A Development of Fuzzy Encoding and Decoding ... - Semantic Scholar

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 57, NO. 4, APRIL 2008

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A Development of Fuzzy Encoding and Decoding Through Fuzzy Clustering Witold Pedrycz, Fellow, IEEE, and José Valente de Oliveira

Abstract—Fuzzy clustering has emerged as a fundamental technique of information granulation. In this study, we introduce and discuss multivariable encoding and decoding mechanisms (referred altogether as a reconstruction problem) expressed in the language of fuzzy sets and fuzzy relations. The underlying performance index associated with the problem helps quantify a reconstruction error that arises when transforming a numeric datum through fuzzy sets (relations) and then reconstructing it into an original numeric format. The clustering platform considered in this study concerns the well-known algorithm of Fuzzy C-Means (FCM). The main design aspects deal with the relationships between the number of clusters versus the reconstruction properties and the resulting reconstruction error. The impact of the fuzzification coefficient on the reconstruction quality is investigated. This finding is of interest, given the fact that predominantly all applications involving FCM use the value of the fuzzification coefficient equal to 2. In light of the completed experiments, we demonstrate that this selection may not be experimentally legitimate. We also carry out a comparative analysis of the reconstruction properties of the Boolean decoding that is induced by the fuzzy partition. Experimental investigations involve selected machine learning data. Index Terms—Boolean reconstruction, encoding and decoding, fuzzification coefficient, Fuzzy C-Means (FCM), fuzzy vector quantization, reconstruction error, reconstruction problem.

I. I NTRODUCTION

I

N VARIOUS constructs of fuzzy sets, we are faced with an important problem of information granulation. Information granules—fuzzy sets and fuzzy relations—are the building blocks of fuzzy models, classifiers, and rule-based systems, just to name a few of the representative constructs [14], [16], [20], [21]. There is a clear (though maybe not fully quantified) impression that the number and character of fuzzy sets being specified prior to the detailed development of the models play a pivotal role in the overall quality of the result (in spite of the specific design objective being emphasized within the design). It has also been fully acknowledged that information granules (fuzzy sets) are formed on the basis of existing numeric evidence. This gave rise to the ever-growing importance of Manuscript received January 24, 2006; revised November 4, 2007. This work was supported by the Natural Sciences and Engineering Research Council (NSERC), the Canada Research Chair, and the Portugese Foundation for Science and Technology (FST). W. Pedrycz is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB TGR 2G7, Canada, and also with the Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland (e-mail: [email protected]). J. Valente de Oliveira is with the Algarve Informatics Lab, Universidade do Algarve, 8000-117 Faro, Portugal (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2007.913809

various fuzzy clustering mechanisms and to the relevance of the Fuzzy C-Means (FCM), in particular [3], [9]. In essence, the FCM optimization approach has become the de facto standard used in the formation of information granules. Just recently, in fuzzy clustering, we have started to acknowledge that to make information granules experimentally legitimate and conceptually (semantically) sound, we should resort to the clustering schemes seamlessly integrating numeric data (experimental component) and existing domain knowledge. This conceptualization is usually referred to as knowledge-based clustering [16]. Interestingly, fuzzy sets and information granulation have started to play a more visible role in the setting of a measurement practice, as evidenced by a number of interesting pursuits reported in recent years [2], [5]–[7], [13], [17]. The appealing facet that comes across quite vividly underlines a relevance of the semantics of fuzzy sets as information granules and a sound rapport of fuzzy sets with real-world problems. The point of departure of our discussion is a well-known scheme of vector quantization, which is a principle that has been with us for many decades [1], [4], [8]. It originated in signal processing and communications and afterward became visible in almost every discipline concerned with information measurement and compression, its retrieval, decoding, and reconstruction. Although there are a number of pursuits in this area (cf. [10]–[12]) in the domain of fuzzy sets, this concept, with a few exceptions (see [18] and [19]), has not been fully explored and embraced as a certain design approach being worth implementing. The objective of this study is to raise awareness about the essence of the encoding and decoding processes completed in the context of fuzzy sets. Assuming that fuzzy sets (relations) are the result of fuzzy clustering and of the FCM in particular, we propose and discuss the schemes of encoding and decoding that explicitly relate to the way in which fuzzy clustering is completed. This helps us take full advantage of the information granules formed so far. In light of the changing information granularity occurring within the encoding-decoding scheme, it is evident that this leads to some deterioration of the original data. We introduce a performance index that quantifies a reconstruction error. For the given clustering environment, we explore possible design aspects and show how the parameters of the FCM help to minimize the values of the performance index. The essential components available therein concern the number of clusters (which impact the detail of the granular representation of data) and the values of the fuzzification coefficient (whose values influence the shape of the membership functions).

