### The use of Kernel set and sample memberships in the identification

### of nonlinear time series

B. Wu, Y.-Y. Hsu

Abstract The problem of system modeling and identifi-cation has attracted considerable attention in the non-linear time series analysis mostly because of a large number of applications in diverse fields like financial management, biomedical system, transportation, ecology, electric power systems, hydrology, and aeronautics. Many papers have been presented on the study of time series clustering and identification. Nonetheless, we would like to point out that in dealing with clustering time series, we should also take the vague case as they belong to two or more classes simultaneously into account. Because many patterns of grouping in time series really are ambiguous, those phenomena should not be assigned to certain specific classes inflexibly. In this paper, we pro-pose a procedure that can effectively cluster nonlinear time series into several patterns based on kernel set. This algorithm also combines with the concept of a fuzzy set. The membership degree of each datum corresponding to the cluster centers is calculated and is used for perfor-mance index grouping. We also suggest a principle for extending the fuzzy set by kernel set and further interpret events in a sensible light through these sets. Finally, the procedure is demonstrated by set off RRI data and its performance is shown to compare favorably with other procedures published in the literature.

Keywords Fuzzy sets, Kernel sets, Clustering, Identification, Nonlinear time series

1

Introduction

Clustering is aimed at organizing and revealing structures within data. Clustering is commonly viewed as an instance of unsupervised learning to cluster a data set into groups of similar individuals. Moreover, the conventional

clustering methods restrict that each point of the data set belongs to exactly class and omit the possibility that they belong to two or more classes simultaneously.

Fuzzy sets originated by Zadeh (1965) gave an idea of uncertainty that is described with a membership function. Since hardly ever any disturbance or noise in the data set can be completely eliminated and some inherent data uncertainly cannot be avoided, the use of fuzzy sets therefore provide a solvent for indistinct parts in the data. Therefore, it is quite natural and useful to apply the idea of fuzzy set theory in cluster analysis.

To exhibit the empirical data in an appropriate class in the application of nonlinear system, researchers in this field have been concerned with clustering techniques combining with fuzzy logic. For instance, Cutsem and Gath (1993) proposed a procedure of fuzzy clustering to detect outliers and robustly estimate parameters. Yoshi-nari, Pedrycz and Hirota (1993) presented fuzzy clustering techniques to construct fuzzy models. Romer et al. (1995) used fuzzy partitions and possibility theory in statistical inference. Cheng et al. (1998) presented a multistage ran-dom sampling fuzzy c-means-based clustering algorithm to partition a data set into c classes. Wu and Hsu (1999) utilized the average of cumulative fuzzy entropy to classify and identity Taiwan’s unemployment structural changes. Barra and Boire (2000) proposed a possibilistic clustering algorithm using fuzzy theory to test on phantom image of normal and Alzheimer’s brains. Besides, Werners (1987), Tseng and Klein (1992), and Yang (1993) conducted more comprehensive research on fuzzy clustering in fuzzy decision analysis.

In the field of time series analysis for system identi-fication, time series are often encountered is important cases where the statistics of the data exhibit nonstation-ary structure. There is a need for techniques to identify the system of some patterns that could influence decision-making. In the absence of a powerful enough algorithm for this in the past, the usual ploy has been to assume the data under consideration we piecewise stationary and apply identification algorithms which were signed for stationary data but whose estimates converge quickly enough that the assumption of piecewise stationary would be badly abused.

Hence, in order to meet real situation, it has been better to employ the concept of ‘‘kernel set’’ instead of ‘‘crisp set’’ in clustering and identification. In this paper, we propose a procedure that can effectively cluster nonlinear time series into several patterns based on kernel set. This paper is organized as follows. In Sect. 2, we introduce definitions

Original paper *Soft Computing 8 (2004) 207–216 Ó Springer-Verlag 2003*

DOI 10.1007/s00500-003-0265-3

Published online: 14 July 2003

B. Wu (*&)*

Department of Mathematical Sciences, National Chengchi University, Wenshan, Taipei 11623, Taiwan, ROC

e-mail: berlin@math.nccu.edu.tw

Y.-Y. Hsu

Department of Applied Mathematics,

National Donghwa University, Hualien 974, Taiwan, ROC

We are grateful to the referees for their careful reading and helpful comments.

and related theorems for clustering and analyzing. In addition, we develop a measuring method for calculating membership degree of each datum corresponding to the cluster centers and further get kernel set from fuzzy set after clustering in Sect. 3. The empirical application and comparative issues for RRI data are covered in Sect. 4. These include computational problems in clustering non-linear time series, and the issue of whether the procedure combining fuzzy set theory has a better performance than other procedures. Sect. 5 gives the conclusion and sug-gestions.

2

The nature of kernel sets 2.1

Fuzzy sets and kernel sets

Though there are huge papers discussing about the fuzzy sets and its applications. Few literatures can be found about the crisp definition of fuzzy set. In this paper, we will present an appropriate definition for it. The following definitions are made in order to formalize and simplify nonlinear time series analysis.

Definition 2.1. [Fuzzy set] Let X ¼ Xf t;t ¼ 1; 2; . . . ; Ng

be a universal set and L ¼ Lf k;k ¼ 1; 2; . . . ; rg be a

sequence of linguistic terms with monotone degree of semantic. For each k ¼ 1; 2; . . . ; r, the fuzzy set Ak

generated by Lk on X is a generalized set, which assigns

each Xt to a value between 0 and 1.

