IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 17, NO. 5 ...

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 17, NO. 5, OCTOBER 2009 1189 α-Plane Representation for Type-2 Fuzzy Sets: Theory and Applications Jerry M. Mendel, Life Fellow, IEEE, Feilong Liu, Member, IEEE, and Daoyuan Zhai, Student Member, IEEE Abstract—This paper 1) reviews the α-plane representation of a type-2 fuzzy set (T2 FS), which is a representation that is compara- ble to the α-cut representation of a type-1 FS (T1 FS) and is useful for both theoretical and computational studies of and for T2 FSs; 2) proves that set theoretic operations for T2 FSs can be computed using very simple α-plane computations that are the set theoretic operations for interval T2 (IT2) FSs; 3) reviews how the centroid of a T2 FS can be computed using α-plane computations that are also very simple because they can be performed using existing Karnik Mendel algorithms that are applied to each α-plane; 4) shows how many theoretically based geometrical properties can be obtained about the centroid, even before the centroid is computed; 5) pro- vides examples that show that the mean value (defuzzified value) of the centroid can often be approximated by using the centroids of only 0 and 1 α-planes of a T2 FS; 6) examines a triangle quasi-T2 fuzzy logic system (Q-T2 FLS) whose secondary membership func- tions are triangles and for which all calculations use existing T1 or IT2 FS mathematics, and hence, they may be a good next step in the hierarchy of FLSs, from T1 to IT2 to T2; and 7) compares T1, IT2, and triangle Q-T2 FLSs to forecast noise-corrupted measurements of a chaotic Mackey–Glass time series. Index Termsα-Plane, centroid, Mackey–Glass time series, quasi-type-2 fuzzy logic systems (Q-T2 FLSs), set theoretic op- erations, type-2 fuzzy sets (T2 FSs). I. INTRODUCTION A LTHOUGH most applications of type-2 fuzzy sets (T2 FSs) 1 use interval T2 FSs (IT2 FSs) (e.g., [9]–[12], [24]–[26], [30], [43], [47], [50], and [51]), recently, there has been a growing interest in using general T2 FSs (e.g., [2]–[5], [7], [8], [16], [22], [23], [44]–[46], and [48]) because they have more design degrees of freedom than IT2 FSs and, therefore, have the potential to outperform a system that uses IT2 FSs. This paper describes a method that represents a T2 FS, which has been recently published in [22] and [23], which is called an α-plane representation, and how it can be used to solve some problems, including what may be the next step in the logical progression from a type-1 fuzzy logic system (T1 FLS) to an interval type-2 FLS (T2 FLS) to a so-called triangle Q-T2 FLS Manuscript received March 23, 2008; revised August 12, 2008, February 8, 2009, and April 7, 2009; accepted April 27, 2009. First published June 5, 2009; current version published October 8, 2009. J. M. Mendel and D. Zhai are with the Signal and Image Processing Insti- tute, Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089-2564 USA (e-mail: [email protected]; [email protected]). F. Liu was with the University of Southern California, Los Angeles, CA 90089-2564 USA. He is now with Chevron Corporation, Richmond, CA 94801 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2009.2024411 1 For many more applications, see [32] and also the T2 Web site http://www. type2fuzzylogic.org. that was proposed in [37]. Although it relies upon the methodol- ogy in [22], [23], and [37], it presents many new results—results that we hope will make general T2 FSs more widely used. This paper is organized as follows. Background materials, in- cluding a brief review of the α-plane representation, are covered in Section II. Computing set theoretic operations for T2 FSs us- ing α-planes is described in Section III. Computing the centroid of a T2 FS using α-planes and obtaining a collection of proper- ties about the shape of the centroid are presented in Section IV. Examples of centroid calculations, as well as some approxima- tions to these calculations, are given in Section V. Triangle Q-T2 FLSs are described in Section VI, which also includes a compar- ison of performances obtained from three kinds of FLSs for one- step prediction of a Mackey–Glass chaotic time series, whose measurements are corrupted by additive noise. Conclusions and some suggestions for further work are given in Section VII. II. BACKGROUND This section begins by reviewing three popular representa- tions for a T2 FS, and then, the new α-plane representation for a T2 FS is reviewed. It also includes some discussions about com- putation when the α-plane representation is used and provides some historical notes about the α-plane representation. A. Popular Representations for a T2 FS The point-valued representation, which is usually the starting point for understanding or describing a general T2 FS, e.g., [17], [30, p. 82], and [34], is one in which the membership function (MF) of ˜ A is specified at every point in its 2-D domain of support, i.e., ˜ A = {((x, u)˜ A (x, u)) |∀x X, u J x [0, 1] } . (1) While useful as a starting point for obtaining the other repre- sentations, (1) does not seem to be useful for much of anything else. The vertical-slice representation focuses on each value of the primary variable x, and expresses (1) (e.g., [17], [30, p. 83], [34], [40], and [52]) as ˜ A = xX µ ˜ A (x)/x (2) µ ˜ A (x)= µ ˜ A (u|x)= u J x [0, 1] f x (u)/u (3) where µ ˜ A (x) is a T1 FS and is called a secondary MF 2 or a ver- tical slice, J x is called the primary membership (or codomain), 2 µ ˜ A (x) is actually a function of secondary variable u, which is why it is also shown in (3) as µ ˜ A (u|x). Because the notation µ ˜ A (x) is already widely used by the T2 FS community, it is not changed here. 1063-6706/$26.00 © 2009 IEEE Authorized licensed use limited to: University of Southern California. Downloaded on October 6, 2009 at 18:14 from IEEE Xplore. Restrictions apply.

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 17, NO. 5, OCTOBER 2009 1189

α-Plane Representation for Type-2 Fuzzy Sets:Theory and Applications

Jerry M. Mendel, Life Fellow, IEEE, Feilong Liu, Member, IEEE, and Daoyuan Zhai, Student Member, IEEE

Abstract—This paper 1) reviews the α-plane representation of atype-2 fuzzy set (T2 FS), which is a representation that is compara-ble to the α-cut representation of a type-1 FS (T1 FS) and is usefulfor both theoretical and computational studies of and for T2 FSs;2) proves that set theoretic operations for T2 FSs can be computedusing very simple α-plane computations that are the set theoreticoperations for interval T2 (IT2) FSs; 3) reviews how the centroid ofa T2 FS can be computed using α-plane computations that are alsovery simple because they can be performed using existing KarnikMendel algorithms that are applied to each α-plane; 4) shows howmany theoretically based geometrical properties can be obtainedabout the centroid, even before the centroid is computed; 5) pro-vides examples that show that the mean value (defuzzified value)of the centroid can often be approximated by using the centroids ofonly 0 and 1 α-planes of a T2 FS; 6) examines a triangle quasi-T2fuzzy logic system (Q-T2 FLS) whose secondary membership func-tions are triangles and for which all calculations use existing T1 orIT2 FS mathematics, and hence, they may be a good next step in thehierarchy of FLSs, from T1 to IT2 to T2; and 7) compares T1, IT2,and triangle Q-T2 FLSs to forecast noise-corrupted measurementsof a chaotic Mackey–Glass time series.

Index Terms—α-Plane, centroid, Mackey–Glass time series,quasi-type-2 fuzzy logic systems (Q-T2 FLSs), set theoretic op-erations, type-2 fuzzy sets (T2 FSs).

I. INTRODUCTION

A LTHOUGH most applications of type-2 fuzzy sets (T2FSs)1 use interval T2 FSs (IT2 FSs) (e.g., [9]–[12],

[24]–[26], [30], [43], [47], [50], and [51]), recently, there hasbeen a growing interest in using general T2 FSs (e.g., [2]–[5],[7], [8], [16], [22], [23], [44]–[46], and [48]) because they havemore design degrees of freedom than IT2 FSs and, therefore,have the potential to outperform a system that uses IT2 FSs.This paper describes a method that represents a T2 FS, whichhas been recently published in [22] and [23], which is called anα-plane representation, and how it can be used to solve someproblems, including what may be the next step in the logicalprogression from a type-1 fuzzy logic system (T1 FLS) to aninterval type-2 FLS (T2 FLS) to a so-called triangle Q-T2 FLS

Manuscript received March 23, 2008; revised August 12, 2008, February 8,2009, and April 7, 2009; accepted April 27, 2009. First published June 5, 2009;current version published October 8, 2009.

J. M. Mendel and D. Zhai are with the Signal and Image Processing Insti-tute, Ming Hsieh Department of Electrical Engineering, University of SouthernCalifornia, Los Angeles, CA 90089-2564 USA (e-mail: [email protected];[email protected]).

F. Liu was with the University of Southern California, Los Angeles, CA90089-2564 USA. He is now with Chevron Corporation, Richmond, CA 94801USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TFUZZ.2009.20244111For many more applications, see [32] and also the T2 Web site http://www.

type2fuzzylogic.org.

that was proposed in [37]. Although it relies upon the methodol-ogy in [22], [23], and [37], it presents many new results—resultsthat we hope will make general T2 FSs more widely used.

This paper is organized as follows. Background materials, in-cluding a brief review of the α-plane representation, are coveredin Section II. Computing set theoretic operations for T2 FSs us-ing α-planes is described in Section III. Computing the centroidof a T2 FS using α-planes and obtaining a collection of proper-ties about the shape of the centroid are presented in Section IV.Examples of centroid calculations, as well as some approxima-tions to these calculations, are given in Section V. Triangle Q-T2FLSs are described in Section VI, which also includes a compar-ison of performances obtained from three kinds of FLSs for one-step prediction of a Mackey–Glass chaotic time series, whosemeasurements are corrupted by additive noise. Conclusions andsome suggestions for further work are given in Section VII.

II. BACKGROUND

This section begins by reviewing three popular representa-tions for a T2 FS, and then, the new α-plane representation for aT2 FS is reviewed. It also includes some discussions about com-putation when the α-plane representation is used and providessome historical notes about the α-plane representation.

A. Popular Representations for a T2 FS

The point-valued representation, which is usually the startingpoint for understanding or describing a general T2 FS, e.g., [17],[30, p. 82], and [34], is one in which the membership function(MF) of A is specified at every point in its 2-D domain ofsupport, i.e.,

A = {((x, u), µA (x, u)) |∀x ∈ X,∀u ∈ Jx ⊆ [0, 1]} . (1)

While useful as a starting point for obtaining the other repre-sentations, (1) does not seem to be useful for much of anythingelse.

