Chapter 10 Estimation of the membership function

C h a p t e r 10

Estimation of the membership function

People think that mathematics is precise and exact. It seems that worldwide scientists and engineers focus their attentions on finding the most exact numbers in their studies with errors smaller than 0.1%, 0.01% and even 0.001%. On the other hand, vague values and fuzzy language are more often used. Words like "tall" in "he is tall", "fat" in "she is fat" are all fuzzy words without clear definitions. 190 cm is definitely tall, but 175 may be tall or may not. Being tall cannot be measured by single index like height, and it is also influenced by one's figure, weight, face or even clothes. Mathematics is not unable to handle fuzzy phenomena. There is a mathematics branch, called fuzzy set theory able to describe such fuzzy things. Fuzzy set theory is introduced into mathematics by Zadeh in 1970s (Zimmermann, 1985). Since then, fuzzy set theory has been applied to language studies, control etc. The most important concept in fuzzy set theory is the membership function. Although fuzzy set theory and statistical estimation are two different branches of mathematics, the former can also be studied by use of the latter. So in this chapter, the method introduced in previous chapters is applied to determining membership functions. The membership function is usually determined by the user in applications. It seems that direct determination of the membership function based on sample data was proposed by Fujimoto et al (1994) and Zong et al (1995). 10.1 Introduction

In traditional set theory, the boundaries between two sets are crisp, meaning that an element is either in set A or set B, but not in both. In Figure 10.1(a) is shown three crisp sets intervals A~,A2,A 3 on the real line, that is, A~ = [0,30], A2 = (30, 70],

A3 =

(70,100]. 237

238

10. Estimation of the membership function

Al

A2 .

.

.

.

A3 .

.

.

.

9.:::. ~ i:::.li: ):i::..:.:.:.i:i:i:i 30 P

X

,,

v

70

(a) Crisp sets

20

40

60

80

(b) Fuzzy sets

Figure 10.1 Crisp sets vs. fuzzy sets If we introduce step functions defined by 1

if

x ~ [0,301

gt,(x) = 0

/f

x ~ (0,301'

1 /f P2(x)= 0 /f

x~(30,701

/13(x) =

10 /f /f

(lO.la)

xr

(10.1b)

xe(70,1001 x ~ (70,100]'

(1O.lc)

then AI, A2 and A 3 may be redefined by /~ = {x" ~,(x) ~ 0} , i= 1,2,3.

(10.2)

In words, ,4, is a collection of those points which do not make the function/~, vanish. We give a special name membership function to/t,, which is characterized by

(1)

/~, (x) > O,

(10.3a)

3

(2/

~ fl, (x) = 1.

(10.3b)

t=l

We do not have any reason to say that membership functions p, must be step functions. In fact, functions satisfying equations (10.3) can be used as membership functions. For example, we may define/1, by

10.1. Introduction

0

x<20

2 ( x - 2 20 0)....

20
1- 2 ( x - 4 0 ) 2 30
20

2 ( x - 27 20 0)

~,(x) =

50 < x <

70
1

x<20

60

l - 2 ( x - 2 0 ; 2 20
0

(lO.4a)

60
0

2 ( x - 24 20 0)

239

(10.4b)

30
0

(x-5o)

2k, 20

/~(x) =

l'

x<50 (10.4c)

-

20' 1

50
These functions are plotted in Figure 10.1 (b). Similar to Equation (10.2), we define three sets, specially written in the form of AI, A2 and _'J3, by _A, = {x : ,u, (x) 4: 0}, i = 1, 2, 3.

(10.5)

They are fuzzy sets. For crisp sets, if point P e A2 , then P r A~ and P r A3 , see figure 10.1(a). But for fuzzy sets, a point can be in two or even more sets

240

10. Estimation of the membership function

simultaneously. For Point P in figure 10.1 (b), both Pl * 0

and P2 s 0 .