Definition 2.2. [Kernel set] Let X ¼ Xf t;t ¼ 1; 2; . . . ; Ng

be a universal set and L ¼ Lf k;k ¼ 1; 2; . . . ; rg be a

sequence of linguistic terms with monotone degree of semantic. For any significant level ak2 ½0; 1,

k ¼ 1; 2; . . . ; r, the kernel set of Ak generated by Lk is

denoted by KerðAkÞ ¼ _

a2KðAkÞja_{ak}

a ðAkÞa; k ¼ 1; 2; . . . ; r : ð1Þ

where KðAkÞ is the level set of Ak, ðAkÞa is a a-cut of Ak,

and _ denotes the finite fuzzy union. Fuzzy set – an extension of kernel set

Let X ¼ Xf t;t ¼ 1; 2; . . . ; Ng be a universal set and

L ¼ Lf k;k ¼ 1; 2; . . . ; rg be a sequence of linguistic terms

with monotone degree of semantic. The fuzzy set Ak

generated by Lk defined on X is an extension of its kernel

set and denoted by Ak ¼ _

a2KðAkÞ

a ðAkÞa; k ¼ 1; 2; . . . ; r : ð2Þ

where KðAkÞ is the level set of Ak, ðAkÞa is a a-cut of Ak,

and _ denotes the finite fuzzy union.

Example 2.1. Let X ¼ fX1;X2;X3;X4g ¼ fLavender;

Eucalyptus; Thyme; Ylangg be a universal set of essential oil for helping sleep and

L ¼ fL1;L2;L3;L4;L5g ¼ fHighly inef f ective;

Inef f ective; Common; Ef f ective; Highlyef f ectiveg be a sequence of linguistic terms with monotone degree of

semantic. After interviewing 10 consumers, we get the membership grades of the foregoing essential oils. The results are showed at Table 1.

By Definition 2.1, the fuzzy set A1generated by L1on X is

A1 ¼ 0:10=X1þ 0:30=X2þ 0:20=X3 :

and the level set of A1 is KðA1Þ ¼ f0:10; 0:30; 0:20g.

Moreover, via its membership grades, the a-cuts of A1

become

ðA1Þ0:10¼ 1=X1þ 1=X2þ 1=X3þ 0=X4 ;

ðA1Þ_{0:30}¼ 0=X1þ 1=X2þ 0=X3þ 0=X4 ;

ðA1Þ_{0:20}¼ 0=X1þ 1=X2þ 1=X3þ 0=X4 :

If we multiply each ðA1Þa by corresponding a value, then

we convert each of the a-cuts to a special fuzzy set, i.e. 0:10 ðA1Þ0:10¼ 0:10=X1þ 0:10=X2þ 0:10=X3þ 0:00=X4 ;

0:30 ðA1Þ0:30¼ 0:00=X1þ 0:30=X2þ 0:00=X3þ 0:00=X4 ;

0:20 ðA1Þ0:20¼ 0:00=X1þ 0:20=X2þ 0:20=X3þ 0:00=X4 :

Under the significant level a1¼ 0:80 and by Definition 2.2,

we can get the kernel set of A1 is

KerðA1Þ ¼ _ a2KðA1Þja0:80

a ðA1Þa¼ / :

and take the finite fuzzy union of above special fuzzy sets, a ðA1Þa, we get

A1¼ _ a2KðA1Þ

a ðA1Þa

¼ ½0:10 ðA1Þ0:10 _ ½0:30 ðA1Þ0:30 _ ½0:20 ðA1Þ0:20

¼ ð0:10 _ 0:00 _ 0:00Þ=X1þ ð0:10 _ 0:30 _ 0:20Þ=X2

þ ð0:10 _ 0:00 _ 0:20Þ=X3þ ð0:00 _ 0:00 _ 0:00Þ=X4

¼ 0:10=X1þ 0:30=X2þ 0:20=X3 :

Similarly, we can get the following Table 2.

Table 1. The membership grades of fXtg for fLkg

L1 L2 L3 L4 L5

X1 0.10 0.35 0.50 0.75 0.65

X2 0.30 0.50 0.60 0.20 0.10

X3 0.20 0.45 0.80 0.50 0.30

X4 0.00 0.10 0.40 0.80 0.92

Table 2. The kernel sets and fuzzy sets for fLkg under the

sig-nificant levels ak¼ 0:80, k ¼ 1; . . . ; 5

Kernel set Fuzzy set

L1 / 0:10=X1þ 0:30=X2þ 0:20=X3 L2 / 0:35=X1þ 0:50=X2þ 0:45=X3þ 0:10=X4 L3 0:80=X3 0:50=X1þ 0:60=X2þ 0:80=X3þ 0:40=X4 L4 0:80=X4 0:75=X1þ 0:20=X2þ 0:50=X3þ 0:80=X4 L5 0:92=X4 0:65=X1þ 0:10=X2þ 0:30=X3þ 0:92=X4 208

