- Email: [email protected]
On a class of fuzzy c-numbers clustering procedures for fuzzy data
sets and systems
ELSEVIER
Fuzzy Sets and Systems 84 (I 996) 49-60
On a class of fuzzy c-numbers clustering procedures for fuzzy data Miin-Shen Drpurtmmt
of Muthrmatics,
Yang *, Cheng-Hsiu
Chwq- Yuun Clwistiun Uniwrsity,
Ko
Chuny-Li.
Tuiwun 32023, ROC
Received February 1994; revised August 1995
Abstract This paper describes a class of fuzzy clustering procedures for fuzzy data. Most fuzzy clustering techniques are designed for handling crisp data with their class memberships using the idea of fuzzy set theory. Here we derive new types of fuzzy clustering procedures in dealing with fuzzy data. These procedures are called fuzzy c-numbers (FCN) clusterings. Specially, we construct these FCNs for U-type, triangular, trapezoidal and normal fuzzy numbers. K~JWOY~S; Cluster analysis; Fuzzy clustering; clustering; Fuzzy c-numbers clustering
Fuzzy data analysis; Fuzzy numbers;
Fuzzy intervals;
Fuzzy c-means
1. Introduction Bellman
et al. [2] and Ruspini
[ 131 opened
the door of research
on fuzzy
clustering
since
Zadeh
[ 191 gave
of works on fuzzy clustering. But most fuzzy clustering techniques are designed for handling exactly numerical (crisp) data with their class memberships using the idea of fuzzy set theory, see a survey of Yang [ 171. How is it about the fuzzy clustering of fuzzy data? There is less work on this topic. In this paper we propose a class of fuzzy objective function based clustering procedures in dealing with fuzzy data. In the literature of fuzzy clustering, fuzzy c-means (KM) clustering procedures, proposed by Dunn [ 1 l] and extended by Bezdek [4], are most used and discussed. See, for example, [6-8, 161, etc. Based on the similar idea of FCMs construction, we give new types of fuzzy clustering procedures for handling fuzzy data. These are called fuzzy c-numbers (FCN) clustering procedures. These FCNs could well be used for clustering fuzzy data. Specially, we shall construct these FCNs for handling LR-type, triangular, trapezoidal and normal fuzzy numbers. Fuzzy numbers are well used to model the fuzziness of data and usually used to represent fuzzy data, see [14]. Section 2 describes a class of FCN clustering procedures on the space of all normalized fuzzy numbers. In Section 3, we construct a class of FCN clusterings in dealing with LR-type fuzzy numbers and intervals. Then we construct these FCNs for triangular and normal fuzzy numbers and also for trapezoidal fuzzy intervals. Finally, we present some numerical examples and make conclusions in Section 4. the original
work
in fuzzy
sets.
After
that
there
are branch
* Corresponding author. 01650114/96/$15.00 Copyright @ 1996 Elsevier Science B.V. All rights reserved PII SOl65-01
14(95)0030X-8
2. A class of FCN
clustering procedures
Fuzzy data is quite a natural type of data, like non-precise data or data with source of uncertainty not caused by randomness. This kind of data is easy to be found in natural language, social science, psychometrics, environmetrics and econometrics, etc. Fuzzy numbers have been used to represent fuzzy data. These are also used to mode1 fuzziness of data; [ 10, 141. Let [w be a one-dimensional Euclidean space with its norm denoted by 11 11.A fuzzy number is an upper semicontinuous convex function X : [w + [0, I] with {.y ??iw IX(x) = I} non-empty. Let .po([w) be the set of all fuzzy numbers X such that the x-cut XX = {X t rWlX(x) > x} is non-empty and compact for each 0 < x d 1. For any X and Y in c9,(iw), define a metric df with d,(X, Y) = SUP,,~, dH(X,, Y,), where dH is the Hausdorff metric with
Puri and Ralescu [ 121 defined d, and claimed that the space (.po( 1w),d, ) is a complete metric space. Let 9 be a subset of .9+,(R) with its metric d,. Let c be a positive integer greater than one. A partition of 9 into c parts can be represented by mutually disjoint sets ??I,. . Cic such that Yr U U Yc = 9 or equivalently by the indicator functions ~1,. ,,uu,,such that p;(X) = l if X E 3 and p,(X) = 0 if X $ % for all X in 9 and for all i = 1,. ..c. We call it as crisp clustering 9 into c clusters by p = (~1,. ,pc). Here p = (pi,. , pc) is called a hard c-partition of the fuzzy data set 9. Now consider an extension to allow l;(X) to be membership functions assuming values in the interval [0, I] such that CFl, pi(X) = 1 for all X in 9. This idea of extension was introduced by Ruspini [ 131. In this case, p = (~1,. . . , pc) is called a fuzzy c-partition of the fuzzy data set 9. Let 9 = {Xi,. . .,X,,} be a set of n fuzzy numbers in 9,,( rW>. We are interested in clustering 9 into c clusters. Based on the similar idea of construction of fuzzy c-means (FCM) clustering procedures, we propose the following objective function:
where m 3 I is called the index of fuzziness and W = (WI,. , W,.) in &o( [w) are called fuzzy c-numbers. We consider the optimization problem of F,&, W) by choosing a fuzzy c-partition p* and a fuzzy c-number W* to minimize the objective function F,(p, W). Then we call this optimization problem a class of fuzzy c-numbers (FCN) clustering procedures. These FCNs are new types of fuzzy clustering procedures designed specially for handling fuzzy data.
3. FCN
clusterings
for LR-type fuzzy numbers and intervals
In Section 2 we gave a class of FCN clustering procedures for the complete metric space (90( [w), 4, ) where pa(rW) is the set of all fuzzy numbers with compact x-cuts. Usually, LR-type fuzzy numbers are most convenient and useful in describing fuzzy data (see [14]). Therefore, we shall focus on the set of LR-type fuzzy numbers. We derive a class of FCNs clustering procedures for these LR-type fuzzy numbers in this section. First, we give its definition as follows. Definition 1 (Zimmermann [20, pp. 62-631). Let L (and R) be decreasing, shape functions from [w+ to [0, l] with L(0) = 1; L(x) < 1 for all x > 0; L(x) > 0 for all x < 1; L( 1) = 0 or (L(x) > 0 for all x and
M.-S. Yang, C.-H. KolFtcy
L(+ca)
= 0). Then a fuzzy number m-x
1 L-
X(x)
(
=
(
51
if for m,a > 0, p > 0 in R,
X is called of LR-type
for x 6 m,
x
)
x - m
R-
Sets and Systems 84 (1996) 49-60
P
for x 3 m. )
where m is called the mean value of X and r and p are called Symbolically, X is denoted by (m, r, IJ)LR.
the left and right spreads,
respectively.
Now let ~LR(K?) denote the set of all LR-type fuzzy numbers, In order to consider a class of FCNs on ILK, we define a new type of distance dLR for any X and Y in ?LR([W) with X = (m,, cz,, /&)LR and Y = (m,., xv, flv),_~ as follows: Y) = (m, - m,)’ + ((mx ~ Ir,) - (nz,. - lx,.))’ + ((m, + r/A) - (m, + ~[j~))~,
&(X where
I
1 I=
L-‘(to)dw
and
I’
s0
.o
Theorem 1. (F&R), Proof. (2) (3) (4)
RR’(cu)dto.
r =
dl,R) is u metric spuce.
(1) VX E .FLR, dLR(X,X) = 0. V distinct X, Y E 9LR, d&X, Y) > 0. (Symmetry) VX, Y E $LR, dLR(X, Y) = d&Y, X). (Triangular inequality) VX Y,Z E .FLR with X = (m,,z,,/j,),,u.
Y = (m,., zy,BJ,)LR, Z = (m,, r_, /I’:)~~.
diR(X, Y) = (m, - mJ.)2 + ((m, - la,) - (m,, - Ix,.))’ + ((m, + r/G) - (rn? + rBJ.>)’ = ((m, - m,) + (m, - m,.))’ + ((m, - Ix,,) - (m; - 1%:) + (m, - /cc_) - (m,. - /xv))’ +((mx + r/h) - (m; + r/C) + (m, + &) =
2
d~,dXZ~ +2{(m,
+
2 d,,(Z
- (m,. + rflj,))2
Y)
mz)(mZ- rn)) + [(m, - Ix,) - (m, - /2=)][(m, - Ix,) - (m,. - Ir,. )]
-
+ Km, + &)
- (m, +
$,)I[(m, + ~/A)- (mJ + rB,,)l}
d &R(X Z) + d&(2, Y) + 2d&X,Z)dLR(Z, = ([email protected]) Thus dAX,
Y) d du(X,Z)
Y)
(by Cauchy-Schwarz
inequality)
+ dLR(Z, Y))‘. + dLR(Z, Y). By (l)-(4),
d,,(X,
Y) is a metric.