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Fig. 1. Encoding and decoding of numeric data through information granules. The optimization mechanisms are invoked at the level of fuzzy clusters (granular representation of data).

This research can shed light on the very nature of the processes of encoding and decoding realized in terms of fuzzy sets. The concept of analog-to-digital (A/D) and digital-toanalog (D/A) conversion is fundamental. The analysis of the associated quantization error of the algorithms existing in this domain has been well developed. One may view the investigations in this study as offering a generalization of such an A/D and D/A mapping scheme (in which we encounter interval form of information granules), which is now realized in terms of fuzzy sets. Although the 1-D case has been studied in [15] with the optimal codebook expressed in the form of triangular fuzzy sets with an 1/2 overlap between neighboring fuzzy sets, the issue of a multivariable encoding-decoding mechanism remains open. Interestingly, the terms of encoding and decoding have been explicitly used in fuzzy sets and fuzzy controllers, in particular. They came under the slightly confusing yet maybe quite descriptive terms fuzzification and defuzzification, respectively. The proposed constructs pertained almost exclusively to a scalar case in which we dealt with the processing of scalar numeric entities. Likewise, there were no fundamental results and design guidelines as to the formation of fuzzy sets. This is somewhat surprising, given the fact that there have been a lot of heuristics developed in this domain. The organization of the material is reflective of the main design issues. In Section II, we formulate the problem and, in this context, carefully revisit the mechanisms of fuzzy clustering (Section III). Afterward, we concentrate on the issue of decoding and representation of the ensuing error (Section IV). Experimental studies are offered in Section V, while some concluding observations are covered in Section VI. In the study, we confine ourselves to the standard notation commonly encountered in fuzzy sets. Data (patterns) are regarded as n-dimensional vectors of real numbers, that is, the kth pattern is denoted by xk ∈ Rn . Capital letters are used to denote fuzzy sets and fuzzy relations. II. F ORMULATION OF THE P ROBLEM Fuzzy clusters are multidimensional information granules that reflect upon the experimental data and lead to their conceptual compression. The representation of any numeric datum in the language of information granules formed through the clustering process can be referred to as encoding (see Fig. 1). The complementary mechanism of conversion that produces numeric results is referred to as decoding. The key objective

is to simultaneously optimize the processes of encoding and decoding (in particular, their underlying parameters) so that the result of decoding is made as close as possible to the original numeric entity that has been originally used at the encoding end of the process. Quite often, one can envision some processing being realized in between the encoding and decoding blocks. There are, however, several important scenarios in which no processing takes place, for example, in cases in which we are interested in compression and decompression of data. In general, we could regard the encoding-decoding problem as an optimization of the clustering mechanisms guided by a minimization of some assumed performance index that quantiˆ from the original numeric entry fies a departure (distance) of x ˆ , with . being processed by the encoder (x), namely x − x a certain distance function. The formulation of the problem captures the essence of the numeric-granular-numeric transformation as being encountered along the overall encodingdecoding processing path. Technically, the underlying architectures of these two phases of encoding and decoding have to be specified before moving forward with any further detailed analysis and possible design guidelines. We confine ourselves to the standard version of the FCM. III. FCM: A N O PTIMIZATION E NVIRONMENT In this section, we briefly review the objective function-based fuzzy clustering with the primary objective to highlight its key features useful when dealing with encoding and decoding of information granules. The FCM comes as a standard mechanism aimed at the formation of “c” fuzzy sets (relations) in Rn . The objective function Q guiding the clustering process is expressed as a sum of the distances of individual data from the prototypes v1 , v2 , . . ., and vc . Q=

c
N


2 um ik xk − vi 

(1)

i=1 k=1

In (1),   denotes a certain distance function, and m stands for a fuzzification factor (coefficient) m > 1.0. The resulting partition matrix is denoted by U = [uik ], i = 1, 2, . . . , c; k = 1, 2, . . . , N. While there is a substantial diversity as far as distance functions are concerned, here, we adhere to a weighted Euclidean distance taking the following form: xk − vi 2 =

n
(xkj − vij )2 j=1

σj2

(2)

with σj being a standard deviation of the jth variable. While not being computationally demanding, this type of distance is still quite flexible. The minimization of Q is realized in successive iterations by adjusting both the prototypes and entries of the partition matrix: min Q(U, v1 , v2 , . . . , vc ). The properties of the optimization algorithm are well documented in the literature [1], [2]. In the context of our investigations, we note that the resulting partition matrix is a clear realization of “c” fuzzy relations with the membership functions U1 , U2 , . . . , Uc