The vertical-slice representation focuses on each value of theprimary variable x, and expresses (1) (e.g., [17], [30, p. 83],[34], [40], and [52]) as

A =∫∀x∈X

µA (x)/x (2)

µA (x) = µA (u|x) =∫∀u∈Jx ⊆[0,1]

fx(u)/u (3)

where µA (x) is a T1 FS and is called a secondary MF2 or a ver-tical slice, Jx is called the primary membership (or codomain),

2µA (x) is actually a function of secondary variable u, which is why it is alsoshown in (3) as µA (u|x). Because the notation µA (x) is already widely usedby the T2 FS community, it is not changed here.

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1190 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 17, NO. 5, OCTOBER 2009

and fx(u) is called the secondary grade. The vertical-slice rep-resentation is extremely useful for computation and can also beused for theoretical studies.

Finally, there is a wavy slice representation (which is alsocalled an embedded T2 FS representation or the Mendel–Johnrepresentation) [34] that is most valuable in theoretical studiesbecause it quickly leads to the structure of the solution to anew problem, after which, practical procedures are developedto compute that solution. An embedded T2 FS Aj

e is a T2 FS thathas only one primary membership at each xi . It is also called awavy slice [34]. The wavy slice representation is given as

A =⋃∀j

Aje . (4)

This representation of a T2 FS in terms of much simpler T2FSs—the embedded T2 FSs—is not recommended for compu-tational purposes, because it may (and usually does) require theexplicit enumeration of a large (referred to as astronomicallylarge) number of embedded T2 FSs to make its use infeasible.

B. α-Plane Representation for a T2 FS

Recently, Liu [22], [23] introduced a horizontal slice repre-sentation for a T2 FS, which is useful not only for theoreticalstudies but for computation as well. Because a horizontal sliceis analogous to the α-cut of a T1 FS, it is called an α-planerepresentation for a T2 FS.

Definition 1: An α-plane for the general T2 FS A, whichis denoted by Aα , is the union of all primary membershipsof A, whose secondary grades are greater than or equal to α(0 ≤ α ≤ 1), i.e.,

Aα = {(x, u), µA (x, u) ≥ α |∀x ∈ X,∀u ∈ Jx ⊆ [0, 1]}

=∫∀x∈X

∫∀u∈Jx

{(x, u) |fx(u) ≥ α}. (5)

Let SA (x|α) denote an α-cut of the secondary MF µA (x),i.e.,

SA (x|α) = [sL (x|α), sR (x|α)]. (6)

Then, another useful way to express Aα is

Aα =∫∀x∈X

SA (x|α)/x =∫∀x∈X

(∫∀u∈[sL (x|α),sR (x|α)]

u

)/x.

(7)Definition 2: The 2-D domain of A, which is called the

footprint of uncertainty (FOU) of A [30, p. 87], is denotedas FOU(A) and is the α = 0 plane, i.e.,

FOU(A) = A0 . (8)

Definition 3: FOU(A) is upper and lower bounded by

FOU(A) and FOU(A), respectively. FOU(A) is called an

upper MF (UMF) for FOU(A), i.e., UMFFOU(A)(x), and

FOU(A) is called a lower MF (LMF) for FOU(A), i.e.,LMFFOU(A)(x) (∀x ∈ X).

Using the concepts of LMF and UMF, Jx in (1) can be ex-pressed as

Jx ={(x, u) : ∀u ∈ [LMFFOU(A)(x),UMFFOU(A)(x)]

}⊆ [0, 1]. (9)

An IT2 FS is completely described by its FOU, but a generalT2 FS is not described by just its FOU.

Definition 4: Let IAα(x, u|α) be a 3-D indicator function for

α-plane Aα , where ∀x ∈ X and ∀u ∈ SA (x|α) and is given as

IAα(x, u|α) =

{1 (x, u) ∈ Aα

0 (x, u) /∈ Aα .(10)

Definition 5: An α-plane FOU, i.e., FOU(Aα ),3 is

FOU(Aα ) = αIAα(x, u|α). (11)

It can also be expressed as

FOU(Aα ) = α

∫∀x∈X

ISA (x|α)(u|α)/x (12)

where [see (6) and (7)]

ISA (x|α)(u|α) ={

1 ∀u ∈ [sL (x|α), sR (x|α)]

0 ∀u /∈ [sL (x|α), sR (x|α)].(13)

Equation (11) is a point-value representation of FOU(Aα ),whereas (12) is a vertical slice representation of FOU(Aα ).Equations (12) and (11) are equivalent because∫

∀x∈X

ISA (x|α)(u|α)/x = IAα(x, u|α). (14)

Definition 6: The α-plane representation (theorem) for Ais [22], [23]

A =⋃

α∈[0,1]

FOU(Aα ). (15)

A proof of this result is given in [22] and [23].The α-plane representation is very interesting because each

FOU(Aα ) can be associated with an IT2 FS of level α, withMF being equal to FOU(Aα ). This suggests that operationsthat involve T2 FSs can be performed using readily availableoperations for IT2 FSs. The procedure for performing the sameis demonstrated in Sections II and III.

Example 1: Fig. 1 depicts the vertical slice representationfor A, i.e., at each sampled value of primary variable x, thesecondary MF, i.e., µA (x), is shown. This figure is easy to draw(3-D figures, such as [23, Fig. 3], are usually not easy to draw)and gives the appearance of a 3-D diagram, but it is on a 2-Dplane, which is why it is called a “2.5-D representation.” Anexample of an α-plane for A is depicted in Fig. 2. It is the darkgray shaded area where the continuous primary variable x is

3FOU(Aα ) is called an “associated T2 FS, A(α)” in [22] and [23]. In thispaper, we prefer FOU(Aα ) since this terminology is already well establishedin the T2 FS field.

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MENDEL et al.: α -PLANE REPRESENTATION FOR TYPE-2 FUZZY SETS: THEORY AND APPLICATIONS 1191

Fig. 1. A 2.5-D representation of the 3-D MF for µA (x, u). The gray shadedarea is the FOU. The black filled-in entities are secondary MFs µA (xi ). Theyare filled in only for the effect and give a 3-D effect to the 2-D figure.

Fig. 2. α-plane for A. It is the dark gray shaded area where the continuousprimary variable x is sampled as shown. Each of the dark black vertical linescorresponds to SA (xi |α).

Fig. 3. Dark gray shaded area is FOU(Aα ) for the example in Fig. 2.

sampled, as shown. Each dark black vertical line correspondsto SA (xi |α). As the sampling of x becomes ever so fine, thesedark black vertical lines fill in to give the shaded area of theα-plane. FOU(Aα ) is depicted in Fig. 3. Unlike Aα , which is

located on the x–u plane in Fig. 2, FOU(Aα ) is the plane Aα

raised off of the x–u plane to level α.

C. Computation

From (4) and (15), it is obvious that α-plane and wavy slicerepresentations for a T2 FS are theoretically equivalent, i.e.,

A =⋃

α∈[0,1]

FOU(Aα ) =⋃∀j

Aje . (16)

Consequently, any computation that involves T2 FSs, which mayhave originally been performed using embedded T2 FSs, neednot be performed this way. Instead, they can be performed usingα-planes; however, unless all of the discretizations required byeach approach are very small, the results will not necessarily beexactly the same.

The wavy slice approach depends upon the discretizationsused for the primary and secondary variables, i.e., the finer thediscretizations, the more embedded T2 FSs there will be. On theother hand, the α-plane approach depends only on the discretiza-tion of α, i.e., the finer the discretizations, the more α-planesthere will be, but this number does not become astronomical.

When computing an α-plane, following two cases are distin-guished.

1) LMFFOU(A)(x) and UMFFOU(A)(x) are prespecified ei-ther numerically or analytically ∀x ∈ X: In this case, (3)is reexpressed as

µA (x) = µA (u|x)

=∫∀u∈[LMFF O U ( A ) (x),UMFF O U ( A ) (x)]

fx(u)/u.

(17)

Because µA (x) is a T1 FS, its α-cut, which is SA (x|α), canbe computed using any existing technique for computingthe α-cut of a T1 FS ∀x ∈ X .

2) The 3-D T2 FS is stored as in (1), e.g., from a previ-ous computation. Extract vertical slices µA (uj |xi), j =1, . . . ,M(Mi), from storage (beginning with x1 and end-ing with xN ). By fitting a spline function to the M(Mi)points, µA (u|xi) is obtained, and u ∈ [u1(xi), uM (xi)].4

µA (u|xi) is an approximation to µA (u|xi). BecauseµA (u|xi) is a T1 FS, its α-cut [which approximates theα-cut of µA (u|xi)] SA (xi |α) can be computed using anyexisting technique for computing the α-cut of a T1 FS∀xi(i = 1, . . . , N).

D. Historical Notes

Zadeh [52] has always been acknowledged to be the origina-tor of a T2 FS (and even higher type FSs). He does not use theterm α-plane, nor does he have an α-plane decomposition fora T2 FS. Chen and Kawase [1] use the term α-cuts, and whilethey do not have a formal representation theorem for a T2 FS,a secondary MF is expressed in [1, Def. 3.2] in terms of

4This step permits the secondary MFs to be sampled at arbitrary values of α.

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1192 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 17, NO. 5, OCTOBER 2009

α-cuts. Surprisingly, they seem to be totally unaware of theT2 FS literature, i.e., they do not refer to any paper about T2FSs (not even [52]). Tahayori et al. [46] state a “representationprinciple” without proof; it is the same as Liu’s α-plane rep-resentation theorem. They do not refer to [1], nor do they usethe term α-plane. Liu [22], [23] was the first to state and provethe α-plane representation theorem and to call it the α-planerepresentation of a T2 FS. He does not refer to [1], [46], or [52].As sometimes happens, the same result is obtained almost atthe same time by a different group of researchers. Although thework by Wagner and Hagras [48] appeared after the first versionof this paper had been submitted, it is still notable. They inde-pendently arrived at the same decomposition as Liu [22], [23]. Infact, a z-slice is the same as an α-plane. They call it a “z-slice”because the three axes for the MF of their T2 FS are labeled asx, y, and z. Their paper contains a representation theorem thatis equivalent to the α-plane representation theorem.

III. SET-THEORETIC OPERATIONS

A. Introduction

Consider two T2 FSs: A and B, i.e.,

A =∫∀x∈X

µA (x)/x =∫∀x∈X

[∫∀u∈J u

x

fx(u)/u

]/x

Jux = {(x, u) : ∀u ∈ [LMFFOU(A)(x),UMFFOU(A)(x)]}

⊆ [0, 1]

(18)and

B =∫∀x∈X

µB (x)/x =∫∀x∈X

[∫∀w∈J w

x

gx(w)/w

]/x

Jwx = {(x,w) : ∀w ∈ [LMFFOU(B )(x),UMFFOU(B )(x)]}

⊆ [0, 1]

.

