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Performance characteristics of the fuzzy sign detector
I
ZZY
sets and systems ELSEVIER
Fuzzy Sets and Systems 74 (1995) 195 205
Performance characteristics of the fuzzy sign detector S u n Y o n g K i m ", I i c k h o S o n g "'*, J a e C h e o l S o n b, S a n g y o u b K i m ~ aDepartment of Electrical Engineering, Korea Advanced Institute of Science and Technology (KA1ST), 373-1 Guseong Dong, Yuseong Gu, Daejeon, 305-701, South Korea bDigital Signal Processing Technology Center, Samsung Electronics Co., LTD., San 14, Nongseo-Ri, Kihung-Eup, Yongin-Kun, Kyungki-Do, 449-900, South Korea cDepartment of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109, USA Received December 1993; revised October 1994
Abstract
In this paper, fuzzy statistical techniques are introduced into the optimum signal detection problems. The likelihood ratio for fuzzy detection of known signals is obtained. As a special case of the fuzzy signal detector, a fuzzy set theoretic approach to sign detection of known signals is considered. Specifically, the test statistic of the fuzzy sign detector for known signals is obtained. Some properties of the fuzzy sign nonlinearity, which constitutes the fuzzy sign detector, are also described. Finally, the performance characteristics of the fuzzy sign detector are investigated and compared to those of the crisp sign detector.
Keywords: Fuzzy sets; Fuzzy test; Fuzzy sign detector
1. Introduction
In signal processing areas, we face many practical problems where imprecise information is common. Based on this observation, applications of fuzzy set theory have been considered in several signal processing problems (e.g., [1, 3]). Although the fuzzy set theory has already been introduced into a signal detection area based on the concept of interval-valued fuzzy sets [5], it is also a reasonable approach to combine the fuzzy testing of statistical hypothesis [6-1 and signal detection theory. The rationale to connect the fuzzy set theory and the signal detection theory may partially be explained as follows. Although we cannot exactly estimate the statistical characteristics of the noise process and it is hard to assume that there is no additional noise such as quantization error and the self-noise of the processor in practice, we usually neglect additional noise in modeling the physical situations. Obviously, we may partially compensate this negligence by estimating the effect of both facts. This method is, however, somewhat cumbersome and time-consuming in general. It would be more convenient and reasonable to rely on the thought that the observations provide us with fuzzy information such as "the real value is just around the * Corresponding author. Tel.: + 82-42-869-3445. Fax: + 82-42-869-3410. E-mail: [email protected]. 0165-0114/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved SSDI 0 1 6 5 - 0 1 1 4 ( 9 4 ) 0 0 3 2 7 - 0
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observed value", and consequently to employ the fuzzy information processing techniques. The objective of this paper is to obtain detection schemes based on the fuzzy testing of statistical hypothesis. Specifically, we consider a fuzzy set theoretic approach to sign detection and obtain the detection schemes having lower sensitivity of performance to small deviation of assumed noise statistic. In this paper, we reconsider the signal detection theory based on the fuzzy set theory and fuzzy tests. We discuss the fuzzy sign detector as a particular example of the application of the fuzzy decision problems in signal detection. A sign detector based on the signs of data is one of the best-known nonparametric detectors. While a rank detector, which is also one of non-parametric detectors, exhibits better performance than the sign detector in many cases, it is considerably more difficult to implement. The sign detector, on the other hand, is usually quite simple to implement. We describe some properties of the fuzzy sign nonlinearity, which constitutes the fuzzy sign detector. They include statistical properties of the fuzzy sign nonlinearity. In order to discuss performance charcteristics of the crisp and fuzzy sign detectors, numerical results are presented under additional noise environments. We show that the performance characteristics of the fuzzy sign detector is strongly related to the incredibility, which is a measure of fuzziness of the observations, and obtain the optimum values of the incredibility under various additional noise environments. 2. Preliminaries
