INTRODUCTION
In the literature, there are several classification and clustering methods
which use exponential membership functions, patterns or fuzzy architecture.
For example, Feng et al. (2009) proposed a new
training algorithm for Hierarchical Hybrid FuzzyNeural Networks (HHFNN) based
on gaussian membership function. Devillez (2004) introduced
fuzzy pattern matching algorithm with exponential function in fuzzy supervised
classification methods in order to design process monitoring of metal cutting
with highspeed machining. McNicholas (2010) suggested
a novel model based classification technique based on parsimonious Gaussian
mixture models. Yang and Bose (2006) proposed automatic
fuzzy membership generation with unsupervised learning in where the proper cluster
is generated and then the fuzzy membership function is generated according to
cluster. Yang and Wu (2006) offered a Possibilistic
Clustering Algorithm (PCA) which results exponential membership function in
order to be robust to noise and outliers. Chen and Chang
(2005) proposed a new method to construct the membership functions of attributes
and generate weighted fuzzy rules from training instances for handling fuzzy
classification problems without any human experts’ intervention. Agrawal
et al. (2007) presented a supervised neural network classification
model based on roughfuzzy membership function, weak fuzzy similarity relation
and backpropagation algorithm. ElAsir and Mamlook (2002)
classified heart electrical axis by using fuzzy logic architecture.
Besides them, there are also some studies that utilize the data or distribution
of data in the evaluation of membership functions, class labels or classification.
For example, Liu et al. (2008) designed a novel
fuzzy membership function to represent the distribution of image samples in
order to promote the classification performances of canonical correlation analysis.
Mansoori et al. (2007) proposed an approach divides
the covering subspace of each fuzzy rule into two subdivisions based on a α
threshold. The splitting threshold for each rule was found by using distribution
of patterns in the covering subspace of that rule. Teng
et al. (2004) chosen a regionbased exponential functions to construct
fuzzy model and introduced an algorithm that partitions input space into several
characteristic regions by using training data. Choi and
Rhee (2009) suggested three novel interval type2 fuzzy membership function
(IT2 FMF) generation methods which based on heuristics, histograms and interval
type2 fuzzy Cmeans. Wu and Chen (1999) proposed a
fuzzy learning algorithm based on the αcuts of equivalence relations.
Chang and Lilly (2004) suggested an evolutionary approach
where rules and membership functions are automatically created and optimized
in evolutionary process of compact fuzzy classification system. This system
is derived from directly data without any a priori knowledge or assumptions
on the distribution of the data. Au et al. (2006)
presented a method to determine the membership functions of fuzzy sets directly
from data. It maximizes the classattribute interdependence in order to improve
the classification results. Simpson (1992) handled a
supervised learning neural network classifier that utilizes fuzzy sets as pattern
classes where minmax points are used. The minmax points are determined by
using the fuzzy minmax learning algorithm that can learn nonlinear class boundaries
in a single pass through the data.
In the current study, the parameter formulas of exponential membership function based on a minimization problem are presented. In the minimization problem, it is tried to reach an exponential membership function such as it takes form regard to the shape of frequency table, in other words histogram of data.
PRELIMINARIES
A fuzzy number A is a fuzzy subset of the real line R with the membership function
μ_{A} which is normal, fuzzy convex, upper semicontinuous, supp
A is bounded, where supp A = cl {x∈R, μ_{A} (x)>0} and
cl is the closer operator. The LRparametric form of A can be written follows
(Nasibov, 2002):
where, A^{α} is an αlevel set of the fuzzy number, L: [0,1]→(∞, ∞) is monotic nondecreasing left continuous, R: [0,1]→(∞, ∞) is monotic nonincreasing right continuous functions of left and right hand sides of the fuzzy number, respectively.
In the current study, the following definition of parametric exponential fuzzy number is considered:
And, the satisfaction of:
are supposed.
Table 1: 
Frequency table 

