Fuzzy relational calculus approach to multidimensional pattern classification

Pattern Recognition 32 (1999) 973 — 995

Fuzzy relational calculus approach to multidimensional pattern classification Tapan Kr. Dindaa, Kumar S. Ray
*, Mihir Kr. Chakrabortya Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Calcutta 700 019, India
Electronics and Communication Sciences Unit, Indian Statistical Institute, 203, Barrackpore Trunk Road, Calcutta 700 035, India Received 24 November 1997; received in revised form; accepted 24 August 1998

Abstract Our aim is to design a pattern classifier using fuzzy relational calculus (FRC) which was initially proposed by W. Pedrycz. In the course of doing this we introduce a new interpretation of multidimensional fuzzy implication (MFI) to represent our knowledge about the training data set. The new interpretation is basically a set of one-dimensional fuzzy implications. The consequences of all one-dimensional fuzzy implications are finally collected through one intersection operator ‘5’. Subsequently, we consider the notion of a fuzzy pattern vector, which is formed by the cylindrical extension of the antecedent part of each one-dimensional fuzzy implication. Thus, we get a set of fuzzy pattern vectors for the new interpretation of MFI and represent the population of training patterns in the pattern space. We also introduce a new approach to the computation of the derivative of the fuzzy max and min functions using the concept of a generalized function. During the construction of the classifier, based on FRC, we use fuzzy linguistic statements (or fuzzy membership function to represent the linguistic statement) to represent the ranges of features (e.g. feature F is small/medium/big, etc. G ∀i) for a population of patterns. Note that the construction of the classifier essential depends on the estimate of a fuzzy relation R between the antecedent part and consequent part of each one-dimensional fuzzy implication. Thus, a set of G fuzzy relations is formed from the new interpretation of MFI. This set of fuzzy relations is termed as a core of the classifier. Once the classifier is constructed the non-fuzzy features of a pattern can be classified. At the time of classification of the test patterns, we use the concept of fuzzy singleton to fuzzify the non-fuzzy feature values of the test patterns. The performance of the proposed scheme is tested on synthetic data. Finally, we use the proposed scheme for the vowel classification problem of Indian languages.  1999 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. Keywords: Multidimensional fuzzy implication (MFI); Fuzzy pattern vector (FPV); Generalized function; Fuzzy relational calculus (FRC); Multidimensional pattern classifier

1. Introduction In real-world pattern classification problems, we face with fuzziness that is connected with diverse facets of *Corresponding author. Fax: 0091 33 556 6680; E-mail: [email protected]

cognitive activity of human being. The sources of fuzziness are related to labels expressed in pattern space as well as to labels of classes taken into account in classification procedures. Although lot of scientific developments have already been made in the area of pattern classification, still the existing techniques of pattern classification are inferior to the human classification processes which perform extremely complex task in classification. Hence,

0031-3203/99/$ — See front matter  1999 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 1 - 3 2 0 3 ( 9 8 ) 0 0 1 3 3 - 2

974

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

we attempt to develop a plausible tool using fuzzy relational calculus (FRC) for modeling and mimicking the cognitive process of human reasoning for pattern classification. The FRC approach to pattern classification can take care of uncertainties in feature values of the patterns under different conditions like measurement error, noise, etc. Though there are several existing approaches to design a classifier using the concept of fuzzy set/fuzzy logic, we mainly pick up the concept proposed by Pedrycz [1] and suitably modify it to incorporate our new concept of the computation of the derivative of the fuzzy max and min functions. To represent the knowledge about the training data set, we consider a new interpretation of multidimensional fuzzy implication (MFI). The new interpretation is basically a set of onedimensional fuzzy implications. The consequences of all one-dimensional fuzzy implication are finally collected through one intersection operator ‘‘5’’. We introduce a notion of fuzzy pattern vector which is formed by the cylindrical extension of the antecedent part of each onedimensional fuzzy implication. Thus, we get a set of fuzzy pattern vectors for the new interpretation of MFI and meaningfully carry out the task of pattern classification using FRC. During the construction of the classifier based on FRC we use fuzzy linguistic statements (or fuzzy membership function to represent the linguistic statement) to represent the ranges of features (e.g. feature F is small/medium/big, etc. ∀i) for a population of patG terns represented by the said fuzzy pattern vector. Note that the construction of the classifier essentially depends on the estimate of a fuzzy relation R between the antecedent part and consequent of each one-dimensional fuzzy implication. Thus, a set of fuzzy relations is formed from the new interpretation of MFI. This set of fuzzy relations is treated as a core of the classifier. Once the classifier is constructed the non-fuzzy features of a pattern can be classified. At the time of classification of the non-fuzzy features of the test patterns, we use the concept of fuzzy singleton to fuzzify the non-fuzzy feature values of the test patterns. The performance of the proposed scheme is tested on synthetic data. Finally, we use the proposed scheme for the vowel classification problem of Indian languages.

2. Brief review of the state of art A general description of the classification problem indicating a role of fuzzy sets has been given in [25]. In [26] Zadeh gave another description of pattern classification which visualizes the need of fuzzy set. A brief list of applications of fuzzy sets in the diverse fields of pattern classification is given in Table 1. Let us start with a conventional approach to pattern classification. In general, the process of pattern classification utilizes a sequence of steps, e.g. data acquisition, feature selection

Table 1 A brief list of applications of fuzzy set in Pattern Classification 1. 2. 3. 4. 5. 6. 7. 8.

Scene analysis [2] Speech recognition [3, 4] Image processing [5—7] Character recognition [8—10] Signal classification [11—16] Recognition of geometric objects [17, 18] Different medical applications [19—22] Intelligent robots (formation of visual recognition channel) [23, 24]

and classification procedure. At the first step data is collected via a set of sensors. Afterwards a classification space is constituted. Usually, this step is involved with the reduction of space dimensionality which has a significant role in the entire classification processes. Statistical pattern classification is mainly concerned with the discriminating power of features ignoring their robustness, easiness of determination, precision extraction, etc. Finally, at the third step, the classifier is constructed by establishing a transformation between classes and features. The transformation may be either Bayesian or nearest-neighbour rule, or nearest prototype (mean classifier), etc. [27].

3. Multidimensional fuzzy implication and the notion of fuzzy pattern vector [28–31, 61] For simplicity of discussion and/or demonstration, let us restrict ourselves to the problem of pattern classification on R. Without lack of generality all the discussions and/or demonstrations will be valid for the problem of pattern classification on RA, c'2. Let us now consider a brief discussion on the correspondence between conventional approach to pattern classification and FRC approach to pattern classification [28—30]. In the conventional approach, the position of each pattern (say P) on the finite range of pattern space is represented by a pattern vector [28—30] and we always try to discriminate among patterns by classifying the pattern vector [28—30]. Therefore, from the given data set (i.e. the training data set), we always know where the patterns are located and then try to separate them by some appropriate decision function. Subsequently, we use the said decision function to classify the test pattern vectors. But, if we go by mimicking the cognitive process of human reasoning for pattern classification then the problem is, from a given set of imprecise observations stated in terms of fuzzy If—Then rules, how to represent the patterns on the pattern space for developing suitable

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

inferencing technique for classification. As the observations are given in terms of fuzzy If—Then rules, it is clear that at every observation (rule) features are given in the form of fuzzy sets defined over the feature universes. Therefore in such a case we can not represent/locate an individual pattern by a vector. Instead, from a given observation (rule) we can represent/locate a collection of population of patterns in an area (in case of R; but a region in general) on the pattern space by a fuzzy pattern vector which is a very natural representation of the antecedent part of a multidimensional fuzzy implication (MFI) [28—30, 61]. Since a multidimensional fuzzy implication (MFI) such as ‘‘if (x is A, y is B) then z is C’’ where A, B, C are fuzzy sets is not merely a collection of one-dimensional implications, a conventional interpretation is usually taken in the multidimensional case. For example, according to the conventional interpretation, the above two-dimensional implication is translated into [61] if x is A and y is B then z is C

(1a)

or if x is A then y is B then z is C.

(1b)

Using the interpretation of Eq. (1a), we have already designed the pattern classifier in [32] where the fuzzy relation R between antecedent parts and consequent part of a rule is obtained logically. According to Tsukamoto [33], a multidimensional fuzzy implication (MFI) can alternatively be interpreted as if x is A then z is C

(2)

and if y is B then z is C and the intersection C5C, where C is the inferred value from the first implication and C is that from the second implication, is taken for the consequence of reasoning. To tackle the pattern classification problem using FRC, we provide the following new interpretation of the multidimensional fuzzy implication (MFI): if x is A then z is C and if y is B then z is C

(3)

and the intersection C5C, where C is the inferred value from the first implication and C is that from the second implication, is taken for the consequence of reasoning. Note that the fuzzy set C< "C5C and C of Eq. (1a) will carry the same defuzzified information which is ultimately needed for classification of patterns/objects [28—30].

975

The essential difference between Tsukamoto model (i.e. Eq. (2)) and the newly proposed models (i.e. Eq. (3)) occurs at the interpretation of the consequent part of each one-dimensional fuzzy implication of the (MFI). According to Tsukamoto, the consequent part of each one-dimensional fuzzy implication (see Eq. (2)) of a MFI is same as the consequent part of the said MFI. Whereas, in the newly proposed model, the consequent part of each one-dimensional fuzzy implication (see Eq. (3)) is different from the consequent part of the MFI. The linguistic connective ‘and’ of Eqs. (2) and (3) has a logical interpretation ‘‘5’’. The fuzzy sets C and C of Eq. (3) are not directly obtainable only from the fuzzy set C of Eq. (1a) which is the consequent part of a MFI. The fuzzy set C is formed depending upon the relative position of the fuzzy pattern vector with respect to the defined cover of the pattern space. On the other hand, when we consider the elements of the fuzzy vector individually, which are also fuzzy sets on the universe of respective feature axis, the cylindrical extension of them (the said fuzzy sets) and the well defined cover of the pattern space provide the information about the fuzzy sets C and C [28—30]. Let us now define the pattern space as the cover of the classes c , i"1, 2, 2 , n, and we consider them to be the G generic elements of the pattern space which is the universe. The possibility of occurrence of different classes of patterns under a particular fuzzy pattern vector [28—30] is represented by a fuzzy set which is defined over the pattern space and which is the consequent part of a MFI. According to the usual interpretation of MFI (i.e. Eq. (1a)) we have to estimate the relation R such that, R : F(º );F(º )PF(C ); and according to the new   AJ?QQ interpretation (i.e. Eq. (3)) we have to estimate two relations R and R as follows: R : F(º )PF(C ) and     AJ?QQ R : F(º )PF(C ).   AJ?QQ From the given If—Then rules once the fuzzy relation/relations (i.e. either R or R and R ) is/are estimated   we can use it further for inferencing for classification of patterns as discussed in the subsequent sections. To estimate the fuzzy relation (R or R , R ) between   antecedent clauses (clause) and consequent clause of a fuzzy implication (or a set of fuzzy implication) we may either go by logical approach [34] or go by FRC approach [35]. In this paper, we will consider FRC approach as proposed by Pedrycz [35] and subsequently modify the said approach to suit our needs for classification. The advantages which we obtain for FRC approach to pattern classification are as follows: 1. We obtain the local description of the pattern space in terms of few quantized zones. Depending upon the need of the problem, we may increase/decrease the granularity of our description of the pattern space by smaller/bigger quantized zones.

