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554 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 14, NO. 3, MAY 2003

Flexible Neuro-Fuzzy SystemsLeszek Rutkowski, Senior Member, IEEE,and Krzysztof Cpalka

Abstract—In this paper, we derive new neuro-fuzzy structurescalled flexible neuro-fuzzy inference systems or FLEXNFIS.Based on the input–output data, we learn not only the parametersof the membership functions but also the type of the systems(Mamdani or logical). Moreover, we introduce: 1) softness tofuzzy implication operators, to aggregation of rules and toconnectives of antecedents; 2) certainty weights to aggregationof rules and to connectives of antecedents; and 3) parameterizedfamilies of T-norms and S-norms to fuzzy implication operators,to aggregation of rules and to connectives of antecedents. Ourapproach introduces more flexibility to the structure and designof neuro-fuzzy systems. Through computer simulations, we showthat Mamdani-type systems are more suitable to approximationproblems, whereas logical-type systems may be preferred forclassification problems.

Index Terms—Certainty weights, logical approach, Mamdaniapproach, neuro-fuzzy inference systems (NFIS).

I. INTRODUCTION

I N the last decade, various neuro-fuzzy systems have beendeveloped (see, e.g., [4]–[7], [10], [19], [21], [26], [27],

[29]–[33], [37]–[39], [41]–[48], [64]–[66]). They combine thenatural language description of fuzzy systems and the learningproperties of neural-networks. Some of them are known in theliterature under short names like ANFIS [20], ANNBFIS [7],DENFIS [23], FALCON [31], GARIC [2], NEFCLASS [38],NEFPROX [37], [38], SANFIS [57] and others. In this paper,we study a wide class of fuzzy systems trained by the backpropagation method. Following other authors we call themneuro-fuzzy inference systems (NFIS). To emphasize theirmain feature-flexibility, we also use name FLEXNFIS.

In the literature to date, two approaches [7], [44], [58], [67]have been proposed to design fuzzy systems.

1) The first approach, called the Mamdani method, uses con-junction for inference and disjunction to aggregate indi-vidual rules. In the Mamdani approach, the most widelyused operators measuring the truth of the relation betweeninput and output are the following:

(1)

and

(2)

or more generally

(3)

Manuscript received June 20, 2002; revised February 18, 2003.The authors are with the Department of Computer Engineering, Technical

University of Czestochowa, Czestochowa 42 200, Poland.Digital Object Identifier 10.1109/TNN.2003.811698

It should be emphasized that formulas (1) and (2) donot satisfy the conditions of fuzzy implication formu-lated by Fodor [11]. We refer to (1) and (2) as to “engi-neering implications” (see Mendel [34], [35]) contrary tothe fuzzy implications satisfying the axiomatic definition(see Definition 1).

The aggregation is performed by an application ofS-norm

(4)e.g.,

(5)

It should also be noted that in most cases the aggrega-tion of rules is performed as a part of defuzzification (see,e.g., [35] and [58]).

2) The second paradigm applies fuzzy implications to infer-ence and conjunction to aggregation.

Definition 1. (Fuzzy Implication):A fuzzy implication is afunction satisfying the following conditions:

(I1) if , then , for all

(I2) if , then , for all

(I3) , for all (falsity implies any-thing).

(I4) , for all (anything impliestautology).

(I5) (Booleanity).Selected fuzzy implications satisfying the above conditions arelisted in Table I. In this table, implications 1–4 are examples ofan S-implication associated with an S-norm

(6)

e.g.,

(7)

For fuzzy systems with a logical implication, the aggregation isrealized by a T-norm

(8)e.g.,

(9)

Neuro-fuzzy inference systems of a logical-type are describedin Section III-B. It should be noted that the aggregation of an-tecedents in each rule is performed by the same formula (8) forboth Mamdani and logical-type systems.

1045-9227/03$17.00 © 2003 IEEE

RUTKOWSKI AND CPALKA: FLEXIBLE NEURO-FUZZY SYSTEMS 555

TABLE IFUZZY IMPLICATIONS

It was emphasized by Yager [64], [65] that “no formal reasonexists for the preponderant use of the Mamdani method in fuzzylogic control as opposed to the logical method other than in-ertia.” Moreover, Yager said [66] that “as a matter of fact theMamdani approach has some disadvantages: its inability to dis-tinguish more specific information in the face of the rules spanthe whole input space.” This statement was an inspiration for usto determine the type of fuzzy inference (Mamdani or logical)in the process of learning. We decided to study the problem, de-spite the widely held belief about the inferiority of the logicalmethod (see Remark 2 in Section III).

In this paper, we present a novel approach to fuzzy modeling.The novelty is summarized as follows.

