Interval Type-2 Complex-Fuzzy Inferential System ― A New Approach to Modeling

Interval Type-2 Complex-Fuzzy Inferential System ― A New Approach to

Modeling

Chunshien Li

Department of Information Management, National Central University

No.300, Jhongda Rd., Jhongli City, Taoyuan County 32001, Taiwan.

[email protected]

ABSTRACT

Fuzzy rationale has been developed to deal with

imprecision in information that happens in the real

world usually. L.A. Zadeh proposed the important

concept of type-2 fuzzy sets 10 years after the

inception of regular fuzzy sets that are also known

as type-1 fuzzy sets. The former uses real-valued

membership degree to describe set-element

relationship, while the latter uses fuzzy set to do so.

The movement from type-1 to type-2 fuzzy sets and

logic is a very important research direction for

fuzzy systems and applications. In the meanwhile,

another critical direction is the research of complex

fuzzy sets (CFSs) and logic to generalize

membership description to complex-valued degrees

so that membership can be widely enriched in the

complex plane. A CFS is also called type-1 CFS. In

this paper, an interval type-2 complex-fuzzy

inferential system is proposed, using interval type-2

complex fuzzy sets (IT2-CFS), each of which is

newly synthesized by two type-1 CFSs. For

optimization, a hybrid-learning method called the

PSO-KFA method is used to equip with a self-

learning ability for the proposed system. Through

experimental results of function approximation, the

proposed approach has shown promising result and

performance.

KEYWORDS

Type-2 fuzzy set, complex fuzzy set, type-2

complex fuzzy set, modeling.

1 INTRODUCTION

Since the advent of fuzzy sets (namely, type-

1fuzzy sets, or T1-FSs in short) by Prof. Zadeh

[1], related studies in fuzzy logic and systems

and fuzzy-based applications have been

prospered in various areas, such as controls,

signal processing, industrial products, modeling,

prediction, and more. In general, the set-

element relationship of a fuzzy set is defined

with a membership function (MF) whose value

is called membership degree, within the real-

valued interval [0,1]. Fuzzy sets are used to

deal with vagueness, imprecision and

uncertainty in signal and information of

communication. Usually, uncertainty or

imprecision of information happens in the real

world. In 1975, L. A. Zadeh presented the

concept of type-2 fuzzy sets (T2-FSs) [2] that

are much different from T1-FSs in the

description of membership. The set-element

membership of a T2-FS is described in terms of

T1-FS, while a T1-FS’s membership is given

by a crisp real-valued value within the unit

interval [0,1]. In principle, a T2-FS can be

defined by two membership functions of T1-

FSs in a two-staged structure. The first

membership function is called the primary

membership function, whose range becomes the

domain of the second membership function,

called the secondary membership function of

the T2-FS. The theory of type-2 fuzzy sets can

be further developed to become a logic

rationale called the T2-FS based fuzzy logic,

which is useful to system-based applications.

The theory of T2-FSs is more powerful than

that of T1-FSs, for the former shows more

general in set-element relationship than the

latter and thus has better capability in dealing

with information uncertainty. Although the

shifting from T1-FSs to T2-FSs is important to

ISBN: 978-1-941968-16-1 ©2015 SDIWC 192

Proceedings of The Fourth International Conference on Informatics & Applications, Takamatsu, Japan, 2015

modeling, they are all on real-valued

membership degrees based on which inference

logic is practiced. However, there is another

new perspective to think about set-element

relationship: the theory of complex fuzzy sets

(CFSs) [3] which are important to research and

applications. Basically, CFSs, also called type-1

CFSs (T1-CFSs), are defined by complex-

valued membership function with some elegant

mathematic form. CFSs are naturally general

fuzzy sets that show richer set-element

information in membership description than

traditional type-1 fuzzy sets. Similarly to T2-

FSs, one can design a type-2 CFS (T2-CFS)

based on the concept of T1-CFS. This paper

reports some primary results of interval type-2

complex fuzzy sets (IT2-CFSs) and application

on function approximation. Using interval type-

2 complex fuzzy sets (IT2-CFSs), a new fuzzy

inferential model, called the interval type-2

complex fuzzy inferential system (IT2-CFIS), is

presented in this paper. An IT2-CFS is a special

fuzzy set constructed with two T1-CFSs whose

membership degrees are complex-valued within

the unit disc of the complex plane. Basically,

the membership function of a T1-CFS

comprises an amplitude function and a phase

function. If the phase function disappears, the

T1-CFS degenerates to become a T1-FS whose

membership function is real-valued within the

unit interval [0,1] only. For this reason, the

class of T1-CFSs can include that of T1-FSs.