976

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

2. For estimating the relation R of a classifier, we do not need to select the representative data set from the given set of data (patterns). Instead, we use the gross property of few populations of given data (patterns) spread over the pattern space by using few fuzzy pattern vectors which describes the overall distribution of patterns in pattern space. 3. We obtain multiple classification which is very natural in case of overlapped classes of patterns. 4. If we try to classify the patterns on RA, c'2, our new interpretation for MFI given by Eq. (3) does not suffer from the curse of large feature dimension. For instance, according to conventional interpretation of MFI, i.e. Eq. (1a), the antecedent clauses of fuzzy If—Then rules will produce a large dimension of relation when ‘‘c’’ becomes large, say even 3, 4, 5, etc. Whereas, according to our new interpretation, i.e. in Eq. (3), we may assume any large value of ‘‘c’’ which does not affect the above said problem of dimensionality. 5. Classifier based on fuzzy If—Then rules learns faster than crisp classifier systems [36]. The reason for this is that fuzzy rules are high-level languages which keeps the cardinality of the term set smaller than crisp rules. Thus, we can accurately describe the decision strategy over the pattern space by a few number of rules.

4. Statement of the problem For the present problem, let us consider the new interpretation of a MFI as indicated by Eq. (3) and the notion of fuzzy pattern vector which is formed by the cylindrical extension of the antecedent part of each implication in Eq. (3) to represent a population of patterns P on the pattern space. Thus, a population of patterns P can be described by linguistic features rather than numerical features. Assume the pattern space F consists of ‘‘c’’ coordinates, i.e., a Cartesian product of the universes º , º , 2 , º of the following form; º"º ;º ;   A   2;º and there exist a set C of finite number of A AJ?QQ classes c , c , 2 , c , i.e., C "+c , c , 2 , c ,, by   L AJ?QQ   L which the finite range of the pattern space is covered. Let a fuzzy set AG3F(º ) is associated with a fuzzy set G CG" L k (c )/c , where k (c ) denotes the strength of H !G H H !G H belongingness (grade of membership) of the population of patterns P to the class c , for j"1, 2, 2 , n. Thus, H there should exist a relation between AG and CG. More precisely, AG and CG are related via certain relation R (i.e. G AGR CG), which at this moment is unknown and which we G have to estimate for the design of the classifier. Now we specify how CG is derived from given AG and R . Such specification is needed for the testing of the G classifier. We may consider either of the two forms of fuzzy relational equations, namely a direct equation CG"AG ° R , G

or an adjoint equation CG"AG R . G

(4)

Eq. (4) can be rewritten, in terms of the membership functions, in the following form: k (c )"max [k (aG) tkR (aG, c )] !G H G H G ?GZ3G for j"1, 2, 2 , n, k (c )"min [k (aG) kR (aG, c )] !G H G H G ?GZ3G for j"1, 2, 2 , n,

(5)

where t stands for triangular norm and stands for its pseudo-complement. Since the pattern space F consists of c features, so when we consider c features for the classification of the pattern, the resulting fuzzy set on C is obtained by AJ?QQ A C" 7 CG G

(6)

with membership values A k (c )" d k (c ) for j"1, 2, 2 , n. ! H !G H G

(7)

The advantage of this explicit form of Eq. (4) of the classifier lies in the fact that having a training set of patterns the fuzzy relation, standing in either of Eq. (5), can be determined. Let us assume that the training set for the ith feature consists of ordered pairs (P , CG ), (P , CG ), 2 , (P , CG )     I I and the classifier relation is supposed to specify a system of equations



AG ° R , G CG" J J AG R , J G

l"1, 2, 3, 2 , k, l"1, 2, 3, 2 , k,

(8)

then the fuzzy relation defined in Eq. (8) is given by I R< " 7 (AG CG), G J J J (9) I R< " 8 (AG;CG). G J J J But the above-mentioned system of equations in Eq. (8) may not possess solutions [1]. In such a case, we cannot expect that the equality AG ° R< "CG or AG R< "CG is still J G J J G J preserved ∀l3+1, 2, 2 , k,. Hence, in this paper, we look for an approximate solution of the system of fuzzy relation equations in Eq. (8) using conventional computational facility.

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

5. Existing method to solve fuzzy relation equation The numerical solution of fuzzy relational equation has been proposed by several researchers [35, 37—47]. In this section we briefly review the method proposed by Pedrycz [35]. Now we focus our attention on max-t composition operator of fuzzy relational equations, which are defined on finite spaces: y"x ° R,

(10)

where ° ,max-t composition operator, x and y are the fuzzy sets defined on the universe of discourses X and ½, respectively, and R is the fuzzy relation on X;½. Let card(X)"m, card(½)"n and also define r "(x , y /x 3X, y 3½), i"1, 2, 2 , m, j"1, 2, 2 , n; GH G H G H then the fuzzy sets x and y and fuzzy relation R are as follows: x"[k (x )] ; 3F(X), V G  K y"[k (y )] ; 3F(½), W H  L

(11)

R"[kR (r )] ; 3F(X;½). GH K L

for j"1, 2, 2 , n,

(13)

where y is the calculated fuzzy set using the Eq. (12) and yJ is the desired fuzzy set. Now the basic problem is to estimate R"[kR (r )] ; GH K L via some given x and y which minimize E defined in Eq. (13) and satisfying +kR (r )(1!kR (r )),*0, ∀i" GH GH 1, 2, 2 , m and j"1, 2, 2 , n. A general method to solve an optimization problem defined above is to solve a set of equations, which form the necessary conditions for a minimum of the error E defined in Eq. (13). Thus, we have *E/*R"[0] ; . K L Now, we discuss the applicability of Newtons method and its simplification. The Newtons iterative scheme for finding the solution of R"[kR (r )] ; is GH K L *E kR (r )Q>"kR (r )Q!a GH GH Q *kR (r ) GH



R

, R

*E "2 +k (y )!k N (y ), P , W H W H GH *kR (r ) GH

(15)

where *k (y ) P " W H , GH *kR (r ) GH

(16)

i.e.

 



K * e +k (x ) t kR (r ), P " V S SH GH *kR (r ) GH S



* K " e +k (x ) t kR (r ),R+k (x ) t kR (r ), V S SH V G GH *kR (r ) GH S$G

(17)

(12)

where t is the triangular-norm operator. Let E be the sum of the square of the error over v"1, 2, 2 , n and is defined by L E" +k (y )!k N (y ),, W T W T T

factor depending on the number of iteration and can be described as a "1/(2.0#sG). i*0 is chosen empirically Q in order to achieve good convergent properties and avoid significant oscillations in the iteration procedure [35]. Now

for i"1, 2, 2 , m and j"1, 2, 2 , n. Thus, depending on t-norm operator in Eq. (12) we get the following two types of problems:

Eq. (10) can be put in the following form: K k (y )" e +k (x ) t kR (r ), W H V G GH G

977

(14)

where i"1, 2, 2 , m and j"1, 2, 2 , n and a is the Q convergent factor and also is an non increasing gain

¹ype I: t-norm operator is ‘‘prod’’. We consider t-norm operator as ‘‘prod’’, then Eq. (12) is written as K k (y )" e +k (x ) ) kR (r ), W H V G GH G

for j"1, 2, 2 , n

(18)

and in this case P in Eq. (17) is determined as GH K   kV (xG) if e +kV (xS) ) kR (rSH),)kV (xG) ) kR (rGH), S$G P " (19) GH  0 otherwise.  for i"1, 2, 2 , m and j"1, 2, 2 , n. ¹ype II: t-norm operator is ‘min’: Again we consider tnorm operator is ‘min’, then Eq. (12) is written as K k (y )" e +k (x )kR (r ), for j"1, 2, 2 , n W H V G GH G

(20)

and in this case P is determined as GH K  1 if e +kV (xS)kR (rSH),)kV (xG)kR (rGH)  S$G P " and k (x )*kR (r ), (21) GH  V G GH  0 otherwise for i"1, 2, 2 , m and j"1, 2, 2 , n.

978

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

Here the derivative of the max-function and min-function used in Eqs. (17), (19) and (21), respectively, are as follows:



1 if w'a, * (wRa)" *w 0 if w(a,

(22)

where a"eK +k (x ) t kR (r ), and w"k (x ) t kR (r ) and S$G V S SH V G GH



1 if z(b, * (zb)" *z 0 if z'b.

(23)

where b"k (x ) and z"kR (r ), which are piecewise V G GH differentiable and is undefined at w"a for max-function in Eq. (22) and z"b for min-function in Eq. (23). Thus, we get some problem in our numerical computation as mentioned in [42]. To overcome this problem we define the derivatives at w"a and z"b, respectively, as follows:



1 if w*a * (wRa)" *w 0 if w(a

(24)

and

In this case, error E is taken by summing over the training set. Thus, Eq. (13) is modified as follows: I L E" +k (y )!k N (y ), W T WJ T J T J

(29)

satisfying +kR (r )(1!kR (r )),*0, i"1, 2, 2 , m and GH GH j"1, 2, 2 , n, where y is the calculated fuzzy set using J Eq. (27) and yJ is the desired fuzzy set. The iterative J scheme of Eq. (14) for finding the relation R remains same. Only the expression a used in Eq. (14) could be Q modified as a "1/(2;k#sG), which depends on the Q number of data k.

