistic variables in more
detail. The linguistic variable “age,” for instance,
may have overlapping categories (members) of
“young,” “very young,” “middle age,” “old,” and
“very old.” Once these categories, or members
are defined, the fuzzy set is obtained, and a membership function is then developed for each member in the set.
Historically, developing the number of members in a set and their associated membership
functions was a task that required expert
knowledge of the application area [2]. A recent
methodology was developed to automatically
generate membership functions by Hong, et al.
[3]. This new methodology was applied to a set
of data used for a speaker-independent voice
recognition application. The following section
gives an overview of fuzzy theory; section 3 describes the algorithm in detail, implementation
and results are discussed in section 4, and conclusion in section 5.
2. FUZZY SETS
Fuzzy logic and fuzzy sets were first introduced by Lotfi A. Zadeh. Fuzzy sets were derived from generalizing the concept of set theory.
Fuzzy sets can be thought of as an extension of
classical sets [4]. In a classical set or crisp set,
the objects in a set are called elements or members of the set. An element x belonging to a set A
is defined as x  A, an element that is not a member in A is noted as x  A. A characteristic function or membership function A(x) is defined as
an element in the universe U having a crisp value
of 1 or 0. For every x  U,
1 for x  A,
0 for x  A.
 A x   
this can also be expressed as  A x   0,1 .
(1)
For the classical set or crisp set, membership
functions take a value of 1or 0, for fuzzy sets, the
membership function can take values in the interval [0,1]. The range between 0 and 1 is referred
to as the membership grade or degree of membership [5]. A fuzzy set A is defined below:
A  x,  A x  | x  A,  A x   0,1
(2)
Where A(x) is a membership function belonging
to the interval [0,1]. Two operations that are
widely used in set theory are the union and intersection. The union and intersection of A and B
can be defined as:
and
 AB  max  A ,  B  ,
(3)
 AB  min  A ,  B  .
(4)
These two operations are used in the algorithm
for deriving the membership functions in the following section.
3.0 GENERATING MEMBERSHIP FUNCTIONS
The method proposed by Hong, et al. [3] automatically generates membership functions. This
method partitions a set of data into classes that
can be used to derive membership functions. The
procedure of the algorithm has several major
steps and consists of clustering the data into classes. The membership functions are then generated from the classes obtained. The algorithm is
described for one parameter, and is detailed below.
Step 1. Given a data set, there are n training
samples. The values for the parameter in question,

X  x1 , x2 ,, xn , are sorted into ascending or

der, denoted as Y  sort[X ]  y1, y2 ,, y n . The
values are sorted to find an association between
adjacent values.
Step 2. The difference between adjacent values
in the sorted data is determined. The difference
obtained will provide a way to calculate the similarity between adjacent values. The difference for
a set of training set data is:
diff i  yi 1  yi , for i=(1,2,3,…n-1),
(5)
where yi and yi+1 are adjacent values in the sorted
data.
Step 3. Find the similarities between adjacent
values. The formula below finds the similarities
between adjacent values and maps them into real
numbers between 0 and 1.
diff i


for diff i  C *  s ,
1 
si   C *  s

 0

otherwise,
(6)
where
diffi is the difference between adjacent data,
s is the standard deviation of diffi, and
C is the control parameter.
The control parameter is used to determine the
shape of the membership function.
Step 4. The data is grouped according to similarities. A threshold value, , divides adjacent values
into classes. The number of classes determines
the number of membership functions. Determination of the number of classes can be summarized
by a rule: If the similarity is greater than the
threshold value, then the two adjacent values belong to the same class, otherwise the values are
divided into different classes. That is, a new class
is created. Expressed as a formula,
IF ( s i   ) THEN y i , y i 1  C i ,
ELSE y i  C i , y i 1  C i 1
(7)
where Ci and Ci+1 denote two distinct classes for
the same input or output parameter.
Step 5. The membership function for each class
is defined. There are several types of membership functions such as triangular, trapezoidal, and
gaussian, to name a few. One of the simplest
membership functions is the triangular membership function, and will be used for the remainder
of the equations. The triangular membership
function for class j consists of three points: the
central vertex point, bj, and the two endpoints, aj
and cj. The central vertex point is determined for
each class and is calculated by the given formula:
bj 
si  si 1
s
 s k 1
   y k 1 * k 2
 y k * s k 1
2
2
s  si 1
s
 s k 1
si  i
   k 2
 s k 1
2
2
yi * si  yi 1 *
(7)
where
j represents the jth class,
k represents the ending data index for this class,
i.e., data yi through yk fall into class j, and
si is the similarity between yi and yi+1.
The endpoints of the membership function, aj and
cj, are obtained using interpolation. The following
equations determine the left and right endpoints.
a j  bj 
cj  bj 
b j  yi
(8)
1   j ( yi )
yk  b j
(9)
1   j ( yk )
Where aj is the left endpoint and cj is the right
endpoint. j(yi) and j(yk) is the membership determined by the formula:
 j  yi    j  yk   minsi , si 1 ,, sk 1  (10)
where k represents the maximum data index value
within the class.
The membership functions for the classes determined by this method are shown below:
Fuzzy
Values
1
······
0
ai
ci
ak
ck
4. IMPLEMENTATION
The algorithm described in the previous section was implemented in C, using equations (5)
through (10). The C program is currently being
tested using the Vowel data set, described below.
Data
The application problem that will serve as a
testbench for this study is Vowel, one of a collection of data sets used as a neural network benchmarks [6]. It is used for speaker-independent
speech recognition of the eleven vowel sounds
from multiple speakers. The Vowel data set used
in this study was originally collected by Deterding
[7], who recorded examples of the eleven steady
state vowels of English spoken by fifteen speakers
for a “non-connectionist” [7] speaker normalization study. Four male and four female speakers
were used to create the training data, and the other
four male and three female speakers were used to
create the testing data. The actual composition of
the data set consists of 10 inputs, which are obtained by sampling, filtering and carrying out linear predictive analysis.
Specifically, the speech signals were low pass
filtered at 4.7 kHz and digitized to 12 bits with a
10 kHz sampling rate. Twelfth order linear predictive analysis was carried out on six 512 sample
Hamming windowed segments from the steady
part of the vowel. The reflection coefficients
were used to calculate 10 log area parameters,
giving a 10 dimensional input space [7]. Each
speaker thus yielded six frames of speech from
eleven vowels. This results in 528 frames from
the eight speakers used for the training set, and
462 frames from the seven speakers used to create
the testing set.
This is a small database, but the values are
diverse and thus Vowel is an excellent testbench
for the module that produces these membership
functions.
The membership functions for each input parameter in the database are obtained using the
code developed to implement the algorithm. Current work is being done in finding a systematic
way of determining the threshold and control parameters. The similarities obtained in section 3
will be used as the starting point in determining
the above parameters.
5. CONCLUSIONS
Fuzzy logic has been used in many applications [8]. One study showed that fuzzy techniques
incorporated into a neural network operation improved its performance [9]. In speakerindependent voice recognition, studies have
shown that there is an improvement in performance compared to a neural network alone [10].
The study being carried out, and presented in this
paper, intends to provide a tool to improve and
potentially automate the selection of membership
functions.
With a module that automatically creates membership functions for a system’s input parameters,
neuro-fuzzy systems that utilize fuzzy sets will be
much more efficient. The author’s current neurofuzzy system requires expert knowledge for its
membership functions [11]. By incorporating the
results of this study, the applications for this system abound.
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