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to 2. Lower values of the fuzzification coefficient produce more Boolean-like shapes of the fuzzy sets, where the regions of intermediate membership values are very much reduced. When we increase the values of m above 2, the resulting membership functions start to become “spiky,” with the values close to 1 in a very close vicinity of the prototypes. Beyond this region, the membership values tend to reduce significantly while they approach the value of 1/c. Next, we concentrate on the detailed realization of the encoding and decoding mechanisms. IV. FCM-B ASED E NCODING AND D ECODING : C OMPUTING D ETAILS AND E VALUATION P ROCEDURE The general scheme portrayed in Fig. 1 is now made more detailed as we elaborate on the specific realization of the encoding and decoding schemes. Those are inherently associated with the functional components generated by the FCM. Let us assume that we are provided with a collection of prototypes (v1 , v2 , . . . , vc ) that are generated by running the FCM method on some experimental data. The algorithm is run for some specific values of the setting—that is, the number of clusters and the value of the fuzzification coefficient—and the values of these parameters will be used in the encoding and decoding processes. A. Encoding For any data (pattern) x (which could be a completely new element or an element being used during the clustering process), we obtain its membership grades to the corresponding clusters. This is a representation of x in terms of the information granules U1 , U2 , . . . , Uc . They are denoted by u1 , u2 , . . . , uc and are the result of the following minimization task: c


2 um i vi − x → Min u1 , u2 , . . . , uc

i=1

subject to Fig. 2. Plots of the membership functions produced by the FCM for n = 2 and selected values of the fuzzification coefficient (m), m = 1.1. (a) m = 2.0. (b) m = 3.0. (c) Prototypes are set up as v1 = [1.1 0.95], v2 = [2.5 1.1], and v3 = [3.4 3.7]. The series of graphs display the cluster associated with the first prototype.

forming the corresponding rows of the partition matrix U, that T T is, U = [UT 1 U2 . . . Uc ]. From the design standpoint, there are several essential parameters of the FCM that impact its usage of the produced results. These parameters concern the number of clusters, the values of the fuzzification coefficient, and a form of the distance function. We will concentrate on the two first alternatives. The choice of the optimal (or plausible) number of clusters is guided by various cluster validity indexes; however, in essence, this number has to reflect the specificity of the problem at hand. The fuzzification coefficient exhibits a significant impact on the form of the developed clusters. Some representative examples are shown in Fig. 2. The commonly used value of m is equal

c


ui = 1.

(3)

i=1

(In essence, we can recognize that this minimization is similar to the one we have encountered when dealing with the original FCM problem and optimizing its objective function.) By solving (3) through the use of Lagrange multipliers, we arrive at the expression of the granular representation of the numeric datum: ui =

c j=1



1 x−vi  x−vj 

. 2  m−1

(4)

The vector of the membership grades u(x) = [u1 u2 . . . uc ] is thus a result of encoding; hence, a numeric datum becomes represented in the language of the information granules. We have used the notation u(x) to highlight that u depends directly

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Fig. 3. Plots of the values of F of the reconstruction error for selected values of m. (a) m = 1.1, (b) m = 2.0, and (c) m = 3.5. The prototypes are given as v1 = [1.1 0.95]T , v2 = [2.5 1.1]T , and v3 = [3.4 3.7]T .

upon the numeric input x it encodes. This representation usually leads to a useful compression effect; instead of transmitting (storing) the original pattern x, we need to store “c-1” numeric values taking the values in the [0, 1] interval (the last coordinate of u could be easily computed given the normalization condition). For instance, if n = 100 and c = 5, we have to store vectors whose lengths amount to 20% of the original vectors. B. Decoding The decoding process relies on two components. As before, we use the prototypes and involve the vector of membership grades u(x). The form of the decoding formula results from the minimization of the following expression (decoding error), which is completed with respect to the result of ˆ: decoding x F(x) =

c


ˆ 2 um i (x)vi − x

(5)

i=1

ˆ is positioned in such a way as to In essence, we require that x minimize the distances from the prototypes; the impact of the membership grades in the overall computing of F is noticeable. Assuming the use of the Euclidean distance in (5) and zeroing the gradient of F(x) that has been computed with respect to the encoded vector, we obtain the following expression for the encoded numeric result of u(x): c 