6. Modified approach to solve the fuzzy relational equation (FRE) We modify the above said approach to solve the FRE by incorporating a rigorous treatment on the computation of the derivative of max-function and min-function indicated in Eqs. (24) and (25), respectively. 6.1. Derivative of max-function



1 if z)b, * (zb)" *z 0 if z'b.

(25)

Both the formulas for the computation of the derivative of the max- and min-functions, as mentioned above, returns either 0 or 1 value of the derivatives. Such twovalued results of the derivatives have got some inherent difficulties mentioned in [42]. To overcome such difficulties there are some propositions in [42]. In the following section we will provide an alternative approach based on generalized function (see Appendix A). The above method for solving fuzzy relational equations can be extended for simultaneous fuzzy relational equations [35] as given below. The simultaneous fuzzy relational equations for given total k sets of data in training set are as follows: y "x ° R, l"1, 2, 2 , k J J

(26)

and their membership functions forms are as follows: K k (y )" e +k (x ) t kR (r ), for j"1, 2, 2 , n, V G WJ H GH G J

(27)

where l"1, 2, 2 , k and x "[k (x )] ; 3F(X), J VJ G  K y "[k (y )] ; 3F(½), J WJ H  L R"[kR (r )] ; 3F(X;½). GH K L

(28)

Let the maximum value of h , i"1, 2, 2 , s, be deterG mined by a function called max-function defined by Q h "eh . K?V G G

(30)

Now our intention is to calculate the derivative of maxfunction defined as above with respect to one of its variables. Hence, we transfer the said max-function of Eq. (30) in the following functional form: G(h , h , h , 2 , h , h )    Q K?V Q , +H(h !h )!1,#G "0, K?V G C G where H(h

(31)

!h ) is the Heaviside function defined by K?V G



1 H(h !h )" K?V G 0

if h 'h , K?V G otherwise,

(32)

and G is the number of h ’s which are equal to h . Also C G K?V it is a constant and independent of h , i"1, 2, 2 , s and G h . K?V Now by using implicit function theorem, we write



*h *G *G K?V"! , *h *h *h P K?V P where r3+1, 2, 2 , s,.

(33)

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

We calculate the partial derivatives *G/*h and P *G/*h , using the derivative of the Heaviside function in K?V Eq. (A.7) as follows; *G "!d(h !h ), K?V P *h P *G Q " d(h !h ), K?V G *h K?V G where d(*) is the Dirac delta function. Using Eqs. (34) and (35) in Eq. (33) we get d(h !h ) *h K?V" K?V P *h Q P d(h !h ) K?V G G 1 if h "h , K?V P N K?V " 0 otherwise,



(35)

*h d(h !h ) KGL" P KGL *h Q P d(h !h ) G KGL G 1 if h "h , KGL P N KGL " 0 otherwise,



(36)

Derivative of min-function

Let the minimum value of h , i"1, 2, 2 , s, be deterG mined by a function called min-function and defined by Q h "dh . (37) KGL G G Now our intention is to calculate the derivative of minfunction defined as above with respect to one of its variables. Hence, we transfer the said max-function of Eq. (37) in the following functional form: ¸(h , h , h , 2 , h , h )    Q KGL Q , +H(h !h )!1,#¸ "0, (38) G KGL C G where ¸ is the number of h ’s which are equal to h . C G KGL Also it is a constant independent of h , i"1, 2, 2 , s, G and h . KGL Now by using implicit function theorem, we write



*¸ *¸ *h KGL "! , (39) *h *h *h P P KGL where r3+1, 2, 2 , s,. We calculate the partial derivatives *¸/*h and P *¸/*h , using the derivative of Heaviside function in KGL Eq. (A1) as follows: *¸ "d(h !h ), P KGL *h P

(41)

Using Eqs. (40) and (41) in Eq. (39) we get (34)

where N _ number of terms h , which are satisfying K?V G the condition h "h , i"1, 2, 2 , s, i.e. N " K?V G K?V Q d(h !h ), which never vanishes because at least G K?V G one of h , i"1, 2, 2 , s, must be equal to h . So G K?V *h /*h in Eq. (36) always exists finitely. K?V P 6.2.

*¸ Q "! d(h !h ). G KGL *h KGL G

979

(40)

(42)

where N _ number of terms h , i"1, 2, 2 , s, which KGL G are satisfying the condition h "h , i.e. N " KGL G KGL Q d(h !h ), which never vanishes because at least G G KGL one of h , i"1, 2, 2 , s, must be equal to h . So G KGL *h /*h in Eq. (42) always exists finitely. KGL P Thus, from the above discussion, we understand that the derivative of max- and min-functions depend on the derivative of the Heaviside function which is discussed, for general readability of the paper, in Appendix A. 6.3. Applicable form of the computation of derivative of max- and min- functions For the sake of implementation of the expression of the derivative of max- and min-functions we approximate the Delta function using a finite pulse shown in Fig. 1. The motivation behind the approximation of the Delta function by a finite pulse is to incorporate the notion of uncertainties built in the given data, which are all attached with fuzzy membership functions indicating their (data) degree of possibilities to take part in any decision making process. Thus if we approximate the Delta function by a finite pulse with width b that means we try to take care of the possibilities of all the data that falls within the range of b in our computation of the derivative of a fuzzy max and min functions. Using these approximation we formulate the approximate derivative of the max- and min-functions, respectively, as follows; 1 if h !h )b, K?V P *h N K?V + K?V (43) *h P 0 otherwise,



where N _ number of terms h which are satisfying the K?V G condition h !h )b, for i"1, 2, 2 , s, and the paraK?V G meter b controls the width of the pulse;



1 *h N KGL + KGL *h P 0

if h !h )b, P KGL

(44)

otherwise,

where N number of terms h which are satisfying the KGL G condition h !h )b, for i"1, 2, 2 , s. G KGL

980

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

Step 1: Start with an initial trial values of R" [kR (r )] ; such that +kR (r )(1!kR (r )),*0 GH K L GH GH for i"1, 2, 2 , m and j"1, 2, 2 , n. Step 2: Set the width of the pulse b, convergent factor i, the error threshold e, and maximum number of iterations s . Set the initial iteration number K?V s"0. Step 3: Set new iteration number s"s#1. Step 4: Using the given fuzzy data x and yJ , evaluate J J y by Eq. (26) and E by Eq. (29). J Step 5: Evaluate *E/*kR (r ), in Eq. (15) using either GH Eq. (46) (for the problem of Type I) or Eq. (48) (for the problem of Type II) for i"1, 2, 2 , m and j"1, 2, 2 , n and a . Q Step 6: Update the values of kR (r ) using the Newton’s GH iterative scheme (see Eq. (14)), Fig. 1. Approximation of Delta function d(x).

Now the expression in Eq. (16) can be rewritten as *+k (x ) t kR (r ), *k (y ) V G GH . W H ) (45) P " GH *+k (x ) t kR (r ), *kR (r ) V G GH GH Comparing Eq. (30) with Eq. (18) we have h "k (y ) K?V W H and h "k (x ) ) kR (r ), i"1, 2, 2 , m, ∀j3+1, 2, 2 , n,. G V G GH Using the Eq. (43) in Eq. (45) where t,‘prod’ we get,



k (x ) V G N K?V P " GH 0

if k (y )!+k (x ) ) kR (r ),)b W H V G GH

(46)

otherwise.

where i"1, 2, 2 , m and j"1, 2, 2 , n. These results are used only for the problem of Type I of Section 5. Comparing Eq. (37) with Eq. (20) we have h " KGL k (x )kR (r ). Now there is only one variable kR (r ) V G GH GH in h as given above so N "1 only when KGL KGL k (x )*kR (r ). Therefore, the derivative V G GH 1 if k (x )*kR (r ), *h V G GH KGL " (47) *kR (r ) 0 otherwise. GH



Using Eq. (47) in Eq. (45) where t,min we get 1   NK?V P " GH   0

if k (y )!+k (x )kR (r ),)b W H V G GH and k (x )*kR (r ), V G GH otherwise,

(48)

where i"1, 2, 2 , m and j"1, 2, 2 , n. These results are used only for the problem of Type II of Section 5.



*E kR (r )Q>"kR (r )Q!a ) GH GH Q *kR (r ) R R GH   where i"1, 2, 2 , m and j"1, 2, 2 , n. Step 7: Now test whether +kR (r )(1!kR (r )),*0 or GH GH not for all i"1, 2, 2 , m and j"1, 2, 2 , n. If not then construct a set of pair of indexes N F " + ( i , j ) / + k R( r )  ( 1 ! k R( r ) ) , ( 0 , GH GH i"1, 2, 2 , m and j"1, 2, 2 , n,. Set +kR (r )(1!kR (r )),"0, ∀(i, j)3NF. GH GH Step 8: Repeat from Step 3 until E(e and/or s"s . K?V 6.5. Illustration of the modified approach to the estimation of R We illustrate the modified method based on the data set (see Table 2) given by Pedrycz [35]. Here we see that card(º )"m "3, and card(º )"m "4. Now     we start with an initial trial values kR (r )"0, GH i"1, 2, 2 , 12 and j"1, 2, 3, 4. The value i"10\ is chosen to ensure good convergence properties. Width of the pulse b"0.05. The error threshold e"10\, and the maximum number of iterations s "500. The values of K?V the error for every 25 step of iteration are displayed in Fig. 2. The solutions of R’s of the problem of Type I of Section 5 are shown in Table 3.

7. Design of the multidimensional classifier based on fuzzy relational calculus (FRC)

6.4. Algorithm for the estimation of R This algorithm gives the step by step calculation for the estimation of R using the modified computational approach.