(19)Mizumoto and Tanaka [40] showed that the union of A and

B is another T2 FS, whose MF can be computed from

A ∪ B ⇔ µA∪B (x, v) =∫∀x∈X

µA∪B (x)/x (20)

µA∪B (x) =∫∀u∈J u

x

∫∀w∈J w

x

fx(u)�gx(w)/(u ∨ w)

= µA (x) µB (x) ∀x ∈ X (21)

where denotes the join operation, and it is performed betweenthe two secondary MFs µA (x) and µB (x). A careful explanationof what (21) means can be found in [17], [30, p. 218], and [32].

In general, evaluating the join is difficult for arbitrary T2FSs. Karnik and Mendel [17] have shown that for n con-vex and normal T1 FSs, F1 , . . . , Fn , which are characterizedby MFs f1(θ), . . . , fn (θ), respectively, where f1(v1) = · · · =fn (vn ) = 1 and the fi(θ) are reordered so that v1 ≤ v2 ≤ · · · ≤vn , the MF ofn

i=1Fi , using maximum t-conorm and either min-imum or product t-norm, can be expressed as (T denotes the

t-norm)

µni = 1 Fi

(θ) =

Tni=1fi(θ), θ < v1

Tni=k+1fi(θ), vk ≤ θ≤ vk+1 , 1≤ k≤n− 1

∨ni=1fi(θ), θ > vn .

(22)It is also well known [40] that the intersection of A and B is

another T2 FS whose MF can be computed from

A ∩ B ⇔ µA∩B (x, v) =∫∀x∈X

µA∩B (x)/x (23)

µA∩B (x) =∫∀u∈J u

x

∫∀w∈J w

x

fx(u)�gx(w)/(u ∧ w)

= µA (x) � µB (x) ∀x ∈ X (24)

where � denotes the meet operation and is performed betweenthe two secondary MFs µA (x) and µB (x) but, this time, for theoperations in the middle of (24). A careful explanation of what(24) means can also be found in [17], [30, p. 219], and [32].

Note that in (24), there are two t-norms: � and ∧. Althoughthey are usually chosen to be the same, yet they do not alwayshave to be the same. See [44] for discussions about this.

In general, evaluating the meet is also difficult for arbi-trary T2 FSs, especially when the product t-norm is used.Karnik and Mendel [17] have also shown that for n con-vex and normal T1 FSs, F1 , . . . , Fn , which are characterizedby MFs f1(θ), . . . , fn (θ), respectively, where f1(v1) = · · · =fn (vn ) = 1 and the fi(θ) are reordered so that v1 ≤ v2 ≤ · · · ≤vn , the MF of �n

i=1Fi using maximum t-conorm and minimumt-norm (for both � and ∧) can be expressed as

µ�ni = 1 Fi

(θ) =

∨n

i=1fi(θ), θ < v1

∧ki=1fi(θ), vk ≤ θ ≤ vk+1 , 1 ≤ k ≤ n − 1

∧ni=1fi(θ), θ > vn .

(25)To date, no formula that is similar to (25) exists for the product

t-norm, which is unfortunate because many applications useproduct t-norm. When all MFs are Gaussian, then Karnik andMendel [17] have an approximation to compute the meet underproduct t-norm, one that leads to another Gausssian MF, so thatthe approximate meet is “reproducing” and can be expanded inmultiargument form.

The usual derivations of (21) and (24) utilize Zadeh’s exten-sion principle [52]. Mendel and John [34] show how (21) and(24) can be easily derived without using the extension principlewhen A and B are expressed using their representation theorem.

More recently, Coupland and John [3]–[5] have shown how tocompute the join and meet of A and B using methods from com-putational geometry (e.g., a modified Weiler–Atherton clippingalgorithm and a Bentley–Ottman plane sweep algorithm). Theirapproach is based on modeling a secondary MF geometricallyas a [4] “set of connected straight line segments that need notbe equally spaced across the domain” and is, thus far, limitedto the minimum t-norm and the maximum t-conorm. They dis-tinguish between a partially discrete T2 FS and a discrete T2FS. A partially discrete T2 FS is one in which the primary vari-able is discrete (sampled) but secondary MFs are continuous,

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whereas a discrete T2 FS is one in which the primary variableand secondary MFs are discrete (sampled). Based on extensivesimulations of a two-rule FLS in which each rule has two an-tecedents, and the secondary MFs are discretized into ten pointsand are also described by two line segments, Coupland and Johnobtain over a “four-and-a-half-fold increase in the inferencingspeed.” They also state “. . . that for any T2 FLS with secondaryMFs with five or more discretizations, using a partially discretemodel would give a faster and more accurate system.”

Even more recently, Greenfield and John [7] presented anoptimized grid method for computing the join and meet, whichis much more computationally efficient than a direct approachto using (22) and (25).

A limitation of the geometric and grid methods is that theydo not obtain closed-form formulas for the join and meet op-erations. Such formulas can be very useful when a T2 FLS isdesigned by optimizing an objective function with respect toMF parameters and explicit derivatives of that function have tobe computed so that optimal values of these parameters can befound.

B. Computing Set Theoretic Operations Using α-Planes

Liu [22], [23] states the following as “properties” with-out proofs5 (A ∪ B)α = Aα ∪ Bα and (A ∩ B)α = Aα ∩ Bα .These properties let one compute A ∪ B and A ∩ B, which isnot mentioned in [22] and [23].

Theorem 1: Let (A ∪ B)α and (A ∩ B)α be the α-planes ofA ∪ B and A ∩ B, respectively. For the minimum t-norm [i.e.,the star in (21) and (24)], it is true that

A ∪ B =⋃

α∈[0,1]

α/(A ∪ B)α

=⋃

α∈[0,1]

α/Aα ∪ Bα =⋃

α∈[0,1]

FOU(Aα ∪ Bα ) (26)

A ∩ B =⋃

α∈[0,1]

α/(A ∩ B)α

=⋃

α∈[0,1]

α/Aα ∩ Bα =⋃

α∈[0,1]

FOU(Aα ∩ Bα ). (27)

To date, comparable results for other star t-norms, such asthe product, are not available; however, any conjunction can beused in the meet, e.g., minimum or product.

The proof of (26) is given in Appendix A. In this proof [andalso for (27)], it is demonstrated that because Aα and Bα areinterval-valued sets (at level α), the calculations of Aα ∪ Bα

(and Aα ∩ Bα ) involve only interval arithmetic and, in fact,reduce to the same computations that already exist for the union[and intersection] of IT2 FSs.

To compute Aα ∪ Bα , for each value of α, the following stepsare followed.

5The containment property Aα 1 ⊆ Aα 2 , if α1 ≥ α2 , is also included in [22]and [23]. This can also be stated as FOU(Aα 1 ) ⊆ FOU(Aα 2 ), if α1 ≥ α2 .It is very easy to visualize the correctness of this result when all secondary MFsare convex and normal T1 FSs.

Fig. 4. (a) α-Planes Aα and Bα . (b) Their union (the outlined region).

1) Compute Aα and Bα , the results of which are two boundedplanes.

2) Determine LMF(Aα ), UMF(Aα ), LMF(Bα ), andUMF(Bα ).

3) Compute Aα ∪ Bα = [LMF(Aα ∪ Bα ),UMF(Aα ∪ Bα )],using the last line of (A12) ∀x ∈ X .

Example 2: Fig. 4(a) depicts Aα and Bα , as well as theirlower and upper MFs. Once the two α-planes are drawn, it isa relatively simple matter to draw the α-plane of Aα ∪ Bα .Just take the maximum value of UMF(Aα ) and UMF(Bα )to find UMF(Aα ∪ Bα ). Similarly, take the maximum value ofLMF(Aα ) and LMF(Bα ) to find LMF(Aα ∪ Bα ). These lowerand upper MFs, as well as the α-plane (A ∪ B)α = Aα ∪ Bα ,are shown in Fig. 4(b).

C. Historical Notes

What seems not to be well known is that in Zadeh’s section[52, pp. 242–247] entitled “Fuzzy sets with fuzzy MFs,” heproves that the intersection of two T2 FSs can be computed usingthe α-level sets of the secondary MFs (he does not call these MFsas “secondary MFs” but, instead, refers to them as “fuzzy MFs”).This is summarized in [52, eq. (3.104)] but only for minimumt-norm; hence, he was the first to recognize and show how

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set-theoretic operations for T2 FSs could be computed using α-cuts of the secondary MFs. How the t-norm between two T2 FSscan be computed using α-cuts of the secondary MFs is shownin [1, Th. 4.2]. This result extended Zadeh’s intersection resultfrom minimum to any t-norm. How to compute the intersectionand union for two T2 FSs using the representation principleis shown in [46, Th. l and Th. 2]. A proof is provided for theintersection (see [46, Th. 1]). They do not refer to Zadeh’s[52] discussions about using α-levels to compute set-theoreticoperations. Wagner and Hagras [48] focus on how to compute settheoretic operations using z-slices. Their approach of provingTheorem 1 uses vertical slices and is very different from ourproof that uses α-cuts.

IV. CENTROID AND ITS PROPERTIES

A. Background

The centroid of a T2 FS, which was developed by Karnik andMendel [18], has turned out to be a very important concept forIT2, as well as general T2 FSs and their associated FLSs. Asan application of the representation theorem, the centroid of A,which is CA (x), is simply the union of the centroids of all theembedded T2 FSs of A, i.e., [18], [30, p. 249]

CA (x) =∫∀θ1 ∈Jx 1

· · ·∫∀θN ∈Jx N

min (fx1 (θ1), . . . , fxN(θN ))/

∑Ni=1 xiθi∑Ni=1 θi

. (28)

In this equation, θi , Jxi, and fxi

(∀i) are associated with µA (xi),and CA (x) is shown as an explicit function of x because thecentroid of each embedded T2 FS falls on the x-axis. Note thatCA (x) is a T1 FS whose domain values are to the right of theslash mark in (28) and range values (i.e., membership grades)are to the left of the slash mark. Hence, notation CA (x) denotesboth the centroid and its MF.

Until very recently, the only way to compute CA (x) was touse the following exhaustive computation procedure.

1) Discretize the x-domain of A into N points x1 , . . . , xN ,as shown in Fig. 1.

2) Discretize each Jxi= [LMFFOU(A)(xi),UMFFOU(A)(xi)]

(the primary memberships of µA (x) at xi) into a suitablenumber of points, say Mi (i = 1, . . . , N ). Chooseθi ∈ Jxi

.3) Enumerate all the embedded T1 sets of A; there will be∏N

i=1 Mi of them.4) Compute the centroid of each enumerated embedded T1

set and assign it a membership grade equal to the minimumof the secondary grades corresponding to that enumeratedembedded T1 set.