0
Theorem 2. (cF&lR), dLR) is complrtr. Proof.
Let (X, : n > l} be a Cauchy
sequence
in $,JR([W) where
XI = (m,, , cc,,, /Lx,1,. . ,X, = (my,, x,~, & 1,. . . ,Jfj = Cm,!,xl!, /?I, 1,. . Then (m, - nzrij< d&%X,)-‘0 m, as n ----fcx). Similarly,
as i,.j *cc.
That is, {m.,,},X_, is a Cauchy sequence
in [w. Then nz,, -
52
M.-S.
Yang, C.-H. KolFuzzy
Sets und Systems 84 (1996)
49-60
That is, {m, - 1xX!,},“=, is a Cauchy sequence in [w. But {m,,}~, is a Cauchy sequence in [w, and 1 is constant. Thus, {xX{,},“=, is also a Cauchy sequence in R and xX,,+ x, as n + XI. Similarly, we have that A,, -pX asn+nc:
-0
as n + cc.
Thus, X, --f X as n --) ca. This completes
1,.
the proof.
0
Let 9LR = {XI,. . .,X,} be a set of 12 LR-type fuzzy numbers in sir with X, = (in,,, oc,,, /I&, j = , n. We want to cluster 6% into c classes. Consider the following FCNs objective functions:
where m 3 1 and p = (PI,. . . ,&) are fuzzy c-partitions and W, = (m),,,, cc>,;,fi,l,,)LR, i = 1,. , c, are fuzzy cnumbers of LR-type in ~LR( R). Then the minimization problem of H&y, W) with respect to fuzzy c-partitions ,U and fuzzy c-numbers of LR-type WI,. . , W,, in p,& R) becomes a class of FCNs for LR-type fuzzy numbers. Consider the Lagrangian L(p, W, i) with L(& W, i) = 2
e /iy(<,)d2,(Xj,
j=lizl
5 /Ai
Wi) - 2 (
- 1
i=l
)
If we have the first derivatives of L(,u, W, i) with respect to all parameters necessary conditions for a minimizer (8, I?) of H,,(,u, W) as follows: i =
equal to zero, then we can get the
l,...,c,
(1.1)
(1.2)
(1.3)
i=
/Yi,(x,) =
l,...,
c;
j=
l,...,
n,
(1.4)
where
d2RCXj, &I = Cm,,- ha,Y + (Cm,,- la, >- (kM;- ld,,,>I2+ (Cm,,+ 4, >- (fiw,+ r&, >j2 Based on these necessary
conditions,
we give the FCN clustering
algorithms
for LR-type fuzzy numbers.
Algorithm (FCN for LR-type jiizzy numbers). (Sl) Fix m > 1; fix c E {2,3 ,..., n - l}; and fix any E > 0. Choose an initial fuzzy c-partition p(O) and choose initial left and right spreads.
kzzy
:SMOIlOJ se pauyap s! (ti)N_& u! N(.‘l) ‘eizu) = x pue N( X~‘Qu) = x SlaquInu aw2wp ayl uayL .slaqi.unu lczzn3 ICWOU 11~30 las aql aq (H)N& la7
leuuou ohzl Icur?10~ (A ‘X)NP
‘30 > x>
Azzn3 ad&x7
02-
103
(A-)dxa=(x)X
‘a.1 ‘N( D ‘w) = x Lq palouap ‘laqurnu Lzzy j~uuou r2 palIt s! x uayl ‘x laqurnu e 103 (&II/( zu 1 x))-)dxa = (x)x = (x)7 31 .slaqurnu Lzzn3 ~eurrou aql uo stvo3 sn Ial MON
‘$P ql!M “ip 30 $uatuaDeIdal aql pue + = 1 = J ~I!M (p.1 )-( 1’1) .sbg ale (4 ‘rf)w~ 30 (A ‘71) laz!uI!u!ur e 103 suog!puo3 hssa~au aql uaqL ‘(~o>Ig u! slaqtunu-2 Iczzy ~e@ue!~~ a.n2 -L(it’d’% c%.u) = ‘4 pw ‘s.xaqurnu kzn3 .n+%vg 30 las wp B s! {“x‘.. . ‘lx} fsuop!ved-3 Iczzn3 ale (W ‘+.. ‘171) = ri : 1 Q zu alaqM
:s~oj103 se (.Lp‘(ti).L&) a3eds 3!.Ilaru alalduro3 aql .103 (,#j ‘11)“~ suo!lXuI3 aAgc?afqo ,ys+~ aql a,wq puoLue!a Lq pasodold SBM q3!qM aX&?ls!p aql 01 .n?p~!s s! alaq (A ‘,!@p leql uoguaw aM
aM MON ‘[6]
‘,((Q/ ,((.‘d$
-
I&@ + (,izLi-
+ oh) -
(“cl:
“w)) + ,(( ,C;C_ xn)f _ (.‘u
+ %u)) + ,(( i;cf -
,“ul) -
_ “1~)) + Z(,im _ xw) =
(“Xl: -
“U)) + Z(~iuI-
Qu) = (A ‘x)$7
:SMOllO3 Se (&L& UO (A ‘J!#p a3uels!p aqi aheq aM aqi 01 2kup.10~~~ waqwnu lczzy .w@ue!q 11” 30 uogDaIlo3 aq$ sluasaldal J( “‘d ‘*‘P‘,‘u) = x put? .I( “c/ “;c ‘Q) = x s.Iaqwnu dzzn3 .u?@ucg OMJ .tap!suoD
‘aloJaq pauyap (A ‘39x727 a~r?wp
qXqM (~)-l,”
a3eds aql30
‘(0 < d)ur < X .I03 ‘(o
~ d
-
U-X
aql30
=
(X)X
I
.nqnE!ue~~le pallet
‘0
s! x
uaql
= (X)J
‘X-I
aJr! )I pue 7 31 ‘g7( d ‘M‘~1) = x laqurnu
.pasn @IO~ILLIO~ lsour a.w slaqtunu
I
t-1 I - lu
Lzzy
‘laqwnu
‘as!Mlaqio ‘[3X>() ~03 aM ‘IxaN
1
Lzzn3 ad&-x7
E .IOJ
waqwnu Lzzy 30 sad& leTDads 0~1 asaql 103 SNS~ aA@ Ip2qs lczzn3 ~XKIOU pue .uqt-@ugl aql ‘slaqurnu Iczzy ad&x7 UI
M.-S.
54
objective
functions
49-60
are
Thus, the necessary and the replacement If the m in X of fuzzy number but a
conditions for a minimizer @, fi) of N,(p, W) are Eqs. ( 1.1 )-( 1.4) with Y = I = fi/2 of d& with d;. Definition 1 is not a real number but an interval [ml, ml] then the fuzzy set X is not a fuzzy interval. Accordingly, a LR-type fuzzy interval can be defined as follows.
Definition 2 (Zimmermann interval
Yang, C.-H. Kol Fuzzy Sets and Systems 84 (1996)
[20, p. 641). X = ( ml. m2, CI,fl)LR with u > 0, fl > 0 is called a LR-type fuzzy
if
I
L-
X(x)
ml --x
( 1
for ml 6 x 6 m2,
1
=
for x d ml,
x
[R(‘y)
forx2mz.
Let S,&(R) be the set of all LR-type fuzzy intervals. For any X T~,~,)LR in 9LR([W), we define a metric df(X, Y) as follows: d?(X, Y) = (ml, - mtJ)’ + (mzY - mz,)2 + ((ml, - ICC,)- (ml_,
tm1x,m2x,t%,bx)LR
&y))2
and
Y
=
(ml,,m2,.,
+((m2~+r~~)-(m2~+rBv))",
where I
I=
.I’
L-‘(o)dco
I
and
J’
R-‘(u)
dw.