In the design of the multidimensional classifier we have basically two phases; namely the learning phase (training phase) where we estimate the fuzzy relation R (based on G the algorithm of Section 6.4) for set of one-dimensional

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

981

Table 2 Fuzzy sets A, A, yJ , l"1, 2, 2 , 8, for the fuzzy system J J J l

Membership values of fuzzy set A J

1 2 3 4 5 6 7 8

0.00 1.00 0.00 0.38 0.37 0.00 0.20 0.75

0.27 0.23 0.05 0.92 0.48 0.90 1.00 0.12

Membership values of fuzzy set A J 0.80 0.00 1.00 0.00 0.00 0.02 0.08 0.00

0.97 0.00 0.08 0.00 0.00 0.00 0.00 0.00

0.11 0.35 0.70 0.02 0.10 0.10 0.12 0.10

Membership values of fuzzy set yJ J 0.00 1.00 0.30 0.15 0.30 0.45 0.50 0.15

0.00 0.00 0.00 0.95 0.90 0.02 0.01 0.80

0.00 0.08 0.00 0.01 0.02 0.00 0.02 0.01

0.35 0.66 0.03 0.12 0.20 0.12 0.13 0.25

0.80 0.30 0.15 0.38 0.37 0.30 0.33 0.75

0.11 0.18 0.70 0.71 0.37 0.36 0.40 0.60

Fig. 2. Error against each 25 iterations.

Table 3 Relation R for the problem Type I 0.00

0.00

0.00

0.00

0.00

0.03

0.03

0.01

0.00

0.45

1.00

0.14

0.02

0.03

0.03

0.03

0.01

0.02

0.03

0.03

0.00

0.01

0.03

1.00

0.08

0.67

0.30

0.18

0.02

0.26

0.70

0.83

0.00

0.00

0.01

0.05

0.01

0.33

1.00

0.80

0.02

0.12

0.43

0.82

0.00

0.00

0.00

0.00

fuzzy implications for the ith feature and the testing phase (classification phase) where we test the performance of the multidimensional classifier using the first equation of Eq. (5) which involves the expression R ∀i3+1, 2, 2 , c,. G

Firstly in the training phase with the training data set, we discretize (quantize) the individual feature axis and the entire pattern space in the following way: Determine the lower and upper bounds of the data of ith feature value. Let f G be the jth data of the ith feature H

982

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

Table 4 Fuzzy sets in F(º ) G uG  k G (uG ) "  k G (uG ) "‚  $

uG  k G (uG ) "  k G (uG ) "‚  $

2

2

2

DG  DG  $

2

2

2

2

2

2

$

$

$

DG IG

k G (uG ) "IG 

k G (uG ) "IG 

2

2

2

(*)

uG KG k G (uG ) " KG k G (uG ) "‚ KG $ k G (uG ) "IG KG

F and d is the length of segmentation along ith feature G G axis. Minimum of the data f G, j"1, 2, 2 , of the ith feature H is f G "min ( f G ). Let rG be the remainder when f G is KGL H H KGL KGL divided by d . Therefore, the lower bound of the ith G feature axis,



fG if rG "0, KGL ¸B " KGL G f G !rG otherwise. KGL KGL

(49)

This ¸B is taken as the ith coordinate of the origin. G Again, maximum of the data f G, j"1, 2, 2 , of the ith H feature is f G "max ( f G ). Let rG be the remainder K?V H H K?V when f G is divided by d . Therefore, the upper bound of K?V G the ith feature axis,



fG ºB " K?V G f G #(d !rG ) K?V G K?V

if rG "0, K?V otherwise.

(50)

Let º be the universe of discourse on the ith feature axis G F then º has m "(ºB !¸B )/d generic elements and G G G G G G

these are uG , j"1, 2, 2 , m , which we define as follows: H G uG " H [¸B #( j!1) ) d , ¸B #j ) d ) for j"1, 2, 2 , (m !1), G G G G G [¸B #(m !1) ) d , ºB ] for j"m . G G G G G (51)



Therefore, the universe on the ith feature axis, º "+uG , uG , 2 , uG ,. Now we define k fuzzy sets on G   KG G º say DG, l"1, 2, 2 , k are in shown in Table 4. So G J G there are k fuzzy If—Then rules as follows: G RG : if F is DG then C is CG 3F(C ) J G J J AJ?QQ where C is the universe of discourse constructed by AJ?QQ all the classes in the pattern space, i.e. C " AJ?QQ +c , c , 2 , c ,. According to the fuzzy implication   L method we write R : F(º )PF(C ). G G AJ?QQ Determination of membership value of the class c 3C when DG is on º is taken in the following ways: N AJ?QQ J G e (k G(uG )), (52) k G (c )" !J N " H *HZ* 2KG+/SGH 3 AN5$8"GJ+ J where FZ(DG) is the zone which represents the tip of the J fuzzy pattern vector as stated in [28—30] and is constructed by the cylindrical extension of the fuzzy set DG where J l"1, 2, 2 , k . G Based on the generated fuzzy rules, as stated above, at the end of training phase we estimate the fuzzy relation R using the algorithm of the Section 6.4. In the same way G we can find all the fuzzy relations R , i"1, 2, 2 , c. Next G we switch over to the testing phase (classification phase).

Fig. 3. Architecture of the multidimensional pattern classifier. (Keys: The fuzzy sets DG, i"1, 2, 2 , c, are the test inputs based on fuzzy singleton. The fuzzy sets R , i"1, 2, 2 , c, are estimate in training phase, CG, i"1, 2, 2 , c, are the output fuzzy sets for the individual G features and the fuzzy set C"7 A CG is the output result of the multidimensional classifier.) G

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

983

At the testing phase (classification phase) we use the first equation of Eq. (5) as stated earlier. The selected features for testing are fuzzified using the concept of the fuzzy singleton as discussed in [48]. Fig. 3 represents the architecture of the multidimensional pattern classifier. The classification results produce the possibility of occurrence of each pattern at different classes in the pattern space. At the time of taking a non-fuzzy decision (i.e. defuzzification) out of this fuzzy classification we proceed as follows: Let CG" L k (c )/c be the resulting fuzzy H !G H H set obtained by the first equation of Eq. (5) of Section 4 for every i"1, 2, 2 , c. Now the fuzzy set when we consider all the features is C"5A CG with membership G value of c in C is given by H A k (c )" d k (c ), j"1, 2, 2 , n. ! H !G H G Now we defuzzify C as follows: First we find L +k (c ),. hgt(C)"max G ! G

(53)

Fig. 4. First synthetic data.

Let DF(C) be set of indexes defined by DF(C)" +i3+1, 2, 2 , n,/k (c )"hgt(C),. If DF(C) is a singleton ! G set of index r the resulting class will be c ; otherwise the P resulting class will be c "Arbitrary +c ,. P G G3"$!

(54)

That means we can go by selecting the class having highest possibility value. In case of tie situation, which normally occurs for the patterns lying in the overlapped zone [28—30], we have to state the equal possibility of a pattern belonging to both the classes. Such a conclusion is quite natural which normally does not exist in the conventional classification approach. In some cases, patterns in the overlapped zone are classified with ‘‘almost equal’’ possibility of occurrence for more than one class. If such a situation is treated as the tie situation mentioned above, we have to select an appropriate threshold which entirely depends on the need of the problem. Thus, from the graded consequence, when we select a single class having the highest membership value, we consider the hard partitioning [28—30] of the pattern space. Whereas from the graded consequence, when we consider the occurrence of multiple classes, we consider the fuzzy partitioning [28—30] of the pattern space.

8. Effectiveness of the proposed method To test the effectiveness of our design as stated in Section 7, we consider the classification of two synthetic data as shown in Figs. 4 and 5. At the time of writing fuzzy If—Then rules for the classifier we may consider

Fig. 5. Second synthetic data.

complete cover of the pattern space (see the discussion of Section 3). But as the consideration of complete cover of the pattern space does not bring any significant change in classification score, for all practical purposes, without loss of generality we consider partial cover of the pattern space.

984

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

8.1. Classification of first synthetic data The first synthetic data as shown in Fig. 4 has two features. For this data we chose suitable length of segmentations d "0.5"d . Therefore, we get, ¸B "0    "¸B by Eq. (49) and ºB "6"ºB by Eq. (50).    Therefore, m "(ºB !¸B )/d "12 and m "      (ºB !¸B )/d "12.    By Eq. (51) we get º "+u, u, 2 , u ,,     º "+u, u, 2 , u ,.     We define k "3 fuzzy sets on º and k "5 fuzzy sets    on º which are shown in Tables 5 and 6, respectively.  Therefore, the fuzzy If—Then rules and their consequent parts are shown in Tables 7 and 8. Now we start with an initial trial values of kR (r )"0.1, i"1, 2, 2 , m and j"1, 2, 2 , n, i"0.5, J J GH b"0.05, e"10\, S "1000 and estimate the relaK?V tions R for l"1, 2 by using the algorithm in Section 6.4 J for the problem of Type I of Section 5. The classification scores are shown in Table 9 for Type I.

Table 7 Antecedent and consequent part of the fuzzy If—Then rules for the first feature Rule Antecedent part

l R  R  R 

Consequent part Membership values of A B C D 1.0 0.7 0.0 1.0 0.7 1.0 0.7 0.7 0.0 0.7 1.0 0.0

F  D  D  D 

E 0.7 1.0 0.7

Table 8 Antecedent and consequent part of the fuzzy If—Then rules for the second feature Rule

Antecedent part Consequent part

l R  R  R  R  R 

F  D  D  D  D  D 

Membership A B 1.0 1.0 1.0 1.0 0.7 1.0 0.1 0.3 0.0 0.0

values C 1.0 1.0 1.0 0.3 0.0

of D 0.0 0.3 1.0 1.0 0.7

E 0.1 0.7 1.0 1.0 1.0

8.2. Classification of second synthetic data For the data shown in Fig. 5, we chose suitable length of segmentations d "0.5"d . Therefore, we get   ¸B "0"¸B by Eq. (49) and ºB "6"ºB by     Eq. (50). Therefore, m "(ºB !¸B )/d "12 and     m "(ºB !¸B )/d "12.    

By Eq. (51) we get º "+u, u, 2 , u ,,     º "+u, u, 2 , u ,.    