Thus P e A! and P ~ A2 from definition (10.5). So fuzzy sets do not have clear boundaries. Detailed presentation of the theory can be found in Zimmermann (1985) One of prominent applications of fuzzy set theory is in the field of quantitative description of language variables due to the fact that language itself is fuzzy. Take age for instance. "Young", "middle-aged" and "old" are such language variables that their boundaries cannot be clearly defined. A person being 30 years old may either be young or be middle-aged. It is hard to draw a clear line between "young" and "middle-aged", so is "middle-aged" and "old". Therefore, ~ = young, A2 = middle- aged and A3 = old are three fuzzy sets. If we want to describe language variables like "young", "middle-aged" and "old" in the framework of crisp set theory, we have to define two critical numbers. For example, 30 years old and 70 years old are defined as the two critical numbers. Below 30 is young and above 30 is middle-aged. Below 70 is middle-aged and above 70 is old. This is in fact what is shown in Figure 10.1 (a) and defined in equation (10.1). Such definition is easy, but a little wired due to the absurdness that a person one day past his thirtieth birthday is middle-aged and another person one day to his thirtieth birthday is young. They are, however, may be only two days different in age. In fuzzy set theory, 30 years old is considered half young and half middle-aged. It may be interpreted at two levels. At the first level, half young and half middle-aged means that he is at the transition stage of life from young to middle-aged. At the second level, half young and half old means that if a survey is made of those 30 years old, half of them may be considered young while half of them are considered middle-aged. In this sense, fuzzy sets with unclear boundaries are in fact more informative and reasonable than crisp sets. This is schematically shown in Figure 10.1 (b) and defined by equation (10.4). Once the rationale for fuzzy sets is justified, the remaining question is how to determine the membership functions. Equation (10.4) and Figure 10. l(b) is in fact a way to define the membership functions for these three fuzzy sets. How to find them? To answer the question is more difficult than to ask. Over the years, several forms of functions have been employed for defining membership functions. Those in Figure 10.1 are two examples. In general, the forms of membership functions may be problem-dependent. Even for the same problem, they are also user-dependent. For the same problem, two users may use quite different forms of membership functions. Take height for instance. In Figure 10.2 are shown two different forms of membership functions for assessing one's height. Three fuzzy sets are defined: "short", "medium" and "tall". One may choose those in figure 10.2 (a) for membership functions and the other may well choose those in Figure 10.2.(b) as membership functions. Using different forms of membership functions will definitely influence the subsequent

10.1. Introduction short

medium

tall

241

short

medium

tall

Height (cm)

Height

(cm) v

150

160

170

180

160

150

170

180

Figure 10.2 Two different membership functions

M

~OO

$

OON

20 (a) randomly prepared triangles

9

30

40

50

60

70

80

(b) fuzzy sample

Figure 10.3 Experiment on the perception of triangular area by a student assessments. Therefore, a systematic methodology is needed to build and determine membership functions on an objective base. Membership functions/-6 (x) may be interpreted as the probability of a point to be in a specific fuzzy set. That is, for a point x, the probability for it to be is p, (x). Note that p~ (x) + kt2(x) + P3 (x) = l, meaning that a point must be in a set. In summary, the membership function lying at the heart of fuzzy set theory serves as two purposes. They define fuzzy sets themselves and they determine the probability of a point belonging to a specific fuzzy set. A membership function is in fact a probability density function determining the possibility for a point to be in a fuzzy set. Therefore, the methods proposed in the previous chapters are also applicable to the determination of the membership functions.

242

10. Estimation of the membershipfunction

By aid of measured entropy analysis, the membership function can be determined in an objective way.

10.2 Fuzzy experiment and fuzzy sample To statistically determine membership functions, we need to design a fuzzy experiment. A fuzzy experiment is slightly different from an ordinary statistical experiment. Fujimoto (1994) designed a fuzzy experiment. Here his experiment procedure is slightly adapted for general purposes.