ˆ= x

um i vi

i=1 c 

i=1

(6) um i

In this expression, each prototype is weighted by the corresponding coordinates of u. The fuzzification coefficient becomes an integral part of this aggregation of the prototypes. As an illustration of the quality of encoding and decoding, let us consider three 2-D clusters (prototypes) and assume several values of the fuzzification coefficient. The series of plots in Fig. 3 shows the values of F treated as a function of x. The plots of the decoding error F (i.e., the differences between the

Fig. 4. Detailed insight into the encoding and decoding involving the resulting constructs of the FCM (prototypes).

original numeric entry and its decoded result viz. reconstructed value) are included in Fig. 4. As intuitively anticipated, the values of F are small in some close neighborhoods of the prototypes and start increasing when moving into the regions where we are distant from any prototype. The distribution of error with respect to the values of the fuzzification coefficient is also worth noting. (We will recall this effect when dealing with the detailed experiments.) The decoding error usually assumes nonzero values, which is quite intuitive since we must have introduced some error by using the granular representation of the numeric data. The nonideal decoding and nonzero decoding error are typical for multivariable cases. While this error could be minimized, it cannot be completely eliminated. However, this is not the case in a 1-D case where x ∈ R. It could be easily demonstrated (cf. [15]) that fuzzy sets with triangular membership functions where each two successive fuzzy sets overlap at the level of 0.5 lead to the zero values of the decoding error. This somewhat explains the popularity of the use of triangular fuzzy sets (even though the concept of the encodingdecoding mechanisms is not widely known in the fuzzy-set community). C. Performance Evaluation of the Encoding-Decoding Scheme So far, we have demonstrated how, for some input x, the reconstruction error can be computed. In general, an overall

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Fig. 5. Plots of V (fuzzy and Boolean decoding) versus the fuzzification coefficient (m) and select values of c. (a) c = 2, (b) c = 3, (c) c = 6, and (d) c = 8. The fuzzy decoding produces lower values of V for some values of m. F-fuzzy decoding, B-Boolean decoding. TABLE I RANGES OF THE FUZZIFICATION COEFFICIENT FOR WHICH THE FUZZY DECODING OUTPERFORMS THE BOOLEAN DECODING

Fig. 6. Optimal values of the fuzzification coefficient for different numbers of clusters.

performance of the reconstruction is more representative of the design of the encoding and decoding schemes. To assess this performance, we consider the dataset for which the clustering has been completed. This gives rise to the following index: V=

N


ˆ k 2 . xk − x

(7)

k=1

(Obviously, one could consider some other sets of data for which the testing of the scheme could be realized.) The optimization of V with respect to the number of clusters and the values of the fuzzification coefficient forms the essence of the design activities of the encoding-decoding tandem V = V(c, m).

An interesting alternative to the fuzzy decoding is a Boolean (two-valued) option. Here, instead of considering all the prototypes, we choose the one for which x is the closest and use this prototype in the decoding process. In other words, we choose the index of the prototype i0 , where the following holds: i0 = arg mini x − vi .

(8)

ˆ = vi0 for all The decoded result is the i0 th prototype x x for which (8) holds. The evaluation of the Boolean decoding is quantified by the same performance index as given by (7). Let us highlight the essence of the overall scheme and allude to the way we will be running the experimental part of the study: the two critical parameters of the FCM such as the number of clusters (implying the level of specificity of the decoded information) and the fuzzification coefficient (affecting the geometry of fuzzy sets generated through the clustering process) play an important role in the mechanisms of encoding and decoding.

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Fig. 7. Plots of V (fuzzy and Boolean decoding) versus the fuzzification coefficient (m) and selected values of c. (a) c = 2, (b) c = 3, (c) c = 6, and (d) c = 8. The fuzzy decoding produces lower values of V for some values of m. TABLE II RANGES OF THE FUZZIFICATION COEFFICIENT FOR WHICH THE FUZZY DECODING OUTPERFORMS THE BOOLEAN DECODING

Fig. 8. Optimal values of the fuzzification coefficient versus different number of clusters.