Table 5 Fuzzy sets in F(º ) for the first synthetic data 

D  D  D 

Membership values of u u u   

u 

u 

u 

u 

u 

u 

u 

u 

u 

0.3 0.0 0.0

0.7 0.3 0.0

0.3 0.7 0.0

0.1 1.0 0.1

0.0 0.7 0.3

0.0 0.3 0.7

0.0 0.1 1.0

0.0 0.0 0.7

0.0 0.0 0.3

0.0 0.0 0.1

0.7 0.0 0.0

1.0 0.1 0.0

Table 6 Fuzzy sets in F(º ) for the first synthetic data 

D  D  D  D  D 

Membership values of u u u   

u 

u 

u 

u 

u 

u 

u 

u 

u 

0.7 0.1 0.0 0.0 0.0

0.3 1.0 0.3 0.0 0.0

0.1 0.7 0.7 0.1 0.0

0.0 0.3 1.0 0.3 0.0

0.0 0.1 0.7 0.7 0.1

0.0 0.0 0.3 1.0 0.3

0.0 0.0 0.1 0.7 0.7

0.0 0.0 0.0 0.3 1.0

0.0 0.0 0.0 0.1 0.7

0.0 0.0 0.0 0.0 0.3

1.0 0.3 0.0 0.0 0.0

0.7 0.7 0.1 0.0 0.0

F 0.1 1.0 1.0

F 0.1 0.7 1.0 1.0 1.0

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

985

Table 9 Classification scores of first synthetic data for the problem of Type I To A

B

C

D

E

No. of data

F

From

I

II

I

II

I

II

I

II

I

II

I

A B C D E F

113 31 0 7 0 0

0 6 0 0 0 0

3 84 27 0 8 3

56 3 11 0 0 0

0 2 52 0 0 6

0 55 19 0 4 13

0 0 0 54 3 0

4 1 0 25 0 0

0 0 0 28 134 27

54 5 0 54 6 67

0 0 0 0 6 79

2 47 49 10 141 35

Total

113

0

84

3

52

19

54

25

134

6

79

35

Score (%) Partitioning

II 116 117 79 89 151 115

Hard

Fuzzy

97.41 71.79 65.82 60.67 88.74 68.70

97.41 74.36 89.87 88.76 92.72 99.13

77.36

90.55

Table 10 Fuzzy sets in F(º ) for the second synthetic data 

D  D  D 

Membership values of u u u   

u 

u 

u 

u 

u 

u 

u 

u 

u 

0.8 0.0 0.0

0.5 0.5 0.0

0.2 0.8 0.0

0.0 1.0 0.2

0.0 0.8 0.5

0.0 0.5 0.8

0.0 0.2 1.0

0.0 0.0 0.8

0.0 0.0 0.5

0.0 0.0 0.2

1.0 0.0 0.0

0.8 0.2 0.0

Table 11 Fuzzy sets in F(º ) for the second synthetic data 

D  D  D  D 

Membership values of u u u   

u 

u 

u 

u 

u 

u 

u 

u 

u 

0.1 0.0 0.0 0.0

0.5 0.1 0.0 0.0

0.1 0.5 0.0 0.0

0.0 1.0 0.0 0.0

0.0 0.5 0.1 0.0

0.0 0.1 0.5 0.0

0.0 0.0 1.0 0.0

0.0 0.0 0.5 0.1

0.0 0.0 0.1 0.5

0.0 0.0 0.0 1.0

0.5 0.0 0.0 0.0

1.0 0.0 0.0 0.0

Table 12 Antecedent and consequent part of the fuzzy If—Then rules for first feature

Table 13 Antecedent and consequent part of the fuzzy If—Then rules for second feature

Rule

Antecedent

Consequent part

Rule

Antecedent part

l R  R  R 

part F  D  D  D 

Membership A 1.0 1.0 0.8

l R  R  R  R 

F  D  D  D  D 

values of B C 0.2 0.2 1.0 1.0 1.0 1.0

We define k "3 fuzzy sets on º and k "4 fuzzy sets    on º which are shown in Tables 10 and 11, respectively.  Therefore, the fuzzy If—Then rules and their consequent parts are shown in Tables 12 and 13.

Consequent part Membership values A B 1.0 0.5 1.0 1.0 1.0 0.0 0.5 0.0

of C 0.0 0.5 1.0 0.0

Now we start with an initial trial values of kR (r )" J GH 0.0, i"1, 2, 2 , m and j"1, 2, 2 , n, i"1.0, b"0.05, J e"10\, S "1000 and estimate the relations R for K?V J l"1, 2 by using the algorithm in Section 6.4 for the

986

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

Table 14 Classification scores of second synthetic data for the problem of Type I From

A B C Total

To A

B

No. of data

C

I

II

I

II

I

II

226 23 22 226

61 64 78 61

40 74 0 74

103 11 17 11

21 0 78 78

123 22 5 5

problem of Type I of Section 5. The classification scores are shown in Table 14 for Type I.

9. Applications After achieving satisfactory results on a synthetic set of data, we apply the proposed design for the vowel classification problems of two Indian languages, namely Telugu and Bengali [32]. In the following subsections, we briefly discuss the experimental procedures for extraction of vowels of each language and the classification results. 9.1. Experiment on the classification of the Telugu vowels A number of discrete phonetically balanced speech samples for the vowels of Telugu language in CNC (consonant-vowel nucleus-consonant) form are selected. A CNC combination is taken because the form of consonants connected to a vowel is responsible for influencing the role and quality of vowels. These speech units are recorded by five informants on an AKAI type recorder. The spectrographic analysis has been done on Kay Sonagraph Model 7029-A, which is a very standard audio frequency spectrum analyzer that produces a permanent record of the spectra of any complex waveform in the range of 5 Hz to 16 kHz. For the present study of vowels, the spectrographic display of frequency versus time (see Fig. 7) has been done for 800 Telugu words uttered by three male informants in the age group of 25—30 yr chosen from 15 educated persons. The total bandwidth of the system is 80 Hz to 8 kHz with a resolution of 300 Hz. The experiment deals with the formant frequencies at the steady state of the Telugu vowels and their variations on different consonantal context and for different speakers. The average positions (see Table 15) of different Telugu vowels with respect to cardinal vowels and their distribution in F !F frequency planes are considered (see   Fig. 6). For further details of the extraction procedures of Telugu vowel see [32]. The present investigation has been carried out with the telugu vowels (listed in Table 15) both short and long. It is well known that the

Score(%) Partitioning

287 97 100

Hard

Fuzzy

78.75 76.29 78.00 78.10

100.00 87.63 83.00 94.01

Table 15 Average formant frequencies of Telugu vowel Phonetic symbol

Average formant frequencies (Hz) F F F   

/*/ / a:/ / i / / i:/ / u —" / / u:/ / e / / e:/ / o —" / / o:/ / ae/

606 710 365 325 370 348 517 470 476 486 575

1473 1240 2116 2260 1066 923 1796 1883 1133 1000 1744

2420 2400 2757 2836 2500 2543 2633 2657 2630 2540 2700

first three formant frequencies carry most of the information regarding the vowel quality. But for all practical purposes of vowel classification, we can use the first two formant frequencies, i.e. F and F .   For these data shown in Fig. 6, we chose suitable length of segmentations d "50 and d "100. Therefore,   we get, ¸B "200 and ¸B "600 by Eq. (49) and   ºB "850 and UB "2600 by Eq. (50). Therefore,   m "(ºB !¸B )/d "13 and m "(ºB !¸B )/        d "20.  By Eq. (51) we get º "+u, u, 2 , u ,,     º "+u, u, 2 , u ,.     We define k "5 fuzzy sets on º and k "7 fuzzy sets    on º which are shown in Tables 16 and 17, respectively.  Therefore, the fuzzy If—Then rules and their consequent parts are shown in Tables 18 and 19. Now we start with an initial trial values of kR (r )" J GH 0.1, i"1, 2, 2 , m and j"1, 2, 2 , n, i"0.5, b"0.05, J e"10\, S "1000 and estimate the relations R for K?V J

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

987

9.2. Experiment on the classification of Bengali vowels The experiment has been conducted on a sample of carefully selected 350 commonly spoken Bengali words so that the vowels can be studied in all possible consonant—nucleus—consonant (CNC) contexts. These words are uttered by 10 male and 10 female educative and phonetically conscious informants drawn from linguists, professor of literature and dons of performing arts. Of all these records, data for three male informants have been chosen on the basis of good and clear spectrographs. The age of the informants varied between 30 and 55 yr. The spectrographic analysis has been done on Kay Sona-graph model 7029-A in the bandwidth of 80 Hz to 8 kHz using a resolution of 300 Hz. The acoustic data, namely the first four formant frequencies and the duration, are derived from the spectrographs. As the vowels are embedded in various consonantal contexts in multisyllabic words, appropriate segmentation procedures are adopted to accurately fix both the transitions and the steady state of the vowels. The details of the segmentation procedure and measurement procedure are same as stated in [32]. Table 21 represents the average formant frequencies for first three formants. For these data shown in Fig. 8, we chose suitable length of segmentations d "50 and d "100. Therefore, we get, ¸B "200 and    ¸B "700 by Eq. (49) and ºB "850 and ºB "2400    by Eq. (50). Therefore, m "(ºB !¸B )/d "13 and     m "(ºB !¸B )/d "17.     By Eq. (51) we get

Fig. 6. Telugu vowels.

º "+u, u, 2 , u ,,     º "+u, u, 2 , u ,.     We define k "5 fuzzy sets on º and k "7 fuzzy sets    on º which are shown in Tables 22 and 23, respectively.  Therefore, the fuzzy If—Then rules and their consequent parts are shown in Tables 24 and 25. Now we start with an initial trial values of kR (r )" J GH 0.0, i"1, 2, 2 , m and j"1, 2, 2 , n, i"0.3, b"0.05, J e"10\, S "1000 and estimate the relations R for K?V J l"1, 2 by using the algorithm in Section 6.4 for the

Fig. 7. Spectrogram of the Telugu word [ti: ka].

l"1, 2 by using the algorithm in Section 6.4 for the problem of Type I of Section 5. The classification scores are shown in Table 20 for Type I.