Theorem 2.1. Let X be a universal set and A be a fuzzy
set on X. Then the complement of fuzzy set, Ac_{, is}

denoted by

Ac ¼ _

ð1aÞ2KðAc_{Þ}ð1 aÞ ðA

c_{Þ}

ð1aÞ : ð3Þ

where KðAc_{Þ is the level set of A}c_{, ðA}c_{Þ}

ð1aÞis a ð1 aÞ-cut

of Ac_{, and _ denotes the finite fuzzy union.}

Proof. For each x 2 X, let a ¼ lAcðxÞ. Then,
l
_
ð1aÞ2KðAcÞð1aÞðA
c_{Þ}
ð1aÞ
ðxÞ
¼ max

ð1aÞ2KðAc_{Þ}lðð1aÞðAcÞð1aÞÞðxÞ

¼ max max
ð1aÞ2KðAc_{Þj}
ð1aÞa
l _{ð1aÞðA}c_{Þ}
ð1aÞ
ð ÞðxÞ;
max
ð1aÞ2KðAc_{Þj}
ð1aÞ>a
l _{ð1aÞðA}c_{Þ}
ð1aÞ
ð ÞðxÞ
!
: ð4Þ

For each ð1 aÞ 2 KðAc_{Þj}

ð1aÞ>a, we have
l_{A}cðxÞ ¼ a < 1 a ;
it implies
l _{ð1aÞðA}c_{Þ}
ð1aÞ
ð ÞðxÞ ¼ 0 : ð5Þ

On the other hand, for each ð1 aÞ 2 KðAc_{Þj}

ð1aÞa, we
have
l_{A}cðxÞ ¼ a 1 a ;
it implies
l _{ð1aÞðA}c_{Þ}
ð1aÞ
ð ÞðxÞ ¼ 1 a : ð6Þ
Eq. (4) follows,
l
_
ð1aÞ2KðAcÞð1aÞðA
c_{Þ}
ð1aÞ
ðxÞ ¼ max
ð1aÞ2KðAc_{Þj}
ð1aÞa
ð1 aÞ
¼ a ¼ lAcðxÞ :
The proof is complete.

Corollary 2.1. Let X be a universal set and A be a fuzzy set on X. For any significant level ð1 a1Þ 2 ½0; 1, the

com-plement of fuzzy set, Ac_{, is denoted by}

Ac ¼ KerðAc_{Þ}

[F _

ð1aÞ2KðAc_{Þj}

ð1aÞ<ð1a1Þ

ð1 aÞ ðAcÞ_{ð1aÞ}
!

: ð7Þ

where KðAc_{Þ is the level set of A}c_{, ðA}c_{Þ}

ð1aÞis a ð1 aÞ-cut

of Ac _{and [}

F denotes the standard fuzzy union.

Proof. For each x 2 X, let 1 a1¼ lAcðxÞ be the signifi-cant level. Then,

l_{A}cðxÞ ¼ l
_
ð1aÞ2KðAcÞð1aÞðA
c_{Þ}
ð1aÞ
ðxÞ ðby Theorem 2:1:Þ
¼ max

ð1aÞ2KðAc_{Þ}lðð1aÞðAcÞð1aÞÞðxÞ

¼ max max
ð1aÞ2KðAc_{Þj}
ð1aÞ<ð1a1Þ
l _{ð1aÞðA}c_{Þ}
ð1aÞ
ð ÞðxÞ;
max
ð1aÞ2KðAc_{Þj}
ð1aÞð1a1Þ
l _{ð1aÞðA}c_{Þ}
ð1aÞ
ð ÞðxÞ
!
¼ max l
_
ð1aÞ2KðAcÞjð1aÞ<ð1a_{1Þ}ð1aÞðA
c_{Þ}
ð1aÞ
ðxÞ;
0
B
B
@
l
_
ð1aÞ2KðAcÞjð1aÞð1a1Þ
ð1aÞðAc_{Þ}
ð1aÞ
ðxÞ
1
C
C
A :
ð8Þ
Since, under the significant level 1 a1,

_
ð1aÞ2KðAc_{Þj}
ð1aÞð1a1Þ
ð1 aÞ ðAc_{Þ}
ð1aÞ ¼ KerðAcÞ : ð9Þ
Eq. (8) follows,
l_{A}cðxÞ ¼ max l
_
ð1aÞ2KðAcÞjð1aÞ<ð1a_{1Þ}ð1aÞðA
c_{Þ}
ð1aÞ
ðxÞ;
0
B
B
@
lKerðAc_{Þ}ðxÞ
1
C
A

¼ max lKerðAc_{Þ}ðxÞ;
0
B
B
@
l
_
ð1aÞ2KðAcÞjð1aÞ<ð1a_{1Þ}ð1aÞðA
c_{Þ}
ð1aÞ
ðxÞ
1
C
C
A :
ðby the commutative lawÞ ð10Þ
By the standard fuzzy union, Eq. (10) follows,

l_{A}cðxÞ ¼ l
KerðAc_{Þ[}_{F} _{_}
ð1aÞ2KðAcÞjð1aÞ<ð1a1Þ
ð1aÞðAc_{Þ}
ð1aÞ
ðxÞ :

The proof is complete.

Example 2.2. From Example 2.1, the complement of fuzzy set A1 generated by L1 on X can be written as:

Ac_{1}¼ 0:90=X1þ 0:70=X2þ 0:80=X3þ 1:00=X4

¼ ð1 0:10Þ=X1þ ð1 0:30Þ=X2

þ ð1 0:20Þ=X3þ ð1 0:00Þ=X4 :

and the level set of Ac

1 is KðAc1Þ ¼ f0:90; 0:70; 0:80; 1:00g.