10.2.1 How large is large? A survey designed to collect the information about people's perception of fuzzy concept like "large" and "small" is to prepare sixty one triangular sheets, the areas of which ranged from 20 cm 2 to 80 cm 2, to say. The side lengths and interior angles of the triangles were randomly determined so that all the triangles had different shapes, as shown in figure 10.3 (a). Three reference triangles, representing "large" with actual area 70, "medium" with actual area 50 and "small" with actual area 25, are also prepared. In the survey, the three reference triangles are shown to the people under test first. Then these triangular sheets are shown to each person in the survey one by one. For each sheet, he or she is requested to assess it using "small", "medium" or "large". After all the sheets are shown to him, a fuzzy sample shown in Figure 10.3 (b) is obtained. In the figure, the horizontal axis is the real triangle area and the vertical axis is the classification result. The experiment may be repeated to another person, and so on. Finally, a fuzzy sample as shown in Figure 10.3 (b) is obtained, from which membership functions are to be estimated. What is remarkable in the figure is that there are overlapping regions among each class. That is, the boundaries among each class are fuzzy. So, the data are called fuzzy data. The fuzzy data roughly indicate the correlation between the linguistic expressions and the physical quantity under consideration. The question asked at the very beginning of this section on "how large is large" can now be answered based on the data in Figure 10.3. For this problem, "large" is a set ranging roughly from 60 to 80 cm 2. It is clear from the figure that triangles with area below 55 cm 2 has never been classified as "large". Thus, it is safe to say that triangles with area above 55 cm 2 are large while those between 55 and 65 cm 2 are in transition stage from "large" to "medium".

10.2.2 Fuzzy data in physical sciences The above example is somewhat arbitrary because the experiment has been performed on humans. In physical sciences, however, such fuzzy data are also

10.2. Fuzzy experiment and fuzzy sample

243

available. Fluid flow in a pipe is a topic which has been of extensive interest in fluid mechanics. A parameter describing the flow state is called Reynolds number Re defined by Re =

Ud

(10.6)

V

where U is maximum flow velocity in the pipe, d is the diameter of the pipe and v is the kinetic viscosity of the fluid in the pipe. For water, v = 10-6 .

Turbulent AAAA AAA

A

90 O 0 0 ( X ~ Laminar v

logRe Turbulent Figure 10.4 Fuzzy data for Laminar and Turbulent flow Reynolds number Re dominates flow state in the pipe. If it is small, the flow is laminar, a state fluid particles smoothly flow in the pipe. If it is large, the flow is turbulent, a state fluid particles irregularly flow in the pipe. It has been experimentally found that transition l~om laminar flow to turbulent flow is not unique. Over a wide range of Reynolds number, the transition may happen depending pipe wall smoothness. If the pipe wall is very smooth, the transition does not occur until Re=40000, while if the pipe wall is rough, the transition may occur around 2100. When many experimental results are plotted in one single figure, we obtain fuzzy data schematically shown in Figure 10.4. Note this figure is schematic rather than physically accurate. Fuzzy set theory also provides us with a new look at some old problems. A rod under compression load P may become unstable suddenly as the load gradually increases. This is a well known buckling problem in mechanics of materials. The critical load at which the rod becomes unstable is theoretically a deterministic value, but in real-world application is random. If many rods of same size and same material are tested, we obtain data showing large scatters

10. Estimation of the membership function

244

around the theoretical critical value of the rod. From the viewpoint of mathematics stood away from physical background, fuzzy data in Figures 10.4 and 10.5 are no different in nature.

Figure 10.5 Fuzzy data for Stable and Unstable rod

Quite some examples in a variety of engineering fields and science disciplines can be given showing similar patterns as in Figures 10.3---10.5. All these phenomena can be treated using fuzzy set theory. But figure 10.3 and Figures 10.4---10.5 are slightly different in that the latter is physics-based phenomena requiring membership functions to be objectively determined while the former is human-related the membership functions of which are better to be determined objectively. This justifies the need to find membership functions in an objective way.

10.2.3 B-spline Approximation of the membership functions In this section, we will determine the membership function through the fuzzy data obtained in the previous section. Suppose the universe of discourse is X. Let x~,(e = 1,2,...,x,) be a sample from X. Further, suppose the fuzzy sets are A, ,

A2 ,

...,

AM and the corresponding membership functions are

/~(x),/u2(x),...,/~ M(x). In the triangle experiment, x, is the area of a triangle and A,, A2 and A3 correspond to "Small", "Medium" and "Large", respectively. As done in the previous Chapters, we again assume that the membership function/1 (x) can be expressed in the form of a linear combination of B-spline functions in the universe of discourse X, i.e.,

10.2. Fuzzy experiment and fuzzy sample

245

~, (x) = a,,B, (x) + a,,B~ (x) +... + a,~B~ (x) ~, (x) = a,, B, (x) + a~B~ (x) +... + a ~ B ~ (x) ...