The set-element relationship by T1-CFSs is

more general than that by T1-FSs, and so the

logical inference based on the former can be

much more powerful than that based on the

latter.

Since the presentation of type-2 fuzzy set, the

research development was proceeding slowly

till the early 1980s. The main results were on

fuzzy-and and fuzzy-or operations of type-2

fuzzy sets. In the mid and late 1980s, interval-

valued fuzzy sets and their studies on fuzzy

inferential system were focused. Especially,

Gorzalczany [4] should be respected as the

research pioneer whose research idea in this

was similar to the concept of footprint of

uncertainty (FOU) of interval type-2 fuzzy sets.

Interval type-2 fuzzy set (IT2-FS) is a simpler

version of type-2 fuzzy set, where the

membership degrees of the secondary

membership function of the former are always

one. Turksen [5] was a contributor to this study,

whose contribution was on the connection

between type-1 and interval type-2 fuzzy sets.

In general, the powerful ability of interval type-

2 fuzzy inference using IT2-FSs stems from the

preservation of information uncertainty in the

inferential process. Due to the complexity of

computation, the research development of type-

2 fuzzy studies was slow in that time. In the late

1990s, some problems of type-2 fuzzy studies

were overcome, such as type-reduction (TR),

computational method for TR and fuzzy

inference based on type-2 fuzzy system. These

results helped in the studies. J. M. Mendel and

his colleagues made excellent contributions on

the type-2 fuzzy theory [6]. Karnik and Mendel

first defined the problem of TR that was critical

for type-2 fuzzy systems. Based on this

foundation, they further established the

computational procedure by which the type-2

fuzzy inference carried on. By TR, one can

obtain a type-1 fuzzy set or an interval which is

a special case of type-1 fuzzy set after the fuzzy

inference process of type-2 fuzzy model. Then,

defuzzification follows to obtain a crisp value

for the model output. These studies laid the

cornerstone for the research of type-2 fuzzy

logic and systems. After that, Karnik and

Mendel presented the famous Karnik-Mendel

(KM) algorithms for type reduction in 2001,

which could calculate the correction interval

after the inference process of an interval type-2

fuzzy model. Mendel’s book for type-2 fuzzy

theory was published in the same year [7]. In

2002, the representation theorem (RT) for type-

2 fuzzy sets was presented by Mendel and John

[8]. Based on RT, a type-2 fuzzy set can be

represented as a collection of simpler type-2

embedded sets. RT is important because join

and meet operations [9][10] of type-2 fuzzy sets

can be operated through type-1 fuzzy sets so

that the computation/analysis can be made

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simple. With RT, a type-2 fuzzy set can be

represented in two ways: one is called the

vertical-slice representation (VSR) that is good

for type-2 fuzzy computation and the other is

named the wavy-slice representation (WSR),

also known as the Mendel-John RT, that is

good for theoretical analysis of type-2 fuzzy

study. Nowadays, type-2 fuzzy has been

prospering in various areas such as control,

medical application, signal processing,

prediction, modeling, and more. The

computational efficiency of TR has been

concerned in the research community. Wu et al.

[11] presented a TR method based on α-planes.

Linda and Manic [12] presented the so-called

Monotone Centroid Flow (MCF) algorithm for

TR, which is also based on α-planes. These

developments above are all concerned in the

real-valued type-2 fuzzy studies. In this paper, I

present some results about complex-valued IT2

fuzzy studies.

The rest of the paper is organized below.

Section 2 presents the methodology used in the

study. Section 3 gives the PSO-KFA machine-

learning method for model optimization.

Section 4 is for experimentation where two

examples of function approximation are used to

test the proposed approach. Finally, the paper is

concluded with some remarks.

2 METHODOLOGY

2.1 Preliminary Background

Interval type-2 fuzzy set which is a simpler

version of real-valued type-2 fuzzy set is

specified in the following form.

�̃� = ∫ ∫ 1/(𝑥, 𝑢)𝑢∈𝐽𝑥⊆[0,1]𝑥∈𝑋

=

∫ [∫ 1/𝑢𝑢∈𝐽𝑥⊆[0,1]

] /𝑥𝑥∈𝑋

. (1)

where 𝑥 is the primary base variable; 𝑢 is the

secondary base variable whose value is

calculated using the primary membership

function and 𝑥; 𝐽𝑥 which is changed according

to 𝑥 is the domain of 𝑢. In general, an interval

type-2 fuzzy set can be defined by two primary

membership functions, that is, the so-called

upper membership function (UMF) and lower

membership function (LMF). They together

form the so-called footprint of uncertainty

(FOU) which is a two-dimensional area of the

x-u plane.