Thus, the ð1 aÞ-cuts of Ac

1 according to its membership

grades become
Ac_{1}
ð10:10Þ¼ 1=X1þ 0=X2þ 0=X3þ 1=X4 ;
Ac_{1}
ð10:30Þ¼ 1=X1þ 1=X2þ 1=X3þ 1=X4 ;
Ac_{1}
ð10:20Þ¼ 1=X1þ 0=X2þ 1=X3þ 1=X4 ;
Ac_{1}
ð10:00Þ¼ 0=X1þ 0=X2þ 0=X3þ 1=X4 :

If we multiply each ðAc

1Þð1aÞ by corresponding ð1 aÞ

value, then we convert each of the ð1 aÞ-cuts to a special fuzzy set, i.e.

ð1 0:10Þ Ac
1
ð10:10Þ
¼ 0:90=X1þ 0:00=X2þ 0:00=X3þ 0:90=X4 ;
ð1 0:30Þ A c_{1} _{ð10:30Þ}
¼ 0:70=X1þ 0:70=X2þ 0:70=X3þ 0:70=X4 ;
ð1 0:20Þ A c_{1} _{ð10:20Þ}
¼ 0:80=X1þ 0:00=X2þ 0:80=X3þ 0:80=X4 ;
ð1 0:00Þ Ac
1
ð10:00Þ
¼ 0:00=X1þ 0:00=X2þ 0:00=X3þ 1:00=X4 :

Since, under the significant level 1 a1¼ 0:80, we can get

Ker A c_{1}

¼ _

ð1aÞ2KðAc 1Þjð1aÞ0:80

ð1 aÞ ðAc_{1}Þ_{ð1aÞ}

¼ ½ð1 0:10Þ A c_{1} _{ð10:10Þ} _ ½ð1 0:20Þ A c_{1} _{ð10:20Þ}
_ ½ð1 0:00Þ ðAc_{1}Þ_{ð10:00Þ}
¼ ð0:90 _ 0:80 _ 0:00Þ=X1þ ð0:00 _ 0:00 _ 0:00Þ=X2
þ ð0:00 _ 0:80 _ 0:00Þ=X3þ ð0:90 _ 0:80 _ 1:00Þ=X4
¼ 0:90=X1þ 0:00=X2þ 0:80=X3þ 1:00=X4 :
and
_
ð1aÞ2K Ac
1
ð Þj_{ð1aÞ<0:80}ð1 aÞ A
c
1
ð1aÞ
¼ ð1 0:30Þ Ac
1
ð10:30Þ
¼ 0:70=X1þ 0:70=X2þ 0:70=X3þ 0:70=X4 :

Finally, by Corollary 2.1, we get

Ac_{1} ¼ KerðAc
1Þ [F _
ð1aÞ2KðAc
1Þjð1aÞ<0:80
ð1 aÞ Ac
1
ð1aÞ
0
@
1
A
¼ 0:90=X1þ 0:70=X2þ 0:80=X3þ 1:00=X4 :

Similarly, we can get the following results.
Ac_{2} ¼ 0:65=X1þ 0:50=X2þ 0:55=X3þ 0:90=X4 ;

Ac_{3} ¼ 0:50=X1þ 0:40=X2þ 0:20=X3þ 0:60=X4 ;

Ac_{4} ¼ 0:25=X1þ 0:80=X2þ 0:50=X3þ 0:20=X4 ;

Ac_{5} ¼ 0:35=X1þ 0:90=X2þ 0:70=X3þ 0:08=X4 :

Theorem 2.2. Let X be a universal set, A and B be two fuzzy sets on X. For any significant levels a1;a22 ½0; 1, the

standard fuzzy union and intersection of A and B are denoted by (i) A [FB ¼ ½ _ a2KðAÞa ðAÞa [F½ _b2KðBÞb ðBÞb ¼ ½KerðAÞ [FKerðBÞ [F _

a2KðAÞj_{a<a1}a ðAÞa

!
"
[F _
b2KðBÞj_{b<a2}b ðBÞb
!#
: ð11Þ
(ii)
A \FB ¼ _
a2KðAÞa ðAÞa
\F _
b2KðBÞb ðBÞb
¼ ½KerðAÞ \FKerðBÞ
[F KerðAÞ \F _
b2KðBÞj_{b<a2}b ðBÞb
!
" #
[F _

a2KðAÞj_{a<a1}a ðAÞa

!

\FKerðBÞ

" #

[F _

a2KðAÞj_{a<a1}a ðAÞa

!
"
\F _
b2KðBÞj_{b<a2}b ðBÞb
!#
: ð12Þ

where KðAÞ is the level set of A, KðBÞ is the level set of B,
ðAÞ_{a} is a a-cut of A, ðBÞbis a b-cut of B, [Fand \Fdenote

the standard fuzzy union and intersection respectively. Proof. (i) For each x 2 X, let lAðxÞ ¼ a1and lBðxÞ ¼ a2be

the significant levels. Then, by the standard fuzzy union, it
implies
l_{A[}_{F}_{B}ðxÞ
¼ maxðlAðxÞ; lBðxÞÞ
¼ max l
_
a2KðAÞaðAÞa
_{ðxÞ; l}
_
b2KðBÞbðBÞb
_{ðxÞ}
0
@
1
A
¼ max max

a2KðAÞlðaðAÞaÞðxÞ; max_{b2KðBÞ}lðbðBÞbÞðxÞ

¼ max "

max max

a2KðAÞj_{a<a1}lðaðAÞaÞðxÞ; max
a2KðAÞja_{a1}
l_{ðaðAÞ}
aÞðxÞ
!
;
max max
b2KðBÞj_{b<a2}lðbðBÞbÞðxÞ;_{b2KðBÞj}max
ba2
l_{ðbðBÞ}
bÞðxÞ
!#
210