Let X = (X, Bx, Fx) be an experiment, where X is a set in the real line ~, Bx is the Borel a-field on X, and Fx belongs to a family of probability distributions A on the measurable space (X, Bx). Suppose that the experimentation from X does not provide exact information, but provides only fuzzy information. For mathematical handling of the fuzzy observations in such circumstances, let us introduce the concepts of the fuzzy information system [2] and sample fuzzy information [2] in Definitions 1 and 2, respectively. Definition 1. A fuzzy information system ~ associated with the experiment X is a fuzzy partition (orthogonal system) of X by means of fuzzy events. Definition 2. A fuzzy random sample of size nfrom ~, denoted by ~n~, is a fuzzy partition of a random sample of size n from X, by means of fuzzy events of the random sample, in such a way that each element called sample fuzzy information in ~")is characterized by means of an n-tuple of fuzzy information, x = (xl, ..., x,), xi ~ ~, i = l,...,n. The probability distribution of a fuzzy event was first introduced in [8], which has been directly extended to the case of the fuzzy random sample as follows. Definition 3. The probability distribution of the fuzzy random sample (~"~is a mapping P from ~"~ to [0, 1] given by
P(x) = fx. pK,(xl) ... I~.(x,)dFx.(Xl, ... ,x,),
(1)
where the integral is the Lebesgue-Stieltjes integral, Fxn(Xl . . . . . x,) is the probability distribution on (X ", Bxo) determined by Fx on (X, Bx), #~, is called the membership function of xi, and X"=Xx
... xX.
n times
Let us introduce a few more definitions for the hypotheses testing with fuzzy observations. The definitions of a fuzzy test, a fuzzy test function, and a power function of the fuzzy test are not quite different from the crisp counterparts. For instance, a rule with which we choose for each sample fuzzy information x in (~n~
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197
between the inferences "accept the null hypothesis" and "accept the alternative hypothesis" with probabilities 1 - ~b(x) and ~b(x), respectively, is called a fuzzy test. In addition, the function ~b from ~'~ to [0, 1], allocating to each x the probability ~b(x) of accepting the alternative hypothesis, is called a fuzzy test function. Definition 4. The power function fl, of the fuzzy test is a mapping from A to [0, 1] and is defined by
fl,(Fx) = ~
dp(x)P(x)
for all Fx e A.
Ke~n~
(2)
Definition 5. Let Ao be the set of distributions in A for which a given null hypothesis is true, and let Foe Ao. Then the value sup FoeAofl,(Fo) is called the size of the fuzzy test. If the size of the fuzzy test is at most equal to ct with ct e [0, 1], then the fuzzy test is said to be a fuzzy test of size ot or a size-orfuzzy test.
3. Fuzzy sign detector for known signals
3.1. Observation model
Let us consider the well-known signal-detection problem which is described as a binary hypotheses testing problem: Ho:
1I/= W/,
(3)
Y/= 0el + IV/,
(4)
versus Hi:
for i = 1 . . . . . n. In the above formulation, e~ is the known signal component, W~ is the purely-additive noise component, and Yi is the observation at the ith sampling instant. The quantity 0 is the amplitude parameter which controls the signal strength. The purely-additive noise components Wi, i = 1, ..., n, are assumed to be independent and identically distributed (i.i.d.) with the common continuous probability density function (p.d.f)f. It is assumed that the p.d.f, f is zero-mean, even-symmetric ( f ( y ) = f ( - y)), and unimodal. Note that the observation model is the same as that in most conventional studies on known-signal detection (e.g., [7]). However, some difference in the amount of fuzziness on sample information exists. That is, in this paper the available probability distribution from the experiment is P(x) induced from H 7= l f(Y~). 3.2. Test statistic
The test statistic for the fuzzy signal detection problem based on the fuzzy set theoretic extension of the Neyman-Pearson criterion is defined in terms of the likelihood ratio. The test statistic can easily be derived to be T(x) = ~" gopt(tq),
(5)
i=1
where gopt(Ki) =
In
P(KiIH1) P(x~[ Ho)
= In I#~,(x)f(x - OeDdx I~K,(x)f (x) dx
(6)
In (6), the function #opt(X~) is called the optimum fuzzy nonlinearity and characterizes the detector structure.
