computational intelligence software for interval type-2 fuzzy logic

Computational IntelligenceSoftware for Interval Type-2Fuzzy LogicOSCAR CASTILLO,1 PATRICIA MELIN,1 JUAN R. CASTRO2

1Tijuana Institute of Technology, Calzada Tecnologico S/N, 22379 Tijuana, Mexico

2UABC University, Calzada Universidad S/N, 22379 Tijuana, Mexico

Received 15 February 2010; accepted 21 December 2010

ABSTRACT: A software tool for interval type-2 fuzzy logic is presented in this article. The software tool includes

a graphical user interface for construction, edition, and observation of the fuzzy systems. The Interval Type-2 Fuzzy

Logic System Toolbox (IT2FLS) has a user-friendly environment for interval type-2 fuzzy logic inference system

development. Tools that cover the different phases of the fuzzy system design process, from the initial description

phase, to thefinal implementationphase, arepresentedaspart of theToolbox. TheToolbox’s best properties are the

capacity to develop complex systems and the flexibility that permits the user to extend the availability of functions

for working with type-2 fuzzy operators, linguistic variables, interval type-2 membership functions, defuzzification

methods, and the evaluation of interval type-2 fuzzy inference systems. The toolbox can be used for educational and

research purposes. �2011 Wiley Periodicals, Inc. Comput Appl Eng Educ; View this article online at wileyonlineli-

brary.com; DOI 10.1002/cae.20522

Keywords: type-2 fuzzy logic; fuzzy systems; Fuzzy Inference System Design

INTRODUCTION

Fuzzy sets were originally presented by Zadeh in 1965 [1,2] to

process/manipulate data and information affected by un-probabil-

istic uncertainty/imprecision. These were designed to mathemat-

ically represent the vagueness and uncertainty of linguistic

problems, thereby obtaining formal tools to work with intrinsic

imprecision in different types of problems [3].

Intelligent systems based on fuzzy logic are fundamental

tools for nonlinear complex system modeling. Fuzzy sets and

fuzzy logic are the basis for fuzzy systems, where their goal

has been to model how the brain manipulates inexact information.

Type-2 fuzzy sets are used for modeling uncertainty and impre-

cision in a better way. These type-2 fuzzy sets were originally

presented by Zadeh in 1975 and are essentially ‘‘fuzzy fuzzy’’ sets

where the fuzzy degree of membership is a type-1 fuzzy set [4–6].

The new concepts were introduced by Karnik and Mendel [7] and

Liang and Mendel [8] allowing the characterization of a type-2

fuzzy set with a superior membership function and an inferior

membership function; these two functions can be represented each

by a type-1 fuzzy set membership function. The interval between

these two functions represents the footprint of uncertainty (FOU),

which is used to characterize a type-2 fuzzy set. Recently, there

have been a lot of applications of type-2 fuzzy systems and for this

reason we considered the need of building a software tool that

could ease the development of type-2 fuzzy systems for real-world

problems [9,10]. Also, there is an imperative need of having this

software tool for teaching graduate and undergraduate students the

basic concepts and methods of type-2 fuzzy logic. For these

reasons, we consider that the interval type-2 fuzzy logic toolbox

makes an important contribution both to education and research of

type-2 fuzzy logic.

Currently, most of the tools for designing fuzzy systems

are focused on conventional type-1 fuzzy logic [9]. However,

for type-2 fuzzy logic there are only a few attempts of providing

tools that can help in the design of type-2 fuzzy systems. The main

reasons for not having widely used tools in this area are that type-2

fuzzy logic has only been used for a few years and also that type-2

fuzzy systems aremore difficult to design due to the higher number

of parameters involved [9,10]. As a consequence, our goal was to

design and develop a friendly and efficient type-2 fuzzy logic

toolbox that could be used both for research and education pur-

poses. The contribution of the research work has been to facilitate

the use of the theory and concepts of type-2 fuzzy logic for both

researchers and students. Regarding researchers, there have been

several type-2 fuzzy systems applications that have been achieved

by using the toolbox. Also, on the educational side there has been a

significant improvement in the level of knowledge of the students

that have used the toolbox, which is appreciated with a performed

evaluation to a sample of students.

Correspondence to O. Castillo ([email protected]).

� 2011 Wiley Periodicals, Inc.

1

INTERVAL TYPE-2 FUZZY SET THEORY

In this section the basic concepts of interval type-2 fuzzy logic

are presented. Also, the main differences with respect to conven-

tional type-1 fuzzy logic are explained. This section is very

important to understand the need of having the software presented

in this article.