1) We propose a new class of NFIS characterized by auto-matic determination of a fuzzy inference (Mamdani/log-ical) in the process of learning. Consequently, the struc-ture of the system is determined in the process of learning.This class is based on the definition of an H-functionwhich becomes a T-norm or S-norm depending on a cer-tain parameter which can be found in the process oflearning. We refer to this class asOR-type fuzzy systems.

2) We developAND-type neuro-fuzzy inference systems bymaking use of the concept of flexible structures studied byYager and Filev [67]. TheAND-type fuzzy inference sys-tems exhibit simultaneously Mamdani and logical typeinferences.

3) We introduce• softness to fuzzy implication operators, to aggrega-

tion of rules and to connectives of antecedents;• certainty weights to aggregation of rules and to con-

nectives of antecedents;• parameterized families of T-norms and S-norms to

fuzzy implication operators, to aggregation of rulesand to connectives of antecedents

TABLE IIAND-TYPE SYSTEM

in both AND-type and OR-type neuro-fuzzy inferencesystems.

4) Through computer simulations we show that Mamdani-type systems are more suitable to approximation prob-lems, whereas logical-type systems may be preferred forclassification problems. Moreover, we observe that themost influential parameters in FLEXNFIS are certaintyweights (introduced in this paper in a novel way) inconnectives of antecedents and in aggregations of rules.They significantly improve the performance of NFIS inthe process of learning.

This paper is organized into eight sections. In the next sec-tion, we discuss and propose various flexibility issues in NFIS.In Section III, a formal description of NFIS is presented, whichalso provides a general architecture [Fig. 2 and formula (44)] ofall systems (flexible and nonflexible) studied in this paper. InSection IV, we introduce an H-function and give a frameworkfor the description, unification and development of NFIS. TheOR-type andAND-type FLEXNFIS are studied in Sections V andVI, respectively. Section VII shows the simulation results andcomparative studies with other neuro-fuzzy systems. Conclu-sions and discussions are drawn in Section VIII.

II. FLEXIBILITY IN NFIS

A. AND-Type Compromise NFIS

Obviously, the Mamdani and the logical systems lead to dif-ferent results and, in the literature, there are no formal proofsas to which of them is superior. Therefore, Yager and Filev [67]proposed to combine both methods. TheAND-type compromiseNFIS is characterized by the simultaneous appearance of Mam-dani-type and logical-type systems. In this paper, we study thefollowing combination of “engineering” and fuzzy implications

(10)

e.g.,

(11)

In Section VI, we develop compromise NFIS based on formula(10). It should be emphasized that parametercan be found inthe process of learning subject to the constraint . InSection VII, based on the input-output data, we learn a systemtype starting from as an initial value. The behavior oftheAND-type compromise NFIS is depicted in Table II.

B. OR-Type NFIS

OR-type NFIS have recently been proposed by Rutkowskiand Cpalka [46], [47]. Depending on a certain parameterthisclass of systems exhibits “more Mamdani” ( )


TABLE IIIOR-TYPE SYSTEM

or “more logical” ( ) behavior. At the bound-aries the system becomes more of a Mamdani-type ( )or logical-type ( ). The definition of OR-type systemsheavily relies on the concept of an H-function (see Section IVand Rutkowski and Cpalka [46], [47]). The H-function exhibitsthe behavior of fuzzy norms. More precisely, it is a T-norm for

and S-norm for . For the H-functionresembles a T-norm and for the H-function resem-bles an S-norm. In a similar spirit, we construct OR-type impli-cations. The parametercan be found in the process of learningsubject to the constraint . In Section VII, basedon the input-output data, we learn a system type starting from

as an initial value. The behavior of theOR-type sys-tems is shown in Table III (see Section IV for details). Observethat this system-contrary to theAND-type system-does not ex-hibit simultaneously Mamdani and logical features. It is strictlyanOR-type system. TheOR-type NFIS are studied in Section V.

C. Soft NFIS

The soft versions of operators (8) and (4) were proposed byYager and Filev [67]. They are defined as follows:

(12)

and

(13)

where .In the same spirit, we define softening of “engineering impli-

cation” (3) by

(14)

and logical “fuzzy implication” (6) by

(15)

where .The soft compromise NFIS are studied in Sections V-B, V-C,

VI-B, and VI-C.

D. NFIS Realized by Parameterized Families ofT-Norms and S-Norms

Most fuzzy inference structures studied in the literature em-ploy the triangular norms shown in Table IV. There is only a littleknowledge within the engineering community about so-calledparameterized families of T-norm and S-norms. They include

TABLE IVBASIC TRIANGULAR NORMS

the Aczél-Alsina, Dombi, Dubois-Prade, Frank, Hamacher,Schweizer-Sklar, Sugeno-Weber, and Yager families [28].