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As a special case of the fuzzy signal detector, let us consider the situation where the available informations 9reater than ½0e~ with the membership are: K_ = "approximately less than ~~ .ve~ and. x + = "'approximately . . . functions x_ and x+ having the following properties: (1) The membership function /~K (x) is monotonically decreasing with l i m x ~ / ~ ( x ) = 0 and limx~-o~#~ ( x ) = 1. (2) The membership function p~.(x) is monotonically increasing with limx~o~/~÷(x)= 1 and limx~_~ pK+(x) = 0. (3) For any x, #~ (x) + / ~ . ( x ) = 1 a n d / ~ (x) = gK+(- x). Typical examples of the membership functions are shown in Fig. 1. In Fig. l(a), the parameter A, which we call the incredibility, is a measure of fuzziness of observations. When the value of A is large, we cannot distinguish between x_ and x +. We consider small values of A assuming that the fuzziness of observations is relatively small. It should be noted that the membership function with A = 0 is for the crisp sign detection problem. The membership functions in Fig. 1(b) are more general than those of Fig. 1(a). In Fig. 1(b), other measures of fuzziness (e.g., [4]) should be used for the fuzziness of observations. The membership functions in Fig. 1(a) are relatively easy to handle mathematically and many cases produces good results.
3.3. Some properties of fuzzy sign nonlinearity In this section, we present several properties of the fuzzy sign nonlinearity. We will denote by 9(x) the fuzzy sign (FS) nonlinearity. From (6) the FS nonlinearity g(x) is given as follows:
g(xi) = [ l n ( f PK (x)f(x - Oei)dx / f l~K ( x ) f ( x ) d x )
for xi = x - ,
(7)
ln(f,~+(x)f(x-Oe,)dx/f~K+(x)f(x)dx)
for x , = x + .
The FS nonlinearity (7) has the following properties. Property 1. 9(x = r~_ ) < 0 and g(x = x+ ) > O. Property 2. g(x = x _ ) = - g ( x = x+).
In other words, the FS nonlinearity is an odd-symmetric function of x. In addition, as a function of 0, the FS nonlinearity has the following properties. Property 3. 9(x = ~:- ) is a decreasing function of O, while O(x = ~c+) is an increasin9 function of O. Property 4. For the membership functions of Fig. 1(a), g(x = ~c_ ) is an increasino function of A, while 9(x = x+ )
is a decreasin9 function of A. For the statistical average and variance of the FS nonlinearity we have Properties 5 and 6. Property 5. E [9(x) l Ho] = - E [ 9 ( x ) I H1] and E[T(x)I Ho] = - E [ T [(x)l Hi ], where E [ ' ] denotes the
expectation. Property 6. VEg(x) l H o ] = V[g(x)lH1]
variance.
and V [ T ( x ) I H o ] = V[T(x)IHx], where V [ ' ] denotes the
S.Y. Kim et al. / Fuzzy Sets and Systems 74 (1995) 195-205
0ei 2
A
0ei -~-
199
0e i T+A
(a) ~t_(x)
~t.(x)
0.5 x
0e 2
(b) Fig. 1. Membership functions of x_ and x+.
Proofs of Properties 1-6 can be found in Appendix A. The FS nonlinearity which constitutes the test statistic of the fuzzy sign detector compensates for the effects of the noise. The observation in Property 3 is quite natural since 0 is directly related to the signal to noise ratio: that is, the confidence of the information increases as 0 increases. In Property 4 we can also make a similar observation. 3.4. Performance characteristics
Here we will consider the performance characteristics of the fuzzy sign detector with the membership functions in Fig. 1(a). Let us investigate performance characteristics of the fuzzy sign detector and compare them to those of the crisp sign detector. The proofs of these theorems are given in Appendix B.
Theorem 1. Let tPa be {xl the subset of the sample space in which the fuzzy sign detector with the incredibility A accepts H1 } and ~Po be {xl the subset of the sample space in which the crisp sign detector accepts H1 }. Then ~eo w_ ~pa, w_ ~pa2for any A1 < A2.