Type-2 Fuzzy Sets Concept

A type-2 fuzzy set [7,8] expresses the non-deterministic truth

degree with imprecision and uncertainty for an element that

belongs to a set. A type-2 fuzzy set, denoted by~~A, is charac-

terized by a type-2 membership function m~~Aðx; uÞ, where

x2X;u2Jux � ½0; 1� and 0 � m~~Aðx; uÞ � 1 aredefined inEquation(1).

~~A ¼ ðx;m~~AðxÞÞjx2X

n o

~~A ¼ ðx; u;m~~Aðx; uÞÞj8x2X; 8u2Jux � ½0; 1�

n o(1)

An example of a type-2 membership function constructed

with the IT2FLS toolbox is composed of a Pi primary and a Gbell

secondary type-1 membership functions, and is depicted in

Figure 1.

If~~A is continuous then it is denoted as in Equation (2).

~~A ¼Zx2X

Zu2Jux�½0;1�

fxðuÞ=u

264

375=x

8><>:

9>=>; (2)

whereR R

denotes the union of x and u. If~~A is discrete then it is

denoted by Equation (3).

~~A ¼Xx2X

m~~AðxÞ=x

( )¼

XNi¼1

XMi

k¼1

fxiðuikÞ=uik" #

=xi

( )(3)

wherePP

denotes the union of x and u. If fxðuÞ ¼ 1;

8u2½Jux ; Jux � � ½0; 1�, the type-2 membership function

m~~Aðx; uÞis expressed by one type-1 inferior membership function,

Jux � mAðxÞ and one type-1 superior, Jux � mAðxÞ (Fig. 2), then it is

called an interval type-2 fuzzy set [8] denoted by Equations (4)

and (5).

~~A ¼ ðx; u; 1Þj8x2X;8u2½m

AðxÞ;mAðxÞ� � ½0; 1�

�(4)

or

~~A ¼Zx2X

Zu2½Jux ;J

u

x ��½0;1�

1=u

2664

3775=x

8>><>>:

9>>=>>;

¼Zx2X

Zu2½m

AðxÞ;mAðxÞ��½0;1�

1=u

264

375=x

8><>:

9>=>; (5)

If~~A is a type-2 fuzzy Singleton, then the membership func-

tion is defined by Equation (6):

m~~AðxÞ ¼ 1=1 si x ¼ x0

1=0 si x 6¼ x0

�(6)

Figure 1 FOU for type-2 membership functions. [Color figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com.]

2 CASTILLO, MELIN, AND CASTRO

Interval type-2 fuzzy sets are the ones that, at themoment, are

mostwidely used in the applications due to their simplicity and less

computer overhead requirements. Generalized type-2 fuzzy sets

are, as of today, too complex to perform the computations and as

consequence are not widely used [9]. Figure 2 shows the FOU for

several types of interval type-2 membership functions.

Fuzzy Set Operations

We can apply some operators to the fuzzy sets, or we can make

some operations between them [4,10,11]. When we apply an

operator to one fuzzy set we obtain another fuzzy set; by the same

manner when we combine an operation with two or more sets we

obtain another fuzzy set. If we have two type-2 fuzzy subsets

identified by the letters~~A and ~~B, associated with a linguistic

variable, then we can define three basic operations: complement,

union and intersection (Table 1).

Fuzzy Inference System

The human knowledge can be expressed as fuzzy rules with the

following form [12]:

IF <fuzzy proposition> THEN <fuzzy proposition>

Figure 2 FOU for Gbell primary interval type-2 membership functions. [Color figure can be viewed in the online issue,

which is available at wileyonlinelibrary.com.]

Table 1 Interval Type-2 Fuzzy Set Operations

Name Operator Operation

UnionR ¼ join ~~A~~B ¼ R

x2Xm~~A

ðxÞm~~BðxÞ

� �¼ R

x2X

Ra2½m~A

ðxÞ_m~BðxÞ;m~AðxÞ_m~BðxÞ�

1=a

24

35=x

8<:

9=;

Intersection ¼ meet ¼ Rx2X

R�2½�

~AðxÞ^�

~BðxÞ;�~AðxÞ^�~BðxÞ�

1=�

24

35=x

8<:

9=;

Negation : :~~A ¼ Rx2X

m~~AðxÞ=x

� �¼ R

x2X

Ra2½1�m~A

ðxÞ;1�m~AðxÞ�

1=a

24

35=x

8<:

9=;

COMPUTATIONAL INTELLIGENCE SOFTWARE 3

The fuzzy propositions are divided into two types, the first

one is called atomic: x is A, where x is a linguistic variable andA is

a linguistic value; the second one is called composite: x is A AND

y is BORz is NOTC, this is a composite atomic fuzzy proposition

with the ‘‘AND,’’ ‘‘OR,’’ and ‘‘NOT’’ connectors, representing

fuzzy intersection, union, and complement, respectively. The

composite fuzzy propositions are fuzzy relationships. The mem-

bership function of the rule IF-THEN is a fuzzy relation deter-

mined by a fuzzy implication operator. The fuzzy rules combine

one or more fuzzy sets of entry, called antecedent, and are associ-

ated with one output fuzzy set, called consequents. The fuzzy sets

of the antecedent are associated with fuzzy operators AND, OR,

NOT, and linguistic modifiers. The fuzzy rules permit expressing

the available knowledge about the relationship between antecedent

and consequents. To express this knowledge completely we nor-

mally have several rules, grouped to form what it is known a rule

base, that is, a set of rules that express the known relationships

between antecedent and consequents. The fuzzy rules are basically

IF <Antecedent> THEN <Consequent> and express a fuzzy

relationship or proposition.

In fuzzy logic the reasoning is imprecise (approximate),

which means that we can infer from one rule a conclusion even

if the antecedent does not comply completely.We can count on two

basic inference methods between rules and inference laws, Gener-

alized Modus Ponens (GMP) [5,6,8,13] and Generalized Modus

Tollens (GMT), that represent the extensions or generalizations of

classic reasoning. The GMP inference method is known as direct

reasoning and is summarized as [14,15]:

whereA, A’,B, and B’ are fuzzy sets of any kind. This relationship is

expressed as B0 ¼ A0�ðA ! BÞ. Figure 3 shows an example of

Interval Type-2 direct reasoning with Interval Type-2 Fuzzy Inputs.

A fuzzy inference system is a rule base system that uses fuzzy

logic, instead of Boolean logic utilized in data analysis [16,17]. Its

basic structure includes four components (Fig. 4):

� Fuzzifier: Translates inputs (real values) to fuzzy values.� Inference System: Applies a fuzzy reasoning mechanism to

obtain a fuzzy output.� Type Defuzzifer/Reducer: The defuzzifier translates one

output to precise values; the type reducer transforms a

Type-2 Fuzzy Set into a Type-1 Fuzzy Set.

Figure 3 Interval type-2 fuzzy reasoning. [Color figure can be viewed in the online issue, which is available at



� Knowledge Base: Contains a set of fuzzy rules, and a mem-

bership functions set known as the database.

The decision process is a task that identifies parameters by the

inference system using the rules of the fuzzy rule base [18–20].

These fuzzy rules define the connection between the input

and output fuzzy variables. A fuzzy rule has the form: IF

<Antecedent> THEN <Consequent>, where antecedent is a

composite fuzzy logic expression of one or more simple fuzzy

expressions connected with fuzzy operators; and the consequent

is an expression that assigns fuzzy values to output variables

[21,22].

INTERVAL TYPE-2 FUZZY SYSTEM DESIGN

The Mamdani and Takagi-Sugeno-Kang (TSK) Interval Type-2

Fuzzy Inference Models [23,24] and the design of Interval Type-2

membership functions and operators are implemented in the

IT2FLS Toolbox (Interval Type-2 Fuzzy Logic Systems) reused

from the Matlab1 commercial Fuzzy Logic Toolbox.

The IT2FLS Toolbox includes a series of folders called

dit2mf, it2fis, it2mf, and it2op (Fig. 5). These folders contain

the functions to create Mamdani and TSK Interval Type-2 Fuzzy

Inference Systems (newfistype2.m), functions to add input–output

variables and their ranges (addvartype2.m), it has functions to add

22 types of Interval Type-2 Membership functions for input–

outputs (addmftype2.m), functions to add the rule matrix (addru-

letype2.m), it can evaluate the Interval Type-2 Fuzzy Inference

Systems (evalifistype2.m), evaluate Interval Type-2 Membership

functions (evalimftype2.m), it can generate the initial parameters

of the Interval Type-2Membership functions (igenparamtype2.m),

it can plot the Interval Type-2 Membership functions with the

input–output variables (plotimftype2.m), it can generate the

solution surface of the Fuzzy Inference System (gensurftype2.m),

it plots the Interval Type-2 membership functions (plot2dtype2.m,

plot2dctype2.m), a folder to evaluate the derivatives of the Interval

Type-2Membership Functions (dit2mf) and a folder with different

and generalized Type-2 Fuzzy operators (it2op, t2op).