It should be noted that these parameterized families includethe triangular norms listed in Table IV. For example, the Dombifamily is defined as follows:

1) the Dombi T-normififif (16)

where stands for the T-norm of a Dombi family pa-rameterized by ;

2) the Dombi S-normififif

(17)where stands for the S-norm of a Dombi family pa-rameterized by .

The parameter can be found in the process oflearning.

Obviously formula (16) defines the “engineering implica-tion.” Combining (6) and (17) we get the fuzzy S-implicationgenerated by the Dombi family

(18)

The NFIS realized by parameterized families of T-norms andS-norms are studied in Sections V-B, V-C, VI-B, and VI-C.

E. NFIS Realized by T-Norms and S-Norms With WeightedArguments

In this paper, we propose the weighted T-norm

(19)Parameters and can be interpreted as antecedents of a rule.The weights and are corresponding certainties (credibil-ities) of both antecedents.


Observe the following.

1) If , then the weighted T-norm (19) is re-duced to the standard T-norm. In the context of linguisticvalues we assign the truth to both antecedentsandof the rule.

2) If , then

(20)

Therefore, the antecedent is discarded since its cer-tainty is equal to zero. Similarly, if then the an-tecedent vanishes

(21)

3) If and then we assume a partialcertainty of antecedents and .

The S-norm corresponding to the T-norm (19) is defined as fol-lows:

(22)

In the same spirit we propose the weighted triangular norms

(23)

and

(24)

to aggregate individual rules in Mamdani-type and logical-typesystems, respectively. The weights and in (19), as wellas and in (23) or (24), can be found in the process oflearning subject to the constraints , , , .In Sections V-C and VI-C we apply the weighted T-norm (19)to a selection of significant inputs, and the weighted S-norm(23) or T-norm (24) to a selection of important rules. The resultsare depicted in the form of diagrams in Section VII (dark areascorrespond to low values of weights and vice versa).

Remark 1: It was pointed out by one of the reviewers thatdesigning of neuro-fuzzy systems should be a compromisebetween accuracy of the model and its transparency. The mea-sure of accuracy is usually the RMSE-criterion (approximationproblems) and percentage of correct or wrong decisions (classi-fication problems). The measure of transparency is the numberand form of fuzzy rules obtained. It was mentioned by severalauthors (see, e.g., [1] and [14]) that the lack of transparency is amajor drawback of many neuro-fuzzy systems. Most designersfocus their effort on approximation accuracy, while the issueof transparency has received less attention. In this context ourmethod of weighted triangular norms seems to be a promisingtool for extracting both transparent and accurate rule-basedknowledge from empirical data. More specifically, diagrams(weights representation) presented in Section VII can be usedfor the analysis and pruning of the fuzzy-rule bases in all thesimulation examples. The FLEXNFIS realized by T-norms andS-norms with weighted arguments are studied in Sections V-Cand VI-C. Note that our application of weights in NFIS isdifferent from those studied in [16], [36], [55], and [69].

III. FORMAL DESCRIPTION OF THENFIS

In this paper, we consider multi-input–single-output fuzzyNFIS mapping , where and .

The fuzzifier performs a mapping from the observed crispinput space to the fuzzy sets defined in . The mostcommonly used fuzzifier is the singleton fuzzifier which maps

into a fuzzy set characterizedby the membership function

ifif

(25)

The fuzzy rule base consists of a collection offuzzy IF-THENrules, aggregated by disjunction or conjunction, in the form

IF is ANDis ANDis

THEN is

(26)

or

IF is THEN is (27)

where , , ,are fuzzy sets characterized by membership

functions , , , whereasare fuzzy sets characterized by membership functions ,respectively, .

The fuzzy inference determines a mapping from the fuzzysets in the input space to the fuzzy sets in the output space

. Each of rules (26) determines a fuzzy set givenby the compositional rule of inference

(28)

where .Fuzzy sets , according to the formula (28), are character-

ized by membership functions expressed by thesup-starcom-position

(29)

where can be any operator in the class of T-norms. It is easilyseen that for a crisp input , i.e., a singleton fuzzifier (25),formula (29) becomes

(30)

where is an “engineering implication” or fuzzy implication.The aggregation operator, applied in order to obtain the fuzzy

set based on fuzzy sets , is the T-norm or S-norm operator,depending on the type of fuzzy implication.

The defuzzifier performs a mapping from a fuzzy setto acrisp point in . The COA (centre of area) method isdefined by the following formula:

(31)


Fig. 1. Illustration of inference based on the binary implication, and Zadeh,product, and Lukasiewicz aggregations.

or by

(32)

in the discrete form, where denotes centres of the member-ship functions , i.e., for

(33)

For other definitions of the defuzzifier, the reader is referred to[7].