0
0
Similar to Theorems
r =
1 and 2 (3LR(F8),dl)
is a complete
Let 9& = {XI,. . . , Xn} be a set of n LR-type j = 1,. . . , n. Consider the objective function
metric space.
fuzzy intervals
in .YL,( [w) with Xj = (ml,,, mix,, cc,, , & )LR,
where m 3 1 and ,n = (tit,..., pc) are fuzzy c-partitions and Wi = (ml,, mzrv,, CL,,,/L,)LR are LR-type c-intervals. The necessary conditions for a minimizer (/I, @) of Z,(,U, W) are the following: ~ ,,,,/ =
~~=,fi~t~)t2m~x, + 4&, - zxcx, 1) i = 2C~=,ci?(xj)
CT,,=
C;=,FYY~j)(hu~, - (ml*, I c&R4>
fuzzy
l,...,c,
’
Zch,))
’
i=l
3.1.) c,
(2.3)
M.-S. Yung, C.-H. KolFu::y
55
Sets and Sysfems 84 (1996) 49-60
(2.4)
(2.5) Based on these necessary conditions, intervals. We note that the trapezoidal LR-type fuzzy intervals. Next, we shall Let X = (ml,m2, x,IJ)LR be a LR-type
we can construct these FCN clustering algorithms for LR-type fuzzy type of fuzzy intervals is the simplest and most used one among consider FCN clusterings for this special type of fuzzy intervals. fuzzy interval. If L and R are of the form
O,
l-x,
otherwise, then X is called a trapezoidal ml --x
1-p X(x)
=
We denote it by X = (ml, m2, x, [j)r,,
for x 6 ml,
CI
for ml d .Y 6 m2,
1 I
fuzzy interval.
x - m2 ~
‘-
forx3mz.
fi
Based on the distance dl defined on -9LR([w) before, we define a distance dn(X, Y) for any two trapezoidal fuzzy intervals X = (ml,,m2,, x,,/&)n and Y = (ml ,,,rnzJ, r,.,by)n in the space yr,(rW) of all trapezoidal fuzzy intervals as follows: d&(X Y) =(ml, +
- ml,.)’ + (mix - m2.V)2+ ((ml, - lx\-) ~ (ml,. - ;x~))’
((m2.\-
+
tDx)
The FCN objective functions intervals in .Fr,(&?) are
-
(m2,.
+
i8,,)12.
&(!A, W) for clustering
Therefore, the necessary conditions for a minimizer and the replacement of df with d$.
4. Numerical
examples
the data set $1
(,& I$) of &(p,
= {XI,.
,X,,} of trapezoidal
W) are Eqs. (2.1))(2.5)
fuzzy
with I’= 1 = i
and conclusions
In Section 3, we have constructed FCN clustering algorithms for LR-type fuzzy numbers and intervals, specially for triangular, normal and trapezoidal types. In this section, we shall give some numerical examples. We numerically pick three data sets: (1) 30 triangular fuzzy numbers (see Table 1 and Fig. 1); (2) 30 normal fuzzy numbers (see Table 2 and Fig. 2); and (3) 30 trapezoidal fuzzy intervals (see Table 3 and Fig. 3). Then we run FCN clustering algorithms for these three data sets, with m = 2, c = 3, and E = 0.00 1. Fuzzy c-partitions c = (PI, G2, fiJ) and FCN centers I? = (et, $2, @j) are shown in Tables I-3 and Figs. 4-6. The results of numerical examples seem to be satisfactory. Note that we have preassumed the number L’of clusters for all these numerical examples. The method that finds optimal c. called cluster validity, is very important.
M.-S.
Yang, C.-H. Kol Fuzy
Table I Data of 30 triangular Triangular
Sets and Systems
fuzzy numbers
fuzzy numbers
Memberships P = (i,>&.&)
(3.34 (9.56 (10.56 (10.89 (13.89 (14.78 (14.90 (15.67 (16.87 (17.45 (19.78 (20.67 (21.45 (22.34 (23.47 (24.67 (25.78 (26.45 (28.34 (32.29 (32.77 (34.88 (35.45 (35.88 (38.88 (40.25 (40.47 (43.56 (43.98 (45.77
(0.053824 (0.010266 (0.005756 (0.003872 (0.001838 (0.007380 (0.006121 (0.011653 (0.024142 (0.029484 (0.034567 (0.027751 (0.018319 (0.007625 (0.001816 (0.002319 (0.018131 (0.027172 (0.099257 (0.5255 I9 (0.582464 (0.811237 (0.862406 (0.877326 (0.998021 (0.995373 (0.997603 (0.942318 (0.932188 (0.894959
1.30) I .OO) 1.93) 1.17) 0.88) 1.21) 0.41) 0.90) 1.85) 1.95) 0.42) 1.10) 1.60) 1.58) 0.51) 1.09) 1.51) 0.92) 0.12) 1.64) 0.47) 0.66) 1.26) 0.16) 0.64) I .7l ) 0.15) 0.63) 1.69) 0.79)
49-60
and their fuzzy 3-partitions
x = (w a, B)r 1.46 0.27 1.95 0.56 0.89 0.12 I.19 1.82 1.90 1.79 1.47 1.34 0.92 0.04 0.81 0.14 0.39 I.61 1.95 1.66 0.63 1.08 1.48 1.79 0.66 0.52 1.95 0.92 1.74 1.71
84 (1996)
Fig. 1. The graph of 30 triangular
0.162549 0.04371 I 0.026166 0.018228 0.011500 0.052909 0.042768 0.090467 0.234564 0.323684 0.672476 0.812497 0.905601 0.969866 0.994055 0.994262 0.962896 0.947361 0.845 105 0.403530 0.351715 0.152717 0.1101 I5 0.097848 0.001499 0.003430 0.001784 0.041239 0.048243 0.073501
0.783627) 0.946023) 0.968078) 0.977900) 0.986661) 0.939712) 0.951 I I I) 0.897879) 0.741294) 0.646832) 0.292957) 0.159753) 0.076079) 0.022509) 0.004128) 0.003419) 0.018973) 0.025467) 0.055638) 0.07095 1) 0.065820) 0.036046) 0.027479) 0.024826) 0.000480) 0.001197) 0.000613) 0.016443) 0.019569) 0.031540)
fuzzy numbers
M.-S.