The Interval Type-2 Fuzzy Inference Systems (IT2FIS) struc-

ture is the MATLAB object that contains all the interval type-2

fuzzy inference system information. This structure is stored inside

each graphics user interface (GUI) tool. Access functions such as

getifistype2 and setifistype2make it easy to examine this structure.

All the information for a given fuzzy inference system is contained

in the IT2FIS structure, including variable names, membership

function definitions, and so on. This structure can itself be thought

of as a hierarchy of structures, as shown in the following diagram

(Fig. 6).

The implementation of the IT2FLS GUI is analogous to the

GUI used for Type-1 FLS in the Matlab1 Fuzzy Logic Toolbox,

thus permitting the experienced user to adapt easily to the use of

IT2FLS GUI. Figures 7 and 8 show the main view of the Interval

Type-2 Fuzzy Systems Structure Editor called IT2FIS (Interval

Type-2 Fuzzy Inference Systems).

SIMULATION RESULTS WITH THE INTERVAL TYPE-2TOOLBOX

We present simulation results for the shower and truck backer–

upper problems with interval type-2 fuzzy logic systems using the

IT2FLS Toolbox. These benchmark problems are available in the

well-known Fuzzy Logic Toolbox ofMatlab (this is a type-1 fuzzy

logic toolbox) and for this reason are a good choice to illustrate the

Figure 4 Type-2 inference fuzzy system structure.

Figure 5 Toolbox folders.


results of type-2 fuzzy logic. We also present results with type-2

fuzzy logic for theMackey–Glass time series forecasting, which is

another benchmark problem.

Shower Control Simulation

The application of the interval type-2 fuzzy control scheme to the

shower gives good control results, which can be noticed in

Figure 9.

Truck Backer–Upper Control Simulation

The application of the interval type-2 fuzzy control scheme to the

truck backer–upper problem gives good control results and in

Figures 10 and 11we show the control results of the car trajectories

for different initial conditions.

The results in Table 2 show a comparative analysis for the

Mackey–Glass chaotic time series. The study considers the use of

intelligent hybrid methods, with neural networks (Mamdani,

TSK), type-1 fuzzy inference systems and genetic algorithms

(neuro-genetic, fuzzy-genetic, and neuro-fuzzy) and an interval

type-2 fuzzy logicmodel, for the implicit knowledge acquisition in

a time series behavioral historical data. To identify the model we

use 5 delays, L5x(t), 6 periods, and 500 training data values to

forecast 500 output values. The IT2FLS system works with 4

inputs, 2 interval type-2 Gaussian membership functions for each

input, 16 rules and 1 output with 16 interval linear functions. The

forecasted root mean square error (RMSE) is 0.0335 (Table 2).

In Table 2, NNFF stands for a neuro-fuzzymodel, CANFIS is

a modified adaptive neuro-fuzzy inference system, NNFF-GA

stands for genetically optimized neuro-fuzzy system,

FLS(TSK)-GA means a Sugeno fuzzy model optimized with a

genetic algorithm, FLS(MAM)-GA means a Mamdani fuzzy

model optimized with a genetic algorithm, and IT2FLS an interval

type-2 fuzzy model developed with the type-2 fuzzy logic toolbox

presented in this article.

Figure 6 Hierarchy of IT2FIS structures diagram.


RELATED WORK

In comparing the developed toolbox (shown as IT2tool in Table 3)

with available software for type-2 and type-1 fuzzy systems in the

literature [25,26],we can find that no other tool exists for designing

and implementing hybrid type-2 fuzzy systems (like neuro-type-2

fuzzy systems). Other tools could be used for the same task, but a

lot of programming would be needed by the user. In Table 3, a

summary of a comparison of the IT2tool, shown in the last column,

with respect to other tools is presented. The comparison is made

based on the main characteristics needed of a good software tool

for designing and implementing type-2 fuzzy systems, this is

indicated in the first column of Table 3. It is easy to notice that

only the proposed IT2tool fulfills all the requirements of an ideal

tool for this purpose. Other software tools that only have some of

the ideal characteristics are not flexible enough for complex and

high-dimensional problems.