Remark 2: Several authors (e.g., Jager [18], Mendel [35])reported problems with the application of logical implicationsto NFIS. A major problem is caused by the indeterminant partof the membership function. We illustrate such a situation inFig. 1, showing the inference for binary implication (7). Theaggregation is performed by making use of Zadeh T-norm,product T-norm and Lukasiewicz T-norm as listed inTable IV. Observe that there is no indeterminancy in the case ofLukasiewicz T-norm applied to aggregation.

The indicated problem can be easily resolved by the applica-tion of a modified center of gravity defuzzifier

(34)

where

(35)

The value describes the indeterminancy that accom-panies the corresponding part of information. It is easily seenthat in order to eliminate the indeterminant part of the member-ship function , the informative part has to be parallelyshifted downward by the value of. Neuro-fuzzy inference sys-tems of a logical-type with defuzzifier (34) have been studied byCzogala and Leski [7].

Depending on implication (30), two types of NFIS can bedistinguished.

A. Nonflexible NFIS: Mamdani-Type

In this approach, the implication (30) is a T-norm (e.g., min-imum, product, Dombi)

(36)

and the aggregated output fuzzy set is given by

(37)Consequently, (32) takes the form

(38)

Obviously, the T-norms used to connect the antecedents in therule and in the “engineering implication” do not have to be thesame. Besides, they can be chosen as differentiable functionslike Dombi families.

Remark 3: If1) the implication is of a Mamdani-type;2) for ;

then formula (38) reduces to the well-known fuzzy systemstudied by Wang [58]

(39)

B. Nonflexible NFIS: Logical-Type

In this approach, the fuzzy implication (30) is an S-implica-tion in the form

(40)

e.g., binary implication (known as the Kleene–Dienes implica-tion)

(41)The aggregated output fuzzy set is given by

(42)

and formula (32) becomes

(43)Now, we generalize both approaches described in points

and and propose a general architecture of NFIS. It is easilyseen that the systems (38) and (43) can be presented in the form

(44)


Fig. 2. General architecture of NFIS studied in this paper (flexible andnonflexible).

where

for Mamdani approach

for logical approach

(45)and

for Mamdani approachfor logical approach

(46)

Moreover, the firing strength of rules is given by

(47)

The general architecture of (44) is depicted in Fig. 2.Remark 4: It should be emphasized that (44) and the scheme

depicted in Fig. 2 are applicable to all the systems, flexible andnonflexible, studied in this paper with different definitions of

and . Nonflexible systems are describedby (44), (45), (46) and (47), whereas flexible systems by (44)and , , defined in Sections V andVI. How we define the aggregation operator andthe implication operator , depends on the particularclass of the system.

Remark 5: It is well known that the basic concept of thebackpropagation algorithm, commonly used to train neural net-works, can be also applied to any feedforward network. Let

and be a sequence of inputs and de-sirable output signals, respectively. Based on the learning se-quence we wish to determineall parameters (including the system’s typeor ) and weightsof NFIS such that

(48)

is minimized, were is given (44). The steepest descent op-timization algorithm can be applied to solve this problem. For

Fig. 3. 3-D plot of function (53).

instance, the parameters, , are trained by theiterative procedure

(49)

Directly calculating partial derivatives in recursion like (49) israther complicated. Therefore, we recall that our NFIS has a lay-ered architecture (Fig. 2) and apply the idea of back propagationto train the system. The exact recursions reflect that idea, how-ever, they are not a copy of the standard backpropagation. Fordetails, the reader is referred to our previous paper [47].

IV. FRAMEWORK FOR DESCRIPTION, UNIFICATION, AND

DEVELOPMENT OFNFIS

In this section the following properties of dual T-norms anddual S-norms will be used

(50)

(51)

Our goal is to find a framework for the description, unificationand development of all systems studied in this paper. We achievethis goal using two definitions (see Rutkowski and Cpalka [46]).

Definition 2. (Compromise Operator):A function

(52)

given by

(53)

is called a compromise operator where and.

Observe that

forforfor

(54)

Obviously, function is a strong negation (see, e.g., [28]) for. The 3-D plot of function (53) is depicted in Fig. 3.

Remark 6: The formula (50) can be rewritten with the nota-tion of definition 2

(55)


for . Apparently

(56)

for .The right-hand sides of (55) and (56) can be written as fol-

lows:

(57)

with or . One may wish to vary the parameterin(57) from 0 to 1. This concept leads to the following definition,allowing us to switch smoothly between S-norm and T-norm:

Definition 3. (H-Function): A function

(58)

given by

(59)is called an H-function where .