The use of the reconstruction criterion is used to optimize these two design components. V. E XPERIMENTAL S TUDIES In this section, we report on a variety of numeric experiments. Our ultimate goal is to quantify the behavior of the clustering mechanism, treated here as the underlying encodingdecoding scheme. In particular, we are interested in the quantification of the impact being brought by the number of clusters and the values of the fuzzification coefficient. The former implies the level of granularity exercised in the problem. We have a good sense what it could bring to the picture; yet, the quantification of this effect requires some careful attention. The fuzzification coefficient influences the shape of the membership functions, and here, this impact is less clearly delineated; if there were some optimal values of m, those perhaps could link to the nature of the data and also relate to the number of information granules. It is also of interest to contrast the

performance of the fuzzy set and set-based encoding-decoding and report some general tendency that may be present in this regard. For a given dataset, the experiments are carried out by systematically sweeping through the values of the number of clusters and the values of the fuzzification coefficient. Start with c = 2, and increment its value. { sweep through the range of values of m (starting from m = 1.05 and increasing it until the value of 4.0 or 5.0 is reached; above these values, the results produced by the FCM are not significantly affected); for the specific values of c and m, run the FCM algorithm, produce prototypes, and assess the quality of the resulting decoding by returning the value of V } For each combination of the number of clusters and the fuzzification coefficient, we end up with different prototypes on which basis the encoding and decoding becomes completed. The experimentation concerns a number of Machine Learning datasets coming from the Machine Learning repository

PEDRYCZ AND VALENTE DE OLIVEIRA: A DEVELOPMENT OF FUZZY ENCODING AND DECODING THROUGH FUZZY CLUSTERING

Fig. 9.

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V versus the fuzzification coefficient. (a) Abalone, c = 8. (b) Glass, c = 2 and 5. (c) Ionosphere, c = 2 and 4.

(see http://www.ics.uci.edu/~mlearn/MLRepository.html). To assure high confidence in the produced results, for each combination of the parameters of the clustering (the number of clusters and the value of the fuzzification coefficient), the experiments were repeated 40 times. The clustering was run for 50 iterations. (The changes in the maximal value of the successive partition matrices are less than 10−5 .) The following datasets were used in the experiments: Boston housing, autompg, abalone, glass, and ionosphere. The average values of the reconstruction error V were reported, and these results are shown for the fuzzy as well as Boolean decoding. A. Boston Housing The values of V for the fuzzy and Boolean decoding are shown in Fig. 5. The collection of plots shown refers to some selected number of clusters. The tendency is evident. The fuzzy decoding shows an obvious advantage over the Boolean decoding in all cases; yet, this advantage happens only for a certain range of values of the fuzzification coefficient. The optimal value of m is not overly critical, but a range of its optimal values are positioned below what is deemed to be a typical value of 2.0. Fig. 5 summarizes the optimal values of the fuzzification coefficients determined for the different number of clusters. The same tendency is again visible. The region where the fuzzy decoding outperforms the Boolean decoding is located at low values of m. Those values are not higher than 1.6. Once the fuzzification coefficient attains higher values, Boolean decoding is a preferred alternative, producing lower values of the decoding error. The superiority of the fuzzy decoding is more

pronounced for the lower number of the clusters. For example, we see a significant difference between the curves for c = 2, while the differences are not that substantial for c = 8. The monotonicity of V is also quite evident for the lower number of clusters used in the granulation of data. Higher values of c lead to a somewhat more complicated behavior of V being treated as a function of m. The dependency is increasing, but the changes are not always the same, and we encounter some intervals of m where their changes do not substantially impact the values of. The optimal value of m assumes the value of 1.25 for c = 2, 3, and 1.20 for the higher number of the clusters. In this sense, the fuzzification coefficient does not change over the variable level of granularity of information brought into the problem (Fig. 6). It is of interest to find the range of values of the fuzzification coefficient where the fuzzy decoding performs better than its Boolean counterpart (compare Fig. 5 and Table I). For this dataset, the range of values of m producing lower values of V in comparison to those produced by the Boolean decoding is located between 1.0 and 1.5. B. Auto-mpg Data In the experiments, we follow the same setup as in the previous dataset and report the results in Figs. 7 and 8 and in Table II. The general predispositions are very much the same as reported in the previous series of experiments. Optimal values of m are located below the value of 2.0, and with the increased number of the clusters, it shifts toward lower values. In the remaining datasets with which we have experimented (Fig. 9), the