A complex fuzzy set (or CFS in short) is

defined by a complex-valued membership

function whose values are within the unit disc

of the complex plane (UDCP). A CFS is viewed

as a type-1 CFS (T1-CFS). Assume that there is

a T1-CFS, denoted as 𝑆 , whose membership

function comprises an amplitude function and a

phase function. The T1-CFS is defined below.

𝑢𝑠(𝑥) = 𝑟𝑠(𝑥) exp(𝑗𝜔𝑠(𝑥)), (2)

where 𝑢𝑠(𝑥) is the complex-valued

membership function; 𝑟𝑠(𝑥) ∈ [0,1] is the

amplitude function; 𝜔𝑠(𝑥) ∈ 𝑅 is the phase

function; 𝑗 = √−1; 𝑥 ∈ 𝑋 is the base variable

(or called numerical variable)。 It is clear that

𝑢𝑠(𝑥) ∈ UDCP , where UDCP ≡ {𝑢 | abs(𝑢) ≤1, 𝑢 ∈ 𝐶} and 𝐶 is the universe of all complex

numbers. If 𝜔𝑠(𝑥) = 0, 𝑢𝑠(𝑥) = 𝑟𝑠(𝑥). That is,

the T1-CFS degenerates to become a regular

T1-FS. This clearly reveals the fact that T1-

CFS includes T1-FS completely. An example of

T1-CFS is shown in Fig. 1. For a T2-CFS can

be constructed with two T1-CFSs in a two-

staged structure, based on this idea, one con

construct a Tn-CFS using n T1-CFSs in an n-

staged structure.

Similarly to a real-valued T2-FS, a T2-CFS is

with the so-called primary and secondary

complex-valued membership functions for the

two-staged structure. If either of the two phase

functions of the primary and secondary

membership functions of the T2-CFS exists, it

is still a type-2 complex fuzzy set. For this

reason, the class of T2-CFSs includes

completely that of T2-FSs. This simply implies

that a real-valued T2-FS for which two phase

functions disappear concurrently is only a

special case of a T2-CFS. A type-2 complex

fuzzy set, �̃�, can be expressed as follows.

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Fig. 1. Illustration of a type-1 complex fuzzy set, where

the real-part and imaginary-part membership dimensions

and the base-variable dimension are shown. If viewing in

the direction of the base-variable dimension, one can see

the complex-valued membership function 𝑢𝑠(𝑥) lying on

the unit disc of the complex plane.

�̃� = ∫ 𝜇�̃�(𝑥)/𝑥𝑥∈𝑋

, or

�̃� = ∫ ∫ 𝜇�̃�(𝑥, 𝑢)/(𝑥, 𝑢)𝑢∈𝐼𝑥𝑥∈𝑋

=

∫ [∫ 𝜇�̃�(𝑢)/𝑢𝑢∈𝐼𝑥

]𝑥∈𝑋

/𝑥,

(3)

where 𝑥 is the primary base variable; 𝑢 is the

secondary base variable whose value is

obtained using the primary complex-valued

membership and 𝑥; 𝐼𝑥 is the domain of 𝑢, 𝐼𝑥 ⊂UDCP。

2.2 Interval Type-2 Complex Fuzzy Set

If the secondary complex-valued membership

function degenerates to become the constant

one, i.e., 𝜇�̃�(𝑥, 𝑢) = 1, then the corresponding

type-2 complex fuzzy set will become a much

simpler type-2 version, called the interval type-

2 complex fuzzy set (IT2-CFS) . An IT2-CFS

can be expressed below.

�̃� = ∫ ∫1

(𝑥,𝑢)𝑢∈𝐼𝑥𝑥∈𝑋= ∫ [∫ 1/𝑢

𝑢∈𝐼𝑥]

𝑥∈𝑋/𝑥, (4)

An IT2-CFS can be constructed using two

complex-valued membership functions (also

known as two primary complex-valued

membership functions): one is called the upper

complex-valued membership function (UMFc)

and the other is called the lower complex-

valued membership function (LMFc). The area

between UMFc and LMFc is called the

complex-valued footprint of uncertainty,

denoted as FOUc, in which there are many

embedded T1-CFSs. FOUc can be expressed

below.

FOUc(�̃�) = ⋃ 𝐼𝑥𝑥∈𝑋 . (5)

If the two phase functions of the UMFc and

LMFc disappear, the FOUc degenerates to

become a real-valued FOU. This implies FOUc

can include FOU completely. IT2-CFS can be

with various forms, one of which is shown in

Fig. 2.