¼ max max
2
6
6
4 l
_
a_{2KðAÞja<a1}aðAÞa
_{ðxÞ;l}
_
a_{2KðAÞjaa1}aðAÞa
_{ðxÞ}
0
B
B
@
1
C
C
A;
max l
_
b_{2KðBÞjb<a}
2
bðBÞb
_{ðxÞ;l}
_
b_{2KðBÞjba}
2
bðBÞb
_{ðxÞ}
0
B
B
@
1
C
C
A
3
7
7
5 :
ð13Þ
Since, under the significant level a1,

_

a2KðAÞja_{a1}

a ðAÞa¼ KerðAÞ : ð14Þ

Similarly, under the significant level a2,

_
b2KðBÞjb_{a2}
b ðBÞ_{b}¼ KerðBÞ : ð15Þ
Eq. (13) follows,
l_{A[}_{F}_{B}ðxÞ
¼ max max l
_
a_{2KðAÞja<a1}aðAÞa
ðxÞ;lKerðAÞðxÞ
0
B
B
@
1
C
C
A;
2
6
6
4
max l
_
b2KðBÞjb<a_{2}bðBÞb
ðxÞ;l_{KerðBÞ}ðxÞ
0
B
B
@
1
C
C
A
3
7
7
5
¼ max
2
6
6

4max lKerðAÞðxÞ;lKerðBÞðxÞ

;
max l
_
a_{2KðAÞja<a1}aðAÞa
ðxÞ;l
_
b_{2KðBÞjb<a}
2
bðBÞ_{b}
ðxÞ
1
C
C
A
0
B
B
@
3
7
7
5 :
ðby the commutative and associative lawsÞ ð16Þ
By the standard fuzzy union, Eq. (16) follows,

l_{A[}_{F}_{B}ðxÞ

¼ max lðKerðAÞ[FKerðBÞÞðxÞ;
0
B
B
@
l
_
a_{2KðAÞja<a1}aðAÞa
[F _
b_{2KðBÞjb<a}
2
bðBÞb
ðxÞ
1
C
C
A
¼ l
KerðAÞ[FKerðBÞ
½ [F _
a_{2KðAÞja<a1}aðAÞa
[F _
b_{2KðBÞjb<a}
2
bðBÞb
ðxÞ
:

The proof is complete.

(ii) For each x 2 X, let lAðxÞ ¼ a1and lBðxÞ ¼ a2be the

significant levels. Then, by the standard fuzzy intersection,
it implies
l_{A\}_{F}_{B}ðxÞ
¼ min lð AðxÞ; lBðxÞÞ
¼ min l
_
a2KðAÞaðAÞa
_{ðxÞ; l}
_
b2KðBÞbðBÞb
_{ðxÞ}
0
@
1
A
¼ min max

a2KðAÞlðaðAÞaÞðxÞ; max_{b2KðBÞ}lðbðBÞbÞðxÞ

¼ min "

max max

a2KðAÞj_{a<a1}lðaðAÞaÞðxÞ;_{a2KðAÞj}max
a_{a1}
l_{aðAÞ}
a
ð ÞðxÞ
!
;
max max
b2KðBÞj_{b<a2}lðbðBÞbÞðxÞ;_{b2KðBÞj}max
b_{a2}
l_{bðBÞ}
b
ð ÞðxÞ
!#
¼ min
2
6
6
4 max l
_
a_{2KðAÞja<a1}aðAÞa
_{ðxÞ; l}
_
a_{2KðAÞjaa1}aðAÞa
_{ðxÞ}
0
B
B
@
1
C
C
A;
max l
_
b_{2KðBÞjb<a}
2
bðBÞb
_{ðxÞ; l}
_
b_{2KðBÞjba}
2
bðBÞb
_{ðxÞ}
1
C
C
A
0
B
B
@
3
7
7
5 :
ð17Þ
Since, under the significant level a1,

_

a2KðAÞja_{a1}

a ðAÞ_{a}¼ KerðAÞ : ð18Þ

Similarly, under the significant level a2,

_
b2KðBÞjba2
b ðBÞb¼ KerðBÞ : ð19Þ
Eq. (17) follows,
l_{A\}_{F}_{B}ðxÞ
¼ min max l
_
a_{2KðAÞja<a1}aðAÞa
ðxÞ; l_{KerðAÞ}ðxÞ
0
B
B
@
1
C
C
A;
2
6
6
4
max l
_
b_{2KðBÞjb<a}
2
bðBÞb
ðxÞ; l_{KerðBÞ}ðxÞ
0
B
B
@
1
C
C
A
3
7
7
5
¼ max min lh _{KerðAÞ}ðxÞ; l_{KerðBÞ}ðxÞ;

min lKerðAÞðxÞ; l
_
b_{2KðBÞjb<a}
2
bðBÞb
ðxÞ
0
B
B
@
1
C
C
A;
211

min l
_
a_{2KðAÞja<a1}aðAÞa
ðxÞ; l_{KerðBÞ}ðxÞ
0
B
B
@
1
C
C
A;
min l
_
a_{2KðAÞja<a1}aðAÞa
ðxÞ;
0
B
B
@
l
_
b_{2KðBÞjb<a}
2
bðBÞb
ðxÞ
1
C
C
A
3
7
7
5 :