(10.7)

/.tM (x) = aM,B , (X) + aM2B 2 (X) +'" + aMNBN (X) In concise form we have N

kt,(x) = Z o t , , B , ( x ) , i= 1,...,M

(10.8)

/=1

where N is the number of B-spline functions which consist of the membership functions, a, are the combination coefficients, and B~ (x) is the B-spline functions of chosen order and is of the following form if order 3 B-spline function is chosen 3 (x., - x) 2H(x,+, B,(x) = (x,-x,_3)~--' ~ +' -X).H(x-x,), w'(x,+,) t=0

(10.9)

3

(lO.lO)

w, (x) = l-[ ( x - x~+,) . t=O

Now we are to determine the parameters a, . From the process of classification, the membership function ,u,(xr) is regarded as same as the probability that a sample point x~ is classified into fuzzy set ~ , that is, Pr[x, ~ ,4,] =/t, (xc).

(10.11)

Therefore, we employ the likelihood analysis for the determination of the membership functions. The probability of the classification event for all the sample points x~(e = 1,2,-..,n, ) is expressed by the following likelihood function.

L= H

FI

FI

xt ~dl

xt eA_2

xr eA_,w

(10.12)

The log-likelihood function is M

/, = logL = ~ ~ log/~,(xr). ~=1x~~4,

(lo.13)

10. Estimation of the membership function

246 M

M

N

N

M

where ~--'p, (x)= Z ~-" o,,,B, (x)= Z ( Z a , , ) B , (x)= 1. t=l

t=l

I=i

/=l

t=l

According to the B-spline function properties we have N

~-" B,(x) = 1.

(10.14)

/=!

From the above two equations the following relationship is obtained, M

~--'c% = 1 , j = 1,...,N.

(10.15)

t=!

A membership function is always greater than or equal to zero. To guarantee this we simply impose the restriction that all parameters a,, are greater than or equal to zero, that is, a',, >0

i= 1,...,M;j = 1,...,N.

(10.16)

Usually, we hope r is a decreasing function of x and /tM(x ) is an increasing function. It is obvious that these are guaranteed by the following equations:

>crl, /+1

t~'l I - -

~

j=l,

"'''

N-l,

C~M~< C~M,~+i , j = 1,..., N - 1.

(10.17a) (10.17b)

Based on the above analyses the best estimates of the unknown parameters a,j must satisfy M

/, = ~ ~ log/.t, (x,.) ~ max

(10.18)

a=i x~ ~d,

subject to M

~ a , , =1 I=!

j=I,...,N,

(10.19a)

10.3. ME analysis a,j >__a~.,+,

t)t~ <_aM.,+' a,, >0

j = 1,...,N-1,

247 (10.19b)

j = 1,...,N-I,

(10.19C)

i= I , . . . , M ; j = 1,...,N.

(10.19d)

This optimization model has a good property. The optimum solution a,jo is unique (the proof is given in appendix 10.A). That is, if we can find a local maximum point a,~0 by some method it must be the global optimum solution. The optimization method like the Flexible Tolerance Method introduced in Himmeiblau (1972) may be employed to solve the maximization problem ( 10.8)~(10.9). 10.3 ME analysis

In the above optimization problem, the optimization parameters are N and

a,j. If N is fixed, this problem can be solved by ordinary nonlinear programming methods. Because N is also an optimization parameter, measured entropy analysis must be used. Consider a fuzzy sample of size n,. n) denotes the number of unknown parameters for the i-th fuzzy set. Without equation (10.15), n) would be equal to N . Equation (10.15) is an interlink among the M fuzzy sets. It is hard to distribute these N equalities among the M fuzzy sets. We may, however, avoid this difficulty using the method in the following. The entropy for i-th fuzzy set is (10.20)

H, = - I,u, (x) log,u, (x)dx.