Fig. 2. An interval type-2 complex fuzzy set, where the

blue-colored primary complex-valued membership

function serves as the upper complex-valued membership

function (UMFc) and the red-colored one is for the lower

complex-valued membership function (LMFc). The area

between UMFc and LMFc is the complex-valued

footprint of uncertainty (FOUc).

2.3 Interval Type-2 Complex-Fuzzy Logic

System

Fig. 3. Interval type-2 complex fuzzy inferential system.

-5

0

5

-1

-0.5

0

0.5

1-1

-0.5

0

0.5

1

base variable, xmembership degree (real-part)

me

mb

ers

hip

de

gre

e (

ima

gin

ary

-pa

rt)

-5

0

5

-1

-0.5

0

0.5

1-1

-0.5

0

0.5

1

base variable, xmembership degree (real-part)

me

mb

ers

hip

de

gre

e (

ima

gin

ary

-pa

rt)

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Rule base:

Suppose we have an interval type-2 complex

fuzzy inferential system (IT2-CFIS), as shown

in Fig. 3, whose rule base has K Takagi-Sugeno

fuzzy If-Then rules, given below.

Rule 𝑘:

If 𝑣1 = �̃�1(𝑘)(𝑥1) and … and 𝑣𝑝 = �̃�𝑝

(𝑘)(𝑥𝑝)

Then 𝑦(𝑘) = 𝑎0(𝑘)

+ 𝑎1(𝑘)

𝑥1 + ⋯+ 𝑎𝑝(𝑘)

𝑥𝑝,

(for 𝑘 = 1,2, … , 𝐾)

(6)

where 𝑣𝑗 is the jth input linguistic variable

for 𝑗 = 1,2, … , 𝑝 ; for 𝑣𝑗 , 𝑥𝑗 ∈ 𝑋𝑗 is its

corresponding base variable and 𝑋𝑗 is the

universe of discourse for 𝑥𝑗 ; �̃�𝑗(𝑘)

is the jth

premise condition of the kth fuzzy rule and it

can be specified using an IT2-CFS for the IT2-

CFIS; 𝑦(𝑘) at the consequence of the kth fuzzy

rule is the rule output that is defined by a linear

function of {𝑥𝑗 , 𝑗 = 1,2,… 𝑝} ; {𝑎𝑗(𝑘)

, 𝑗 =

0,1,2, … 𝑝} is the parameters of the consequent

linear function. Not that these parameters can

be complex-valued for the IT2-CFIS.

Inferential process of IT2-CFIS:

Suppose that at time t the IT2-CFIS receives an

input vector whose components are {𝑥𝑗(𝑡), 𝑗 =

1,2, … 𝑝} . These input components can be

mapped into their corresponding universes,

respectively. Then, the fuzzy rules are activated

by the input vector. The inferential process is

given below to specify how the IT2-CFIS

computes to obtain its model output.

For the kth fuzzy rule, when 𝑥𝑗 = 𝑥𝑗(𝑡) , the

�̃�𝑗(𝑘)

can calculate a complex-valued

membership degree interval, given below.

𝐼𝑥𝑗(𝑡) ≡ [𝜇�̃�𝑗

(𝑘) (𝑥𝑗(𝑡)) , 𝜇�̃�𝑗

(𝑘) (𝑥𝑗(𝑡))]𝐶

, (7)

𝐼𝑥𝑗(𝑡) ⊂ UDCP , for 𝑗 = 1,2, … 𝑝 , where [𝑎, 𝑏]𝐶represents the complex-valued interval that is

formed by the complex numbers a and b whose

values are in the unit disc of the complex plane;

𝜇�̃�𝑗

(𝑘)(. ) and 𝜇�̃�𝑗

(𝑘)(. ) are the LMFc and UMFc

of �̃�𝑗(𝑘)

, respectively. Unlike the case by

traditional fuzzy inference, the firing strength

of the kth fuzzy rule of the IT2-CFIS is not a

single value but an interval, called the complex-

valued firing-strength interval (CFI), defined

below.

𝐹(𝑘)(x⃑(𝑡)) ≡ [𝛽(𝑘)(x⃑(𝑡)), 𝛽(𝑘)

(x⃑(𝑡))]𝐶

⊂ UDCP, (8)

where

x⃑(𝑡) = [𝑥1(𝑡) 𝑥2(𝑡) … 𝑥𝑝(𝑡)], (9)

𝛽(𝑘)(x⃑(𝑡)) = ∏ 𝜇�̃�𝑗

(𝑘) (𝑥𝑗(𝑡))𝑝𝑗=1 , (10)

𝛽(𝑘)

(x⃑(𝑡)) = ∏ 𝜇�̃�𝑗

(𝑘) (𝑥𝑗(𝑡))𝑝𝑗=1 , (11)

for 𝑘 = 1,2, … , 𝐾.