Mathematically, this means (index k denotes the kth embed-ded FS) that

CA (x) = {(ζk , (min{fxi(θi), i = 1, . . . , N})k )}

∏Ni = 1 Mi

k=1 (29)

ζk =

(∑Ni=1 xiθi∑Ni=1 θi

)k

. (30)

Note that if two or more embedded T1 FSs have the samecentroid (ζk ), the one with the largest value of (min{fxi

(θi), i =1, . . . , N})k is kept.

As explained earlier, the representation theorem leads to thestructure of a solution of a problem, but it should not be used foractual computational purposes. The procedure just described isimpractical for computation because it requires

∏Ni=1 Mi cen-

troid calculations, and this number will, in general, be impracti-cally large. Hence, if the centroid is to be computed for a generalT2 FS, a practical computational procedure must be found to dothis.

Comment: Such a practical computational procedure has al-ready been found for IT2 FSs and is the so-called KarnikMendel (KM) algorithms6 [16], [18], [30, pp. 258–259], [36],[49].

A recent approach for computing CA (x) based on randomlysampling embedded T2 FSs and computing their centroids [8]claims to give rise to a significant reduction in the time orresources needed to perform type reduction (TR). Greenfieldet al. provide examples (for four different primary MFs anddifferent discretizations) that demonstrate that the number ofrandomly selected embedded sets only marginally affect thedefuzzified value [i.e., the mean value of CA (x)]. Excellentresults have been obtained for as few as ten randomly selectedembedded T2 FSs. It is surprising that such a small numberof randomly chosen embedded T2 FSs can lead to such goodresults, and it awaits a theoretical explanation.

Even more recently, Coupland [2] recommends the use of thex-coordinate of the geometric centroid of the 3-D MF of A.This approach goes directly from A to a number rather than to aT1 FS; hence, it does not provide a measure of the uncertaintyin A. Additionally, if the T2 FS reduces to a T1 FS (when alluncertainty disappears), then this x-coordinate does not reduceto the correct centroid of the T1 FS.

Finally, John and Czarnecki [15] and Lucas et al. [25] haveproposed the computation CA (x) as the centroids of all ver-tical slices (each of which is a T1 FS). This ad hoc methodleads to a centroid whose domain of nonzero centroid valuesalways equals the domain of nonzero values of the primaryvariable of A, regardless of the rest of the geometry of theT2 FS. This does not seem to be a correct measure of theuncertainty in A, because the domain of nonzero values ofCA (x) should depend upon the rest of the geometry of theT2 FS.

B. Computing the Centroid Using α-planes

Liu [22], [23] proved that CA (x) can be computed using theα-plane decomposition of A, i.e.,

CA (x) =⋃

α∈[0,1]

α/CAα(x) (31)

6Because of the iterative natures of the KM algorithms, there may be acomputational bottleneck in real-time applications of IT2 FSs, e.g., in fuzzylogic control [11].

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where CAα(x) is the centroid of the α-plane Aα . Because each

Aα is an interval-valued set, its centroid is an interval set, i.e.,

CAα(x) = [lAα

, rAα]. (32)

Wu and Mendel [50] have demonstrated that the centroid is ameasure of the uncertainty of an IT2 FS. Since a general T2 FSis the union of its α-plane FOUs, each of which is an IT2 FS (atlevel α), CA (x) can be used as a measure of the uncertainty ofa general T2 FS.

Equation (31) is not only an interesting theoretical result, butit also provides an entirely new way to compute CA (x), whichis given as follows.

1) Decide on how many α-planes will be used, where α ∈[0, 1]. Call that number k; its choice will depend on theaccuracy that is required. Regardless of k, α = 0 and α =1 must always be used.

2) For each α, compute Aα .3) Compute CAα

(x) using two KM algorithms [18], [30,pp. 258–259], [49]. The accuracy of this result will dependupon the discretization of the primary variable.

4) Repeat steps 2) and 3) for the k values of α chosen instep 1).

5) Bring all of the k CA (x) together using (31) to obtainCA (x).

If parallel processing is available, all of this can be performedusing 2k processors.

Computational costs for this α-plane approach to compute thecentroid are described in [22] and [23, pp. 2233–2234], whereit is shown that if there are N samples for primary variablex and M samples for secondary variable u, then the computa-tional complexity of the exhaustive computations is O(MN N),whereas if one uses α-planes and it takes n iterations for aKM algorithm to converge, then the computational complexityof the α-plane approach is O(Nnk). Because of the super-exponential convergence property of a KM algorithm [36], nis quite small, and for accuracies of 10−2 , it is usually lessthan seven; therefore, the computational complexity of the α-plane approach is actually O(Nk). Since Nk is linear in N andMN N is exponential in N , the α-plane approach for comput-ing the centroid is computationally much more efficient than theexhaustive approach. Additionally, because computing CAα

(x)for each of the k values of α can be done in parallel, as can eachof the two KM algorithms, if 2k parallel processors are used,then the computational complexity of the α-plane approach isonly O(N).

As explained in [22] and [23], computing CA (x) by this newmethod is equivalent to computing a fuzzy weighted average(FWA). See [24] for discussions about how to compute the FWAusing KM algorithms. Because this connection is not needed forthe rest of this paper, it is not explored further herein.

C. Properties of the Centroid

The α-plane decomposition of A provides a very valuableway to think about the shape of CA (x), and in this section, it isshown that this shape is predictable before CA (x) is computed.

To begin, recall that FOU(Aα ) can be associated with an IT2FS whose secondary grade equals α. Although it is not possi-ble to obtain a closed-form formula for CAα

(x), Mendel andWu [39] have obtained closed-form formulas for uncertaintybounds for the two endpoints of the centroid of an IT2 FS,such as CAα

(x). These formulas show that the length of CAα(x)

increases (decreases) as the area of FOU(Aα ) increases (de-creases). This fact is used below.

In the rest of this section, it is assumed that all secondaryMFs of A are monotonically nondecreasing for u ∈ [0,m],monotonically nonincreasing for u ∈ [m, 1], and normal. Thesupport of a secondary MF is u ∈ [a, b], where 0 ≤ a ≤ mand m ≤ b ≤ 1. It may also happen that µA (x) = 1 form1 ≤ m ≤ m2 .

Based on viewing A in terms of its α-planes, it is straightfor-ward to draw the following conclusions about T1 FS CA (x).

1) Maximum uncertainty about A occurs for its α = 0 planeAα=0 .

2) Minimum uncertainty about A occurs for its α = 1 planeAα=1 .

3) When αi ≥ αj , then CAα i(x) ⊆ CAα j

(x).4) CA (x) is first nondecreasing and then nonincreasing.5) When all secondary MFs are normal at exactly one point,

then Aα=1 is a function (i.e., not a plane) so that CA 1(x)

is a single point (i.e., not an interval).6) When all secondary MFs are normal triangles, then CA (x)

is triangle-looking. Its base is computed as CA 0(x), and

its apex is computed as CA 1(x). The latter will be a

single point [see item 5)]; however, its sides may notbe straight lines, and hence, CA (x) is called “triangle-looking.”

7) When some or all of the secondary MFs are normaltrapezoids, then Aα=1 is a plane, and CA 1

(x) is aninterval.

8) When some or all of the secondary MFs are (normal)trapezoids, then CA (x) is trapezoid-looking. Its base iscomputed as CA 0

(x), and its top is computed as CA 1(x).

The latter will be an interval [see item 7)]; however, itstwo sides may not be straight lines, and hence, CA (x) iscalled “trapezoid-looking.”

These properties about the shape of CA (x) will be confirmedby means of examples in Section V.

Next, consider a general T2 FS that is totally symmetrical.Definition 7: A general T2 FS A is said to be totally symmet-

rical if 1) FOU(A) is symmetrical about primary variable x atx = m and 2) all of its secondary MFs are symmetrical.

The most widely used symmetrical secondary MFs to date aretriangles since they seem to represent the next logical extensionfrom IT2 FSs to full-blown general T2 FSs [2], [5], [8], [45].An example of such a totally symmetrical T2 FS is depicted inFig. 5.

Theorem 2: If A is totally symmetrical, then the centroid ofA, which is CA (x), is symmetrical about x = m, and the meanvalue (i.e., the defuzzified value) of CA (x) equals m.

Proof: See Appendix A. �If one is planning to use only the defuzzified value of a totally

symmetrical T2 FS, then to carry out T2 computations is a

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Fig. 5. Totally symmetrical T2 FS whose FOU has a trapezoidal UMF and atriangular LMF and whose secondary MFs are symmetrical triangles.

wasted effort, because the same result could have been obtainedby using T1 calculations.

For a totally symmetrical T2 FS A, an excellent approxima-tion to CA (x) can be obtained by computing the centroid ofthe two α-planes CA 0

(x) and CA 1(x) and then by connecting

the left endpoint of CA 0(x) to the left endpoint of CA 1

(x), theleft endpoint of CA 1

(x) to the right endpoint of CA 1(x), and the

right endpoint of CA 1(x) to the right endpoint of CA 0

(x). Theresulting function will either be a triangle or a trapezoid. Thiscan save an enormous amount of computation.

Interestingly enough, a nonsymmetrical T2 FS will occur fora Mamdani T2 FLS when two or more rules are fired [30, p. 309].A totally symmetrical T2 FS can occur only when a single ruleis fired. This has already been pointed out for the case of IT2FSs in [31] and is also valid for general T2 FSs because theFOU for such an FS [i.e., FOU(Aα=0)] is the same as the FOUof an IT2 FS.

V. CENTROID EXAMPLES

The purpose of this section is to provide some examples forthe centroid for two very important classes of secondary MFs,which are triangles and trapezoids, because to us, they seem tobe the first practical choices for extension of IT2 FSs to generalT2 FSs.

In the examples provided shortly, the following hold.1) The domain of the primary variable for A is x ∈ [0, 10],

and x is uniformly sampled using 10 000 samples suchthat xi+1 − xi = 10−3 .

2) For α-planes, α is also uniformly sampled so that α has∆ + 1 values equal to 0, 1/∆, 2/∆, . . . , (∆ − 1)/∆, 1. InExamples 3 and 4, ∆ = 100, whereas in Example 5, ∆ isvariable. Additional results are given in [22] and [23].

3) For each α-plane, CAα(x) is computed by using two KM

algorithms.Example 3 (Gaussian LMF and UMF, and trapezoidal sec-

ondary MFs)7: FOU(A)8 is shown in Fig. 6(a), for which both

7The same FOU but with triangle secondary MFs whose apex location w(x) ∈[0, 1] is randomly chosen at each x ∈ [0, 10] is used in [22] and [23].