There are other educational software tools available for fuzzy

logic [30], but in most of the cases the software is only for type-1

fuzzy logic or is focused on a particular area of application. The

proposed type-2 fuzzy logic toolbox is for any type of application

and can be used for type-1 or type-2 fuzzy logic.

VALIDATION OF THE SOFTWARE

The validation of the developed software has been done by com-

paring the results of the toolbox against the results of other authors

with the same benchmark problems [31–33]. The type-2 fuzzy

logic results have been compared in several papers by the authors

Figure 7 IT2FIS Editor. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 8 Interval Type-2 MF’s Editor. [Color figure can be viewed in the online issue, which is available at



[34–38]. In fact, for this reason the type-2 fuzzy logic toolbox has

been requested by many researchers and professors of different

countries, interested in teaching or doing research with the toolbox.

On other hand, with respect to the education process, we can

say that the toolbox has been in use for the last four semesters (of

2009 and 2010) with students of a graduate course at the Masters

Degree in Computer Science with great success. The graduate

course is called ‘‘Fuzzy Systems’’ and a total 100 students have

used the toolbox with great success. This course is a basic part of

the Master degree in Computer Science at Tijuana Institute of

Technology because the area of research of the graduate program is

on Computational Intelligence.

With the goal of measuring the performance of the proposed

software for type-2 fuzzy logic (IT2tool) a questionnaire was

Figure 9 Temperature interval type-2 fuzzy control. [Color figure can be viewed in the online issue, which is available at


Figure 10 Interval type-2 fuzzy control for trajectory 1. [Color figure can be viewed in the online issue, which is

available at wileyonlinelibrary.com.]


applied to a sample of n ¼ 30 graduate students that have used the

tool for a semester in a course of fuzzy systems. The students were

also providedwith the Fuzzy Logic toolbox ofMatlab, so that they

could be able to compare the characteristics and performance of

both tools in solving problems using fuzzy logic. The results of the

questionnaire are summarized in Table 4, where the nine questions

are indicated in the first column of this table. The students were

asked to provide a ranking from 1 to 10 to the response to each of the

questions. The averages of questions’ responses of the 30 students

for each of the tools are indicated in the second and third columns

of Table 4, respectively. The summary of the statistics (average and

standard deviations) is indicated at the bottom of Table 4.

The t-statistic value of 4.75 for the difference ofmeans is also

shown at the bottom of Table 4, which indicates that there is

significant difference between the results of our software tool

(IT2tool) those ones provided by Matlab with at least 99% con-

fidence. In other words, the user’s satisfaction with the developed

tool (IT2tool) is significantly better than with the Fuzzy

Logic toolbox of Matlab. Finally, from Table 4 we can notice that

in all the questions the response was in favor (on average) of the

developed tool (IT2tool).

Finally, for measuring the knowledge improvement of the

students when using the software tool, we use a metric based on

two factors: accuracy of the student’s achieved solutions and time

required by the student to develop the solutions to the problem. The

Figure 11 Interval type-2 fuzzy control for trajectory 2. [Color figure can be viewed in the online issue, which is

available at wileyonlinelibrary.com.]

Table 2 Forecasting of Time Series

Methods

Mackey–Glass

RMSE trn/chk epochs cpu(s)

NNFF 0.0595 500/500 200 13.36

CANFIS 0.0016 500/500 50 7.34

NNFF-GA 0.0236 500/500 150 98.23

FLS(TKS)-GA 0.0647 500/500 200 112.01

FLS(MAM)-GA 0.0693 500/500 200 123.21

IT2FLS 0.0335 500/500 6 20.23

Table 3 Comparison With Different Fuzzy Logic Software

Characteristics

Mendel

[27]

Ozek

Toolbox [28]

Matlab

[29] IT2tool

Graphic interface No Yes Yes Yes

Hybrid algorithms No No Yes Yes

Type-2 Fuzzy Logic Yes Yes No Yes

Neuro-Fuzzy Type-2 No No No Yes

Easy to use No Yes Yes Yes

Efficiency (in time) No Yes Yes Yes

Easy to add new methods Yes Yes No Yes

Quick to shows results No No No Yes

Table 4 Summary of the Questionnaire Applied to Students

Question IT2tool Matlab Fuzzy

User interface 9.2560 8.8221

Method for showing results 9.3333 8.7350

Method to input data 9.1251 8.6725

Efficiency (in time) 9.0550 8.4322

Accuracy 9.5275 9.0531

Comparison between both tools 9.4188 8.6337

Comparison with other software 9.6713 8.9572

Global evaluation 9.9071 9.2879

Recommending the tool to others 9.2781 8.6841

Average 9.3970 8.8090

Standard deviation 0.2700 0.2560

t Statistic 4.75


metric can be expressed as given by the following equation:

K ¼ aAþ bT (7)

where K is the knowledge of the student, A is the accuracy of the

achieved solution by the student t, T is the time required by the

student to develop the solution to the problem. In this equation, all

variables are considered to be normalized in the range from0 to 10,

to represent an expert evaluation of the achieved result by the

student. The coefficients a and b represent the weights assigned to

each of the factors involved in evaluating the knowledge acquired

by the student. In Table 5 the results achieved by students before

and after learning to use the software tool are presented. These

results were obtained by giving 30 students the same five bench-

mark problems to solve. Two of the problems are of time series

prediction (Mackey–Glass and Dow Jones time series). The other

three problems are of intelligent control (Shower Control, Truck

Backer–upper control, and Ball and Beam control). From the

statistics of Table 5, in particular the t value, we can conclude

that there exists statistical evidence of significant difference bet-

ween the average knowledge before and after learning to use the

software tool proposed in this paper. In other words, the software

tool provides significant additional knowledge to the students

based on this statistical evidence. This conclusion is achieved

with more than 98% of confidence because the t value if of 3.11.

CONCLUSIONS

The results in the interval type-2 fuzzy control case studies of the

shower and the truck backer–upper have similar results to the type-

1 fuzzy control with moderate uncertain footprints. To better

characterize the interval type-2 fuzzy models we need to generate

more case studies with better knowledge bases for the proposed

problems, therefore classifying the interval type-2 fuzzy model

application strengths.

The design and implementation done in the IT2FLS Toolbox

are potentially important for research in the interval type-2 fuzzy

logic area, thus solving complex problems on the different applied

areas.

The main contribution of the article has been to facilitate the

use of the theory and concepts of type-2 fuzzy logic for both

researchers and students. On the educational side there has been a

significant improvement in the level of knowledge of the students

that have used the toolbox, which is appreciated with a performed

evaluation to a sample of students.

Our future work is to improve the IT2FLS Toolbox with a

better GUI, to have compiled code, and integrate a learning

technique Toolbox to optimize the knowledge base parameters

of the interval type-2 fuzzy inference system, and design interval

type-2 fuzzy neural network hybrid models.

REFERENCES

[1] L. A. Zadeh, Fuzzy sets, Inf Control 8 (1965), 338–353.

[2] L. A. Zadeh, Outline of a new approach to the analysis of complex

systems and decision processes, IEEE Trans Syst Man Cybern 3

(1973), 28–44.

[3] L. A. Zadeh, The concept of a linguistic variable and its applica-

tion to approximate reasoning, parts 1, Inf Sci 8 (1975), 199–249.

[4] L. A. Zadeh, Fuzzy logic, Computer 1 (1988), 83–93.

[5] L. A. Zadeh, Knowledge representation in fuzzy logic, IEEE Trans

Knowl Data Eng 1 (1989), 89–100.

[6] L. A. Zadeh, Fuzzy logic ¼ computing with words, IEEE Trans

Fuzzy Syst 2 (1996), 103–111.

[7] N. N. Karnik and J. M. Mendel, An introduction to type-2 fuzzy

logic systems, University of Southern California, Los Angeles, CA,

1998.

[8] Q. Liang and J. Mendel, Interval type-2 fuzzy logic systems: Theory

and design, IEEE Trans Fuzzy Syst 8 (2000), 535–550.

[9] O.Castillo andP.Melin, Type-2 fuzzy logic: Theory and applications,

Springer-Verlag, Heidelberg, Germany, 2008.

[10] J.M.Mendel, Uncertain rule-based fuzzy logic systems: Introduction

and new directions, Prentice-Hall, NJ, 2001.

[11] M.Mizumoto andK. Tanaka, Some properties of fuzzy sets of type-2,

Inf Control 31 (1976), 312–340.

[12] L.-X. Wang, Adaptive fuzzy systems and control: Design and

stability analysis, Prentice Hall, Englewood Cliffs, NJ, 1994.

[13] R. R. Yager, On the implication operator in fuzzy logic, Inf Sci 31

(1983), 141–164.