The rule output of the kth rule is still an IT2-

CFS, expressed below.

𝜇�̃�(𝑘)(𝑦) = ∫ 1/𝑏(𝑘)

𝑏(𝑘)∈[𝑏(𝑘),𝑏(𝑘)

]𝐶

(12)

where

𝑏(𝑘) ≡ 𝛽(𝑘)(x⃑(𝑡)) ∙ 𝑦(𝑘)

={∏ 𝜇�̃�𝑗

(𝑘) (𝑥𝑗(𝑡))𝑝𝑗=1 } ∙ {𝑎0

(𝑘)+ ∑ 𝑎𝑖

(𝑘)𝑥𝑝

𝑝𝑖=1 },

(13)

𝑏(𝑘)

≡ 𝛽(𝑘)

(x⃑(𝑡)) ∙ 𝑦(𝑘)

={∏ 𝜇�̃�𝑗

(𝑘) (𝑥𝑗(𝑡))𝑝𝑗=1 } ∙ {𝑎0

(𝑘)+ ∑ 𝑎𝑖

(𝑘)𝑥𝑝

𝑝𝑖=1 }.

(14)

The model output by the IT2-CFIS is obtained

by combining all the inferential rule outputs,

expressed below.

𝜇�̃�(𝑦) = ⋃ 𝜇�̃�(𝑘)(𝑦)𝐾𝑘=1 = ∫ 1/𝑏

𝑏∈[𝑏, 𝑏 ]𝐶, (15)

where

𝑏 ≡ ⋁ 𝑏(𝑘)𝐾𝑘=1 , (16)

𝑏 ≡ ⋁ 𝑏(𝑘)

𝐾𝑘=1 . (17)

ISBN: 978-1-941968-16-1 ©2015 SDIWC 196


And, ⋁ represents the fuzzy-union operation.

Basically, the output by the IT2-CFIS is an

interval type-2 complex fuzzy set, as stated in

equation (15). This may need the computation

of type reduction (TR) to obtain an interval,

with which the computation of defuzzification

can be further applied to obtain a crisp value

that can be complex-valued for practical

application. As stated before, the purpose of TR

is to obtain an interval, expressed below.

𝑌𝑇𝑅 = [𝑎(x⃑), 𝑏(x⃑)] ≡ [𝑎, 𝑏]𝐶 (18)

For simple calculation, the output by IT2-CFIS

that is a complex-valued interval [𝑎, 𝑏]𝐶 is

computed with the Liang-Mendel unnormalized

method [13] [14].

𝑎 = ∑ 𝛽(𝑘) ∙ {𝑎0(𝑘)

+ ∑ 𝑎𝑖(𝑘)

𝑥𝑝𝑝𝑖=1 }𝐾

𝑘=1 , (19)

𝑏 = ∑ 𝛽(𝑘)

∙ {𝑎0(𝑘)

+ ∑ 𝑎𝑖(𝑘)

𝑥𝑝𝑝𝑖=1 }𝐾

𝑘=1 , (20)

Using equations (19) and (20), a complex-

valued crisp value can be obtained by the

simple defuzzification method, given below.

𝑑 = (𝑎 + 𝑏)/2. (21)

The defuzzified model output, 𝑑 , can be

expressed as follows.

𝑑 = �⃗⃗� �⃗⃗� , (22)

where

�⃗⃗� = [𝜙1x⃑𝑎 𝜙2x⃑𝑎 ⋯ 𝜙𝐾 x⃑𝑎]1×(𝐾∙(𝑝+1)),

𝜙𝑘 = (𝛽(𝑘) + 𝛽(𝑘)

)/2 for k=1,2,…,K, (23)

x⃑𝑎 = [1 𝑥1 𝑥2 … 𝑥𝑝], (24)

�⃗⃗� = [�⃗⃗� (1) �⃗⃗� (2) ⋯ �⃗⃗� (𝐾)]T,

which is a (𝐾 ∙ (𝑝 + 1)) × 1 vector, (25)

�⃗⃗� (𝑘) = [𝑎0(𝑘)

𝑎1(𝑘)

⋯𝑎𝑝(𝑘)

],

𝑘 = 1,2, … , 𝐾. (26)

2.4 PSO-KFA Method for Machine Learning

The set of parameters of IT2-CFIS, denoted as

W, is composed of two subsets: the subset of

the premise parameters denoted as WIf and the

subset of the consequent parameters denoted as

WThen. Namely, one can express their relation

in the following equation.