8This FOU is representative of the one that might have been obtained bycomputing the union of two fired-rule output sets in a T2 FLS.

UMFFOU(A)(x) and LMFFOU(A)(x) are the maximum of twoGaussian functions, i.e.,

UMFFOU(A)(x) = max{

exp[− (x − 3)2

8

]

0.8 exp[− (x − 6)2

8

]}(33)

LMFFOU(A)(x) = max{

0.5 exp[− (x − 3)2

2

]

0.4 exp[− (x − 6)2

2

]}. (34)

In this example, each secondary MF of A is chosento be a trapezoid whose base equals UMFFOU(A)(x) −LMFFOU(A)(x) and whose top is defined by left and rightendpoints, which are EPl(x) and EPr (x), and both are pa-rameterized as

EPl(x) = LMFFOU(A)(x)

+ 0.6w[UMFFOU(A)(x) − LMFFOU(A)(x)] (35)

EPr (x) = UMFFOU(A)(x)

− 0.6(1 − w)[UMFFOU(A)(x) − LMFFOU(A)(x)]

(36)

where w = 0, 0.25, 0.5, 0.75, and 1. These secondary MFs aredepicted in Fig. 6(b) for x = 2. Therefore, A is defined by (33)–(36).

The centroids of A, which are T1 FSs, are depicted in Fig. 7for the five values of w. Note that when α = 0, CAα = 0

(x) =[3.1556, 5.6748] (this is the support of all five centroids), andtherefore, the mean value of the two endpoints of CAα = 0

(x),i.e., m(CAα = 0

(x)), is 4.4152.Table I provides numerical details for this example for the five

values of w. First, it provides the two endpoints of the top of each“trapezoidal-looking” centroid when α = 1. Then, it providesthe mean value of each centroid m(CA (x)) (i.e., the defuzzifiedvalue of A), after which, it provides the difference and thepercentage difference between m(CAα = 0

(x)) and m(CA (x)).Surprisingly, the percentage differences are less than 1% for allfive values of w.

Each of the centroids in Fig. 7 looks symmetrical; however,they are not exactly symmetrical, as can be seen from Table Iby comparing m(CAα = 0

(x)) with m(CA (x)).Each centroid was then approximated by a trapezoid that

connected only CAα = 0(x) and CAα = 1

(x).9 Table I also pro-

vides the mean value of these approximate centroids m(CA (x)),as well as the differences and percentage differences betweenm(CA (x)) and m(CA (x)). Very surprisingly, the percentage

9To compute the centroid of a trapezoid, first, shift the trapezoid by m aboutthe origin, i.e., its top must be situated about the origin (it is not necessary for itto be symmetric about the origin), and call the resulting coordinates (−bl , 0),(br , 0), (−al , 1), and (ar , 1). Compute CTrap ezoid/orign = 1/3[ar −al + br − bl + (al bl − ar br )/(ar + al + br + bl )], and then, computeCTrap ezoid = CTrap ezoid/orign + m.

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Fig. 6. (a) FOU for Example 3. (b) Secondary MFs at x = 2 for five values of w.

Fig. 7. CA (x) for Example 3 when w = 0, 0.25, 0.5, 0.75, and 1.

differences between m(CA (x)) and m(CA (x)) for all five val-ues of w are less than 1/2%.

These results demonstrate that a very good approximation tothe centroid can be obtained by computing centroids for onlytwo α-planes: α = 0 and α = 1.

Example 4 (Piecewise linear LMF and UMF, and trianglesecondary MFs)10: FOU(A) is shown in Fig. 8(a), for whichUMFFOU(A)(x) and LMFFOU(A)(x) are both the maximum oftwo piecewise linear functions, i.e., (footnote 8 also applies tothis FOU)

UMFFOU(A)(x) = max

{[ (x − 1)/2, 1 ≤ x ≤ 3(7 − x)/4, 3 ≤ x ≤ 70, otherwise

]

[ (x − 2)/5, 2 ≤ x ≤ 6(16 − 2x)/5, 6 ≤ x ≤ 80, otherwise

]}(37)

10The same FOU and secondary MFs are used in [22] and [23], but resultsare provided only for w = 0.

LMFFOU(A)(x) = max

{[ (x − 1)/6, 1 ≤ x ≤ 4(7 − x)/6, 4 ≤ x ≤ 70, otherwise

]

[ (x − 3)/6, 3 ≤ x ≤ 5(8 − x)/9, 5 ≤ x ≤ 80, otherwise

]}. (38)

In this example, each secondary MF is chosen to be a tri-angle of height one whose base equals UMFFOU(A)(x) −LMFFOU(A)(x) and whose apex location Apex(x) is parame-terized as

Apex(x) = LMFFOU(A)(x)

+ w[UMFFOU(A)(x) − LMFFOU(A)(x)] (39)

where w = 0, 0.25, 0.5, 0.75, and 1. These secondary MFs aredepicted in Fig. 8(b) for x = 2. Therefore, A is defined by (37)–(39).

The centroids of A, which are T1 FSs, are depicted in Fig. 9for the five values of w. Note that when α = 0, CAα = 0

(x) =[3.6605, 4.9917] (this is the support of all five centroids), andtherefore, m(CAα = 0

(x)) = 4.3261.Table II provides numerical details of this example for the

five values of w. First, it provides the apex location CAα = 1(x)

of each of the “triangle-looking” centroids. Then, it provides themean value of centroid m(CA (x)) (i.e., the defuzzified valueof A), and the difference and percentage difference betweenm(CAα = 0

(x)) and m(CA (x)). Observe that m(CAα = 0(x)) −

m(CA (x)) is less than 1% for all five values of w, which is inagreement with Example 3.

Each of the centroids in Fig. 8 looks symmetrical; however,they are not exactly symmetrical, as can be seen from Table IIby comparing m(CAα = 0

(x)) with m(CA (x)).Each centroid was then approximated by a triangle that con-

nected CAα = 0(x) and CAα = 1

(x). Table II also provides the

mean value of these approximate centroids,11 i.e., m(CA (x)),as well as the differences and percentage differences betweenm(CA (x)) and m(CA (x)). Observe that the percentage differ-ences between m(CA (x)) and m(CA (x)) for all five values

11The centroid of triangle {(l, 0), (m, h), (r, 0)} is CTriangle = 1/3(m +l + r). This formula can be used for any triangle, i.e., it does not have to beshifted to the origin.

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TABLE IEXAMPLE 3 RESULTS

Fig. 8. (a) FOU for Example 4. (b) Secondary MFs at x = 2 for five values of w.

Fig. 9. CA (x) for Example 4 when w = 0, 0.25, 0.5, 0.75, 1.

of w are less than 1/2%, which is also in agreement withExample 3.

This example again demonstrates that a very good approxi-mation to the centroid can be obtained by computing centroidsfor only two α-planes: α = 0 and α = 1.

Example 5 (Greenfield et al. [8] FOUs): This example ismotivated by examples that are given in [8]; however, becauseformulas for its FOUs are not included in [8], we created fiveFOUs that resemble its FOUs, and, as in [8], we only use sym-metrical triangle secondary MFs.12 The primary focus of this

12This means that at each sampled value of the primary variable, the secondaryMF is symmetrical over the associated primary membership.

example is to examine convergence behaviors of the centroidand its defuzzified value. Results are also provided in Tables IVand V that are analogous to those in Table II for this example’sfive FOUs.

Pictures of the five FOUs (left-shoulder, interior (sym-metrical) triangle, interior (nonsymmetrical) Gaussian, right-shoulder, and a broad right-shoulder FOUs) are shown in col-umn 2 of Table III. Plots of the centroids of the five T2 FSsare shown in column 3. Observe (as in Example 4) that be-cause the secondary MFs are triangles, all the centroids have atriangle-looking shape. Additionally, as predicted by Theorem2, the symmetrical interior triangle FOU, with its symmetricalsecondary triangle MFs, has a centroid that is symmetrical aboutthe symmetry value for its primary variable (at x = 4).

A complete study of the convergence behavior of the centroidcan be done for each of its 101 α-cuts. We show such conver-gence results in column 4 of Table III only for convergence tothe centroid of the α = 0.5 plane. Each of these figures con-tain three curves: one each for the convergence behaviors of thelower (left) and upper (right) ends of the centroid and anotherfor the average value of these two endpoints.

When the [0, 1] range for α is discretized, the α = 0.5 planeis sometimes included in the set of discretized values of α, andsometimes it is not. When it is included, then the true value forthe centroid of the α = 0.5 plane is computed. When it is notincluded (e.g., as would be the case when the [0, 1] range isdiscretized into the six values 0, 0.2, 0.4, 0.6, 0.8, and 1), thenthe centroid of the α = 0.5 plane is approximated by using thecomputed centroid of the α-plane that is just larger than α = 0.5(for the example of six α-panes, this would be the centroid ofthe α = 0.6 plane). This choice is based on the formal definitionof an α-plane.

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TABLE IIEXAMPLE 4 RESULTS

TABLE IIIGRAPHICAL RESULTS FOR EXAMPLE 5

Therefore, by “convergence,” we mean to find how manydiscretized values of α are needed before there is no appreciabledifference between the true value for the centroid of the α = 0.5plane and its approximated value when the α = 0.5 plane isapproximated by using the computed centroid of the α-plane,which is just larger than α = 0.5.

All figures in column 4 of Table III exhibit a transientresponse, which is somewhat oscillatory, and then a steady-state solution, which is indicative of convergence. Convergenceseems to occur for approximately 20 α-planes for the cen-troid endpoints but for less than ten α-planes for their averagevalues.

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TABLE IVEXAMPLE 5 INTERMEDIATE RESULTS

The last column of Table III shows the convergence of thedefuzzified value of the T2 FS as a function of α. This value isof great importance in applications of T2 FSs (e.g., fuzzy logiccontrol) and is obtained by computing the centroid of figureslike those in the third column of Table III but computed for avariable number of α-planes ranging from 2 to 101. It seemsthat convergence occurs for between 5 and 10 α-planes, regard-less of the shape of the FOU. Interestingly, convergence of thedefuzzified value of each T2 FS occurs much sooner than doesconvergence of its centroid. This observation is consistent witha similar observation given first in [8] and awaits a theoreticalexplanation.

Table IV provides some intermediate results that are used inTable V. Table V demonstrates that good approximations to thecentroid are not always obtained by computing centroids foronly two α-planes: α = 0 and α = 1. Less than 1/2% errorsare still obtained for the interior triangle, interior Gaussian, andright-shoulder FOUs; however, larger errors are obtained for theleft-shoulder and broad right-shoulder FOUs. Hence, some (butnot all) shoulder FOUs are associated with larger approximationerrors.