Yang, C.-H. KolFuzzy
Sets and Systems
Table 2 Data of 30 normal fuzzy numbers Normal fuzzy numbers
Memberships ,i = (li,.&.&)
(3.34 (9.56 (10.56 (10.89 (13.89 (14.78 (14.90 (15.67 (16.87 (17.45 (19.78 (20.67 (21.45 (22.34 (23.47 (24.67 (25.78 (26.45 (28.34 (32.29 (32.77 (34.88 (35.45 (35.88 (38.88 (40.25 (40.47 (43.56 (43.98 (45.77
(0.052776 (0.010453 (0.005433 (0.004165 (0.0021 I I (0.006725 (0.007730 (0.013574 (0.024704 (0.029617 (0.032598 (0.025541 (0.018094 (0.008668 (0.001229 (0.002806 (0.017180 (0.034095 (0.126292 (0.532052 (0.591466 (0.819127 (0.865102 (0.896609 (0.997832 (0.996388 (0.994773 (0.941722 (0.933137 (0.892500
1.03) 1.58) 0.63) 1.79) 1.21) 0.65) 1.17) 1.60) 1.71) 0.94) I .27) 0.59) 1.85)
1.26) 1.71) 0.56) 0.92) I .4l ) 0.79) 0.90) I .90) 0.96) I .95)
0.161975 0.045076 0.025265 0.019887 0.013612 0.048591 0.056698 0.111784 0.250473 0.337137 0.715614 0.832123 0.905182 0.963927 0.996056 0.992900 0.963980 0.935302 0.809626 0.396175 0.342551 0.145569 0.107457 0.08 I767 0.001636 0.002676 0.003862 0.041523 0.047433 0.074909
0.785249) 0.944471) 0.969303) 0.975948) 0.984277) 0.944685) 0.935572) 0.874642) 0.724824) 0.633246) 0.251788) 0.142336) 0.076724) 0.027405) 0.002715) 0.004293) 0.018840) 0.030603) 0.064082) 0.071773) 0.065983) 0.035304) 0.027441) 0.021624) 0.000531) 0.000936) 0.001365) 0.016755) 0.019430) 0.032591)
h
0.8
49-60
and their fuzzy 3-partitions
x = (WU)h 1.46) I .09) 0.82) 1.82) 1.19) 1.48) 0.64)
84 (1996)
Ii
0.6
Fig. 2. The graph of 30 normal fuzzy numbers
57
M.-S
Yang. C.-H. Ku I Fuzzy Srts and Systems
84 11996) 49-60
Table 3 Data of 30 trapezoidal fuzzy intervals and theirfuzzy 3-partitions Trapezoidalfuzzy intervals
Memberships
x = (ml,mz,KP)n
/i =
(/i,./T2./ij)
(3.34 5.34 1.46 1.30)
(0.054128 0.164056 0.781816)
(9.56 II.36 0.27 1.00)
(0.011139 0.047289 0.941572)
(10.56 13.79 1.95 1.93)
(0.004207 0.020202 0.975591)
(10.89 13.24 0.56 1.17)
(0.003397 0.016318 0.980285)
(13.89 15.25 0.89 0.88)
(0.001020 0.006212 0.992768)
(14.78 16.34 0.12 1.21)
(0.005757 0.040316 0.953927)
(14.90 16.89 I.19 0.41)
(0.006329 0.044972 0.948698)
(15.67 17.02 1.82 0.90)
(0.009451 0.071329 0.919219)
(16.87 17.54 1.90 1.85)
(0.018075 0.158532 0.823393)
(17.45 18.14 1.79 1.95)
(0.023898 0.234053 0.742049)
(19.78 22.38 1.47 0.42)
(0.032126 0.733173 0.234700)
(20.67 23.57 1.34 1.10)
(0.022702 0.868264 0.109033)
(21.45 23.67 0.92 1.60)
(0.016320 0.918735 0.064945)
(22.34 24.57 0.04 1.58)
(0.006287 0.975737 0.017976)
(23.47 25.47 0.81 0.51)
(0.001246 0.995979 0.002775)
(24.67 25.25 0.14 1.09)
(0.002726 0.992315 0.004959)
(25.78 27.88 0.39 1.51)
(0.020890 0.957697 0.021413)
(26.45 28.34 1.61 0.92)
(0.029728 0.942755 0.027517)
(28.34 30.56 1.95 0.12)
(0.118667 0.820002 0.061331)
(32.29 35.01 1.66 1.64)
(0.585195 0.348492 0.066313)
(32.77 34.67 0.63 0.47)
(0.592451 0.342142 0.065407)
(34.88 36.89 1.08 0.66)
(0.824787 0.141108 0.034105)
(35.45 37.87 1.48 1.26)
(0.887354 0.089429 0.023217)
(35.88 37.89 1.79 0.16)
(0.891492 0.086082 0.022426)
(38.88 40.56 0.66 0.64)
(0.997580 0.001830 0.000590)
(40.25 41.78 0.52 1.71)
(0.996313 0.002733 0.000954)
(40.47 42.35 1.95 0.15)
(0.996267 0.002767 0.000966)
(43.56 45.79 0.92 0.63)
(0.936249 0.045360 0.018391)
(43.98 45.67 1.74 1.69)
(0.932316 0.048068 0.019617)
(45.77 47.56 1.71 0.79)
(0.892947 0.074712 0.032341)
0.6:
0.4:
0.2:
t i Fig. 3. The graph of 30 trapezoidal fuzzy intervals
M.-S.
Yang, C.-H. KolFuzzy
[12.773
1.115 1.185] [24.087
Sets and Systems 84 (1996)
0.914
1.0501
[39.521
49-60
1.283 0.870]
1.0
0.8
0.6
0.4 0.2 _-L
I--_II
10
20
Fig. 4. FCN centers for 30 triangular
[12.705
1.2161
[23.964
fuzzy numbers.
1.2291
[39.472
I.2411
1.0
,
0.8
i
0.4 0.2:
10
20
40
30
50
Fig. 5. FCN centers for 30 normal fuzzy numbers.
[12.890 1.0
14671
I. 137 1.2101
123.972
5.9186 0.905 1
1.040]
[39.401
41.332
1287
0.8751
'1
0.6 1 0.4
0.2
1
--
I 30
Fig. 6. FCN centers for 30 trapezoidal
40
fuzzy intervals.
50
60
M.-S.
Yang, C.-H. Ko I Fuzzy Sets and Systems 84 (1996)
49-60
The diverse validity criteria have been investigated in literatures (see [ 171). These criteria could be also used in these FCN clustering procedures. We have proposed a class of FCN clustering procedures in this paper. Most fuzzy clustering procedures are designed for handling crisp data. Based on these new procedures, we can cluster fuzzy data. Three numerical examples have been given. In fact, these FCN can be applied in diverse areas (see, for example, [ 181). Finally, let us look at asymptotic behaviors of these FCN clustering procedures. By following Bezdek et al. [3, 51 and Yang [15], we can create numerical convergence theorems of FCNs which are similar to these of FCMs. These theorems give us that all the limit points of sequences generated by FCN clustering algorithms are fixed points and that these fixed points are stationary points of the FCN objective function H,,,(p, W). These numerical convergence theorems shall be the important theoretical views on the FCN clustering algorithms.
Acknowledgements We are grateful to the referees for their suggestions and comments. We gratefully acknowledge that this research was supported in part by the National Science Council of the Republic of China under Grant NSC 83-0208-M-033-014.
Reference [I] M. Albrecht, Approximation of functional relationships to fuzzy observations, Fuzzy Sets and Systems 49 (1992) 301-305. [2] R. Bellman, R. Kalaba and L.A. Zadeh, Abstraction and pattern classification, J. Math. Anal. Appl. 13 (1966) l-7. [3] J.C. Bezdek, A convergence theorem for the fuzzy ISODATA clustering algorithms, IEEE Trans. Pattern Anal. Machine Intelligence 2 (1980) l-8. [4] J.C. Bezdek, Pattern Recognition with Fuzzy Objectiae Function Algorithms (Plenum Press, New York, 1981). [5] J.C. Bezdek, R.J. Hathaway, M.J. Sabin and W.T. Tncker, Convergence theory for fuzzy c-means: counterexamples and repairs, IEEE Trans. System Man Cybernet. 17 (1987) 873-877. [6] R. Cannon, J. Dave and J.C. Bezdek, Efficient implementation of the fuzzy c-means clustering algorithms, IEEE Trans. Pattern Anal. Machine Intelligence 8 (1986) 248-255. [7] K.P. Chan and Y.S. Cheung, Clustering of cluster, Pattern Recognition 25 (1992) 21 I-217. [8] J.W. Davenport, J.C. Bezdek and R.J. Hathaway, Parameter estimation for finite mixture distributions. Comput. Math. Appl. 15 (1988) 819-828. [9] P. Diamond, Fuzzy least squares, Inform. Sci. 46 (1988) 141-l 57. [IO] D. Dubois and H. Prade, Fusq~ Sets and Systems: Theory and Applications (Academic Press, New York, 1980). [I I] J.C. Dunn, A fuzzy relative of the ISODATA process and its use in detecting compact, well-separated clusters, J. Cybernet. 3 (1974) 32-57. [I21 M.L. Puri and D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 4099422. [I31 E. Ruspini, A new approach to clustering, Inform. Control 15 (1969) 22-32. [I41 S.F. Schnatter. On statistical inference for fuzzy data with applications to descriptive statistics, Fuzzy Sets and Systems 50 (1992) 143-165. [I51 M.S. Yang, Convergence properties of the generalized fuzzy c-means clustering algorithms, Compur. Math. Appl. 25 (1993) 3-l I. [I61 M.S. Yang, On a class of fuzzy classification maximum likelihood procedures, Fu-_z_vSets and Systems 57 (1993) 365-375. [I71 M.S. Yang, A survey of fuzzy clustering, Math. Comput. Modelling 18 (1993) l-16. [I 81 MS. Yang and C.H. Ko, On cluster-wise fuzzy regression analysis, submitted. [I91 L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338-353. [20] H.J. Zimmermann, Fuzzy Set Theory and ifs Applications (Kluwer, Dordrecht, 1991).