The corresponding asymptotically unbiased estimator of the measured entropy

t

3n t

ME, = - I/~, (x[ c?) log/t, (x[ c?)dx + - ~ 2 n,

9

(10.21)

Then the total measured entropy should be M

MI

ME=~-'H,,=, =-)-",_,

3 nf /~,(xl c~)log/~,(xl c~)dx+~,=, n,

(10.22)

248

10. Estimation o f the membership function

From the definition, the second term become

Z' 3 ,=l nf M

M

t

2=

2

3 (M-1)N

n,

2

(10.23)

/7

Therefore, the best N solves the minimization problem M

ME = - ~

,=l

q,

~fl

3 (M-1)N

, (xl&)log ~,, (x [ d')dx + -

2

n

(10.24)

If AIC is used, M

AIC = - m a x {L} + ~ n't = - m a x {L} + N ( M - 1) ~ min

(10.25)

t=l

where the number of free parameters n t = M x N - N

because there are N

equality constraints in the above model. In the actual optimization process, the best N which makes M E or AIC minimum is numerically searched by the following procedure. First, a positive integer N is assumed. Then the maximum likelihood analysis is carried out to obtain the best estimates of a , . The value of M E or AIC is calculated. Next, changing N gradually, and the corresponding a,j and MEs or AICs are calculated. By the comparison of these ME or AIC values, the best N is found.

10.4 Numerical Examples Example 10.1 Triangular area problem Figure 10.6 (a) shows the fuzzy data obtained from the experiment described in section 10.2. Using these data, we may perform the ME analysis and the likelihood analysis using the method presented before. The B-spline functions of order 3 are used. For the optimization of the likelihood function, the Flexible Tolerance Method (FTM) as outlined in the appendix to this chapter is used. In many nonlinear programming methods a considerable portion of the computational time is spent on satisfying rather rigorous constraint requirements. However, the FTM does not satisfy the constraints first. But, the constraints are gradually satisfied as the search process proceeds toward the true solution. Tables 10.1 gives the results of the analysis. From the table, the minimum M E value is obtained if eleven B-splines are used, and minimum AIC value is

10. 4 Numerical Examples

249

obtained if eight B-splines are used. The estimated membership functions based on minimum ME and minimum AIC are plotted in Figures 10.6 (b) and (c). The results obtained from ME analysis and AIC analysis are quite consistent although the numbers of B-splines in the two analyses are different.

Table 10.1 Results for three fuzzy sets

N

-L

ME

AIC

6 7 8 9 10 11 13 12 14 15

19.7 16.7 14.5 15.5 14.3 13.6 13.1 14.2 13.1 13.2

5.54 4.11 2.95 3.80 3.19 2.52* 3.00 3.27 2.65 3.18

33.65 32.69 32.50* 35.57 36.26 37.63 41.13 40.22 43.09 45.20

L

(a) fuzzy data

M

S

1

,,N

tmO0

m

O0N

9

20

an

30

9

40

50

|

60

9

70

80

Area .

=

(c)A]C

(b) ME

I--4

..Q

0.5

E 0.5

0

0 20

30

40

50 Area

60

70

80

20

30

40

50 60 Area

70

80

Figure 10.6 Estimation of the membership functions for the problem of triangular areas

Example 10.2 Analysis of the fuzzy data offive classifications In shipbuilding industry, welding is a very important process, taking more than 40% workloads. Welding quality is directly related to the life of a ship.

10. Estimation of the membershipfunction

250

Thus, welding quality must be controlled within allowable errors. One factor controlling welding quality is called misalignment 8 as schematically shown in Figure 10.7. The figure shows that two vertical plates are welded to a horizontal plate. The two vertical plates are required to be in one line. This is difficult because a welding worker cannot see the upper plate as he is welding the lower plate to the horizontal plate, and he cannot see the lower plate as he is welding upper plate to the horizontal plate. So after the welding work, a surveyor must do quality examination. The examination results are classified into five classes: "Very Good" (VG), "Good" (G), "Medium" (M), "Bad" (B) and "Very Bad" (VB). Because of several factors involved, the quality assessment is not a simple correlation, but a complicated link as shown in Figure 10.7 (a), in which is given one hundred sample points to show the correlation between misalignment d and quality assessments. VG

(a)fuzzydata

G '

~

!,ql-- 8

8: misalignment t: thickness

9 9 u~m 00

M

9 9

Oll~mN

9 t

DIID OIIIDO

O mOtlO0

D

b" 0

1

b) ME

0.5

1

1.5

0.5

1

1.5

1

0.5

0.5

0

0 0.5

1

1.5

0

Figure 10.7 Five classifications of misalignment Using fuzzy data in Figure 10.7 (a), the membership functions are estimated. Order 3 B-spline functions have been in the analysis. Table 10.2 gives the results of the analysis.