VI. TRIANGLE Q-T2 FLS

A. Introduction

A T2 FLS requires the following chain of computations [16],[19], [30, pp. 287–298]: fuzzification, inference, type-reduction(TR), and defuzzification. Just as there are many kinds of de-fuzzifiers for a T1 FLS, there are a comparable number of TRsfor a T2 FLS. For T1 and T2 FLSs, centroid defuzzification orcentroid TR are costly operations because multiple fired rulesmust first have their fired-rule output fuzzy sets combined by theunion operation. This has led to other kinds of defuzzification(TR) methods, such as height or center of sets.

It is well known [35] that an IT2 FLS can be viewed as a verylarge collection of T1 FLSs. Using the α-plane representationof a general T2 FS, it should now be clear that a general T2 FLScan be viewed as a very large collection of IT2 FLSs with oneIT2 FLS for each value of α.

Let the union of fired-rule-output FSs be called A. Then,centroid TR [16], [30, pp. 265–267] is equivalent to computingCA (x), which can be done by using (31) and (32).

Here, we suggest a different approach to a general T2 FLS,which is based on some of the observations about the centroidthat have been made in Section IV-C. The importance of theseobservations, especially Observation 6, is that they indicate theshape of the centroid. Therefore, even without performing the

T2 inference and union computations, if all secondary MFs aretriangles, then the centroid will be triangle-looking.

B. Architecture for a Triangle Q-T2 FLS

A new architecture for a T2 FLS was proposed in [37], andit is depicted in Fig. 10. This T2 FLS could be called a twoα-plane T2 FLS or a triangle Q-T2 FLS.13 It is much simplerthan a full-blown T2 FLS because it does not involve any T2calculations. A triangle Q-T2 FLS may be a good next stepin the hierarchy of FLSs from T1 to IT2 to T2. Of course, ifhigh accuracy is important, then more than two α-planes can beused; however, Example 4 has shown that fairly high accuracyof the defuzzified output can often be achieved by using onlytwo α-planes.

The triangle Q-T2 FLS in Fig. 10 would be used when allsecondary MFs are triangles. It combines a T1 FLS and an IT2FLS where the coupling of these FLSs occurs during overalldefuzzification. The defuzzified output of the T1 FLS acts asthe apex location of the assumed TR triangle and the TR outputof the IT2 FLS acts as the base of that triangle. The overalloutput of this triangle Q-T2 FLS is the centroid of that triangle,i.e.,

y(x) =13[yL (x|α = 0) + yR (x|α = 0) + y1(x|α = 1)

].

(40)The MF parameters of the T1 and IT2 FLSs are optimized

simultaneously, and because of (40), this is a coupled optimiza-tion and design. Note that Starczewski [45] already suggestedthat for triangle secondary MFs, one needs to keep track of onlythree points.

C. Design Requirement for the Triangle Q-T2 FLS

There is a novel aspect to the parameters of any triangle Q-T2 FLS that must be mentioned. Karnik and Mendel [16] andMendel [30, p. 11] imposed the following design requirementon any IT2 FLS that has been maintained to date by all thosewho have used an IT2 FLS: When all sources of uncertaintydisappear, a T2 FLS must reduce to a comparable T1 FLS. Thisdesign requirement is maintained for the triangle Q-T2 FLS.

D. Reparameterization that Satisfies the Design Requirement

One way to satisfy this design requirement is by couplingthe MF parameters of the T1 and IT2 FLSs as follows. LetθQ-T2 denote the MF parameters of the triangle Q-T2 FLSi.e., θQ-T2 = col(θT 1 |θIT2), where θT 1 and θIT2 denote theparameters of the T1 and IT2 components of the triangle Q-T2FLS, respectively.

Let θjT 1,i(q) denote the qth T1 MF parameter14 for the ith

antecedent and the jth rule (i = 1, . . . , p, j = 1, . . . , M , andq = 1, . . . , Q) of the T1 FLS component of the triangle Q-T2 FLS and θj

IT2,i(lq ) and θjIT2,i(rq ) denote the left and right

endpoints of a comparable parameter of an antecedent FOU forthe IT2 FLS component of the triangle Q-T2 FLS (in which case,

13A trapezoid Q-T2 FLS is proposed in Section VII.14If, e.g., the T1 MF is a Gaussian, then Q = 2.

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TABLE VEXAMPLE 5 RESULTS

Fig. 10. Triangle Q-T2 FLS.

we shall say that the T1 MF parameter has been intervalized),where it is required (by us) that

θjIT2,i(lq ) ≤ θj

T 1,i(q) ≤ θjIT2,i(rq ). (41)

If all uncertainty disappears, then we will also require thatθj

IT2,i(lq ) = θjIT2,i(rq ) = θj

T 1,i(q) so that the design require-ment is satisfied.

In a similar manner, let φjT 1 denote the centroid of the conse-

quent for the jth rule (j = 1, . . . ,M ) for the T1 FLS componentof the triangle Q-T2 FLS, and φj

IT2(l) and φjIT2(r) denote the

left and right endpoints of the centroid of the consequent FOU15

for the IT2 FLS component of the triangle Q-T2 FLS, where itis required (by us) that

φjIT2(l) ≤ φj

T 1 ≤ φjIT2(r). (42)

If all uncertainty disappears, then we will also require thatφj

IT2(l) = φjT 1 = φj

IT2(r); therefore, the design requirement isagain satisfied.

One way by which (41) and (42) can be satisfied is if

θjT 1,i(q) = θj

IT2,i(lq ) +wj

i (θq )∑Mj=1 wj

i (θq )[θj

IT2,i(rq ) − θjIT2,i(lq )]

(43)

φjT 1 = φj

IT2(l) +wj (φ)∑M

j=1 wj (φ)[φj

IT2(r) − φjIT2(l)]. (44)

When all sources of uncertainty disappear, θjIT2,i(rq ) =

θjIT2,i(lq ) and φj

IT2(r) = φjIT2(l); therefore, (43) and (44) be-

come θjT 1,i(q) = θj

IT2,i(lq ) = θjIT2,i(rq ) and φj

T 1 = φjIT2(l) =

φjIT2(r), as required.Normalization of the weights in (43) and (44) by their de-

nominator sum of the weights guarantees that the normalized

15There is no q dependency on the consequent parameter (s), because for aT1 FS, the centroid is a scalar, and for an IT2 FS, it has scalar left and rightendpoints.

weights are between 0 and 1 (for a fixed antecedent and acrossall M rules). A different normalization in which the denomina-tor sums the weights simultaneously across all antecedents andrules is also possible. For this paper, we use the normalizationin (43) and (44).

It may happen that only some of the Q T1 MF parametersare intervalized, e.g., when a Gaussian FOU is obtained from aT1 Gaussian MF and only the mean of the latter is intervalized.For the T1 MF parameters that are not intervalized, one can stillconceptually begin with (43), but then, one sets wj

i (θq ) = 0 andθj

IT2,i(lq ) = θjT 1,i(q).

By means of (43) and (44), the T1 component parameters ofthe triangle Q-T2 FLS have been reparameterized. Instead of

viewing θjT 1,i(q) and φj

T 1 as its design parameters,√

wji (θq )

and√

wj (φ) are now viewed as its design parameters. Thesquare roots of the weights, and not the weights, are used asthe design parameters, because the weights must be positive,and this can always be guaranteed by using the square roots ofthe weights and then by squaring them up to get the weights.

Once√

wji (θq ),√

wj (φ), θjIT2,i(lq ), θj

IT2,i(rq ), φjIT2(l), and

φjIT2(r) have been determined, θj

T 1,i(q) and φjT 1 can be com-

puted by using (43) and (44), respectively.

E. Forecasting of a Chaotic Time Series

T1 and IT2 FLSs have been extensively used in time-seriesforecasting, e.g., [13], [14], [29], and [38]. The most widely usedtime series for such studies is the Mackey–Glass chaotic timeseries. It has become one of the benchmark problems for time-series forecasting in both the neural network and fuzzy logicfields and is the time series that we use to evaluate the perfor-mance of our triangle Q-T2 FLS. We compare the performanceof this FLS with T1 and IT2 FLSs.

1) Forecasting of Time Series: Let s(k) (k = 1, 2, . . . , N )be a time series. Measured values of s(k) are denoted as z(k),

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Fig. 11. Mackey–Glass chaotic time series. (a) Noise-free data s(2000), s(2001), . . . , s(3000). (b) One realization of the noise-corrupted data, z(2000),z(2001), . . . , z(3000), when SNR = 10.

where

z(k) = s(k) + n(k), k = 1, . . . , N (45)

and n(k) denotes measurement errors, i.e., noise. The problemof forecasting a time series is the following: Given a windowof p past measurements of s(k), namely, z(k − p + 1), z(k −p + 2), . . . , z(k), determine an estimate of a future value ofs, s(k + l). The variables p and l are fixed positive integers, andin this paper, p = 4 and l = 1.

FLS forecasters are based on rules that have p antecedentsand one consequent. Three methods for extracting rules fromdata are described in [30, pp. 112–114]. Here, “method 3” isused, in which the architecture of an FLS forecaster is firstestablished, and then data are used to simultaneously optimizeall of the FLS forecaster’s MF parameters. In order to establishthe architecture of the FLS forecaster, the following are fixed:number of rules, number of rule antecedents, shapes of theantecedent and consequent MFs (for T1 FSs) or FOUs (for IT2FSs), t-norm, kind of fuzzification, and kind of defuzzifier fora T1 FLS forecaster or TR for an IT2 FLS forecaster. Detailsabout all of these will be given shortly.

2) Mackey–Glass Chaotic Time Series: In 1977, Mackeyand Glass [27] published an important paper in which they “as-sociate the onset of disease with bifurcations in the dynamics offirst-order differential delay equations that model physiologicalsystems,” and [27, eq. (4b)] has been known as the Mackey–Glass equation. It is a nonlinear delay differential equation, oneform of which is

ds(t)dt

=0.2s(t − τ)

1 + s10(t − τ)− 0.1s(t). (46)

For τ > 17, (47) is known to exhibit chaos. We chose τ = 30.In our simulations, (46) was converted to a discrete-time equa-

tion by using Euler’s method with a step size equal to one [42],

i.e., (46) was simulated as

s(k + 1) = s(k) +[

0.2s(k − 30)1 + s10(k − 30)

− 0.1s(k)]

. (47)

The 31 initial values of s(k) (k ≤ 30) were chosen randomly.It was also assumed that the sampled time series s(k) is cor-rupted by uniformly distributed, stationary, zero-mean additivenoise n(k).