ðby the distributive lawÞ ð20Þ
By the standard fuzzy intersection, Eq. (20) follows,
l_{A\}_{F}_{B}ðxÞ

¼ max l_{ð}_{KerðAÞ\}_{F}_{KerðBÞ}_{Þ}ðxÞ;

l
KerðAÞ\F _
b_{2KðBÞjb<a}
2
bðBÞb
ðxÞ;
l
_
a_{2KðAÞja<a1}aðAÞa
\FKerðBÞ
ðxÞ;
l
_
a_{2KðAÞja<a}
1
aðAÞa
\F _
b_{2KðBÞjb<a}
2
bðBÞb
ðxÞ
1
C
C
A :ð21Þ
By the standard fuzzy union, Eq. (21) follows,

l_{A\}
FBðxÞ ¼ l
KerðAÞ\FKerðBÞ
½ [F KerðAÞ\F _
b2KðBÞjb<a_{2}bðBÞb
:
[F _
a_{2KðAÞja<a}
1
aðAÞa
\FKerðBÞ
[F _
a_{2KðAÞja<a1}aðAÞa
\F _
b2KðBÞjb<a_{2}bðBÞb
_{ðxÞ}
:

The proof is complete.

Example 2.3. From Example 2.1, the standard fuzzy union and intersection of A3, A5can be written, respectively, as:

A3[FA5 ¼ 0:65=X1þ 0:60=X2þ 0:80=X3þ 0:92=X4 :

A3\FA5 ¼ 0:50=X1þ 0:10=X2þ 0:30=X3þ 0:40=X4 :

and the level set of A3 is KðA3Þ ¼ f0:50; 0:60; 0:80; 0:40g,

the level set of A5is KðA5Þ ¼ f0:65; 0:10; 0:30; 0:92g Thus,

the a-cuts of A3 and A5according to their membership

grades, respectively, become

ðA3Þ0:50¼ 1=X1þ 1=X2þ 1=X3þ 0=X4 ;
ðA3Þ0:60¼ 0=X1þ 1=X2þ 1=X3þ 0=X4 ;
ðA3Þ0:80¼ 0=X1þ 0=X2þ 1=X3þ 0=X4 ;
ðA3Þ_{0:40}¼ 1=X1þ 1=X2þ 1=X3þ 1=X4 ;
ðA5Þ0:65¼ 1=X1þ 0=X2þ 0=X3þ 1=X4 ;
ðA5Þ0:10¼ 1=X1þ 1=X2þ 1=X3þ 1=X4 ;
ðA5Þ0:30¼ 1=X1þ 0=X2þ 1=X3þ 1=X4 ;
ðA5Þ0:92¼ 0=X1þ 0=X2þ 0=X3þ 1=X4 :

If we multiply each ðAkÞa by corresponding a value,

k ¼ 3; 5, then we convert each of the a-cuts to a special fuzzy set, i.e.

0:50 ðA3Þ0:50
¼ 0:50=X1þ 0:50=X2þ 0:50=X3þ 0:00=X4 ;
0:60 ðA3Þ0:60
¼ 0:00=X1þ 0:60=X2þ 0:60=X3þ 0:00=X4 ;
0:80 ðA3Þ0:80
¼ 0:00=X1þ 0:00=X2þ 0:80=X3þ 0:00=X4 ;
0:40 ðA3Þ0:40
¼ 0:40=X1þ 0:40=X2þ 0:40=X3þ 0:40=X4 ;
0:65 ðA5Þ_{0:65}
¼ 0:65=X1þ 0:00=X2þ 0:00=X3þ 0:65=X4 ;
0:10 ðA5Þ_{0:10}
¼ 0:10=X1þ 0:10=X2þ 0:10=X3þ 0:10=X4 ;
0:30 ðA5Þ0:30
¼ 0:30=X1þ 0:00=X2þ 0:30=X3þ 0:30=X4 ;
0:92 ðA5Þ0:92
¼ 0:00=X1þ 0:00=X2þ 0:00=X3þ 0:92=X4 :

Since, under the significant levels ak¼ 0:80, k ¼ 3; 5, we

can get KerðA3Þ ¼ _ a2KðA3Þja0:80 a ðA3Þa ¼ 0:80 ðA3Þ0:80 ¼ 0:00=X1þ 0:00=X2þ 0:80=X3þ 0:00=X4 : KerðA5Þ ¼ _ a2KðA5Þja0:80 a ðA5Þa ¼ 0:92 ðA5Þ0:92 ¼ 0:00=X1þ 0:00=X2þ 0:00=X3þ 0:92=X4 : and _ a2KðA3Þja<0:80 a ðA3Þa ¼ 0:50 ðA3Þ0:50 _ 0:60 ðA3Þ0:60 _ 0:40 ðA3Þ0:40 ¼ ð0:50 _ 0:00 _ 0:40Þ=X1þ ð0:50 _ 0:60 _ 0:40Þ=X2 þ ð0:50 _ 0:60 _ 0:40Þ=X3þ ð0:00 _ 0:00 _ 0:40Þ=X4 ¼ 0:50=X1þ 0:60=X2þ 0:60=X3þ 0:40=X4 : 212