Table 16 Fuzzy sets in F(º ) for Telugu vowel 

D  D  D  D  D 

Membership values of u u u   

u 

u 

u 

u 

u 

u 

u 

u 

u 

u 

0.1 0.0 0.0 0.0 0.0

0.5 0.5 0.0 0.0 0.0

0.1 1.0 0.1 0.0 0.0

0.0 0.5 0.5 0.0 0.0

0.0 0.1 1.0 0.1 0.0

0.0 0.0 0.5 0.5 0.0

0.0 0.0 0.1 1.0 0.1

0.0 0.0 0.0 0.5 0.5

0.0 0.0 0.0 0.1 1.0

0.0 0.0 0.0 0.0 0.5

0.0 0.0 0.0 0.0 0.1

0.5 0.0 0.0 0.0 0.0

1.0 0.1 0.0 0.0 0.0

988

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

Table 17 Fuzzy sets in F(º ) for Telugu vowel 

D  D  D  D  D  D  D 

Membership values of u u u u u     

u 

u 

u 

u 

u 

u 

u 

u 

u 

u 

u 

u 

u 

u 

u 

0.1 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.5 0.5 0.0 0.0 0.0 0.0

0.0 0.1 1.0 0.1 0.0 0.0 0.0

0.0 0.0 0.5 0.5 0.0 0.0 0.0

0.0 0.0 0.1 1.0 0.0 0.0 0.0

0.0 0.0 0.0 0.5 0.1 0.0 0.0

0.0 0.0 0.0 0.1 0.5 0.0 0.0

0.0 0.0 0.0 0.0 1.0 0.0 0.0

0.0 0.0 0.0 0.0 0.5 0.1 0.0

0.0 0.0 0.0 0.0 0.1 0.5 0.0

0.0 0.0 0.0 0.0 0.0 1.0 0.0

0.0 0.0 0.0 0.0 0.0 0.5 0.1

0.0 0.0 0.0 0.0 0.0 0.1 0.5

0.0 0.0 0.0 0.0 0.0 0.0 1.0

0.0 0.0 0.0 0.0 0.0 0.0 0.5

0.0 0.0 0.0 0.0 0.0 0.0 0.1

0.5 0.0 0.0 0.0 0.0 0.0 0.0

1.0 0.1 0.0 0.0 0.0 0.0 0.0

0.5 0.5 0.0 0.0 0.0 0.0 0.0

0.1 1.0 0.1 0.0 0.0 0.0 0.0

Table 19 Antecedent and consequent part of the fuzzy If—Then rules for second feature

Table 18 Antecedent and consequent part of the fuzzy If—Then rules for first feature Rule l R  R  R  R  R 

l

Antecedent part F 

Consequent part Membership values of a e i o

u

*

D  D  D  D  D 

0.0 0.0 0.5 1.0 1.0

1.0 1.0 0.1 0.0 0.0

0.0 0.5 1.0 1.0 1.0

0.1 1.0 1.0 1.0 0.5

1.0 1.0 0.5 0.0 0.0

Rule

0.5 1.0 1.0 1.0 0.0

R  R  R  R  R  R  R 

Antecedent part F 

Consequent part Membership values of a e i o

u

*

D  D  D  D  D  D  D 

0.1 1.0 1.0 1.0 0.0 0.0 0.0

1.0 1.0 1.0 0.1 0.0 0.0 0.0

0.0 0.1 1.0 1.0 1.0 0.1 0.0

0.0 0.1 0.5 1.0 1.0 1.0 1.0

0.0 0.0 0.0 0.0 0.5 1.0 1.0

1.0 1.0 1.0 0.1 0.0 0.0 0.0

Table 20 Classification scores of Telugu vowel for the problem of Type I From

a e i o u * Total

To a

e

i

o

I

II

I

II

I

II

I

II

I

II

72 3 0 8 0 27 72

7 1 0 23 1 6 7

4 172 43 6 1 27 172

4 14 90 0 1 3 14

0 9 90 0 0 0 90

0 73 42 0 0 0 42

4 5 0 89 35 4 89

9 8 0 24 76 2 24

0 5 0 13 75 0 75

0 5 0 62 34 1 34

problem of Type I of Section 5. The classification scores are shown in Table 26 for Type I. )

10. Conclusion In this paper we consider a new interpretation (i.e. Eq. (3)) of MFI and introduce a notion of fuzzy pattern

No. of data

*

u

I

II 3 6 0 0 1 8 8

63 99 1 7 0 54 54

83 200 133 116 112 66

Score(%) Partitioning Hard

Fuzzy

86.75 86.00 67.67 76.72 66.96 12.12 71.27

95.18 93.00 99.25 97.41 97.32 93.94 95.92

vector which is formed by the cylindrical extension of the antecedent part of each one-dimensional implication of Eq. (3) and which represents a population of training patterns. We develop a new approach to the computation of the derivative of the fuzzy max- and min-functions. A detail design of the multidimensional pattern classifier based on FRC is developed and very promising results are obtained. Detail treatment on fuzzy pattern

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

dimensional classifier does not suffer from the curse of high dimensionality of patterns.

Table 21 Average formant frequencies of Bengali vowel Phonetic symbol / / / / / / /

u/ o/ */ */ ae/ e/ i/

Average formant frequencies (Hz) F F F    327 438 626 695 681 374 304

935 1015 1095 1326 1663 1935 2095

989

2198 2308 2391 2424 2320 2410 2565

vector will be reported elsewhere. A neural net version of the present design to estimate the fuzzy relation R (for G classification problem) would be the scope for future work. In present design of study we have only considered the problem of Type I. Similar results are obtainable for the problem of Type II. The present design of the multi-

Appendix A A.1. Good function Definition A.1. A function c(x) is said to be a good function if it is differentiable infinitely often everywhere on R and if





dIc(x) "0 lim xP dxI V for every integer k*0 and every integer r*0. Example A.1. Show that the function c(x)"e\V‚ is a good function. Solution. The function c(x)"e\V‚ is differentiable everywhere on R. Now c (x)"!2xc(x). 

Fig. 8. Bengali vowels.

990

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

Table 22 Fuzzy sets in F(º ) for Bengali vowel 

D  D  D  D  D 

Membership values of u u u   

u 

u 

u 

u 

u 

u 

u 

u 

u 

u 

0.3 0.0 0.0 0.0 0.0

0.7 0.7 0.1 0.0 0.0

0.3 1.0 0.3 0.0 0.0

0.1 0.7 0.7 0.1 0.0

0.0 0.3 1.0 0.3 0.0

0.0 0.1 0.7 0.7 0.1

0.0 0.0 0.3 1.0 0.3

0.0 0.0 0.1 0.7 0.7

0.0 0.0 0.0 0.3 1.0

0.0 0.0 0.0 0.1 0.7

0.0 0.0 0.0 0.0 0.1

0.7 0.1 0.0 0.0 0.0

1.0 0.3 0.0 0.0 0.0

Table 23 Fuzzy sets in F(º ) for Bengali vowel 

D  D  D  D  D  D  D 

Membership values of u u u u    

u 

u 

u 

u 

u 

u 

u 

u 

u 

u 

u 

u 

u 

0.3 0.0 0.0 0.0 0.0 0.0 0.0

0.3 1.0 0.3 0.0 0.0 0.0 0.0

0.1 0.7 0.7 0.1 0.0 0.0 0.0

0.0 0.3 1.0 0.3 0.0 0.0 0.0

0.0 0.1 0.7 0.7 0.1 0.0 0.0

0.0 0.0 0.3 1.0 0.3 0.0 0.0

0.0 0.0 0.1 0.7 0.7 0.1 0.0

0.0 0.0 0.0 0.3 1.0 0.3 0.0

0.0 0.0 0.0 0.1 0.7 0.7 0.1

0.0 0.0 0.0 0.0 0.3 1.0 0.3

0.0 0.0 0.0 0.0 0.1 0.7 0.7

0.0 0.0 0.0 0.0 0.0 0.3 1.0

0.0 0.0 0.0 0.0 0.0 0.1 0.7

0.0 0.0 0.0 0.0 0.0 0.0 0.1

0.7 0.1 0.0 0.0 0.0 0.0 0.0

1.0 0.3 0.0 0.0 0.0 0.0 0.0

0.7 0.7 0.1 0.0 0.0 0.0 0.0

Table 24 Antecedent and consequent part of the fuzzy If—Then rules for first feature

Table 25 Antecedent and consequent part of the fuzzy If—Then rules for second feature

Rule

Rule

l R  R  R  R  R 

Antecedent part F 

Consequent part Membership values of a e ae i o

u

*

l

D  D  D  D  D 

1.0 0.7 0.0 0.0 0.0

1.0 1.0 0.7 0.0 0.0

1.0 0.7 0.0 0.0 0.0

R  R  R  R  R  R  R 

1.0 1.0 1.0 1.0 0.3

0.0 0.7 1.0 1.0 0.7

0.0 0.3 1.0 1.0 1.0

0.0 0.0 0.7 1.0 1.0

Using Liebnitz theorem we get, c (x)"!2+xc (x)# L L\ (n!1)c (x),. L\ Using recursive formula we get c (x)"dIc(x)/dxI" I C (x)c(x) where C (x) is the kth degree of polynomial in I I x. Therefore we have





dIc(x) lim xP " lim "C (x)c(x)" I>P dxI V V " lim "C (x) e\V‚". I>P V Now by applying L-Hospital rule we get the above limit P0 as "x"PR. Hence c(x) is a good function.

Antecedent part F 

Consequent part Membership values of a e ae i o

u

*

D  D  D  D  D  D  D 

1.0 1.0 1.0 0.3 0.0 0.0 0.0

0.0 0.0 0.0 0.7 1.0 1.0 1.0

0.0 0.0 0.0 0.0 0.3 1.0 1.0

1.0 1.0 0.1 0.0 0.0 0.0 0.0

1.0 1.0 0.7 0.0 0.0 0.0 0.0

0.7 1.0 1.0 0.3 0.0 0.0 0.0

0.0 0.0 0.7 1.0 1.0 1.0 0.7

Similarly, we see that the function xNe\V‚ (p is a positive integer) is a good function. But a function defined by



c(x)"

0 if x)0, e\V‚ if x'0.

is not good function because c(x) has no derivative at x"0. A good function has the following properties: 1. If c (x) and c (x) are good then c (x)#c (x) and     c (x)c (x) are also good.   2. If c(x) is good then dc(x)/dx is also good.