[14] R. R. Yager, A characterization of the fuzzy extension principle,

Fuzzy Sets Syst 18 (1986), 205–217.

[15] E. H.Mamdani and S. Assilian, An experiment in linguistic synthesis

with a fuzzy logic controller, Int J Man Mach Stud 7 (1975), 1–13.

[16] E. H. Mamdani, Applications of fuzzy logic to approximate reasoning

using linguistic synthesis, IEEE Trans Comput 26 (1977), 1182–1191.

[17] M. Sugeno and G. T. Kang, Structure identification of fuzzy model,

Fuzzy Sets Syst 28 (1988), 15–33.

[18] M. Sugeno, Industrial applications of fuzzy control, Elsevier Science,

Amsterdam, Holland, 1985.

[19] Y. Tsukamoto, An approach to fuzzy reasoning method. In: M. M.

Gupta, R. K. Ragade, R. R. Yager (Eds.), Advances in fuzzy set

theory and applications, Elsevier, North-Holland, Amsterdam, 1979,

pp 137–149.

[20] R. R. Yager, On a general class of fuzzy connectives, Fuzzy Sets Syst

4 (1980), 235–242.

[21] E. H. Mamdani, Advances in the linguistic synthesis of fuzzy con-

trollers, Int J Man Mach Stud 8 (1976), 669–678.

Table 5 Experimental Results of Evaluating Knowledge Improvement With the Software Tool

Benchmark problem

Knowledge improvement

Average accuracy

(before)

Average time

(before)

K

(before)

Average accuracy

(after)

Average time

(after)

K

(after)

Mackey–Glass time series 8.15 8.33 8.24 9.55 9.60 9.575

Dow Jones time series 7.20 7.84 7.52 8.72 8.96 8.84

Shower control 7.45 8.25 7.85 9.15 9.35 9.25

Truck Backer–upper control 9.25 9.15 9.20 9.72 9.84 9.78

Ball and Beam control 6.95 7.47 7.21 8.24 9.28 8.76

Average 8.004 Average 9.241

Standard deviation 0.770 Standard deviation 0.446

Statistic t ¼ 3.11


[22] T. Takagi and M. Sugeno, Fuzzy identification of systems and its

applications to modeling and control, IEEE Trans Syst Man Cybern

15 (1985), 116–132.

[23] J. R. Castro, O. Castillo, P. Melin, and A. Rodriguez-Diaz, Building

fuzzy inference systems with a new interval type-2 fuzzy logic

toolbox, Trans Comput Sci 1 (2008), 104–114.

[24] R. Sepulveda, O. Castillo, P. Melin, A. Rodriguez-Diaz, and O.

Montiel, Experimental study of intelligent controllers under uncer-

tainty using type-1 and type-2 fuzzy logic, Inf Sci 177 (2007), 2023–

2048.

[25] J. R. Castro, O. Castillo, and P. Melin, An interval type-2 fuzzy logic

toolbox for control applications, Proceedings of FUZZ-IEEE,

London, 2007, pp 1–6.

[26] P. Melin and O. Castillo, Hybrid intelligent systems for pattern

recognition, Springer-Verlag, Heidelberg, Germany, 2005.

[27] J. M. Mendel, Type-2 fuzzy logic software, available at: http://sipi2-

b.usc.edu/�mendel/software/.

[28] M. B. Ozek and Z. H. Akpolat, A software tool: Type-2 fuzzy logic

toolbox, Comput Appl Eng Educ 16 (2008), 137–146.

[29] Logic. Fuzzy, Toolbox of Matlab, available at: http://www.math-

works.com/products/fuzzylogic/.

[30] M. A. Akcayol, C. Elmas, O. A. Erdem, and M. Kurt, An educational

tool for fuzzy logic controller and classical controllers, Comput Appl

Eng Educ 12 (2004), 126–135.

[31] O. Mendoza, P. Melin, and G. Licea, Fuzzy inference systems type-1

and type-2 for digital images edges detection, EngLett 15 (2007), 45–

52.

[32] O. Mendoza, P. Melin, and G. Licea, Interval type-2 fuzzy logic for

edges detection in digital images, Int J Intell Syst 24 (2009), 1115–

1133.

[33] C. Leal-Ramirez, O. Castillo, P. Melin, and A. Rodriguez-Diaz,

Simulation of the bird age-structured population growth based on

an interval type-2 fuzzy cellular structure, Inf Sci 181 (2011), 519–

535.