Corollary 1. For A ~ < A 2 and a preassigned size at, we have El'~bcrisp(~C)I Ho']/> El'q~fuz~y(t¢)I Ho] 14=a, 1> El-~bfu~y(~C)I Ho-] la=a2 and
E[~bcri~p(X)I HI] i> E[q~fu~y(X)]H~] I~=~,/> E[Oroz~y(,c)l H , ] I~=~,
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where (~crisp(/~) and (])fuzzy(X) a r e the crisp and fuzzy sign test functions, respectively, and E l . ] means the expectation with respect to the noise p.d.f, including additional noise. Corollary 1 follows immediately from Theorem 1. Thus for a preassigned size, which may not be equal to the false alarm probability, the false alarm and detection probabilities of the crisp sign detector are greater than those of the fuzzy sign detector in practical noise environments. Corollary 1 suggests that the difference between a preassigned size and the false alarm probability resulting from the effect of practical noise including additional noise can be diminished by employing the fuzzy set theoretic approach. In other words, since the incredibility decreases the false alarm probability and additional noise increases the false alarm probability, we can find a value of the incredibility for which the false alarm probability equals to the preassigned ~ in practical noise environments including additional noise. As examples, the false alarm and detection probabilities of the crisp and fuzzy sign detectors in various cases are given in Tables 1-4 when additional noise has normal distributions. It should be remarked that the crisp and fuzzy sign detectors in Tables 1-4 are designed for a preassigned size under the assumption that no additional noise exists. We assumed that the desired size ~ = 0.002, e~ = 1, i = 1, ..., 20, the sample size n = 20, and W~ ~N(0, 1), that is, the noise component is normally distributed with zero mean and unit variance. From Tables 1-4, we see that the crisp sign detector provides better performance than the fuzzy sign detector when no additional noise exists. The performance of the fuzzy sign detector, however, is closer to the expected performance than that of the crisp sign detector when additional noise exists. That is, the false alarm Table 1 Comparison of the false alarm probabilities and detection probabilities of the fuzzy sign detector and the crisp sign detector when 0 = 1 Fuzzy sign detector Additional Noise
Crisp sign detector A=0.5
No Noise N(0, 1) N(0,2) N(0,3)
A=I.0
A=2.0
Pfa
Pd
Pfa
lPd
-'°fa
Pd
Pfa
Pd
0.0020 0.0093 0.0170 0.0238
0.7530 0.5672 0.4761 0.4220
0.0016 0.0077 0.0144 0.0203
0.7301 0.5405 0.4496 0.3963
8.6E-4 0.0048 0.0086 0.0123
0.6390 0.4424 0.3564 0.2764
1.7E-4 0.0011 0.0023 0.0035
0.4525 0.2701 0.2024 0.1673
Table 2 Comparison of the false alarm probabilities and detection probabilities of the fuzzy sign detector and the crisp sign detector when 0 = 2 Fuzzy sign detector Additional Noise
Crisp sign detector d = 0.5
No Noise N(0,1) N (0, 2) N (0, 3)
A = 1.0
A = 2.0
Pfa
Pd
Pfa
Pd
Pfa
Pd
Pfa
Pd
0.0020 0.0318 0.0812 0.1308
0.9999 0.9993 0.9971 0.9933
0.0013 0.0222 0.0593 0.0983
0.9999 0.9984 0.9937 0.9870
3.7E-4 0.0096 0.0299 0.0542
0.9999 0.9966 0.9876 0.9756
9.0E-6 6.2E-4 0.0029 0.0066
0.9980 0.9682 0.9187 0.8692
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201
Table 3 Comparison of the false alarm probabilities and detection probabilities of the fuzzy sign detector and the crisp sign detector when 0 = 3 Fuzzy sign detector Additional Noise
Crisp sign detector A = 0.5
No Noise N(0, 1) N(0,2) N(0,3)