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

991

Table 26 Classification of Bengali vowel for the problem of Type I From

To a

e

ae

i

o

No. of data

*

u

I

II

I

II

I

II

I

II

I

II

I

II

I

II

a e ae i o u *

33 3 0 0 0 0 0

36 24 0 0 0 1 0

36 66 28 2 0 0 0

33 3 0 0 0 0 0

0 3 28 6 0 0 0

0 42 37 52 0 0 0

0 0 9 50 0 0 0

0 3 28 9 19 0 0

0 0 0 3 36 0 0

0 0 0 0 0 17 0

0 0 0 0 0 67 11

0 0 0 0 3 14 45

0 0 0 0 0 14 45

0 0 0 0 14 49 11

Total

33

36

66

3

28

37

50

9

36

0

67

14

45

11

3. If c(x) is good then c(ax#b) where a and b are real constants, is also a good. 4. It is not always true that the integral of good function is good, e.g.



V

e\R‚ dtP0 as xPR.

\

Score(%) Partitioning

69 72 65 61 36 81 56

Hard

Fuzzy

47.83 91.67 43.08 81.97 100.00 82.72 80.36

100.00 95.83 100.00 96.72 100.00 100.00 100.00

73.86

98.86

Definition A.3. A sequence +c (x), of good functions is L said to be regular if for every given good function c(x), such that



 lim c (x)c(x) dx L L \ exists and is finite,

A.2. Fairly good function Definiton A.2. A function t(x) is said to be a fairly good function if it is differentiable infinitely often everywhere on R and if there is some fixed N such that





dIt(x) "0 lim x\, dxI V for every integer k*0. Most simple example of the fairly good function is eGV but eV is not a fairly good function. A fairly good function has the following properties: 1. If t (x) and t (x) are fairly good then t (x)#t (x)     and t (x)t (x) are also fairly good.   2. If t(x) is fairly good then dt(x)/dx is also fairly good. 3. If t(x) is fairly good then t(ax#b) where a and b are real constants, is also a fairly good.





  lim c (x)c(x) dx" c(x) dx. L L \ \ Example A.2. Show that the sequence +e\V‚L, of good functions is regular. Solution: Now we have the inequality, "e\V‚L!1""



 

V 2x V 2x x e\V‚Ldx ( dx" . n n n  

Then







 e\V‚Lc(x) dx" (e\V‚L!1)c(x) dx \ \



#

 \

c(x) dx.

Now A.3. Generalized function



 \

Generalized function is a extension of the good function in following way: Let c (x) and c(x) be the good L functions then c (x)c(x) is also a good function by propL erties of good function and

since







c (x)c(x) dx exists. L \

 

(e\V‚L!1)c(x) dx (



x/n"c(x)"dxP0

\

as nPR



\

x"c(x)"dx(R because c(x) is a good function.

992

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

Hence





  lim e\V‚Lc(x) dx" c(x) dx L \ \

If the above regular sequences are equivalent then RHS of the above limits are equal; consequently, (A.1)

As the limit exists and is finite for every good function c(x) , so the given sequence is regular. Definition A.4. Two regular sequences which give the same limit are said to be equivalent, i.e. +c (x), and L +c (x), are regular, they are equivalent if and only if L









lim c (x)c(x) dx" lim c (x)c(x) dx. L L L \ L \ Example A.3. Show that two sequences +e\V‚L, and +e\V‚L, of good function are regular and equivalent. Solution: By replacing n by 2n in Eq. (A.1) of Example A.2 we get





  e\V‚Lc(x) dx" c(x) dx lim L \ \





  P g (x)c(x) dx"P g (x)c(x) dx iff g (x)"g (x).     \ \ (A.2) Proposition A.1. ¹he sequence +(n/n e\LV‚, is regular and define a generalized function denoted by d(x) such that



 P d(x)c(x) dx"c(0). \ Proof. Now we have



 

 e\LV‚c(x) dx"c(0) e\LV‚dx \ \  # +c(x)!c(0),e\LV‚dx. \ Again by substituting z"nx we get











1   e\X dz" e\LV‚dx" e\Xz\dz (n  \  (nz



 " lim e\V‚Lc(x) dx. L \ Thus by Definition A.3 the sequence +e\V‚L, is regular and by Definition A.4 the sequences +e\V‚L, and +e\V‚L, are equivalent.

(A.3)



1 C(1/2)" " (n

n . n

(A.4)

Also by the mean value theorem of integral calculus we have





Definition A.5. An equivalence class of regular sequence is a generalized function.

V dc(x) dx )M"x". (A.5) dx  Here c(x) is the good function so dc(x)/dx is also good and is bounded by M (say). Using Eq. (A.5), we get

A conventional notation is to write g as a generalized function associated with the equivalence class of which +c (x), is a typical member, L



"c(x)!c(0)""







 lim c (x)c(x) dx"P g(x)c(x) dx. L L \ \ The arrow emphasizes that a limiting process is involved and that the quantity on the right-hand side is not an ordinary integral. Later on, when certain properties have been established, it will be found reasonable to drop the arrow. Let +c (x), and +c (x), be two regular sequences definL L ing the generalized function g and g , respectively. Thus  



(c(x)!c(0)) e\LV‚dx

\





 M M xe\LV‚dx" [!e\LV‚]" .  n n  Using Eqs. (A.4) and (A.6) in Eq. (A.3) we get )2M

  

\

(A.6)




n M e\LV‚c(x) dx"c(0)#O n (n

and taking limit we get

 

  lim c (x)c(x) dx"P g (x)c(x) dx L  L \ \

 n lim e\LV‚c(x) dx"c(0). n L \ Therefore, the given sequence is regular by Definition A.3 and defines a generalized function d(x) such that

  lim c (x)c(x) dx"P g (x)c(x) dx. L  L \ \

 lim L \

 

 

 



n  e\LV‚c(x) dx"P d(x)c(x) dx"c(0). n \

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

Definition A.6. The sequence +dc (x)/dx, is regular and L defines a generalized function which will be denoted by g(x) and called the derivative of g. The sequence +dc (x)/dx, is regular because L  dc (x) L c(x) dx"[c(x) c (x)] L \ dx \  dc(x) ! c (x) dx. L dx \ But c (x)c(x) being the good function, vanishes as L xP$R and dc(x)/dx is a good function so that









 dc (x)  dc(x) L c(x) dx"!P g(x) lim dx. dx dx L \ \ This also demonstrates that equivalent +c (x), give L equivalent +dc (x)/dx, and may also be written as L   dc(x) P g(x)c(x) dx"!P g(x) dx. dx \ \





A. 4. Calculation of the derivative of the Heaviside function H(x) The Heaviside function is first defined at Section 6.1 (see Eq. (32)). Our objective is to calculate the derivative of the said Heaviside function. By using Definition A.6 we have





  P H(x)c(x) dx"!P H(x)c(x) dx \ \ since c(x) is good so c(x) H(x) is a good function which vanishes as xP$R. Hence





  P H(x)c(x) dx"!P c(x) dx"c(0#). \ > Since c(x) is continuous everywhere on [!R, R] so c(0#)"c(0)"c(0!). Using Proposition A.1 and Eq. (A.2) we get







 H(x)c(x) dx"c(0)"P d(x)c(x) dx. \ \ NH(x)"d(x).

(A.7)

We use this result in Sections 6.1 (see Eqs. (34) and (35)) and 6.2 (see Eqs. (40) and (41)).

References [1] W. Pedrycz, Fuzzy sets in pattern recognition methodology and methods, Pattern Recognition 23 (1/2) (1990) 121—146.

993

[2] T.L. Huntsberger, Ch. Rangarajan, S.N. Jayarammurthy, Representation of uncertainty in computer vision using fuzzy sets. IEEE Trans. Comput. 2 (1986) 145—156. [3] R. Di Mori, Computerized Models of Speech using Fuzzy Algorithms, Plenum Press, New York, 1983. [4] Di Mori, P. Laface, Use of fuzzy algorithms for phonetic and phonetic labeling of continuous speech, IEEE Trans. Pattern Anal. Mach. Intell. 2 (1980) 136—148. [5] D. Dubois, M.C. Jaulent, Some techniques for extracting fuzzy regions, Proc. 1st IFSA Congr. vol. II, Mallorca, Spain, 1—6 July 1985. [6] K. Hirota, K. Iwami, W. Pedrycz, FCM-AD (Fuzzy Cluster Means with Additional Data) and its application to aerial images, II IFSA Congr. Tokyo, vol. II, 1987, pp. 729—732. [7] T.L. Huntsberger, C.L. Jacobs, R.L. Canon, Iterative fuzzy image segmentation, Pattern Recognition 18 (1985) 131—138. [8] W.J.M. Kickert, H. Kopplear, Application of fuzzy set theory to syntactic pattern recognition of handwritten capitals, IEEE Trans. Systems Man Cybernet. 6 (1976) 148—151. [9] P. Siy, C.S. Chen, Fuzzy logic for handwritten numerical character recognition, IEEE Trans. Systems Man Cybernet. 6 (1974) 570—575. [10] M. Shimura, Applications of fuzzy set theory to pattern recognition, J. JAACE 19 (1975) 243—248. [11] G. Bortolan, R. Degani, Ranking of fuzzy alternatives in electrocardiography, in: E. Sanchez, M.M. Gupta (Eds), Fuzzy Information, Knowledge Representation and Decision, Analysis, Pergamon Press, Oxford, 1983, pp. 397—402. [12] G. Bortolan, R. Degani, K. Hirota, W. Pedrycz, Classification of ECG signals— utilization of fuzzy pattern matching, Proc. Int. Workshop Fuzzy Systems Appl. Iizuka, vol. 88, 1988. [13] R. Degani, G. Bortolan, Computerized electrocardiogram diagnosis: fuzzy approach, Encyclopedia of Systems and Control, Pergamon Press, Oxford, 1987. [14] R. Degani, G. Bortolan, Fuzzy numbers in computerized electrocardiography. Fuzzy Sets and Systems 24 (1987) 345—362. [15] W. Pedrycz, A. Gacek, Feature selection for ECG signal classification with the aid of qualitative and quantitative criteria, 7th Hungarian Conf. Biomed. Engng. Esztergom, Hungary, 16—18 September 1987. [16] W. Pedrycz, ECG Signal classification with the aid of linguistic classifier, XIV Int. Conf. Med. Biomed. Engg., Espoo, 11—16 August 1985. [17] E.T. Lee, Proximity measure for the classification of geometric figures, J. Cybernet. 2 (1972) 43—59. [18] L.A. Zadeh, K.S. Fu, K. Tanaka, M. Shimura, (Eds.), Fuzzy Sets and their Applications to Cognitive and Decision Processes, Academic Press, New York, 1975. [19] S. Watanabe, Pattern Recognition: Human and Mechanical, Wiley, New York, 1985. [20] A. Kumar, A real-time system for pattern recognition of human sleep stages by fuzzy systems analysis, Pattern Recognition 9 (1977) 43—46. [21] L. Saitta, P. Tarasso, Fuzzy characteristics of coronary disease, Fuzzy Sets and Systems 5 (1981) 245—258.