W = WIf ∪ WThen. (27)

I design a new hybrid machine learning method

that involves the famous particle swarm

optimization (PSO) and the well-known

Kalman filtering algorithm (KFA) to adjust W

of the proposed IT2-CFIS model. The PSO and

KFA are used in a cooperatively hybrid way to

evolve WIf and WThen, respectively, during the

parameter-learning phase for the proposed

model.

PSO is a method that utilizes a swarm-based

intelligence for optimization. Assume that for

PSO there are birds (or called particles) in a

swarm with size n, each of which searches for

better solution during the process of

optimization, where two types of information

are used by each particle: the swarm’s best

location and the particle’s best location. Each

particle’s location is viewed as a potential

solution. This is an excellent strategy that can

avoid of falling into local optima during the

search for better solution. Each particle is with

the information of velocity and location, which

are updated using the following equations.

𝑖(𝑘 + 1) =𝜔𝑖(𝑘) + 𝑐1𝜌1(𝐩𝐛𝐞𝐬𝐭𝑖(𝑘) − 𝐋𝑖(𝑘)) +

𝑐2𝜌2(𝐠𝐛𝐞𝐬𝐭(𝑘) − 𝐋𝑖(𝑘)),

(28)

𝐋𝑖(𝑘 + 1) = 𝐋𝑖(𝑘) + 𝑖(𝑘 + 1), (29)

where i(k)=[i,1(k), i,2(k),…, i,Q(k)]T is the

velocity of the ith particle at iteration k;

Li(k)=[ li,1(k), li,2(k),…, li,Q(k)]T is the location

of the ith particle at iteration k; { 𝜔,c1,c2} are

the PSO parameters; {1, 2} are random

numbers uniformly distributed in the unit

interval [0,1]; 𝐩𝐛𝐞𝐬𝐭𝑖(𝑘) is the best location of

the ith particle at iteration k so far; 𝐠𝐛𝐞𝐬𝐭(𝑘) is

ISBN: 978-1-941968-16-1 ©2015 SDIWC 197


the best location of the whole swarm at iteration

k so far. Note that iteratively all the particles’

best locations compete one another to become

the swarm’s best location.

The method of Kalman filtering is a general

method that can be used in smoothing, filtering

and prediction for signals in a process, either

stationary or non-stationary. The method is also

called the Kalman filtering algorithm (KFA).

The evolution of the parameters for

optimization of the proposed model can be

viewed as a dynamic process, for which the

KFA can be applied. A Kalman process can be

expressed by the following equations [15].

𝓧(𝑘 + 1) = 𝓖(𝑘)𝓧(𝑘) + 𝓗(𝑘)𝓾(𝑘) + 𝔀(𝑘), (30)

𝔂(𝑘) = 𝓒(𝑘)𝓧(𝑘) + 𝛜(𝑘), (31)

where 𝓧(𝑘) is the state of the process at

iteration k; 𝓾(𝑘) is the input vector to the

process; 𝓖(𝑘) is the state matrix; 𝓗(𝑘) is the

input matrix; 𝔀(𝑘) is the process noise; 𝔂(𝑘)

is the output vector of the process; 𝓒(𝑘) is the

output matrix; 𝛜(𝑘) is the measurement noise.

The state 𝓧 can be regarded as the vector �⃗⃗� that

is formed by the consequent parameters of the

proposed model. By the method of KFA, the

target is to obtain the optimal estimated state,

denoted as �̃�, so that 𝐏(𝑘) = 𝐸[e(𝑘)e∗(𝑘)] isminimized, where e(𝑘) is the estimated error of

the state at iteration k and * indicates the

operation of conjugate transpose. For KFA, the

vector �⃗⃗� at iteration k+1 is updated with the

equation given below.

�⃗⃗� (𝑘 + 1) = �⃗⃗� (𝑘) + 𝐊𝑒(𝑘 + 1) ∙

{𝔂(𝑘 + 1) − �⃗⃗� (𝑘 + 1)�⃗⃗� (𝑘)}, (32)

where

𝐊𝑒(𝑘 + 1) =𝐏(𝑘)�⃗⃗� ∗(𝑘+1)

𝜎2𝐈+�⃗⃗� (𝑘+1)𝐏(𝑘)�⃗⃗� ∗(𝑘+1) , (33)

𝐏(𝑘 + 1) = 𝐏(𝑘) − 𝐊𝑒(𝑘 + 1) ∙

�⃗⃗� (𝑘 + 1)𝐏(𝑘), (34)

�⃗⃗� ∗ = congugate transpose of �⃗⃗� , (35)

𝔂(𝑘 + 1) = desired output (target). (36)

The machine learning procedure of the

proposed PSO-KFA method is given below.