A = 1.0
A = 2.0
Pfa
Pd
Pf~
Pd
Pfa
Pd
Pfa
Pd
0.0020 0.0638 0.1852 0.3035
0.9999 0.9999 0.9999 0.9999
0.0011 0.0453 0.1431 0.2458
0.9999 0.9999 0.9999 0.9999
2.0E-4 0.0172 0.0714 0.1416
0.9999 0.9999 0.9999 0.9999
9.5E-7 4.9E-4 0.0042 0.0126
0.9999 0.9999 0.9997 0.9988
Table 4 Comparison of the false alarm probabilities and detection probabilities of the fuzzy sign detector and the crisp sign detector when 0 = 5 Fuzzy sign detector Additional Noise
Crisp sign detector A=0.5
No Noise N(0, 1) N(0,2) N(0,3)
A=I.0
Pfa
Pd
Pfa
Pd
Pfa
Pd
Pfa
Pd
0.0020 0.0770 0.2528 0.4314
0.9999 0.9999 0.9999 0.9999
0.0011 0.0587 0.2184 0.3941
0.9999 0.9999 0.9999 0.9999
1.7E-4 0.0284 0.1397 0.2863
0.9999 0.9999 0.9999 0.9999
1.0E-7 6.5E-4 0.0114 0.0449
0.9999 0.9999 0.9999 0.9999
Table 5 Relationship between the optimum values of the incredibility and additional noises Optimum value of the incredibility Additional noise
No Noise N(0, 0.2) N(0, 0.4) N(0, 0.6) N(0, 0.8) N(0, 1.0) N(0, 1.2) N(0,1.4) N(0,1.6) N(0,1.8) N(0, 2.0) N(0, 2.2) N(0, 2.4) N(0, 2.6) N(0, 2.8) N 10, 3.0)
A=2.0
0=1
0=2
0=3
0.000 0.344 0.679 0.996 1.293 1.572 1.835 2.088 2.333 2.572 2.807 3.041 3.274 3.508 3.743 3.980
0.000 0.344 0.682 1.005 1.311 1.600 1.873 2.136 2.392 2.624 2.861 3.095 3.328 3.561 3.794 4.028
0.000 0.344 0.687 1.021 1.341 1.644 1.920 2.201 2.459 2.706 2.947 3.183 3.416 3.647 3.878 4.109
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202
probabilities of the fuzzy sign detector are more similar to the preassigned ones than those of the crisp sign detector under additional noise environments. It should be remarked that the numerical results in Tables 1-4 are obtained with an arbitrarily chosen values of the incredibility; in other words, we may partially alleviate the effect of additional noise by just assigning an appropriate value to the incredibility without estimating the statistical characteristics of additional noise.
Theorem 2. For any zero-mean symmetric additional noise there exists a fuzzy sign detector whose false alarm probability equals to the preassigned size. Theorem 2 implies that there exists an optimum fuzzy sign detector of which performance is the same as that of the crisp sign detector with exact information on additional noise. When additional noise is normally distributed with zero-mean, the values of the incredibility of optimum fuzzy sign detectors whose false alarm probabilities equal to the preassigned size are given in Table 5.
4. Concluding remarks In this paper, we reconsidered the signal detection theory based on the fuzzy set theory, As a special case, the fuzzy sign detector was obtained, and some properties of the fuzzy sign non-linearity were described. We showed that the fuzzy sign detector has lower sensitivity of performance to small deviation of assumed noise statistic. The incredibility, which is a measure of fuzziness of the observations, has been shown to be able to control the false alarm probability when additional noise exists. The same procedure can be applicable to the random signal detection problem which is now under investigation. It is shown that the performance characteristics of the fuzzy signal detectors are strongly related to the incredibility of the available information. It seems to be an important problem to choose the incredibility for a given noise characteristic, which is also under investigation. An adaptive fuzzy signal detection scheme seems to be possible and is also under investigation.