_ a2KðA5Þja<0:80 a ðA5Þa ¼ 0:65 ðA5Þ0:65 _ 0:10 ðA5Þ0:10 _ ½0:30 ðA5Þ0:30 ¼ ð0:65 _ 0:10 _ 0:30Þ=X1þ ð0:00 _ 0:10 _ 0:00Þ=X2 þ ð0:00 _ 0:10 _ 0:30Þ=X3þ ð0:65 _ 0:10 _ 0:30Þ=X4 ¼ 0:65=X1þ 0:10=X2þ 0:30=X3þ 0:65=X4 :

Finally, by Theorem 2.2 (i), we get A3[FA5¼ KerðA½ 3Þ [FKerðA5Þ [F _ a2KðA3Þja<0:80 a ðA3Þa [F _ a2KðA5Þja<0:80 a ðA5Þa ¼ 0:65=X1þ 0:60=X2þ 0:80=X3þ 0:92=X4 :

Similarly, by Theorem 2.2 (ii), we get A3\FA5¼ KerðA½ 3Þ \FKerðA5Þ [F KerðA3Þ \F _ a2KðA5Þja<0:80 a ðA5Þa [F _ a2KðA3Þja<0:80 a ðA3Þa \FKerðA5Þ [F _ a2KðA3Þja<0:80 a ðA3Þa \F _ a2KðA5Þja<0:80 a ðA5Þa ¼ 0:50=X1þ 0:10=X2þ 0:30=X3þ 0:40=X4 :

Definition 2.3. [The Crisp set] Let X ¼ fXt;t ¼ 1; 2; . . . ; Ng be a universal set and

L ¼ fLk;k ¼ 1; 2; . . . ; rg be a sequence of linguistic terms

with monotone degree of semantic. For each k ¼ 1; 2; . . . ; r; the crisp set Hk generated by Lk on X is a set, which

contains all elements Xt map to the value 1.

Note. For any fuzzy set A on X and the corresponding membership function lA. If the significant level a ¼ 1 and

lAðxÞ ¼ 1 for each x 2 X, then A and KerðAÞ are equal.

Moreover, A is called a crisp set on X. 3

Sample memberships estimation with respect to the kernel set

3.1

How to measure the membership for a sample with respect to its cluster center

It is interesting to see how far for a sample (sets) away from the kernel set in the sampling survey or time series analysis. Especially in the pattern identification process-ing, when we have several typical patterns (here means kernel set of the object), we are eager to know how far are the data different from the proposed type.

3.2

How to get the kernel set after clustering

In order to solve this problem, we will use the sample memberships with respect to the kernel set to estimate their distance. Firstly, we have to construct the kernel set under the features of the samples we have gathered. Which means we will learn the pattern from the experiments or the experience as the neural network did. An integrated procedure for deriving a kernel set with samples pattern is presented as follows:

A procedure for kernel set construction Step 1. For time series fXitgni¼1, do i ¼ 1; . . . ; n.

Step 2. Input the time series fXitg. Find the cluster center

Ci for fXitg.

Step 3. Let lti be the degree of membership of each

ob-servation of fXitg to the cluster center Ci. Compute

the membership lti by

l_{ti}¼ 1
Xit Ci

k k; t ¼ 1; . . . ; N :

where k k denote Euclidean distance and if lti>1,

then let lti¼ 1.

Step 4. Constructing the fuzzy set Ai by its memberships.

Step 5. Choose a proper significant level a for Ai, and

decide the kernel set KerðAiÞ of Ai.

Step 6. The kernel set learned from these samples will be KerðAÞ ¼ [n

i¼1FKerðAiÞ.

After we decide the kernel set form the samples we have known, we will compute the sample memberships. And under the significant level a, we will see how many data exceed the threshold value that will give us a useful suggestion for the final decision-making.

4

Applications with the RRI data

The data analyzed here comes from the ICU of Taipei Vet-erans General Hospital, 2001. The data records RRI of the dead patients and survival patients for the first four days of ICU. The RRI data for each patient is measured with 30 minutes. By discarding the first 100 observation, we analysis the 101 to 600 observations from each patient which con-tains about 1800–3000 observations of RRI. The purpose of this study is to extract features and identify nonlinear time series for the ICU patients. Figure 1a and b plot respectively the dead and survival patients’ RRI. For the 500 observa-tions, we can find the cluster centers for each data set. Now, under the significant level a ¼ 0:9, we can construct our kernel sets by the proposed procedures in the Sect. 3. In following, the dead patients’ and the survival patients’ cluster centers, radiuses of kernel set and ratios are showed in Table 3.

The kernel set learned from the dead patients is KerðDÞ ¼ [4

i¼1FKerðDi2Þ. Then, we can give the following

testing-hypothesis procedure:

H0: the data belongs to the dead patterns.

H1: the data doesn’t belong to the dead patterns.

Decision rule: for the new sample KerðDnewÞ, under the

significant level aD, if there exist i, such that

KerðDnewÞ=KerðDi2Þ > 1 aD then we accept H0.

Other-wise, we reject H0.