Step 1: Prepare training data (TD) for the

proposed model, TD = {(x⃗ 𝑖 , 𝑡𝑖), 𝑖 =1,2, … , 𝑛} , where (x⃗ 𝑖 , 𝑡𝑖) is the ith data pair

with the paired form of (input, target).

Step 2: Use the mean-squared error (MSE) as

the performance index during the training

process of the proposed model. MSE =1

𝑛∑ 𝑒𝑖𝑒𝑖

∗𝑛𝑖=1 , where 𝑒𝑖 = (𝑡𝑖 − 𝑑𝑖) ; 𝑑𝑖 is the

output by the proposed IT2-CFIS to which the

x⃗ 𝑖 is inputted; 𝑡𝑖 is the target for 𝑑𝑖.

Step 3: Start the PSO algorithm, where the

location of each particle represents a position

vector that is formed by the premise

parameters of the IT2-CFIS.

Step 4: For each PSO particle, with the input

vector x⃗ 𝑖, calculate the firing-strength interval

of each fuzzy rule.

Step 5: Apply the KFA to update the

consequent parameters of the IT2-CFIS, with

which the output interval by the proposed

model can be calculated. By defuzzification, a

crisp output 𝑑𝑖 is obtained, with which the

error 𝑒𝑖 = (𝑡𝑖 − 𝑑𝑖) is calculated.

Step 6: Repeat Steps 4 and 5 until all training

data are used up. With the errors {𝑒𝑖 , 𝑖 =1,2, … , 𝑛} , calculate MSE for this PSO

particle. Update its 𝐩𝐛𝐞𝐬𝐭 if better MSE is

found.

Step 7: Go back to Step 4 for next PSO particle

until going through all the PSO particles.

Among all the 𝐩𝐛𝐞𝐬𝐭s of the PSO particles,

select the best one as the 𝐠𝐛𝐞𝐬𝐭. And, update

the premise parameters using the 𝐠𝐛𝐞𝐬𝐭. Step 8: Update the location of each PSO

particle, using its 𝐩𝐛𝐞𝐬𝐭 and the swarm’s

𝐠𝐛𝐞𝐬𝐭 so far.

Step 9: Stop if any of stopping conditions is

satisfied. Otherwise, go back to Step 4 and the

procedure goes on.

ISBN: 978-1-941968-16-1 ©2015 SDIWC 198


3 EXPERIMENTATION

To test the proposed approach, the problem of

function approximation is applied, where the

proposed IT2-CFIS is given to fit the data that are

sampled from a target function. Two examples are

given. For the first example, a one-dimension sine

wave function is used to serve as target function.

And, the two-dimension sinc function is used to

serve as target function for the second example.

Example 1. The one-dimensional sine wave

function that serves as a target function for the

proposed IT2-CFIS is given below.

𝑦 = 0.1 + 1.2𝑥 + 2.8𝑥 ∙ sin(4𝜋𝑥2), (37)

where 0 ≤ 𝑥 ≤ 1. A proposed IT2-CFIS model is given, whose rule

base has nine T-S fuzzy If-Then rules. Each fuzzy

rule is with one input and one output. The rules

are expressed as "If 𝑣1 = �̃�1(𝑘)(𝑥1), then 𝑦(𝑘) =

𝑎0(𝑘)

+ 𝑎1(𝑘)

𝑥1, " 𝑘=1,2,…,9, where �̃�1(𝑘)(𝑥1) is a

IT2-CFS that describes the premise condition of

the kth rule. The universe of discourse for 𝑥1 is

divided by nine IT2-CFSs {�̃�1(𝑘)(𝑥1), 𝑘 =

1,2, … ,9}, each of which is synthesized with 2

Gaussian type-1 complex fuzzy sets [16]

separated horizontally. Fifty data pairs, denoted

as {(𝑥𝑖 , 𝑦𝑖), 𝑖 = 1,2, … ,50} , are sampled

uniformly in 0 ≤ 𝑥 ≤ 1 from the target

function for the training of the IT2-CFIS model.

Another 101 data pairs are sampled for testing.