Appendix A. Proofs of properties Proof of Property 1. Under the assumption that 0 > 0, since P(x_ I Ho) > P(x_ I H~) and P(x+ I Ho) < P(x+ IH1), we have g(x_ ) < 0 and g(x+ ) > O. [] Proof of Property 2. Since P ( x - I H o ) = P ( x + I H 1 ) and P ( x + l H o ) = P ( x - I H 1 ) , we have g ( r _ ) = -g(x+). [] Proof of Property 3. Since P(x_ I Ho) is an increasing function of 0 and P(x_ I H t ) is a decreasing function of 0, it is easy to see that g(~:_ ) is a decreasing function of 0, and that g(x +) is an increasing function of 0. [] Proof of Property 4. First we show that P(~c+ [HI) is a decreasing function of A and P(x_ I H1) is an increasing function of A. It is clear to see that, for A 1 > 0, P(tc+ {H,)14=o - P(x+ {H,){a=~,
S.Y. Kim et al. / Fuzzy Sets and Systems 74 (1995) 195-205
203
__~l)f(y_~)dy_f_° ~(l+ Y'~rf JJV--Oe,'J~dy _ ~ ) f ( y --~-)dy Oe,\ - Jo [a' -2\ l f l --~)f(Y +Oe,\__~)dy
>
O, where we used a change of variables. That is, P(x+ IH~) is maximum at A = 0. Now for A~ < A2: P(K+ I H1)Ia=A, - P(x+ I H~)la=a2
e(x+ I H1)ld=d, } - TOe,\ )dy
{P(K+ I nl)l~ = 0 - P(r+ I H,)ld=a2} -- {P(x+ I Hx)la=o 4:1(1
>
>
-
lYl~f(y-~)dy-f]'
~(1-1~)f(y
- N + a,/ \
O.
Similarly, it can be shown that P(x_ I H~) is an increasing function of A. It thus follows that a decreasing function of A and g(x_) is an increasing function of A. []
g(x+) is
Proof of Property 5. We have
E[g(x)l H~] = g(x+)P(x+I H~) + g(x_)P(x_ I H1) = g(x+) {P(x+ I H ~ ) - P(x_ Inx)} = g ( x + ) { P ( x - I H o ) - P(x+ I Ho)} -- - {g(~c+)P(~+ I Ho) +
g(~c_)P(x_I Ho)}
= - E[g(K) I Ho]. Similarly, we can show that E[T(K)I Ho] = -
E[T(x)I H~].
[]
Proof of Property 6. We have E [ g 2 ( x ) l H 1 ] = g2(x+)e(x+ IH1) + g 2 ( x - ) P ( x -
H1)
= g2(x+){P(x+ IH1) + P(x_ IH~) = g2(t¢+)
= g2(x+)P(x+ IHo) + g 2 ( x - ) P ( x = E[g2(x)l
Ho)
Ho].
Since E 2 [g(x)l Ho] = E 2 [g(x)l HI ] from Property 5, we have V [g(x)l Ho] = can show that V [ T ( x ) I H o ] = V[T(x)IH1]. []
V[g(x)IHI]. Similarly, we
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S.Y. Kim et al. / Fuzzy Sets and Systems 74 (1995) 195 205
Appendix B. Proofs of theorems Proof of Theorem 1. Let u+ and u_ be the n u m b e r of K+'s and x ' s
in the observation x, respectively, F = {r j} be the ordered sample space with elements arranged in the decreasing order of corresponding value of the FS test statistic or equivalently in the decreasing order of the value (u + - u_ ), and zj be t h e j t h element o f F . Since P(x+ I Ho) is an increasing function of A, P(x÷ I Ho) + P(~:- I Ho) = 1, and 0 ~< P ( x t Ho) ~< ½, we see that P ( x _ I H o ) P ( x + I Ho) is an increasing function of A. N o w we see that P(xl Ho) can be expressed as follows: P ( K t H o ) = f i P(~clIHo)
i=1
= t l q U+ i = l { P ( x - I H o ) P ( x + l H o ) } F l i = ~U U, P ( x - I H o ) ,
for u_ > u + ,
HT=~{P(x-IHo)P(~+IHo)}FIT--+~" P ( x + l H o ) ,
(B.1)
for u_ ~
F r o m (B. 1) we can easily see that P(x[ Ho) is an increasing function of A when u ~< u + and P(xl Ho) can be an increasing or a decreasing function of A when u_ > u+. N o w let F+ and F_ be the subset of the sample space in which P(x[ Ho) is an increasing function of A and the subset of the sample space in which P(xl Ho) is a decreasing function of A, respectively. Then we can see that F+ is the same as {zi[ 1 ~ j ~< # ( F ÷ )} and F_ is the same as {r~ I # (F + ) + 1 ~ 0, when d l < d2. Thus we see that t / , _~ ~ 2 for A1 < A2. Since A can range through [0, Go), we finally have ~o -~ ~A for A > 0. As an example, when 0 = 2, ei = 1, i = 1 . . . . . 4, n = 4, A~ = 0.5, A2 = 1, and W~ is normally distributed, P (xl Ho) and )~lj= 1 [P(z~I Ho) for I in [1, # (F)] are obtained and tabulated in Table 6. [ ]