EXPONENTIAL MEMBERSHIP FUNCTION GENERATION BASED ON FREQUENCIES
Sometimes the data distribution may be skewed, so the usage of following exponential
approach to data distribution may give more weak results which is used by Nasibov
and Ulutagay (2010):
Considering this, parametric exponential approximation to data distribution is handled in the current study and the following frequency table is utilized to catch the shape of X_{1}, X_{2}, …, X_{N} data. In Table 1, f values show the frequencies of each class where the number of classes is k.
In the evaluation of parametric exponential membership function, the LR parametric form in Eq. 1 is considered. For the left and right side shape of exponential membership function:
equalities are taken into account in objective function (8), respectively. In here, M denotes the midpoint of class interval with maximum percentage:
Before minimization of objective function in Eq. 8, the following prior procedure is applied in order to keep normal condition of fuzzy number at 1 level:
• 
Assign 1 membership level to midpoint of class interval with
maximum percentage 
• 
Normalize other percentage with: 
where, p_{m} is maximum percentage.
After that the unknown parameters of exponential fuzzy number s_{L}, σ, s_{R}, β in Eq. 2 are found by minimization of Eq. 8.
Theorem: To obtain fuzzy parametric exponential number which minimize the objective function in Eq. 8 the parameters of membership function in Eq. 2 must be as following:
Note: In Theorem:
and
conditions must be satisfied.
Proof: To evaluate the unknown parameters of parametric exponential
membership function which has got membership level close to normalized
levels, the objective function in Eq. 10 can be minimized
with respect to unknown s_{L}, σ, s_{R}, β parameters.
With this aim:
equalities must be satisfied.
To find s_{L} and σ, the following ones have to be solved sequentially.
From here:
and
equalities can be received, respectively.
At the end:
and
can be reached. From here, σ can be written as following:
In a similar way, by solving the following equalities sequentially:
the following ones can be obtained in order:
By transforming, β can be written as:
which ends proof.
BISPECTRAL INDEXES’ CLASSIFICATION BASED ON PARAMETRIC EXPONENTIAL MEMBERSHIP FUNCTIONS
The Bispectral Index (BIS) is a parameter derived from the electroencephalograph
(EEG) (Ozgoren et al., 2008). It correlate with
increasing sedation and loss of awareness so its monitoring may help to assess
the hypnotic component of anesthesia, reduce drug consumption and shorten recovery
times (Kreuer et al., 2001; Gan
et al., 1997). It is scaled from 100 (awake patient) to 0 (no cortical
activity).
In the current section, BIS index classification is handled and the same data
used in Nasibov and Ulutagay (2010) are performed, except
one. To observe the efficiency of the proposed parametric exponential membership
function definition in a classification problem, it is wanted to be compared
with Eq. 5. Accordingly, all data sets are merged and the
values of each sedation stages are regrouped for evaluation of fuzzy numbers.
Then, the following membership functions of each sedation stage are calculated
by using Eq. 5 for comparison:
Table 2: 
Frequency table of first sedation stage 

Bold class denotes the class interval with maximum percentage 
Table 3: 
Frequency table of second sedation stage 

Bold class denotes the class interval with maximum percentage 
After that, the one of the possible frequency tables of sedation stages can be generated as given in Table 26 to apply proposed theorem. In these tables, the frequencies and percentages of bispectral index classes for each sedation stages are shown. The highlighted class in each table denotes the class interval with maximum percentage (p_{m}). The midpoints of these classes are M values in parametric exponential fuzzy numbers of sedation stages.
Accordingly, the following parametric exponential membership functions are found by using proposed theorem:
Table 4: 
Frequency table of third sedation stage 

Bold class denotes the class interval with maximum percentage 
Table 5: 
Frequency table of fourth sedation stage 

Bold class denotes the class interval with maximum percentage 
Table 6: 
Frequency table of fifth sedation stage 

Bold class denotes the class interval with maximum percentage 
In the generation of membership function for 5th sedation stage, the constraints in Eq. 12 is not satisfied, so a curve that has 0.5 membership level at 85% percentages of data is fitted.
The shapes of membership functions evaluated by proposed theorem and Eq.
5 can be seen in Fig. 15, where blue
curve lines represent exponential fuzzy numbers evaluated by offered theorem
while black curve lines represent exponential fuzzy numbers evaluated by Eq.
5.
The classification procedures are applied on 21 data sets with maximum level criteria in order to see the usage of proposed parametric exponential fuzzy number in classification problem. Accordingly, the results of Classification Accuracies (CA) of data sets in Table 7 are obtained where classification accuracy denotes the ratio of correct estimated point number in each data set.
The pairedt test is applied in order to test whether the mean of classification accuracies based on membership functions evaluated by offered Theorem is greater than the one based on Eq. 5, or not.
The results shown in Table 8 are obtained by using Minitab program. At the end of the analysis, it is concluded that the mean of classification accuracies of based on membership functions evaluated by offered theorem is greater than the one based on Eq. 5 in bispectral index data sets (α = 0.10).
Table 8: 
The minitab output of pairedt test of correct point numbers 

Paired T for TheoremFormula (5). 95% lower bound for mean
difference: 0,001251, Ttest of mean difference = 0 (vs>0): Tvalue
= 1.67, pvalue = 0.055 
CONCLUSIONS
In the literature, there are researches underlined the fact that the parameters
of membership functions have important role on classification accuracies. Considering
exponential membership functions in the classification problems, in the current
study, the formulas of exponential membership functions are evaluated via a
minimization problem. In the minimization problem, the objective function is
defined regard to percentages of frequency table in order to form a fuzzy number
similar with the histogram of data.
In the last part of the study, the efficiency of membership function is tested in the classification problems of bispectral indexes which are measurements of brain activity. At the end of the analysis, the offered parameter formulas are found useful in the increasing of classification accuracies in the data sets.
ACKNOWLEDGMENT
The datasets used in the experiments were provided by Dr. M. Ozgoren.