[34] N. R. Cazarez-Castro, L. T. Aguilar, and O. Castillo, Fuzzy logic

control with genetic membership function parameters optimization

for the output regulation of a servomechanism with nonlinear back-

lash, Expert Syst Appl 37 (2010), 4368–4378.

[35] J. R. Castro, O. Castillo, P. Melin, and A. Rodriguez-Diaz, A hybrid

learning algorithm for a class of interval type-2 fuzzy neural net-

works, Inf Sci 179 (2009), 2175–2193.

[36] R. Martinez-Soto, O. Castillo, and L. T. Aguilar, Optimization of

interval type-2 fuzzy logic controllers for a perturbed autonomous

wheeled mobile robot using genetic algorithms, Inf Sci 179 (2009),

2158–2174.

[37] O. Castillo, L. T. Aguilar, N. R. Cazarez, and S. Cardenas, Systematic

design of a stable type-2 fuzzy logic controller, Appl Soft Comput 8

(2008), 1274–1279.

[38] R. Sepulveda, O. Castillo, P. Melin, and O. Montiel, An

efficient computational method to implement type-2 fuzzy logic in

control applications, analysis and design of intelligent systems using

soft computing techniques 2007, Springer-Verlag, Heidelberg,

Germany, pp 45–52.

BIOGRAPHIES

Oscar Castillo holds the Doctor in Science

degree (Doctor Habilitatus) in Computer

Science from the Polish Academy of Sciences

(with the Dissertation ‘‘Soft Computing and

Fractal Theory for Intelligent Manufacturing’’).

He is a Professor of Computer Science in

the Graduate Division, Tijuana Institute of

Technology, Tijuana, Mexico. In addition, he

is serving as Research Director of Computer

Science and head of the research group on

fuzzy logic and genetic algorithms. Currently, he is Vice-President of

HAFSA (Hispanic American Fuzzy Systems Association) and President

Elect of IFSA (International Fuzzy Systems Association). Prof. Castillo is

also Chair of the Mexican Chapter of the Computational Intelligence

Society (IEEE). He belongs to the Mexican Research System with level

II. He belongs to the Technical Committee on Fuzzy Systems of IEEE,

where he is the Chair of the Task Force on FuzzyHardware and also belongs

to the Task Force on ‘‘Extensions to Type-1 Fuzzy Systems’’. He is also a

member of NAFIPS, IFSA and IEEE. His research interests are in Type-2

Fuzzy Logic, Fuzzy Control, Neuro-Fuzzy and Genetic-Fuzzy hybrid

approaches. He has published over 90 journal papers, 5 authored books,

20 edited books, and 200 papers in conference proceedings.

Patricia Melin holds the Doctor in Science

degree (Doctor Habilitatus D.Sc.) in Computer

Science from the Polish Academy of Sciences

(with the Dissertation ‘‘Hybrid Intelligent

Systems for Pattern Recognition using Soft

Computing’’). She is a Professor of Computer

Science in the Graduate Division, Tijuana Insti-

tute of Technology, Tijuana, Mexico, since

1998. In addition, she is serving as Director of

Graduate Studies in Computer Science and head

of the research group on Computational Intelligence (2000-present).

Currently, she is President of HAFSA (Hispanic American Fuzzy Systems

Association and is the founding Chair of the Mexican Chapter of the IEEE

Computational Intelligence Society. She is Chair of the Task Force on

Hybrid Intelligent Systems ofNeuralNetworks Technical Committee of the

IEEE Computational Intelligence Society. She is member of NAFIPS,

IFSA, and IEEE. She belongs to the Mexican Research System with level

II. Her research interests are in Type-2 Fuzzy Logic, Modular Neural

Networks, Pattern Recognition, Neuro-Fuzzy and Genetic-Fuzzy hybrid

approaches. She has published over 90 journal papers, 5 authored books, 15

edited books, and 200 papers in conference proceedings.

Juan R. Castro holds the Ph.D. in

Computer Science from UABC University in

Tijuana, Mexico (2009) and the Masters

Degree in Computer Science from Tijuana

Institute of Technology (2005). Currently he

is Professor of Computer Science in the School

of Engineering of UABC University, where he

lectures at the undergraduate and graduate

levels. His research areas interests are in

type-2 fuzzy logic, genetic fuzzy systems,

type-2 neuro-fuzzy neural networks, and new techniques for time series

prediction and intelligent control.


computational intelligence software for interval type-2 fuzzy logic

Documents