Observe thatforforfor

(60)

It is easily seen that for the H-function resemblesa T-norm and for the H-function resembles anS-norm.

Example 1. (An Example of H-Function):We will show howto switch smoothly from T-norm to S-norm by making use ofdefinition 3. Let and the standard min-norm and max-conorm are chosen

(61)

(62)

The H-function generated by formulas (61) and (62) takes theform

(63)

and varies form (61) to (62) asgoes from zero to one.In Fig. 4, we illustrate function (63) for , ,

, , .Example 2. (An Example of H-Implication):In this example,

we illustrate how an H-implication based on definition threechanges from “engineering implication” (1) to fuzzy implica-tion (7). Let

(64)

and

(65)

Fig. 4. 3-D plot of function (63) for (a)� = 0:00. (b)� = 0:15. (c)� = 0:50.(d) � = 0:85. (e)� = 1:00.

Fig. 5. 3-D plot of function (66) for (a)� = 0:00. (b)� = 0:15. (c)� = 0:50.(d) � = 0:85. (e)� = 1:00.

Then

(66)

goes from (64) to (65) as varies from zero to one.In Fig. 5, we illustrate function (66) for , ,

, , .Remark 7: It is easily seen that the nonflexible NFIS given

by formulas (45)–(47) can be alternatively presented by making


use of definition 3 with or , shown in (67)–(69) atthe bottom of the page.

V. OR-TYPE FLEXNFIS

The OR-type NFIS are based on definition 3 of the H-func-tion. All the systems in this section are described by a generalformula (44), see remark 4, with various definitions of ,

and .

A. Basic NFIS:OR-Type

The basic neuro-fuzzy system of anOR-type is given as fol-lows:

(70)

(71)

(72)

Observe that system (70)–(72) is Mamdani-type for , likeMamdani-type for , undetermined for ,like logical-type for and logical-type for .It is worth noticing that parametercan be learned and conse-quently the type of the system can be determined in the processof learning.

B. Soft NFIS:OR-Type

In this section we propose soft NFIS based on soft fuzzynorms (12) and (13). These systems are characterized by

1) soft strength of firing controlled by parameter;2) soft implication controlled by parameter ;3) soft aggregation of rules controlled by parameter .

Moreover, we assume that fuzzy norms (and H-function) in theconnection of antecedents, implication and aggregation of rulesare parameterized by parameters, , , respectively.We use notation to indicate parameterized familiesanalogously to (16) and (17).

The soft compromise NFIS of anOR-type are defined as fol-lows:

(73)

(74)

(75)Observe that system (73)–(75) is

1) soft Mamdani-type NFIS for ;2) soft logical-type NFIS for ;3) soft like Mamdani-type NFIS for ;4) soft like logical-type NFIS for ;5) undetermined for .

C. Weighted Soft NFIS:OR-Type

We insert weights to the antecedents and to the aggregationoperator of the rules in systemORII:

1) , , ;2) , .

Consequently, we get the weighted soft NFIS of anOR-type

(76)

(77)

(78)

In the ORIII system we use parameterized families and

parameterized families with weights analogously to (19).

(67)

for Mamdani approach

for logical approach(68)

for Mamdani approachfor logical approach

(69)


More specifically, in (76) and (78), we use the following defini-tion:

(79)

where

(80)

VI. AND-TYPE NFIS

In this section, we studyAND-type neuro-fuzzy inference sys-tems. They will be presented in two alternative forms: by usingT and S-norms or by using an H-function with or .

A. Basic NFIS:AND-Type

The basic neuro-fuzzy inference systems of anAND-type em-ploy combinations of “engineering” and fuzzy implication, see,e.g., (10) and (11). The systems are given by the formula:

(81)

(82)

(83)

or

(84)

(85)

(86)

It is easily seen that the above system is of a Mamdani-type forand logical-type for .

B. Soft NFIS:AND-Type

In this section, we propose soft compromise NFIS based onsoft fuzzy norms (12) and (13).

The soft compromise NFIS of anAND-type are given by

(87)

(88)

(89)

Formulas (87)–(89) describe the soft compromise NFIS –AND-type in terms of parameterized families and . Al-ternatively, this system can be presented by making use of theH-function definition

(90)

(91)


(92)

C. Weighted Soft NFIS:AND-Type

Introducing weights to soft NFIS we get the weighted softNFIS of anAND-type

(93)

(94)

(95)

Alternatively, it can be expressed by

(96)

(97)

TABLE VEXPERIMENTAL RESULTS

TABLE VICOMPARISONTABLE

(98)


Fig. 6. Weights representation in the Glass Identification problemw 2

[0; 1], w 2 [0; 1], i = 1; . . . ; 9, k = 1; . . . ; 2 (dark areas correspond tolow values and vice versa).