Similarly, the kernel set learned from the survival pa-tients is KerðSÞ ¼ [4

i¼1FKerðSi2Þ, and the testing-hypothesis

procedures:

H0: the data belongs to the survival patterns.

H1: the data doesn’t belong to the survival patterns.

Decision rule: for the new sample KerðSnewÞ, under the

significant level aD, if there exist i, such that KerðSnewÞ=

KerðSi2Þ > 1 aDthen we accept H0. Otherwise, we reject

H0.

Now, we examine two new RRI samples of patients from ICU. First, we can find the cluster centers for each data set and construct their kernel sets under the significant level a¼ 0:9. Then we can find radiuses of kernel sets for these samples. Finally, we can get their patterns according to their features by above the testing-hypothesis procedures.

Fig. 1. Plots of RRI for dead and survival patients

For these two samples, we get the cluster centers 592.624 and 761.658 respectively. We can calculate the membership for each observation via the distance between observation and its cluster center. Under the significant level a ¼ 0:9, if the membership of observation is lager than 0.9, then this observation is a member of the kernel set. Therefore, the results of two new samples are showed in Table 4.

From Table 4 we can find that:

(1) For two new samples, the radiuses of kernel sets are 0.624 and 0.658 respectively.

(2) For new sample one, the ratio of observations which belongs to its kernel set and total observations is 0:094ð¼ 47=500Þ, which is larger than 0.05. This indicates that the patient has some features of dead patients.

(3) From Table 3, we obtain the significant level

aD¼ 0:05. Under this condition, there exists D32 and

D42 such that the ratio of observations which belongs

to its kernel set and KerðDi2Þ is lager than 0.95,

i ¼ 3; 4. By way of decision rule, the patient will not be surviving.

(4) For new sample two, the ratio of observations which belongs to its kernel set and total observations is 0:034ð¼ 17=500Þ, which is smaller than 0.05. This in-dicates that the patient has some features of survival patients.

(5) Under the significant level aD ¼ 0:05, there exists S22

such that the ratio of observations which belongs to its kernel set and KerðS22Þ is lager than 0.95. By way of

decision rule, the patient will be alive.

5

Conclusion

In the medical science analysis discussed above the time series data have the uncertain property. If we use the conventional clustering methods to analyze these data, it will not solve the orientation problem. The contribution of

Table 4. The sample memberships and kernel set for RRI of new samples The sample memberships and kernel set for

RRI of new sample one (Cluster center: 592.624, a ¼ 0:9)

The sample memberships and kernel set for RRI of new sample two (Cluster center: 761.658, a ¼ 0:9)

Data Memberships Is a member of

the kernel set?

Data Memberships Is a member of the

kernel set? 1 569 0.042 no 751 0.094 no 2 573 0.051 no 740 0.046 no 3 571 0.046 no 760 0.603 no 4 529 0.016 no 734 0.036 no 5 622 0.034 no 730 0.032 no 6 598 0.186 no 729 0.031 no 7 609 0.061 no 718 0.023 no 8 614 0.047 no 713 0.021 no 9 608 0.065 no 708 0.019 no 10 605 0.081 no 741 0.048 no .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . 491 591 0.616 no 737 0.041 no 492 591 0.616 no 725 0.027 no 493 591 0.616 no 722 0.025 no 494 595 0.421 no 717 0.022 no 495 598 0.186 no 721 0.025 no 496 598 0.186 no 721 0.025 no 497 595 0.421 no 717 0.022 no 498 592 1.000 yes 725 0.027 no 499 598 0.186 no 736 0.039 no 500 596 0.296 no 728 0.030 no Total 47ð0:094 > 0:05Þ 17ð0:034 0:05Þ

Table 3. The dead and the survival patients’ cluster centers, radiuses and ratios

Patient Cluster center Radius of kernel set Its ratio KerðDi2Þ j j= Dj i2j Patient Cluster center Radius of kernel set Its ratio KerðSi2Þ j j= Sj i2j D12 623.734 0.734 0.348 S12 750.546 0.546 0.078 D22 976.018 1.018 0.118 S22 850.132 0.868 0.006 D32 883.592 0.592 0.088 S32 561.882 0.882 0.066 D42 651.442 0.558 0.046 S42 667.570 0.570 0.060

Average of KerðDj i2Þj= Dj i2j _¼¼0:15 Average of KerðSj i2Þj= Sj i2j _¼¼0:05

this paper is that it provides a new method to cluster and identify time series. In this paper, we presented a proce-dure that can effectively cluster nonlinear time series into several patterns based on kernel set. The proposed algorithm also combines with the concept of a fuzzy set. We have demonstrated how to find a kernel set to help to cluster nonlinear time series into several patterns.

Our algorithm is highly recommended practically for clustering nonlinear time series and is supported by the empirical results. A major advantage of this framework is that our procedure does not require any initial knowledge about the structure in the data and can take full advantage of much more detailed information for some ambiguity.

However, certain challenging problems still remain open, such as:

(1) Since hardly ever any disturbance or noise in the data set can be completely eliminated, therefore, for the case of interacting noise, the complexity of multivar-iate filtering problems still remains to be solved. (2) The convergence of the algorithm for clustering and

the proposed statistics have not been well proved, although the algorithms and the proposed statistics are known as fuzzy criteria. This needs further investigation.

Although there remain many problems to be overcome, we think fuzzy statistical methods will be a worthwhile approach and will stimulate more future empirical work in time series analysis.

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