ELSEVIER
Fuzzy Sets and Systems 84 (I 996) 49-60
On a class of fuzzy c-numbers clustering procedures for fuzzy data Miin-Shen Drpurtmmt
of Muthrmatics,
Yang *, Cheng-Hsiu
Chwq- Yuun Clwistiun Uniwrsity,
Ko
Chuny-Li.
Tuiwun 32023, ROC
Received February 1994; revised August 1995
Abstract This paper describes a class of fuzzy clustering procedures for fuzzy data. Most fuzzy clustering techniques are designed for handling crisp data with their class memberships using the idea of fuzzy set theory. Here we derive new types of fuzzy clustering procedures in dealing with fuzzy data. These procedures are called fuzzy c-numbers (FCN) clusterings. Specially, we construct these FCNs for U-type, triangular, trapezoidal and normal fuzzy numbers. K~JWOY~S; Cluster analysis; Fuzzy clustering; clustering; Fuzzy c-numbers clustering
Fuzzy data analysis; Fuzzy numbers;
Fuzzy intervals;
Fuzzy c-means
1. Introduction Bellman
et al. [2] and Ruspini
[ 131 opened
the door of research
on fuzzy
clustering
since
Zadeh
[ 191 gave
of works on fuzzy clustering. But most fuzzy clustering techniques are designed for handling exactly numerical (crisp) data with their class memberships using the idea of fuzzy set theory, see a survey of Yang [ 171. How is it about the fuzzy clustering of fuzzy data? There is less work on this topic. In this paper we propose a class of fuzzy objective function based clustering procedures in dealing with fuzzy data. In the literature of fuzzy clustering, fuzzy c-means (KM) clustering procedures, proposed by Dunn [ 1 l] and extended by Bezdek [4], are most used and discussed. See, for example, [6-8, 161, etc. Based on the similar idea of FCMs construction, we give new types of fuzzy clustering procedures for handling fuzzy data. These are called fuzzy c-numbers (FCN) clustering procedures. These FCNs could well be used for clustering fuzzy data. Specially, we shall construct these FCNs for handling LR-type, triangular, trapezoidal and normal fuzzy numbers. Fuzzy numbers are well used to model the fuzziness of data and usually used to represent fuzzy data, see [14]. Section 2 describes a class of FCN clustering procedures on the space of all normalized fuzzy numbers. In Section 3, we construct a class of FCN clusterings in dealing with LR-type fuzzy numbers and intervals. Then we construct these FCNs for triangular and normal fuzzy numbers and also for trapezoidal fuzzy intervals. Finally, we present some numerical examples and make conclusions in Section 4. the original
work
in fuzzy
sets.
After
that
there
are branch
* Corresponding author. 01650114/96/$15.00 Copyright @ 1996 Elsevier Science B.V. All rights reserved PII SOl65-01
14(95)0030X-8
2. A class of FCN
clustering procedures
Fuzzy data is quite a natural type of data, like non-precise data or data with source of uncertainty not caused by randomness. This kind of data is easy to be found in natural language, social science, psychometrics, environmetrics and econometrics, etc. Fuzzy numbers have been used to represent fuzzy data. These are also used to mode1 fuzziness of data; [ 10, 141. Let [w be a one-dimensional Euclidean space with its norm denoted by 11 11.A fuzzy number is an upper semicontinuous convex function X : [w + [0, I] with {.y ??iw IX(x) = I} non-empty. Let .po([w) be the set of all fuzzy numbers X such that the x-cut XX = {X t rWlX(x) > x} is non-empty and compact for each 0 < x d 1. For any X and Y in c9,(iw), define a metric df with d,(X, Y) = SUP,,~, dH(X,, Y,), where dH is the Hausdorff metric with
Puri and Ralescu [ 121 defined d, and claimed that the space (.po( 1w),d, ) is a complete metric space. Let 9 be a subset of .9+,(R) with its metric d,. Let c be a positive integer greater than one. A partition of 9 into c parts can be represented by mutually disjoint sets ??I,. . Cic such that Yr U U Yc = 9 or equivalently by the indicator functions ~1,. ,,uu,,such that p;(X) = l if X E 3 and p,(X) = 0 if X $ % for all X in 9 and for all i = 1,. ..c. We call it as crisp clustering 9 into c clusters by p = (~1,. ,pc). Here p = (pi,. , pc) is called a hard c-partition of the fuzzy data set 9. Now consider an extension to allow l;(X) to be membership functions assuming values in the interval [0, I] such that CFl, pi(X) = 1 for all X in 9. This idea of extension was introduced by Ruspini [ 131. In this case, p = (~1,. . . , pc) is called a fuzzy c-partition of the fuzzy data set 9. Let 9 = {Xi,. . .,X,,} be a set of n fuzzy numbers in 9,,( rW>. We are interested in clustering 9 into c clusters. Based on the similar idea of construction of fuzzy c-means (FCM) clustering procedures, we propose the following objective function:
where m 3 I is called the index of fuzziness and W = (WI,. , W,.) in &o( [w) are called fuzzy c-numbers. We consider the optimization problem of F,&, W) by choosing a fuzzy c-partition p* and a fuzzy c-number W* to minimize the objective function F,(p, W). Then we call this optimization problem a class of fuzzy c-numbers (FCN) clustering procedures. These FCNs are new types of fuzzy clustering procedures designed specially for handling fuzzy data.
3. FCN
clusterings
for LR-type fuzzy numbers and intervals
In Section 2 we gave a class of FCN clustering procedures for the complete metric space (90( [w), 4, ) where pa(rW) is the set of all fuzzy numbers with compact x-cuts. Usually, LR-type fuzzy numbers are most convenient and useful in describing fuzzy data (see [14]). Therefore, we shall focus on the set of LR-type fuzzy numbers. We derive a class of FCNs clustering procedures for these LR-type fuzzy numbers in this section. First, we give its definition as follows. Definition 1 (Zimmermann [20, pp. 62-631). Let L (and R) be decreasing, shape functions from [w+ to [0, l] with L(0) = 1; L(x) < 1 for all x > 0; L(x) > 0 for all x < 1; L( 1) = 0 or (L(x) > 0 for all x and
M.-S. Yang, C.-H. KolFtcy
L(+ca)
= 0). Then a fuzzy number m-x
1 L-
X(x)
(
=
(
51
if for m,a > 0, p > 0 in R,
X is called of LR-type
for x 6 m,
x
)
x - m
R-
Sets and Systems 84 (1996) 49-60
P
for x 3 m. )
where m is called the mean value of X and r and p are called Symbolically, X is denoted by (m, r, IJ)LR.
the left and right spreads,
respectively.
Now let ~LR(K?) denote the set of all LR-type fuzzy numbers, In order to consider a class of FCNs on ILK, we define a new type of distance dLR for any X and Y in ?LR([W) with X = (m,, cz,, /&)LR and Y = (m,., xv, flv),_~ as follows: Y) = (m, - m,)’ + ((mx ~ Ir,) - (nz,. - lx,.))’ + ((m, + r/A) - (m, + ~[j~))~,
&(X where
I
1 I=
L-‘(to)dw
and
I’
s0
.o
Theorem 1. (F&R), Proof. (2) (3) (4)
RR’(cu)dto.
r =
dl,R) is u metric spuce.
(1) VX E .FLR, dLR(X,X) = 0. V distinct X, Y E 9LR, d&X, Y) > 0. (Symmetry) VX, Y E $LR, dLR(X, Y) = d&Y, X). (Triangular inequality) VX Y,Z E .FLR with X = (m,,z,,/j,),,u.
Y = (m,., zy,BJ,)LR, Z = (m,, r_, /I’:)~~.
diR(X, Y) = (m, - mJ.)2 + ((m, - la,) - (m,, - Ix,.))’ + ((m, + r/G) - (rn? + rBJ.>)’ = ((m, - m,) + (m, - m,.))’ + ((m, - Ix,,) - (m; - 1%:) + (m, - /cc_) - (m,. - /xv))’ +((mx + r/h) - (m; + r/C) + (m, + &) =
2
d~,dXZ~ +2{(m,
+
2 d,,(Z
- (m,. + rflj,))2
Y)
mz)(mZ- rn)) + [(m, - Ix,) - (m, - /2=)][(m, - Ix,) - (m,. - Ir,. )]
-
+ Km, + &)
- (m, +
$,)I[(m, + ~/A)- (mJ + rB,,)l}
d &R(X Z) + d&(2, Y) + 2d&X,Z)dLR(Z, = ([email protected]) Thus dAX,
Y) d du(X,Z)
Y)
(by Cauchy-Schwarz
inequality)
+ dLR(Z, Y))‘. + dLR(Z, Y). By (l)-(4),
d,,(X,
Y) is a metric.