TABLE VIIEXPERIMENTAL RESULTS

VII. PERFORMANCEEVALUATION

We have conducted very extensive simulations in order to testour new structures, find an appropriate system type and comparethe results with other authors. The results are summarized anddiscussed in Section VIII. We present 11 simulations by making

TABLE VIIICOMPARISONTABLE

TABLE IXEXPERIMENTAL RESULTS

use of commonly known benchmarks. Each of the simulationsis designed in the same fashion:

1) In the first experiment, based on the input-output data, welearn the parameters of the membership functions and asystem type of theOR I neuro-fuzzy inferencesystem. It will be seen that the optimal values of, deter-mined by a gradient procedure, are either zero or one.

2) In the second experiment, we learn the parameters of themembership functions and a system type of theAND I neuro-fuzzy inference system. It will be seen thatoptimal values of , determined by a gradient procedure,are either zero or one.


TABLE XCOMPARISONTABLE

Fig. 7. Weights representation in the Iris problemw 2 [0; 1], w 2

[0; 1], i = 1; . . . ; 4, k = 1; . . . ; 2 (dark areas correspond to low values andvice versa).

3) In the third experiment, we learn the parameters of themembership functions of theOR I/AND I neuro-fuzzy in-ference systems choosing values ofand as oppositeof those obtained in 1) and 2). Obviously, we expect aworse performance of both systems. Note thatOR-typeare equivalent toAND-type systems if or

.4) In the fourth experiment, we learn the parameters of the

membership functions, system type of theORIIneuro-fuzzy inference system and soft parameters

, , . Moreover, we learnparameters , , ofthe Dombi norm used for the connection of antecedents,implication and aggregation of rules, respectively.

5) In the fifth experiment, we learn the same parameters as inthe fourth experiment and, moreover, the weights

, , , in the antecedentsof rules and weights , , of theaggregation operator of the rules. In all diagrams (weightsrepresentation) we separate , ,

, from , , by avertical dashed line.

TABLE XIEXPERIMENTAL RESULTS

TABLE XIICOMPARISONTABLE

The parameters learned in experiments 1)-5) can be determinedby standard recursive gradient procedures with the constraintslisted above. In order to avoid arduous gradient calculations, wehave developed [47] a universal network trainer that can tune theparameters and weights of FLEXNFIS based on their architec-tures. The idea of the trainer comes from the backpropagationmethod. It should be noted that Gaussian membership functionsare used in all the experiments.

A. Glass Identification

The Glass Identification problem [56] contains 214 instancesand each instance is described by nine attributes (RI: refractiveindex, Na: sodium, Mg: magnesium, Al: aluminum, Si: silicon,K: potassium, Ca: calcium, Ba: barium, Fe: iron). All attributesare continuous. There are two classes: window glass and non-window glass. In our experiments, all sets are divided into alearning sequence (150 sets) and testing sequence (64 sets). Thestudy of classification of types of glass was motivated by crim-inological investigation. At the scene of the crime, the glass left


Fig. 8. Weights representation in the HANG problemw 2 [0; 1], w 2


TABLE XIIIEXPERIMENTAL RESULTS

can be used as evidence if it is correctly identified. The experi-mental results are depicted in Table V, Table VI, and Fig. 6.

B. Ionosphere

This radar data was collected by a system in Goose Bay,Labrador [56]. This system consists of a phased array of 16 high-

TABLE XIVCOMPARISONTABLE

Fig. 9. Weights representation in the Modeling of Box and Jenkins GasFurnace problemw 2 [0; 1], w 2 [0; 1], i = 1; . . . ; 6, k = 1; . . . ; 4(dark areas correspond to low values and vice versa).

frequency antennas with a total transmitted power in the orderof 6.4 kW. The targets were free electrons in the ionosphere.The database is composed of 34 continuous attributes plus theclass variable, using 351 examples. In our experiments, all setsare divided into a learning sequence (246 sets) and testing se-quence (105 sets), , and after learning all weightsare equal to one. The experimental results are depicted in Ta-bles VII and VIII.

C. Iris

The Iris data [56] is a common benchmark in classificationand pattern recognition studies. It contains 50 measurements offour features (sepal length in cm, sepal width in cm, petal lengthin cm, petal width in cm) from each of three species: iris setosa,iris versicolor, and iris virginica. In our experiments, all sets aredivided into a learning sequence (105 sets) and testing sequence(45 sets). The experimental results are depicted in Table IX,Table X, and Fig. 7.