Because s(k) is not a zero-mean signal, SNR (in decibels) forthe Mackey–Glass time series is

SNR = 10 log10(1/N)∑N

k=1 s2(k)σ2

n

. (48)

When SNR is fixed ahead of time, then (48) can be used tocompute σ2

n so that a correctly sized n(k) is simulated.Fig. 11(a) depicts our Mackey–Glass chaotic time series

s(2000), s(2002), . . . , s(3000) and Fig. 11(b) depicts one re-alization of the noise-corrupted data z(2000), z(2002), . . . ,z(3000) when SNR = 10 dB.

3) FLS Forecasters: In all of our FLS forecasters, the fol-lowing were used:

1) four antecedents per rule, namely z(k − 3), z(k − 2),z(k − 1), and z(k), to predict z(k + 1);

2) two fuzzy sets for each antecedent (as in [13], [21], and[30]) so that the number of rules is 24 = 16;

3) Gaussian MFs for T1 FSs and Gaussian FOUs with un-certain mean but certain standard deviation for IT2 FSs;

4) product t-norm;5) singleton fuzzification;6) center-of-sets defuzzification for the T1 FLS forecaster

and center-of-sets TR for the IT2 FLS forecaster.The outputs from the T1, IT2, and triangle Q-T2 FLS fore-

casters are denoted by zT 1(k + 1|θT 1), zIT2(k + 1|θIT2), andzQ-T2(k + 1|θQ-T2), respectively. The T1 FLS forecaster has

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144 parameters (θT 1)16 that have to be optimized, the IT2 FLSforecaster has 224 parameters (θIT2)17 that have to be optimized,and the triangle Q-T2 FLS forecaster has 144 + 224 = 368 pa-rameters (θQ-T2) that have to be optimized.

Because formulas for the FLS forecasters have appeared in anumber of places in the literature, they are not repeated here,18

e.g., for the T1 FLS forecaster, see [30, Ch. 5], and for the IT2FLS forecaster, see [30, Ch. 10]. Of course, both sets of formulasare used for the triangle Q-T2 FLS forecaster.

4) Optimization: The FLS forecasters are based on N =3004 time points, but their optimal designs were based onlyon z(2000), z(2001), . . . , z(3003). During the optimization ofeach FLS forecaster, each training and testing set of data wascomposed of the five measurements {z(k − 4), z(k − 3), z(k −2), z(k − 1) : z(k)}. For training, 500 datasets were used,where k = 2004, 2005, . . . , 2503, and for testing, another 500datasets were used, where k = 2504, 2505, . . . , 3003.

All MF parameters were optimized using the method of steep-est descent that was applied to an objective function duringtraining that is associated with the respective FLS forecaster.The three objective functions that were used are (k = 2004,2005, . . . , 2503) as follows.

T1 FLS forecaster:

JT 1(θT 1) =12[zT 1(k|θT 1) − z(k)]2 . (49)

IT2 FLS forecaster:

JIT2(θIT2) =12[zIT2(k|θIT2) − z(k)]2 . (50)

Triangle Q-T2 FLS forecaster:

JQ-T2(θQ-T2) =12[zIT2(k|θQ-T2) − z(k)]2 . (51)

After each epoch of training (j), the performance of each fore-caster was evaluated by using the testing data and the followingroot-mean-squared errors (RMSE):

RMSET 1(θjT 1) ≡ RMSET 1(j)

=1

500

3003∑k=2504

[zT 1(k|θjT 1) − z(k)]2

(52)

RMSEIT2(θjIT2) ≡ RMSEIT2(j)

=1

500

3003∑k=2504

[zIT2(k|θjIT2) − z(k)]2

(53)

16Each antecedent MF has a mean and standard deviation, and each con-sequent has a center of gravity; hence, there are (4 × 2 + 1) × 16 = 144parameters.

17Each antecedent FOU has a standard deviation and two mean endpointparameters, and each consequent has a two centroid parameters; hence, thereare (4 × 3 + 2) × 16 = 224 parameters.

18Wu and Mendel [51] have formulas for (nonsingleton) T1 and IT2 FL rule-based classifiers; however, these formulas can also be used to implement T2 FLSand IT2 FLS forecasters. They are easily reduced to the singleton fuzzificationsituation. The paper also has detailed formulas to compute the derivatives of allMF parameters for both kinds of forecasters.

RMSEQ-T2(θjQ-T2) ≡ RMSEQ-T2(j)

=1

500

3003∑k=2504

[zQ-T2(k|θjQ-T2)− z(k)]2 .

(54)

Let θ denote a generic symbol that represents all parametersthat are to be optimized for the generic objective function J bythe following steepest descent algorithm:

θnew = θold − λ∂J

∂θ

∣∣∣∣θo ld

(55)

where λ > 0 is the learning parameter, and ∂J/∂θ is computedas

∂J

∂θ=

∂J

∂z(k|θ)∂z(k|θ)

∂θ. (56)

The calculation of ∂J/∂θ depends on the specific FLS fore-caster and requires a very careful use of the chain rule.

5) Designs: Our designs of the three FLSs proceeded as fol-lows. Six epochs were run for each design. The T1 FLS wasdesigned first. To begin, λ in (55) was set equal to 0.2. Aslong as RMSET 1(j + 1) < RMSET 1(j) (j = 1, . . . , 5), thisvalue of λ remained unchanged. If, however, RMSET 1(j + 1)> RMSET 1(j), then λ was made smaller. Examining the meanvalue of RMSET 1(j) for the six epochs, the winning T1 designwas the one that had the smallest mean RMSET 1(j). Values ofits parameters, as well as λ, were then used to initialize the IT2FLS design.

The IT2 FLS was designed next. As long as RMSEIT2(j +1) < RMSEIT2(j), the starting value of λ remained unchanged.If, however, RMSEIT2(j + 1) > RMSEIT2(j), then λ wasagain made smaller. Examining the mean value of RMSEIT2(j)for the six epochs, the winning IT2 design was the one that hadthe smallest mean RMSEIT2(j). Values of its parameters, aswell as λ, were then used to initialize the triangle Q-T2 FLSdesign.

The triangle Q-T2 FLS was designed last.19 As longas RMSEQ-T2(j + 1) < RMSEQ-T2(j), its starting value ofλ remained unchanged. If, however, RMSEQ-T2(j + 1) >RMSEQ-T2(j), then λ was again made smaller. Examining themean value of RMSEQ-T2(j) for the six epochs, the winningtriangle Q-T2 FLS design was the one that had the smallestmean RMSEQ-T2(j).

6) Results: Twenty Monte Carlo simulations and designswere performed for each FLS forecaster. For each FLS design,its 20 Monte Carlo RMSE values are statistically independent;hence, we were able to use the bootstrap method to get moreaccurate estimates of the mean and standard deviations of eachFLS RMSE. We used 1000 bootstrap realizations. Our resultsare summarized in Tables VI and VII. Table VI provides the rawvalues for the mean and standard deviation for the three FLSforecasters. Table VII provides comparative results.

From Table VII, the following can be observed:

19The weights in (43) and (44) were initialized randomly as independentuniformly distributed random variables over [0, 1].

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TABLE VIMEAN AND STANDARD DEVIATION OF RMSET 1 , RMSEIT2 , AND

RMSEQ -T2

TABLE VIIPERCENTAGE IMPROVEMENT OF MEAN RMSE FROM ONE DESIGN TO ANOTHER

1) The IT2 and triangle Q-T2 provide rather modest perfor-mance improvements over the T1 forecaster.

2) The performance improvement for the IT2 forecaster getssmaller as SNR decreases.

3) The performance improvement for the triangle Q-T2 fore-caster first decreases for higher SNRs (10, 8, and 6 dB)but then increases for lower SNRs (4, 2, and 0 dB).

4) For higher SNRs, the maximum improvement over theT1 forecaster is from the IT2 forecaster; however, forlower SNRs, there is a reversal of this, and the maximumimprovement over the T1 forecaster is from the triangle Q-T2 forecaster. This suggests that a Q-T2 forecaster shouldbe considered for low-SNR environments.

7) Remark: TR for an IT2 FLS uses the iterative KM algo-rithms, and as pointed out in the comment in Section IV-A, thismay cause a computational bottleneck, even for real-time appli-cations of the Q-T2 FLSs. A solution to this is to replace the TRstep in the triangle Q-T2 FLSs by the Wu–Mendel uncertaintybounds [50], [33]. Using these bounds directly as the architec-ture for an IT2 FLS has already been reported in [11] and [26],where it has been demonstrated that very good performance canbe achieved by doing this.

VII. CONCLUSIONS

This paper has reviewed the α-plane representation of a T2FS, which is comparable to the α-cut representation of a T1 FS,and is useful for both theoretical and computational studies ofand for T2 FSs. It has introduced a novel 2.5-D representationfor a T2 FS, which allows people to easily communicate geo-metrically about a T2 FS. It has provided a proof that one cancompute the union and intersection of T2 FSs using α-planes,thereby providing a novel way to carry out set theory operationsfor general T2 FSs. The α-plane computations are very simplebecause they use existing closed-form formulas for set-theoreticoperations of IT2 FSs. It has also reviewed how the centroid ofa T2 FS can be computed using α-plane computations. Thesecomputations are also very simple because they can be per-

formed using existing KM algorithms that are applied to eachα-plane and lend themselves to massive parallel processing.

By using an existing theoretical fact proven in [39], this paperhas also shown that it is possible to deduce many useful andnovel properties about the shape of CA (x), even before thecentroid is actually computed. Eight such properties are stated,and a few of these have only been observed previously fromsome simulations [22], [23].

Centroid examples have shown that the mean value (defuzzi-fied value) of the centroid can be approximated, often veryaccurately, by just using the centroids of the α = 0 and α = 1planes of a T2 FS. An open issue is a theoretical study, backedup by supporting simulations, about the effects of discretizationsof the primary variable and α on the calculation of the centroid.Of course, undersampling should always be avoided.

The centroid examples, as well as one of the centroid prop-erties, have led us to propose a novel triangle Q-T2 FLS whosesecondary MFs are triangles. This triangle Q-T2 FLS involvesno T2 calculations; all calculations use existing T1 or IT2 FSmathematics. We suggest that this Q-T2 FLS may be a goodnext step in the hierarchy of FLSs from T1 to IT2 to T2.

Our application of forecasting a chaotic time series when itsmeasurements are corrupted by noise has shown that for verylow SNRs, the triangle Q-T2 FLS notably outperforms a T1FLS; however, the same does not occur for higher SNRs. Muchmore work remains to be done on applying such FLSs.