Proof of Theorem 2. Since the nonlinearities of the crisp and fuzzy sign detectors take a form of the likelihood ratio, we can implement the crisp sign detector with the performance preassigned if we k n o w the exact probabilities of K+ and K for A = 0 under Ho. U n d e r the assumption that the noise p.d.f, is even symmetric a b o u t the origin, we have P(x+ [Ho) + P ( x _ l H o ) = 1, P ( x _ I H 1 ) + P ( x + I H 1 ) = 1, and P(~c I H o ) = P ( x + I H 1 ) for any A ~>0. Therefore if we k n o w the value of P ( x IHo), we can also find the values of P(K+ IHo), P(~c [H1), and P(x+ IH1). Consequently if P(K [ Ho) obtained without considering additional noise for A = A 1 is the same as the exact probability of x for A = 0 under Ho when the information on additional noise is available, the decision rule of the fuzzy sign detector with A = A 1 is the same as that of the o p t i m u m crisp sign detector under additional noise. In addition P(K_ I Ho) is a continuous and decreasing function of A, and the exact probability of x_ for A = 0 under Ho when additional noise exists is lower than P ( x _ I Ho)l A=o. Then we see that there exists
Table 6 P0c] Ho) and yj P(rjl Ho) when A = 0.5 and A = 1 P(~c]Ho)
All x~'s are h-+ Three x~'sare ~+ and one h-~is hTwo xi's are x+ and two xi's are x One xl is x+ and three xi's are • All xi's are ~:
1 4 6 4 1
)~ j P(rjl Ho)
A=0.5
A=I
A=0.5
A=I
8.059E-4 3.977E-3 1.963E-2 9.687E-2 4.780E-1
1.453E-3 5.988E-3 2.468E-2 1.018E-I 4.195E-1
8.059E-4 1.671E-2 1.345E-1 5.220E-1 1
1.453E-3 2.541E-2 1.735E-1 5.807E-1 1
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205
a value of A # 0 for which the probability P(x_ IHo) obtained without considering additional noise is equal to the exact probability of x_ for A = 0 under H0, from which it follows that there exists an optimum value of A for a preassigned performance when zero-mean additional noise exists. []
Acknowledgement The authors wish to thank the anonymous reviewers for their invaluable comments and suggestions. This research was supported in part by the Non-Directed Research Fund, Korea Research Foundation, in 1993, for which the authors would also like to express their thanks.
References [-1] K. Arakawa, Y. Arakawa and H. Harashima, Digital signal processing using fuzzy logic for biomedical signals, Proc. lnternat. Conf. on Fuzzy Logic, Neural Networks, Iizuka, Japan (1990) 95-98. [2] M.R. Casals, M.A. Gil and P. Gil, On the use of Zadeh's probabilistic definition for testing statistical hypotheses from fuzzy information, Fuzzy Sets and Systems 20 (1986) 175-190. [3] M.R. Civanlar and H.J. Trussell, Digital signal restoration using fuzzy sets, IEEE Trans. Acoust. Speech Signal Process. ASSP-34 (1986) 919-936. [4] A. De Luca and S. Termini, A definition ofa nonprobabilistic entropy in the setting fuzzy sets theory, Inform. and Control 20 (1972) 301 312. [-5] A. Dziech and M.B. Gorzalczany, Decision making in signal transmission problems with interval-valued fuzzy sets, Fuzzy Sets and Systems 23 (1987) 191-203. I'6] J.C. Son, I. Song and H.Y. Kim, A fuzzy decision problem based on the generalized Neyman-Pearson criterion, Fuzzy Sets and Systems 47 (1992) 65-75. 1'7] I. Song and S.A. Kassam, Locally optimum detection of signals in a generalized observation model: the known signal case, IEEE Trans. Inform. Theory IT-36 (1990) 502 515. [8] L.A. Zadeh, Probability measures of fuzzy events, J. Math. Anal. Appl. 23, (1968) 421-427.