994

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

[22] M.A. Woodbury, J. Clive, Clinical pure types as fuzzy partition, J Cybernet. 3 (1974) 111—121. [23] K. Hirota, Fuzzy robot vision and fuzzy controlled robot, in: I.B. Turksen (Ed.), NATO ASI CIM, Springer, Berlin, 1988. [24] A. Seif, J. Aguilar-Martin, Multi-group classification using fuzzy correlation, Fuzzy Sets and Systems 3 (1980) 109—122. [25] R.E. Bellman, R. Kalaba, L.A. Zadeh, Abstraction and pattern classification, J. Math. Anal. Appl. 13 (1966) 1—7. [26] L.A. Zadeh, Fuzzy sets and their applications to classification and clustering, in: J. Van Ryzin (Ed.), Classification and Clustering, Academic Press, New York, 1977, pp. 251—299. [27] J.T. Tou, R.C. Gonzalez, Pattern Recognition Principles, Addison-Wesley, Reading, MA, 1974. [28] T.Kr. Dinda, K.S. Ray, Fuzzy relational calculus approach to multidimensional pattern classification, Internal Research Report, Electronics and Communication Sciences Unit, Indian Statistical Institute, 203, B. T. Road, Calcutta-700 035, India, 1998. [29] J. Ghoshal, Neuro fuzzy reasoning for pattern classification and object recognition, Ph.D. Thesis, Indian Statistical Institute, Calcutta-700 035, India, 1998. [30] K.S. Ray, J. Ghoshal,‘‘Neuro-genetic approach to multidimensional fuzzy reasoning for pattern classification’’, Fuzzy Sets and Systems 1999, in press. [31] K.S. Ray, J. Ghoshal, ‘‘Neurofuzzy approach to pattern recognition’’, Neural Networks 10 (1) (1997) 161—182. [32] K.S. Ray, J. Ghoshal, Approximate reasoning approach to pattern recognition, Fuzzy Sets and Systems 77 (1996) 125—150. [33] Y. Tsukamoto, An approach to fuzzy reasoning method, in: M.M. Gupta, R.K. Ragade, R.R. Yager (Eds.), Advance in Fuzzy Set Theory and Applications North-Holland, Amsterdam, 1979, pp. 137—149. [34] L.A. Zadeh, Theory of application reasoning, in: J.E. Hayes, D. Michie, L.I. Mikulich (Eds.), Machine Intelligence, Ellis Horwood, Chichester, UK, pp. 149—194. [35] W. Pedrycz, Numerical and applications aspects of fuzzy relational equations, Fuzzy Sets and Systems 11 (1983) 1—18. [36] G. Schulz, Fuzzy Rule Based Expert Systems and Genetic Machine Learning, Physica-verlag, Germany, 1995. [37] H.-F. Wang, Numerical analysis on fuzzy relation equations with various operators, Fuzzy Sets and Systems 53 (1993) 155—166. [38] W. Pedrycz, Approximate solution of Fuzzy relational equations, Fuzzy Sets and Systems 28 (1988) 183—201. [39] W. Pedrycz, Applications of fuzzy relational equations for methods of reasoning in presence of fuzzy data, Fuzzy Sets and Systems 16 (1985) 163—174. [40] W. Pedrycz, Processing of relational structures: fuzzy relational equations, Fuzzy Sets and Systems 40 (1991) 77—106. [41] W. Pedrycz, On generalized fuzzy relational equations and their applications, J. Math. Anal. Appl. 107 (1985) 520—536.

[42] N. Ikoma, W. Pedrycz, K. Hirota, Estimation of fuzzy relational matrix by using probabilistic descent method, Fuzzy Sets and Systems 57 (1993) 335—349. [43] H. Hellendoorn, The generalized modus ponens considered as a fuzzy relation, Fuzzy Sets and Systems 48 (1992) 29—48. [44] A. Dinola, W. Pedrycz, S. Sessa, E. Sanchez, Fuzzy relation equations theory as a basis of fuzzy modeling: An overview, Fuzzy Sets and Systems 40 (1991) 415—429. [45] S. Gottwald, Approximately solving fuzzy relation equations: some mathematical results and some heuristic proposals, Fuzzy Sets and Systems 66 (1994) 175—193. [46] WU Wangming, Fuzzy reasoning and fuzzy relational equations, Fuzzy Sets and Systems 20 (1986) 67—78. [47] W. Pedrycz, Genetic algorithms for learning in fuzzy relation structures, Fuzzy Sets and Systems 69 (1995) 37. [48] C.C. Lee, Fuzzy logic control systems: fuzzy logic controller — Parts I & II, IEEE Trans. Systems Man Cybernet 20 (2) 1990. [49] R. Fuller, H.J. Zimmerman, Fuzzy reasoning for fuzzy mathematical programming problems, Fuzzy Sets and Systems 60 (1993) 121—133. [50] M.K. Chakraborty, Some aspect of [0, 1] fuzzy relation and a few suggestions towards its use, in: Approximate Reasoning in Expert Systems, North-Holland, Amsterdam, 1985 pp. 139—157. [51] S.V. Ovchinnikov, T. Riera, On fuzzy classifications, Fuzzy Sets and Systems 28 (1990) 119—132. [52] L.A. Zadeh, Similarity relations and fuzzy orderings, Inform. Sci. 3 (1971) 177—200. [53] K. Hirota, W. Pedrycz, Analysis and synthesis of fuzzy systems by the use of probabilistic sets, Fuzzy Sets and Systems 10 (1983) 1—13. [54] Y. Yoshinari, W. Pedrycz, K. Hirota, Construction of fuzzy models through clustering techniques, Fuzzy Sets and Systems 54 (1993) 157—165. [55] J. Jacas, J. Recasens, Eigen vectors and generators of a fuzzy relations, IEEE Trans. Syst. Man. Cyber. 4 (10) (1992) 687—694. [56] T. Arnold, S. Tano, A rule based method to calculate exactly widest solution sets of a max-min fuzzy relation uniquely, Fuzzy Sets and Systems 64 (1994) 39—58. [57] Nikolai, P. Boboshin, D.G. Naryshkin, On identification of multidimensional fuzzy systems, Fuzzy Sets and Systems 35 (1990) 325—331. [58] A. Blanco, M. Delgado, I. Requena, Identification of fuzzy relational equations by fuzzy neural network, Fuzzy Sets and Systems 71 (1995) 215—226. [59] L. Gorzabz, A. Marin, Extending fuzzy relations, Fuzzy Sets and Systems 69 (1995) 157—169. [60] M. Mizumoto, Extended fuzzy reasoning, in: M.M. Gupta, A. Kandel, W. Bandler, J.B. Kiszka (Eds.), Approximate Reasoning in Expert Systems. North-Holland, Amsterdam, 1985, pp. 71—85. [61] M. Sugeno, T. Takagi, Multidimensional fuzzy reasoning, Fuzzy Sets and Systems 9 (1983) 313—325.

T. Kr. Dinda et al. / Pattern Recognition 32 (1999) 973—995

995

About the Author—TAPAN Kr. DINDA was born in 1965. He received his B.Sc.(Hons.) degree in Mathematics in 1991 and M.Sc. degree in Applied Mathematics in 1993 from Vidyasagar University, India. Currently, he is a Senior Research Fellow, working towards the Ph.D. degree at University of Calcutta, India. He is a member of Indian Unit for Pattern Recognition and Artificial Intelligence (IUPRAI) and Indian Society for Fuzzy Mathematics and Information Processing (ISFUMIP). His research interest includes Pattern Recognition, Fuzzy Sets and Systems, Genetic Algorithms, Neural Networks. About the Author—KUMAR S. RAY was born in 1955. He completed his B.E. degree in 1977 at Calcutta University and his M.Sc. degree in 1980 at the University of Bradford, U.K. and his Ph.D. degree at Calcutta University in 1987. He was a visiting faculty member of the University Texas at Austin under a United Nations Development Program (UNDP) fellowship in 1990. Dr. Ray was a member of task force committee of the Government of India, Department of Electronics (DoE) for the application of AI in power plants. He is the founder member of Indian Society for Fuzzy Mathematics and Information Processing (ISFUMIP) and a member of Indian Unit for Pattern Recognition and Artificial Intelligence (IUPRAI). In 1991, he was the recipient of the K. S. Krishnan memorial award for the best system oriented paper in Computer Vision. He has 50 journal publications to his credit. He is the co-author of two edited volumes of North-Holland publications. At present Dr. Ray is a Professor of Electronics and Communication Sciences Unit, Indian Statistical Institute, Calcutta. Dr. Ray’s field of interest are Control Theory, Computer Vision, AI, Fuzzy Reasoning, Neural Networks, Genetic Algorithms and Qualitative Physics. About the Author—MIHIR Kr. CHAKRABORTY was born in 1945. He has received the M. Sc. degree in Pure Mathematics in 1966 from Calcutta University and the Ph.D. degree in 1976 from Kalyan University, India. He was a visiting scholar in the Department of Mathematics and Statistics, University of Regina, Canada in the early part of 1985. In the early seventies he has published papers on fixed point theory. Dr. Chakraborty is the organizer of Indian Society of Fuzzy Mathematics and Information Processing (ISFUMIP), member of Indian Association for Research in Computer Science, TIFR and member of the editorial board of Journal of Rough Sets and Data Analysis. His current interest includes Fuzzy Set Theory, Linear Logic, Multivalued Logic, Modal Logic, Formalized Languages, Rough Set Theory and Topology.