0
Theorem 2. (cF&lR), dLR) is complrtr. Proof.
Let (X, : n > l} be a Cauchy
sequence
in $,JR([W) where
XI = (m,, , cc,,, /Lx,1,. . ,X, = (my,, x,~, & 1,. . . ,Jfj = Cm,!,xl!, /?I, 1,. . Then (m, - nzrij< d&%X,)-‘0 m, as n ----fcx). Similarly,
as i,.j *cc.
That is, {m.,,},X_, is a Cauchy sequence
in [w. Then nz,, -
52
M.-S.
Yang, C.-H. KolFuzzy
Sets und Systems 84 (1996)
49-60
That is, {m, - 1xX!,},“=, is a Cauchy sequence in [w. But {m,,}~, is a Cauchy sequence in [w, and 1 is constant. Thus, {xX{,},“=, is also a Cauchy sequence in R and xX,,+ x, as n + XI. Similarly, we have that A,, -pX asn+nc:
-0
as n + cc.
Thus, X, --f X as n --) ca. This completes
1,.
the proof.
0
Let 9LR = {XI,. . .,X,} be a set of 12 LR-type fuzzy numbers in sir with X, = (in,,, oc,,, /I&, j = , n. We want to cluster 6% into c classes. Consider the following FCNs objective functions:
where m 3 1 and p = (PI,. . . ,&) are fuzzy c-partitions and W, = (m),,,, cc>,;,fi,l,,)LR, i = 1,. , c, are fuzzy cnumbers of LR-type in ~LR( R). Then the minimization problem of H&y, W) with respect to fuzzy c-partitions ,U and fuzzy c-numbers of LR-type WI,. . , W,, in p,& R) becomes a class of FCNs for LR-type fuzzy numbers. Consider the Lagrangian L(p, W, i) with L(& W, i) = 2
e /iy(<,)d2,(Xj,
j=lizl
5 /Ai
Wi) - 2 (
- 1
i=l
)
If we have the first derivatives of L(,u, W, i) with respect to all parameters necessary conditions for a minimizer (8, I?) of H,,(,u, W) as follows: i =
equal to zero, then we can get the
l,...,c,
(1.1)
(1.2)
(1.3)
i=
/Yi,(x,) =
l,...,
c;
j=
l,...,
n,
(1.4)
where
d2RCXj, &I = Cm,,- ha,Y + (Cm,,- la, >- (kM;- ld,,,>I2+ (Cm,,+ 4, >- (fiw,+ r&, >j2 Based on these necessary
conditions,
we give the FCN clustering
algorithms
for LR-type fuzzy numbers.
Algorithm (FCN for LR-type jiizzy numbers). (Sl) Fix m > 1; fix c E {2,3 ,..., n - l}; and fix any E > 0. Choose an initial fuzzy c-partition p(O) and choose initial left and right spreads.
kzzy
:SMOIlOJ se pauyap s! (ti)N_& u! N(.‘l) ‘eizu) = x pue N( X~‘Qu) = x SlaquInu aw2wp ayl uayL .slaqi.unu lczzn3 ICWOU 11~30 las aql aq (H)N& la7
leuuou ohzl Icur?10~ (A ‘X)NP
‘30 > x>
Azzn3 ad&x7
02-
103
(A-)dxa=(x)X
‘a.1 ‘N( D ‘w) = x Lq palouap ‘laqurnu Lzzy j~uuou r2 palIt s! x uayl ‘x laqurnu e 103 (&II/( zu 1 x))-)dxa = (x)x = (x)7 31 .slaqurnu Lzzn3 ~eurrou aql uo stvo3 sn Ial MON
‘$P ql!M “ip 30 $uatuaDeIdal aql pue + = 1 = J ~I!M (p.1 )-( 1’1) .sbg ale (4 ‘rf)w~ 30 (A ‘71) laz!uI!u!ur e 103 suog!puo3 hssa~au aql uaqL ‘(~o>Ig u! slaqtunu-2 Iczzy ~e@ue!~~ a.n2 -L(it’d’% c%.u) = ‘4 pw ‘s.xaqurnu kzn3 .n+%vg 30 las wp B s! {“x‘.. . ‘lx} fsuop!ved-3 Iczzn3 ale (W ‘+.. ‘171) = ri : 1 Q zu alaqM
:s~oj103 se (.Lp‘(ti).L&) a3eds 3!.Ilaru alalduro3 aql .103 (,#j ‘11)“~ suo!lXuI3 aAgc?afqo ,ys+~ aql a,wq puoLue!a Lq pasodold SBM q3!qM aX&?ls!p aql 01 .n?p~!s s! alaq (A ‘,!@p leql uoguaw aM
aM MON ‘[6]
‘,((Q/ ,((.‘d$
-
I&@ + (,izLi-
+ oh) -
(“cl:
“w)) + ,(( ,C;C_ xn)f _ (.‘u
+ %u)) + ,(( i;cf -
,“ul) -
_ “1~)) + Z(,im _ xw) =
(“Xl: -
“U)) + Z(~iuI-
Qu) = (A ‘x)$7
:SMOllO3 Se (&L& UO (A ‘J!#p a3uels!p aqi aheq aM aqi 01 2kup.10~~~ waqwnu lczzy .w@ue!q 11” 30 uogDaIlo3 aq$ sluasaldal J( “‘d ‘*‘P‘,‘u) = x put? .I( “c/ “;c ‘Q) = x s.Iaqwnu dzzn3 .u?@ucg OMJ .tap!suoD
‘aloJaq pauyap (A ‘39x727 a~r?wp
qXqM (~)-l,”
a3eds aql30
‘(0 < d)ur < X .I03 ‘(o
~ d
-
U-X
aql30
=
(X)X
I
.nqnE!ue~~le pallet
‘0
s! x
uaql
= (X)J
‘X-I
aJr! )I pue 7 31 ‘g7( d ‘M‘~1) = x laqurnu
.pasn @IO~ILLIO~ lsour a.w slaqtunu
I
t-1 I - lu
Lzzy
‘laqwnu
‘as!Mlaqio ‘[3X>() ~03 aM ‘IxaN
1
Lzzn3 ad&-x7
E .IOJ
waqwnu Lzzy 30 sad& leTDads 0~1 asaql 103 SNS~ aA@ Ip2qs lczzn3 ~XKIOU pue .uqt-@ugl aql ‘slaqurnu Iczzy ad&x7 UI
M.-S.
54
objective
functions
49-60
are
Thus, the necessary and the replacement If the m in X of fuzzy number but a
conditions for a minimizer @, fi) of N,(p, W) are Eqs. ( 1.1 )-( 1.4) with Y = I = fi/2 of d& with d;. Definition 1 is not a real number but an interval [ml, ml] then the fuzzy set X is not a fuzzy interval. Accordingly, a LR-type fuzzy interval can be defined as follows.
Definition 2 (Zimmermann interval
Yang, C.-H. Kol Fuzzy Sets and Systems 84 (1996)
[20, p. 641). X = ( ml. m2, CI,fl)LR with u > 0, fl > 0 is called a LR-type fuzzy
if
I
L-
X(x)
ml --x
( 1
for ml 6 x 6 m2,
1
=
for x d ml,
x
[R(‘y)
forx2mz.
Let S,&(R) be the set of all LR-type fuzzy intervals. For any X T~,~,)LR in 9LR([W), we define a metric df(X, Y) as follows: d?(X, Y) = (ml, - mtJ)’ + (mzY - mz,)2 + ((ml, - ICC,)- (ml_,
tm1x,m2x,t%,bx)LR
&y))2
and
Y
=
(ml,,m2,.,
+((m2~+r~~)-(m2~+rBv))",
where I
I=
.I’
L-‘(o)dco
I
and
J’
R-‘(u)
dw.