I O.4. Numerical examples

251

Table 10.2 Results for five fuzzy sets N -L Me AIC N -L Me AIC 4 82.4 1.73 98.4 10 56.3 1.49 98.3 5 71.7 1.68 93.7 11 55.5 1.54 101.5 6 67.0 1.48 91.5 12 55.0 1.64 105.0 7 62.0 1.57 91.9 13 55.1 1.71 109.1 8 57.0 1.45 92.0" 14 51.7 1.73 109.7 9 56.1 1.49 94.0 15 53.4 1.78 115.4 The minimum ME value is obtained at N=8 and the minimum AIC is given as N=6. Figures 10.7 (b) and (c) compare the membership functions for the two cases. Example 10.3 Sample Size Influences In this example, the sample size influences of fuzzy data on the forms of membership functions are discussed. Suppose membership functions of the forms shown in Figure 10.8 (a) are given, representing three fuzzy sets A~, d2 and A3 . Then 100, 200 and 300 pairs of uniform random numbers (u,, v,)are generated in the range 0 _ ~/, (u,), then u, e ,~; If vr <_ll,(u,,) < ~t,(ur), then u, e A,. If v~ >/,t, (u~,), then u s is rejected;

1

1 ~..~(b) :o ,o, ~--r---,x s ,,: kT(.r\~.

0

0.5 !

0

2

4

6

8

10

0

, ~. u

2

V.

~., \~

~,.

" 9 ~ ' , ~ " Y , ' s9 clue9 e~

4

6

8

Figure 10.8 Given membership functions (a) and random numbers (b)

10

10. Estimation of the membershipfunction

252

n~ assumes three values, 100, 200 and 300. The three corresponding fuzzy samples are given in the left figures in Figure 10.9. Using the method presented in this Chapter, the membership functions corresponding to the three generated samples can be estimated. The results for ME values are given in Table 10.3 and plotted in Figure 10.9. In the Table, only ME results are given and AIC values are neglected.

Table 10.3 Sample size influences (a):Sample size:100

(b) Sample size:200

-L 39.5 34.5 35.1 33.8 32.0 32.0 32.0 32.0 31.5 31.4 31.6

-L 73.4 69.0 69.2 68.7 68.1 68.3 67.5 67.6 66.9 66.9 67.5

N 5 6 7 8 9 10 11 12 13 14 15

ME 5.70 4.07 4.50 4.14 4.24 3.76 3.86 3.82 3.83 3.72 3.90

ME 5.63 3.76 4.03 3.84 3.62 3.72 3.67 3.63 3.65 3.64 3.63

(c):Sample size:300 -L ME 123.1 5.60 107.9 3.77 109.9 4.12 107.3 3.88 107.2 3.54 127.4 3.65 106.4 3.58 106.7 3.61 105.9 3.69 105.4 3.68 106.5 3.69

For n - 1 0 0 , minimizing ME yields N-14. The profiles of the membership functions as shown in Figure 10.9 (a) are complicated because too many Bsplines are used. If sample size is over-small, statistical fluctuations have significant influence on the shapes of the membership functions. For n, - 2 0 0 and n , - 3 0 0 , minimizing ME yields same results at N-9. The estimated membership functions are shown in Figures 9 (b) and 9(c), which look much better than Figure 10.9 (a). Figures 10.9 (b) and (c) do not exhibit significant differences, showing the convergence as sample size increases. Both are close to the given membership functions as shown in Figure 10.8.

253

10.5. Concluding remarks

(a) n, = 1O0 9

M

9 1 4 9

~

ooq~oIN

OlBO, 9149149

9

9

0.5

-

0

2

-

-

4

9

-

6

9

-

O

10

0

2

4

6

8

10

0

2

4

6

8

10

0

2

4

6

8

10

260

(b),,,

9

M

-

8

9

Ill;

Oe

. . . . . . . . . . . . . . . .

9149149149

qlDO 9

0.5

.

0

.

.

.

2

.