With the PSO-KFA method and the training

data, the proposed model is trained repeatedly

up to 500 epochs. With MSE, the cost function

is designed. For the PSO, the swarm size is

given to be 15; 𝜔=0.85; 𝑐1 = 𝑐2 = 2. For each

PSO particle, initial location and velocity are

given randomly. For the KFA, 𝐏(0) = 1E10 ×I and 𝜎 = 1 . After learning, the experimental

results are given below. The performance by the

proposed approach is compared to that by other

approaches. The results indicate the proposed

approach performs better than the compared

ones.

Table 1. Performance comparison (sine wave function)

Method Epochs MSE

MLP [17] - 6.57 × 10−3

SIANN [17] - 1.76 × 10−3

IT2CFIS

(proposed, training phase) 500 1.52 × 10−7

IT2CFIS

(proposed, testing phase) 500 1.46 × 10−7

Fig. 4. Model response by the IT2-CFIS trained with 500

epochs. (Sine wave function) (Testing with101 data pairs). As

shown, the two curves are hardly identified, for the target

function and the model output are almost coincided.

Example 2. The two-dimensional sinc function

that serves as a target function for the proposed

IT2-CFIS is given as follows.

𝑧 = {

1, 𝑖𝑓 𝑟 = 0

𝑠𝑖𝑛(𝜋𝑟)

𝜋𝑟, otherwise

, (38)

where

𝑟 =√𝑥2 + 𝑦2

2,

−1 ≤ 𝑥 ≤ 1 and − 1 ≤ 𝑦 ≤ 1.

A IT2-CFIS is given, whose rule base has thirty T-

S fuzzy rules with two inputs and one output. The

premise complex fuzzy sets are designed using the

same way as in the previous example. The first

input variable has six IT2-CFSs and the second

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

x

y

Testing phase,MSE=1.4622e-07

target

model output (real part)

ISBN: 978-1-941968-16-1 ©2015 SDIWC 199


input variable has five IT2-CFSs. Each of them

is designed using the same technique as in the

previous example. For the training of the

proposed model, four hundred data pairs are

sampled uniformly from the target function,

denoted as {(𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖), 𝑖 = 1,2, … ,400}.Another 2500 data pairs are sampled for testing.

The PSO-KFA is used to evolve the proposed

model repeatedly up to fifty epochs. The settings

for the PSO and the KFA are the same as in the

previous example. In the following are the

experimental results by the proposed approach,

whose performance is also compared to that by its

real-valued counterpart (denoted as IT2-RFIS),

showing that the proposed approach performs

better than its real-valued counterpart.

(a)

(b)

(c)

Fig. 5. (a) Target function. (b) Model response by the IT2-

CFIS trained with 50 epochs. (c) Error surface. (2D sinc

function) (Testing with 2500 data pairs).

Table 2 Performance comparison (2D sinc function)

Method Epochs MSE

IT2-RFIS (training phase) 50 5.62 × 10−12

IT2-RFIS (testing phase) 50 1.11 × 10−9

IT2-CFIS (proposed, training phase) 50 2.71 × 10−12

IT2-CFIS (proposed, testing phase) 50 2.55 × 10−12

4 CONCLUSION

The proposed interval type-2 complex fuzzy

inferential system (IT2-CFIS) has been presented,

which integrates the rationales of type-2 fuzzy sets

and complex fuzzy sets and fuzzy inferential logic

to forge a new computing tool for modeling. This

paper has reported some primary results of IT2-

CFIS for modeling. Through the experimental

data, the IT2-CFIS has shown very good capability

in modeling. The concept of interval type-2

complex fuzzy sets in its first advent has been

presented in this paper. This idea has been

successfully used in the IT2-CFIS. Moreover, the

PSO-KFA is also a newly devised method for

parameter learning, which has been demonstrated

successfully in the experiments where the

proposed IT2-CFIS has been used for modeling

with good learning efficiency. This gives a very

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Testing phase, target

y

z

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Testing phase, model output (amplitude),MSE=2.552e-12

y

z

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1-1

-0.5

0

0.5

1

x 10-5

x

Testing phase,MSE=2.552e-12

y

err

or

ISBN: 978-1-941968-16-1 ©2015 SDIWC 200


promising motivation for further study of the IT2-

CFIS. The experimental results show that the

proposed approach performs better than the

compared approaches.

ACKNOWLEDGMENTS

This research work was supported by the

Ministry of Science and Technology, Taiwan,

ROC, under the Grant NSC102-2221-E-008-

098.

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Interval Type-2 Complex-Fuzzy Inferential System ― A New Approach to Modeling

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