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results are similar and allow us to draw some general observations. The optimal values of m and mopt recorded for these datasets are all very much below 2. We have the following values: abalone c = 8, mopt = 1.4; glass mopt = 1.15 (c = 3, 4, 5) and 1.1 (c = 6); and ionosphere mopt = 1.1 (c = 2, 3) and mopt = 1.15 (c = 4). Typically, fuzzy decoding outperforms its Boolean counterpart in these regions. Furthermore, the advantages of using the fuzzy decoding are reported to be more profound when dealing with a lower number of clusters.

one should become cognizant that the possible advantages of fuzzy clustering do not result automatically because of the use of the concepts of fuzzy sets; rather, these advantages must be actively exploited through prudent optimization. While this study has focused on FCM-based implementation issues, the encoding and decoding occurring between numeric and granular information form a general paradigm of processing and reconstruction of information. Obviously, the detailed realization of the encoding and decoding should be reflective of the original mechanisms of fuzzy clustering applied to the granulation of information.

VI. C ONCLUSION We have formulated and discussed a problem of a granular reconstruction applied to numeric data. The underlying architecture involves two functional modules realizing encoding and decoding between the formats of numeric and granular information. Given the essential implementation of the granulation mechanism realized by means of the FCM, we consistently observed the following three dominant tendencies on how its parameters affect the resulting reconstruction quality. • The number of clusters affects the reconstruction error: The higher the number of clusters, the lower the reconstruction error. Nevertheless, the most significant reduction in the reconstruction error happens when moving from very low values of clusters to some higher values. For already-high numbers of information granules, the effect is less visible. Although the findings along this line are not surprising, the resulting quantification of the entire phenomenon is of interest; the experiments have demonstrated this tendency. • What is very surprising and could be regarded to be of significant practical relevance is the solid experimental evidence that the optimal values of the fuzzification coefficient are typically lower than the commonly used value of 2.0. This sheds light on the optimization facet of information granules and indicates that they need not be that “fuzzy,” as implied by the value of “m” set up to 2.0, where the associated membership functions resemble a Gaussian-like shape. • Two reconstruction mechanisms were introduced and experimented with. One is typical for the “soft” style of aggregation of the prototypes. The other exhibits a Boolean character and adopts a “winner takes all” strategy by reconstructing a numeric datum on a basis of the single closest prototype. The results show that there could be an evident advantage of exploiting the fuzzy partition over using the Boolean one. Yet, the improvement could be seen for some range of the values of the fuzzification coefficient (and those ranges are typically located below the commonly and unjustifiably used value of 2.0). For the higher values of m, the Boolean style of decoding performs better. In this sense, it is important to acknowledge the superiority of fuzzy encoding-decoding scheme over its Boolean counterpart, but this statement cannot be used blindly (as it does not always hold). In this sense,

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PEDRYCZ AND VALENTE DE OLIVEIRA: A DEVELOPMENT OF FUZZY ENCODING AND DECODING THROUGH FUZZY CLUSTERING

Witold Pedrycz (M’88–SM’94–F’99) received the M.Sc., Ph.D., and D.Sci. degrees from the Silesian University of Technology, Gliwice, Poland. He is a Professor and Canada Research Chair (CRC) of Computational Intelligence with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada. He is also with the Polish Academy of Sciences, Systems Research Institute, Warsaw, Poland. His research interests encompass computational intelligence, granular computing, fuzzy modeling, knowledge discovery and data mining, fuzzy control (including fuzzy controllers), pattern recognition, knowledge-based neural networks, relational computing, and software engineering. He has published numerous papers in these areas. He is also author of 11 research monographs and over 250 journal papers. Dr. Pedrycz has been a member of numerous program committees of IEEE conferences in the area of computational intelligence, granular computing, fuzzy sets, and neurocomputing. He currently serves as an Associate Editor for the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, the IEEE TRANSACTIONS ON NEURAL NETWORKS, and the IEEE TRANSACTIONS ON FUZZY SYSTEMS. He is also Editor-in-Chief of Information Sciences and the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A. He is a recipient of the Norbert Wiener Award and the K. S. Fu Award from the NAFIPS.

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José Valente de Oliveira received the “Licenciado,” M.Sc. (in 1992), and Ph.D. degrees in electrical and computer engineering from the IST, Technical University of Lisbon, Portugal. He is an Assistant Professor with the Faculty of Science and Technology of the University of Algarve, Portugal. He is also Director of the UALGiLAB at the University of Algarve Informatics Lab., Faro, Portugal. Dr. Valente de Oliveira is Associate Editor of the Journal of Intelligent and Fuzzy Systems and coeditor of the book Advances in Fuzzy Clustering and Its Applications.