TABLE XVEXPERIMENTAL RESULTS

TABLE XVICOMPARISONTABLE

D. Modeling of a Static Nonlinear Function (HANG)

In this example, a double-input and single output static func-tion is chosen to be a target system for the new fuzzy modelingstrategy. This function is represented as

(99)

From the evenly distributed grid point of the input rangeof the preceding equation, 50 training data pairs were

Fig. 10. Weights representation in the nonlinear dynamic plant problemw 2 [0; 1], w 2 [0; 1], i = 1; . . . ; 2, k = 1; . . . ; 5 (dark areascorrespond to low values and vice versa).

TABLE XVIIEXPERIMENTAL RESULTS

obtained. The experimental results are depicted in Table XI,Table XII, and Fig. 8.

E. Modeling of Box and Jenkins Gas Furnace

The Box and Jenkins Gas Furnace data consists of 296 mea-surements of a gas furnace system: the input measurement


TABLE XIXEXPERIMENTAL RESULTS

TABLE XXCOMPARISONTABLE

is the gas flow rate into the furnace and the output measure-ment is the CO concentration in outlet gas. The sampling in-terval is 9 s. The experimental results are depicted in Table XIII,Table XIV, and Fig. 9.

TABLE XVIIICOMPARISONTABLE

Fig. 11. Weights representation in the Pima Indians Diabetes problemw 2

[0; 1], w 2 [0; 1], i = 1; . . . ; 8, k = 1; . . . ; 2 (dark areas correspond tolow values and vice versa).

F. Nonlinear Dynamic Plant

We consider the second-order nonlinear plant studied byWang and Yen [61]

(100)

with

(101)The goal is to approximate the nonlinear component

of the plant with a fuzzy model.In [61], 400 simulated data were generated from the plantmodel (101). Starting from the equilibrium state (0, 0), 200samples of identification data were obtained with a random


Fig. 12. Weights representation in the Rice Taste problemw 2 [0; 1],w 2 [0; 1], i = 1; . . . ; 5, k = 1; . . . ; 5 (dark areas correspond to lowvalues and vice versa).

TABLE XXIEXPERIMENTAL RESULTS

input signal uniformly distributed in [ 1.5, 1.5], followedby 200 samples of evaluation data obtained using a sinusoidalinput signal . The experimental results aredepicted in Table XV, Table XVI, and Fig. 10.

TABLE XXIICOMPARISONTABLE

Fig. 13. Weights representation in the Monk 1 problemw 2 [0; 1],w 2


G. Pima Indians Diabetes

The Pima Indians Diabetes data [56] contains two classes,eight attributes (number of times pregnant, plasma glucose con-centration in an oral glucose tolerance test, diastolic blood pres-sure (mm Hg), triceps skin fold thickness (mm), 2-h serum in-sulin (mu U/ml), body mass index [weight in kg/(height in m)),diabetes pedigree function, age (years)]. We consider 768 in-stances, 500 (65.1%) healthy and 268 (34.9%) diabetes cases.All patients were females at least 21 years old, of Pima Indianheritage. In our experiments, all sets are divided into a learningsequence (576 sets) and testing sequence (192 sets). The exper-imental results are depicted in Table XVII, Table XVIII, andFig. 11.

H. Rice Taste

The Rice Taste data contains 105 instances and each instanceis described by five attributes: flavor, appearance, taste, sticki-ness, toughness, and overall evaluation. In simulations the input-output pairs of the rice taste data were normalized in the in-terval [0, 1]. The experimental results are depicted in Table XIX,Table XX, and Fig. 12.

I. The Three Monk’s Problems

The Three Monk’s Problems [56] are artificial, small prob-lems designed to test machine learning algorithms. Each of the


TABLE XXIIIEXPERIMENTAL RESULTS

TABLE XXIVCOMPARISONTABLE

three monks problem requires determining whether an objectdescribed by six features (head shape, body shape, is smiling,holding, jacket color, has tie) is a monk or not.

There are 432 combinations of the six symbolic attributes.In the first problem (Monk1), 124 cases were randomly se-lected for the training set, in the second problem (Monk2) 169cases, and in the third problem (Monk3) 122 cases, of which5% were misclassifications introducing some noise in the data.The experimental results are depicted in Table XXI, Table XXII,and Fig. 13 (Monk1), Table XXIII, Table XXIV, and Fig. 14(Monk2), Table XXV, Table XXVI, and Fig. 15 (Monk3).

J. Wine Recognition

The Wine data [56] contains the chemical analysis of 178wines grown in the same region of Italy but derived from threedifferent vineyards. The 13 continuous attributes available forclassification are: alcohol, malic acid, ash, alcalinity of ash,magnesium, total phenols, flavanoids, nonflavanoid phenols,



TABLE XXVEXPERIMENTAL RESULTS

proanthocyanins, color intensity, hue, OD280/OD315 of dilutedwines and proline. In our experiments all sets are divided into alearning sequence (125 sets) and testing sequence (53 sets). Theexperimental results are depicted in Table XXVII, Table XXIII,and Fig. 16.