0
0
Similar to Theorems
r =
1 and 2 (3LR(F8),dl)
is a complete
Let 9& = {XI,. . . , Xn} be a set of n LR-type j = 1,. . . , n. Consider the objective function
metric space.
fuzzy intervals
in .YL,( [w) with Xj = (ml,,, mix,, cc,, , & )LR,
where m 3 1 and ,n = (tit,..., pc) are fuzzy c-partitions and Wi = (ml,, mzrv,, CL,,,/L,)LR are LR-type c-intervals. The necessary conditions for a minimizer (/I, @) of Z,(,U, W) are the following: ~ ,,,,/ =
~~=,fi~t~)t2m~x, + 4&, - zxcx, 1) i = 2C~=,ci?(xj)
CT,,=
C;=,FYY~j)(hu~, - (ml*, I c&R4>
fuzzy
l,...,c,
’
Zch,))
’
i=l
3.1.) c,
(2.3)
M.-S. Yung, C.-H. KolFu::y
55
Sets and Sysfems 84 (1996) 49-60
(2.4)
(2.5) Based on these necessary conditions, intervals. We note that the trapezoidal LR-type fuzzy intervals. Next, we shall Let X = (ml,m2, x,IJ)LR be a LR-type
we can construct these FCN clustering algorithms for LR-type fuzzy type of fuzzy intervals is the simplest and most used one among consider FCN clusterings for this special type of fuzzy intervals. fuzzy interval. If L and R are of the form
O,
l-x,
otherwise, then X is called a trapezoidal ml --x
1-p X(x)
=
We denote it by X = (ml, m2, x, [j)r,,
for x 6 ml,
CI
for ml d .Y 6 m2,
1 I
fuzzy interval.
x - m2 ~
‘-
forx3mz.
fi
Based on the distance dl defined on -9LR([w) before, we define a distance dn(X, Y) for any two trapezoidal fuzzy intervals X = (ml,,m2,, x,,/&)n and Y = (ml ,,,rnzJ, r,.,by)n in the space yr,(rW) of all trapezoidal fuzzy intervals as follows: d&(X Y) =(ml, +
- ml,.)’ + (mix - m2.V)2+ ((ml, - lx\-) ~ (ml,. - ;x~))’
((m2.\-
+
tDx)
The FCN objective functions intervals in .Fr,(&?) are
-
(m2,.
+
i8,,)12.
&(!A, W) for clustering
Therefore, the necessary conditions for a minimizer and the replacement of df with d$.
4. Numerical
examples
the data set $1
(,& I$) of &(p,
= {XI,.
,X,,} of trapezoidal
W) are Eqs. (2.1))(2.5)
fuzzy
with I’= 1 = i
and conclusions
In Section 3, we have constructed FCN clustering algorithms for LR-type fuzzy numbers and intervals, specially for triangular, normal and trapezoidal types. In this section, we shall give some numerical examples. We numerically pick three data sets: (1) 30 triangular fuzzy numbers (see Table 1 and Fig. 1); (2) 30 normal fuzzy numbers (see Table 2 and Fig. 2); and (3) 30 trapezoidal fuzzy intervals (see Table 3 and Fig. 3). Then we run FCN clustering algorithms for these three data sets, with m = 2, c = 3, and E = 0.00 1. Fuzzy c-partitions c = (PI, G2, fiJ) and FCN centers I? = (et, $2, @j) are shown in Tables I-3 and Figs. 4-6. The results of numerical examples seem to be satisfactory. Note that we have preassumed the number L’of clusters for all these numerical examples. The method that finds optimal c. called cluster validity, is very important.
M.-S.
Yang, C.-H. Kol Fuzy
Table I Data of 30 triangular Triangular
Sets and Systems
fuzzy numbers
fuzzy numbers
Memberships P = (i,>&.&)
(3.34 (9.56 (10.56 (10.89 (13.89 (14.78 (14.90 (15.67 (16.87 (17.45 (19.78 (20.67 (21.45 (22.34 (23.47 (24.67 (25.78 (26.45 (28.34 (32.29 (32.77 (34.88 (35.45 (35.88 (38.88 (40.25 (40.47 (43.56 (43.98 (45.77
(0.053824 (0.010266 (0.005756 (0.003872 (0.001838 (0.007380 (0.006121 (0.011653 (0.024142 (0.029484 (0.034567 (0.027751 (0.018319 (0.007625 (0.001816 (0.002319 (0.018131 (0.027172 (0.099257 (0.5255 I9 (0.582464 (0.811237 (0.862406 (0.877326 (0.998021 (0.995373 (0.997603 (0.942318 (0.932188 (0.894959
1.30) I .OO) 1.93) 1.17) 0.88) 1.21) 0.41) 0.90) 1.85) 1.95) 0.42) 1.10) 1.60) 1.58) 0.51) 1.09) 1.51) 0.92) 0.12) 1.64) 0.47) 0.66) 1.26) 0.16) 0.64) I .7l ) 0.15) 0.63) 1.69) 0.79)
49-60
and their fuzzy 3-partitions
x = (w a, B)r 1.46 0.27 1.95 0.56 0.89 0.12 I.19 1.82 1.90 1.79 1.47 1.34 0.92 0.04 0.81 0.14 0.39 I.61 1.95 1.66 0.63 1.08 1.48 1.79 0.66 0.52 1.95 0.92 1.74 1.71
84 (1996)
Fig. 1. The graph of 30 triangular
0.162549 0.04371 I 0.026166 0.018228 0.011500 0.052909 0.042768 0.090467 0.234564 0.323684 0.672476 0.812497 0.905601 0.969866 0.994055 0.994262 0.962896 0.947361 0.845 105 0.403530 0.351715 0.152717 0.1101 I5 0.097848 0.001499 0.003430 0.001784 0.041239 0.048243 0.073501
0.783627) 0.946023) 0.968078) 0.977900) 0.986661) 0.939712) 0.951 I I I) 0.897879) 0.741294) 0.646832) 0.292957) 0.159753) 0.076079) 0.022509) 0.004128) 0.003419) 0.018973) 0.025467) 0.055638) 0.07095 1) 0.065820) 0.036046) 0.027479) 0.024826) 0.000480) 0.001197) 0.000613) 0.016443) 0.019569) 0.031540)
fuzzy numbers
M.-S.