.

i

4

6

i

0

i

8

l0

(c) h, 300 M

9

9

__L_ZL~2~

-

Dim

0.5 u l n ~ o g n ~ o l N

9

0

2

-

o

D

i

4

i

i

6

i

9 ,

8

i

10

0

Figure 10.9 Given membership functions (right) and fuzzy data (left)

10.5 Concluding Remarks In this chapter, a probabilistic model that can determine the forms of the membership functions based on experimental data is presented. In the method, the membership functions are approximated by a linear combination of B-spline functions. The best number of B-splines which compose the membership functions under consideration is determined by minimizing ME. And the best combination coefficients are determined probabilistically based on the likelihood analysis.

10. Estimation of the membershipfunction

254

The features of the membership functions presented in this paper can be characterized as follows: (1)

The membership functions can be automatically determined from the fuzzy data by the proposed method. No prior knowledge of the form of the membership functions is necessary in the estimation. (2) The method works well irrelevant to the number of classifications. Also, numerical calculations are easily performed because the optimum solution is unique. (3) If the sample size of fuzzy data becomes larger, the estimated membership function becomes more accurate.

Appendix

255

Appendix: Proof of uniqueness of the optimum solution Consider the following nonlinear programming problem.

f (X,,X2,''',Xn) --~ max

(10.A.1)

subject to the following constraints, g,(xi,x2,...,x,) >_0, i = 1,2,...,m,

(lO.a.2)

xj > 0, j = 1,2,.--,n.

(10.a.3)

The Lagrangian is m

L = f(x, ,x2," ",x,,) + ~ u,g, (x,,xz," ",x,,).

(10.A.4)

t=i

If x ~ is a local optimum solution in the above problem, then the sufficient condition for x ~ to be the global maximum point is:

L(x,u~ <_L(x~ u~ ~ 8L(x~176(x,-x, 0 ). a=l

(10.A.5)

~X I

where u ~ satisfies

(aL)

,

OL >O for allu,>O; u, ~u, =0' i = l , . . , m. Ou,

(10.A.6)

The Lagrangian for our problem is M

N

M

Z Z I=! x~ eA.s

I=1

t=!

2•

E .~=1

in which g, 's are constraints given by the following equations.

(10.A.7)

256

10. Estimation o f the membership function

g, (ail,...,aqu) = a'l/-al.j+ I > 0;s = 1 , . . . , N - 1 g, (OtMi ,''''aMN

) = ~[M,l+l

--O[U,I

~-- 0

(10.A.8)

s=N-I+j,j=I,2,...,N-1 ( ~ "- ( ~ I

I , " " " , O(IN , ~ Z I , " " " , ~ Z N

, " " " , ~Mi

, OIM Z , " " " , ~ M N

)

According to Taylor expansion of multivariate function we have M N 0L(o~o) t(~) -- t(~ o) + Z.__.Z,~. o~., (~,, _~o)

|M

2

N

M

t=l /=1

N

O~L(OotO+(l_O)~)

o

o

(a,, - a,, )(a,, - ~, )

(10.A.9)

= I=1

U N,~., OL

1 U N U ~

= L(~~

+ ZEZZ

82L Aot,jAotkt 8a,s Oakl

where 0 < 0 < 1 Because, Bj

2•

(10.A.10)

aL(~)_ Z Z +~, + Z ..h. in which i f (s - j, i = 1)or(s = N - 1 + j, i = M ) i f (s = j + 1, i = 1)or(s - N + j, i= M ) .

h, = 0~,, -

otherwise

So

e~t(a) =_ ~ OCr,lOOtkt 82L(a)

~ = 0

Oa, Oa~

~,~

B,& It 2,

fork = i (10.A.I 1) fork ~ i

Therefore, the third term on right hand side of equation (10.A.9) becomes

Appendix IMNM

N

257

02L

~Aa,jAau 2

t=l

/=1

1M =

= N

i=1 N

~2/-

Aa',1Aa,i I=! I=1

_

1 M

u u B1BIAc%A~.

-

2 Z Z Z Z I=i xt~ 4 I=!

= - ~ ,:, _ 1

2

I=!

(10.A.12)

1"/i

~ ( : 8,A~,,)(Z,:,~,A~,,) Z 7(

8,A~,, <0

2 ,--t x,~4

From equations (10.A.9) and (10.A.12), the following relationship is obtained. M

N

t(oO < L(cr ~ + ~ ~ aL zXa,,. ,-I i-1 c3a~,j

(10.A.13)

The above equation indicates that if a ~ is a local optimum solution it must be the global solution.