Mendel and Liu [37] have also proposed a trapezoidal Q-T2FLS that is depicted in Fig. 12. It combines two IT2 FLSs, wherethe coupling of these FLSs occurs during overall defuzzification.The TR output of the α = 0 plane IT2 FLS acts as the base ofthe assumed TR trapezoid, and the TR output of the α = 1 planeIT2 FLS acts as the top of that trapezoid. The overall output ofthis trapezoidal Q-T2 FLS is the centroid of that trapezoid, i.e.,

y(x) =13

× 2[yR (x|α = 1)− yL (x|α = 1)]+[yR (x|α = 0)− yL (x|α = 0)][yR (x|α = 1)− yL (x|α = 1)]+[yR (x|α = 0)− yL (x|α = 0)]

.

(57)

The MF parameters of the two IT2 FLSs would be optimizedsimultaneously, and because of (57), this is a coupled optimiza-tion and design.

Examining Q-T2 FLSs for specific applications (with andwithout TR) and comparing them with IT2 FLSs (with andwithout TR) and full-blown T2 FLSs needs to be done, andthese should be the subjects of future studies.

Applying the α-plane representation of a T2 FS to other FSand FL problems may also be very promising, e.g., computinguncertainty measures for general T2 FSs. Additionally, it is veryimportant to show how TR methods such as height and centerof sets (which, in practice, are more widely used than centroidTR) can be implemented using α-planes. Note that these TRmethods are different from centroid TR in that they do not beginwith a single T2 FS.

Finally, we conjecture that just as it is now possible to expressa T2 FS whose MF is 3-D as the union of 2-D α-planes, each

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MENDEL et al.: α -PLANE REPRESENTATION FOR TYPE-2 FUZZY SETS: THEORY AND APPLICATIONS 1205

Fig. 12. Trapezoidal Q-T2 FLS.

of which is an IT2 FS, a type-3 FS whose MF is 4-D can beexpressed as the union of 3-D α-surfaces (which have yet to bedefined), each of which is a T2 FS. This kind of progressivedecomposition20 may eventually lead to a general theory andrepresentation of type-m fuzzy sets and systems.

APPENDIX

PROOFS OF THEOREMS

A.1: Proof of Theorem 1

Because the proof of (27) is similar to the proof of (26),the details of the proof of only (26) are provided here, and forconvenience, (26) is repeated here as

A ∪ B =⋃

α∈[0,1]

α/(A ∪ B)α

=⋃

α∈[0,1]

α/Aα ∪ Bα =⋃

α∈[0,1]

FOU(Aα ∪ Bα ). (A1)

To prove all of the parts of (A1), the following must first beproved:

(A ∪ B)α = Aα ∪ Bα . (A2)

To begin, the first part of (21) is used (in the equations givennext, it is understood that they apply ∀x ∈ X), i.e.,

µA∪B (x) =∫∀u∈J u

x

∫∀w∈J w

x

fx(u)�gx(w)/(u ∨ w)

=∫∀u∈J u

x

∫∀w∈J w

x

fx(u)�gx(w)/v (A3)

where

v ≡ u ∨ w. (A4)

Note that v is the primary membership variable that is associatedwith the T1 FS µA∪B (x) [which is short for µA∪B (v|x)].

Using the definition of an α-cut and (A3), the α-cut of the sec-ondary MF µA∪B (x), which is(µA∪B (x))α , must be composedof all v, say vα , for which

fx(u)�gx(w) ≥ α ∀u ∈ Jux , ∀w ∈ Jw

x . (A5)

For the minimum t-norm, (A5) is satisfied if and only if{fx(u) ≥ α ∀u ∈ Ju

x

gx(w) ≥ α ∀w ∈ Jwx .

(A6)

20Perhaps, there is also a different progressive decomposition for verticalslices.

Let Jux,Aα

and Jwx,Bα

denote the primary memberships (both

of which are the interval sets) of x in Aα and Bα , respectively,i.e.,

Jux,Aα

≡ (µA (x))α ≡[lαf (x), rα

f (x)]

(A7)

Jwx,Bα

≡ (µB (x))α ≡[lαg (x), rα

g (x)]. (A8)

From Definition 1 of an α-plane, it follows from (A6) thatfor (µA∪B (x))α {∀u ∈ Ju

x,Aα

∀w ∈ Jwx,Bα

(A9)

and therefore, using interval arithmetic (e.g., [30, p. 225])

(µA∪B (x))α = vα = u ∨ w| ∀u∈J ux , A α

∀w∈J wx , B α

=[lαf (x), rα

f (x)]∨[lαg (x), rα

g (x)]. (A10)

Consequently, using (7) and (A10), it follows that

(A ∪ B)α =∫∀x∈X

SA∪B (x|α)/x =∫∀x∈X

(µA∪B (x))α/x

=∫∀x∈X

∫∀v∈u∨w=[lαf (x)∨lαg (x),rα

f (x)∨rαg (x)]

v/x

=∫∀x∈X

∫∀v∈[lαf (x)∨lαg (x),rα

f (x)∨rαg (x)]

(x, v). (A11)

Next, we focus on Aα ∪ Bα . Each α-plane can be consideredas an IT2 FS at level α. At each x, Aα is characterized by itslower and upper MFs, which are lαf (x) and rα

f (x), respectively,

and Bα is characterized by its lower and upper MFs, which arelαg (x) and rα

g (x), respectively. Using [30, Th. 7-1] for the unionof two interval-valued sets, it is known that at each x [also using(A7) and (A8)]

Aα ∪ Bα =∫∀x∈X

[(µA (x))α ∪ (µB (x))α

]/x

=∫∀x∈X

[Ju

x,Aα∪ Jw

x,Bα

]/x

=∫∀x∈X

∫∀v∈[lαf (x)∨lαg (x),rα

f (x)∨rαg (x)]

(x, v). (A12)

By comparing (A11) and (A12), the truth of (A2) has beenproved.

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1206 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 17, NO. 5, OCTOBER 2009

Finally, the truth of the right-hand side of (A1) is obviousbecause α/Aα ∪ Bα denotes the plane Aα ∪ Bα at height α,and this plane is FOU(Aα ∪ Bα ).

This completes the proof of (26).For completeness, note that

Aα ∩ Bα =∫∀x∈X

∫∀v∈[lαf (x)∧lαg (x),rα

f (x)∧rαg (x)]

(x, v). (A13)

A.2: Proof of Theorem 2

The proof of this theorem makes very extensive use of thefollowing result for a symmetrical IT2 FS that is proved in [31]:If IT2 FS A is symmetrical (i.e., FOU(A) is symmetrical aboutx = m), then the centroid of A, which is CA (x), is symmetricalabout x = m, and the mean value (i.e., the defuzzified value) ofCA (x) equals m.

Proof: Because A is a totally symmetrical T2 FS, it fol-lows from Definition 7 that each of its α-plane FOUs, which isFOU(Aα ), is symmetrical about x = m. Using the aforemen-tioned result for an IT2 FS, it follows that for each α, CAα

(x) issymmetrical about x = m. Consequently, CA (x), as computedby (31) and (32), is also symmetrical about x = m. Finally, themean value of such a symmetrical function equals m. �

Note that Theorem 2 is also true when FOU(A) is symmet-rical about x = m; the secondary MFs of A to the left and tothe right of x = m (e.g., at x = m − x′ and x = m + x′) arethe same, but these secondary MFs may be different at differentvalues of x′. In this hypothetical situation, the α-plane will alsobe symmetrical about x = m.

ACKNOWLEDGMENT

The authors would like to thank the reviewers of this paperfor many very perceptive and useful comments, which we hopehave led to a better paper.

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Jerry M. Mendel (S’59–M’61–SM’72–F’78–LF’04) received the Ph.D. degree in electrical engi-neering from the Polytechnic Institute of Brooklyn,Brooklyn, NY.

Since 1974, he has been with the University ofSouthern California, Los Angeles, where he is cur-rently a Professor of electrical engineering and sys-tems architecting engineering. He has authored orcoauthored more than 480 technical papers and hasauthored and/or edited eight books, including the Un-certain Rule-based Fuzzy Logic Systems: Introduc-

tion and New Directions (Prentice-Hall, 2001). His current research interestsinclude type-2 fuzzy logic systems and their applications to a wide range ofproblems, including smart oil field technology and computing with words.

Prof. Mendel is a Distinguished Member of the IEEE Control Systems So-ciety and a Fellow of the International Fuzzy Systems Association (2009). Hewas the President of the IEEE Control Systems Society in 1986. He is a memberof the Administrative Committee of the IEEE Computational Intelligence Soci-ety and was also the Chairman of its Fuzzy Systems Technical Committee. Hereceived the 1983 Best Transactions Paper Award from the IEEE Geoscienceand Remote Sensing Society, the 1992 Signal Processing Society Paper Award,the 2002 IEEE TRANSACTIONS ON FUZZY SYSTEMS Outstanding Paper Award,the 1984 IEEE Centennial Medal, an IEEE Third Millennium Medal, the FuzzySystems Pioneer Award (2008) from the IEEE Computational Intelligence So-ciety, and the Pioneer Award from the IEEE Granular Computing Conference,May 2006, for Outstanding Contributions in Type-2 Fuzzy Systems.

Feilong Liu (S’06–M’08) received the B.S. degreefrom Chinese Northeastern University, Shengyang,China, in 1995, the M.S. degree from South ChinaUniversity of Technology, Guangzhou, China, in2000, and the Ph.D. degree from the University ofSouthern California, Los Angeles, in 2008, all in elec-trical engineering.

He is currently an Optimization Engineer withChevron Corporation, Richmond, CA, where he isactively involved in applying all kind of advancedtechniques to smart oilfields to improve work effi-

ciency and help onsite engineers make better decisions. His current researchinterests include computational intelligence theory and applications, artificialintelligence and machine learning, signal processing and time–frequency anal-ysis, estimation theory, and pattern recognition and applying these technologiesto smart oilfields.

Dr. Liu is a member of the Society of Petroleum Engineers (SPE). He receivedsecond place at the 2007 SPE Western Region Ph.D. Student Paper Contest in2007 and a summer research grant from the IEEE Computational IntelligenceSociety in 2006.

Daoyuan Zhai (S’08) was born in Guiyang, China.He received the B.S. degree in telecommunicationengineering from Beijing University of Posts andTelecommunications, Beijing, China, in 2006 and theM.S. degree in electrical engineering from the Uni-versity of Southern California, Los Angeles, in 2007,where he is currently working toward the Ph.D. de-gree in electrical engineering.

His current research interest includes type-2 fuzzylogic theory, signal processing, pattern classifica-tion, and smart oil field technologies for water–flood

management.

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