TABLE XXVICOMPARISONTABLE



K. Wisconsin Breast Cancer Data

The Wisconsin Breast Cancer data [56] contains 699 in-stances (of which 16 instances have a single missing attribute)and each instance is described by nine attributes (clumpthickness, uniformity of cell size, uniformity of cell shape,marginal adhesion, single epithelial cell size, bare nuclei,bland chromatin, normal nucleoli, mitoses). We removed those16 instances and used the remaining 683 instances. In ourexperiments, all sets are divided into a learning sequence (478sets) and testing sequence (205 sets). The experimental resultsare depicted in Table XXIX, Table XXX, and Fig. 17.

VIII. F INAL REMARKS

In this paper, we have presented new neuro-fuzzy structures.They are characterized as follows

1) TheAND-type system is a combination, controlled by pa-rameter , of Mamdani-type and logical-type sys-tems. In the process of learning only one type of systemis established ( or ).

2) TheOR-type system is “more Mamdani” ( ) or“more logical” ( ). In the process of learningone gets or .

3) TheOR-type is equivalent to theAND-type system if(Mamdani-type) or (logical-type).

TABLE XXVIIEXPERIMENTAL RESULTS

TABLE XXVIIICOMPARISONTABLE

4) TheOR-type system is less complicated than theAND-typesystem from a computational point of view. Both systemsproduce the same type of inference (Mamdani or logical)in the process of learning.

5) The most influential parameters are certainty weights, , and

, . They significantly im-prove the performance of the system in the process oflearning.

6) The influence of soft parameters , ,on the performance of the system varies

depending on the problem.The main advantage of our approach is the possibility oflearning a system type. The results of simulations are given inTable XXXI.


Fig. 16. Weights representation in the Wine Recognition problemw 2

[0; 1], w 2 [0; 1], i = 1; . . . ; 13, k = 1; . . . ; 3 (dark areas correspondto low values and vice versa).

TABLE XXIXEXPERIMENTAL RESULTS

We conclude that Mamdani-type systems are more suitable toapproximation problems, whereas logical-type systems may bepreferred for classification problems. It should be emphasizedthat the results in simulations A, B, C, D, G, H, I, and J outper-form the best results known in the literature, although it was notthe main goal of our paper.

TABLE XXXCOMPARISONTABLE

Fig. 17. Weights representation in the Wisconsin Breast Cancer problemw 2 [0; 1], w 2 [0; 1], i = 1; . . . ; 9, k = 1; . . . ; 2 (dark areascorrespond to low values and vice versa).

TABLE XXXIRESULTS OFSIMULATIONS

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Leszek Rutkowski(M’92–SM’94) was born in Wroclaw, Poland, in 1952. Hereceived the M.Sc., Ph. D., and D.Sc. degrees in 1977, 1980, 1986, respectively,all from the Technical University of Wroclaw, Wroclaw, Poland.

Since 1980, he has been with the Technical University of Czestochowa,Czestochowa, Poland, where he is currently a Professor and Chairman of theComputer Engineering Department. From 1987 to 1990, he held a visitingposition in the School of Electrical and Computer Engineering, OklahomaState University, Stillwater. His research interests include neural networks,fuzzy systems, computational intelligence, pattern recognition, and systemsidentification. He has published more than 100 technical papers including 14in various series IEEE TRANSACTIONS. He is the author of the books,AdaptiveFilters and Adaptive Signal ProcessingandNew Soft Computing TechniquesFor System Modeling, Pattern Classification and Image Processingto bepublished by Springer, and coauthor of two others,Neural Networks, GeneticAlgorithms and Fuzzy SystemsandNeural Networks for Image Compression.He is also President and Founder of the Polish Neural Networks Society.He organized and served as General Chair of the Polish Neural NetworksConferences held in 1996, 1997, 1999, 2000, and 2002.

Dr. Rutkowski is an Associate Editor of the IEEE TRANSACTIONS ONNEURAL

NETWORKS.

Krzysztof Cpalka was born in Czestochowa, Poland, in 1972. He receivedthe M.Sc. degree in electrical engineeringand the Ph.D. degree (with honors)in computer engineering, both from the Technical University of Czestochowa,Poland, in 1997and 2002, respectively.

Since 2002, he has been an Assistant Professor in the Department of Com-puter Engineering, Technical University of Czestochowa, Czestochowa, Poland.His research interests include fuzzy systems, neural networks, evolutionary al-gorithms, and artificial intelligence. He published over 20 technical papers injournals and conference proceedings.

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