Yang, C.-H. KolFuzzy
Sets and Systems
Table 2 Data of 30 normal fuzzy numbers Normal fuzzy numbers
Memberships ,i = (li,.&.&)
(3.34 (9.56 (10.56 (10.89 (13.89 (14.78 (14.90 (15.67 (16.87 (17.45 (19.78 (20.67 (21.45 (22.34 (23.47 (24.67 (25.78 (26.45 (28.34 (32.29 (32.77 (34.88 (35.45 (35.88 (38.88 (40.25 (40.47 (43.56 (43.98 (45.77
(0.052776 (0.010453 (0.005433 (0.004165 (0.0021 I I (0.006725 (0.007730 (0.013574 (0.024704 (0.029617 (0.032598 (0.025541 (0.018094 (0.008668 (0.001229 (0.002806 (0.017180 (0.034095 (0.126292 (0.532052 (0.591466 (0.819127 (0.865102 (0.896609 (0.997832 (0.996388 (0.994773 (0.941722 (0.933137 (0.892500
1.03) 1.58) 0.63) 1.79) 1.21) 0.65) 1.17) 1.60) 1.71) 0.94) I .27) 0.59) 1.85)
1.26) 1.71) 0.56) 0.92) I .4l ) 0.79) 0.90) I .90) 0.96) I .95)
0.161975 0.045076 0.025265 0.019887 0.013612 0.048591 0.056698 0.111784 0.250473 0.337137 0.715614 0.832123 0.905182 0.963927 0.996056 0.992900 0.963980 0.935302 0.809626 0.396175 0.342551 0.145569 0.107457 0.08 I767 0.001636 0.002676 0.003862 0.041523 0.047433 0.074909
0.785249) 0.944471) 0.969303) 0.975948) 0.984277) 0.944685) 0.935572) 0.874642) 0.724824) 0.633246) 0.251788) 0.142336) 0.076724) 0.027405) 0.002715) 0.004293) 0.018840) 0.030603) 0.064082) 0.071773) 0.065983) 0.035304) 0.027441) 0.021624) 0.000531) 0.000936) 0.001365) 0.016755) 0.019430) 0.032591)
h
0.8
49-60
and their fuzzy 3-partitions
x = (WU)h 1.46) I .09) 0.82) 1.82) 1.19) 1.48) 0.64)
84 (1996)
Ii
0.6
Fig. 2. The graph of 30 normal fuzzy numbers
57
M.-S
Yang. C.-H. Ku I Fuzzy Srts and Systems
84 11996) 49-60
Table 3 Data of 30 trapezoidal fuzzy intervals and theirfuzzy 3-partitions Trapezoidalfuzzy intervals
Memberships
x = (ml,mz,KP)n
/i =
(/i,./T2./ij)
(3.34 5.34 1.46 1.30)
(0.054128 0.164056 0.781816)
(9.56 II.36 0.27 1.00)
(0.011139 0.047289 0.941572)
(10.56 13.79 1.95 1.93)
(0.004207 0.020202 0.975591)
(10.89 13.24 0.56 1.17)
(0.003397 0.016318 0.980285)
(13.89 15.25 0.89 0.88)
(0.001020 0.006212 0.992768)
(14.78 16.34 0.12 1.21)
(0.005757 0.040316 0.953927)
(14.90 16.89 I.19 0.41)
(0.006329 0.044972 0.948698)
(15.67 17.02 1.82 0.90)
(0.009451 0.071329 0.919219)
(16.87 17.54 1.90 1.85)
(0.018075 0.158532 0.823393)
(17.45 18.14 1.79 1.95)
(0.023898 0.234053 0.742049)
(19.78 22.38 1.47 0.42)
(0.032126 0.733173 0.234700)
(20.67 23.57 1.34 1.10)
(0.022702 0.868264 0.109033)
(21.45 23.67 0.92 1.60)
(0.016320 0.918735 0.064945)
(22.34 24.57 0.04 1.58)
(0.006287 0.975737 0.017976)
(23.47 25.47 0.81 0.51)
(0.001246 0.995979 0.002775)
(24.67 25.25 0.14 1.09)
(0.002726 0.992315 0.004959)
(25.78 27.88 0.39 1.51)
(0.020890 0.957697 0.021413)
(26.45 28.34 1.61 0.92)
(0.029728 0.942755 0.027517)
(28.34 30.56 1.95 0.12)
(0.118667 0.820002 0.061331)
(32.29 35.01 1.66 1.64)
(0.585195 0.348492 0.066313)
(32.77 34.67 0.63 0.47)
(0.592451 0.342142 0.065407)
(34.88 36.89 1.08 0.66)
(0.824787 0.141108 0.034105)
(35.45 37.87 1.48 1.26)
(0.887354 0.089429 0.023217)
(35.88 37.89 1.79 0.16)
(0.891492 0.086082 0.022426)
(38.88 40.56 0.66 0.64)
(0.997580 0.001830 0.000590)
(40.25 41.78 0.52 1.71)
(0.996313 0.002733 0.000954)
(40.47 42.35 1.95 0.15)
(0.996267 0.002767 0.000966)
(43.56 45.79 0.92 0.63)
(0.936249 0.045360 0.018391)
(43.98 45.67 1.74 1.69)
(0.932316 0.048068 0.019617)
(45.77 47.56 1.71 0.79)
(0.892947 0.074712 0.032341)
0.6:
0.4:
0.2:
t i Fig. 3. The graph of 30 trapezoidal fuzzy intervals
M.-S.
Yang, C.-H. KolFuzzy
[12.773
1.115 1.185] [24.087
Sets and Systems 84 (1996)
0.914
1.0501
[39.521
49-60
1.283 0.870]
1.0
0.8
0.6
0.4 0.2 _-L
I--_II
10
20
Fig. 4. FCN centers for 30 triangular
[12.705
1.2161
[23.964
fuzzy numbers.
1.2291
[39.472
I.2411
1.0
,
0.8
i
0.4 0.2:
10
20
40
30
50
Fig. 5. FCN centers for 30 normal fuzzy numbers.
[12.890 1.0
14671
I. 137 1.2101
123.972
5.9186 0.905 1
1.040]
[39.401
41.332
1287
0.8751
'1
0.6 1 0.4
0.2
1
--
I 30
Fig. 6. FCN centers for 30 trapezoidal
40
fuzzy intervals.
50
60
M.-S.
Yang, C.-H. Ko I Fuzzy Sets and Systems 84 (1996)
49-60
The diverse validity criteria have been investigated in literatures (see [ 171). These criteria could be also used in these FCN clustering procedures. We have proposed a class of FCN clustering procedures in this paper. Most fuzzy clustering procedures are designed for handling crisp data. Based on these new procedures, we can cluster fuzzy data. Three numerical examples have been given. In fact, these FCN can be applied in diverse areas (see, for example, [ 181). Finally, let us look at asymptotic behaviors of these FCN clustering procedures. By following Bezdek et al. [3, 51 and Yang [15], we can create numerical convergence theorems of FCNs which are similar to these of FCMs. These theorems give us that all the limit points of sequences generated by FCN clustering algorithms are fixed points and that these fixed points are stationary points of the FCN objective function H,,,(p, W). These numerical convergence theorems shall be the important theoretical views on the FCN clustering algorithms.
Acknowledgements We are grateful to the referees for their suggestions and comments. We gratefully acknowledge that this research was supported in part by the National Science Council of the Republic of China under Grant NSC 83-0208-M-033-014.
Reference [I] M. Albrecht, Approximation of functional relationships to fuzzy observations, Fuzzy Sets and Systems 49 (1992) 301-305. [2] R. Bellman, R. Kalaba and L.A. Zadeh, Abstraction and pattern classification, J. Math. Anal. Appl. 13 (1966) l-7. [3] J.C. Bezdek, A convergence theorem for the fuzzy ISODATA clustering algorithms, IEEE Trans. Pattern Anal. Machine Intelligence 2 (1980) l-8. [4] J.C. Bezdek, Pattern Recognition with Fuzzy Objectiae Function Algorithms (Plenum Press, New York, 1981). [5] J.C. Bezdek, R.J. Hathaway, M.J. Sabin and W.T. Tncker, Convergence theory for fuzzy c-means: counterexamples and repairs, IEEE Trans. System Man Cybernet. 17 (1987) 873-877. [6] R. Cannon, J. Dave and J.C. Bezdek, Efficient implementation of the fuzzy c-means clustering algorithms, IEEE Trans. Pattern Anal. Machine Intelligence 8 (1986) 248-255. [7] K.P. Chan and Y.S. Cheung, Clustering of cluster, Pattern Recognition 25 (1992) 21 I-217. [8] J.W. Davenport, J.C. Bezdek and R.J. Hathaway, Parameter estimation for finite mixture distributions. Comput. Math. Appl. 15 (1988) 819-828. [9] P. Diamond, Fuzzy least squares, Inform. Sci. 46 (1988) 141-l 57. [IO] D. Dubois and H. Prade, Fusq~ Sets and Systems: Theory and Applications (Academic Press, New York, 1980). [I I] J.C. Dunn, A fuzzy relative of the ISODATA process and its use in detecting compact, well-separated clusters, J. Cybernet. 3 (1974) 32-57. [I21 M.L. Puri and D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 4099422. [I31 E. Ruspini, A new approach to clustering, Inform. Control 15 (1969) 22-32. [I41 S.F. Schnatter. On statistical inference for fuzzy data with applications to descriptive statistics, Fuzzy Sets and Systems 50 (1992) 143-165. [I51 M.S. Yang, Convergence properties of the generalized fuzzy c-means clustering algorithms, Compur. Math. Appl. 25 (1993) 3-l I. [I61 M.S. Yang, On a class of fuzzy classification maximum likelihood procedures, Fu-_z_vSets and Systems 57 (1993) 365-375. [I71 M.S. Yang, A survey of fuzzy clustering, Math. Comput. Modelling 18 (1993) l-16. [I 81 MS. Yang and C.H. Ko, On cluster-wise fuzzy regression analysis, submitted. [I91 L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338-353. [20] H.J. Zimmermann, Fuzzy Set Theory and ifs Applications (Kluwer, Dordrecht, 1991).