ZZY
sets and systems ELSEVIER
Fuzzy Sets and Systems 74 (1995) 195 205
Performance characteristics of the fuzzy sign detector S u n Y o n g K i m ", I i c k h o S o n g "'*, J a e C h e o l S o n b, S a n g y o u b K i m ~ aDepartment of Electrical Engineering, Korea Advanced Institute of Science and Technology (KA1ST), 373-1 Guseong Dong, Yuseong Gu, Daejeon, 305-701, South Korea bDigital Signal Processing Technology Center, Samsung Electronics Co., LTD., San 14, Nongseo-Ri, Kihung-Eup, Yongin-Kun, Kyungki-Do, 449-900, South Korea cDepartment of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109, USA Received December 1993; revised October 1994
Abstract
In this paper, fuzzy statistical techniques are introduced into the optimum signal detection problems. The likelihood ratio for fuzzy detection of known signals is obtained. As a special case of the fuzzy signal detector, a fuzzy set theoretic approach to sign detection of known signals is considered. Specifically, the test statistic of the fuzzy sign detector for known signals is obtained. Some properties of the fuzzy sign nonlinearity, which constitutes the fuzzy sign detector, are also described. Finally, the performance characteristics of the fuzzy sign detector are investigated and compared to those of the crisp sign detector.
Keywords: Fuzzy sets; Fuzzy test; Fuzzy sign detector
1. Introduction
In signal processing areas, we face many practical problems where imprecise information is common. Based on this observation, applications of fuzzy set theory have been considered in several signal processing problems (e.g., [1, 3]). Although the fuzzy set theory has already been introduced into a signal detection area based on the concept of interval-valued fuzzy sets [5], it is also a reasonable approach to combine the fuzzy testing of statistical hypothesis [6-1 and signal detection theory. The rationale to connect the fuzzy set theory and the signal detection theory may partially be explained as follows. Although we cannot exactly estimate the statistical characteristics of the noise process and it is hard to assume that there is no additional noise such as quantization error and the self-noise of the processor in practice, we usually neglect additional noise in modeling the physical situations. Obviously, we may partially compensate this negligence by estimating the effect of both facts. This method is, however, somewhat cumbersome and time-consuming in general. It would be more convenient and reasonable to rely on the thought that the observations provide us with fuzzy information such as "the real value is just around the * Corresponding author. Tel.: + 82-42-869-3445. Fax: + 82-42-869-3410. E-mail: [email protected]. 0165-0114/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved SSDI 0 1 6 5 - 0 1 1 4 ( 9 4 ) 0 0 3 2 7 - 0
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S.Y. Kim et al. / Fuzzy Sets and Systems 74 (1995) 195-205
observed value", and consequently to employ the fuzzy information processing techniques. The objective of this paper is to obtain detection schemes based on the fuzzy testing of statistical hypothesis. Specifically, we consider a fuzzy set theoretic approach to sign detection and obtain the detection schemes having lower sensitivity of performance to small deviation of assumed noise statistic. In this paper, we reconsider the signal detection theory based on the fuzzy set theory and fuzzy tests. We discuss the fuzzy sign detector as a particular example of the application of the fuzzy decision problems in signal detection. A sign detector based on the signs of data is one of the best-known nonparametric detectors. While a rank detector, which is also one of non-parametric detectors, exhibits better performance than the sign detector in many cases, it is considerably more difficult to implement. The sign detector, on the other hand, is usually quite simple to implement. We describe some properties of the fuzzy sign nonlinearity, which constitutes the fuzzy sign detector. They include statistical properties of the fuzzy sign nonlinearity. In order to discuss performance charcteristics of the crisp and fuzzy sign detectors, numerical results are presented under additional noise environments. We show that the performance characteristics of the fuzzy sign detector is strongly related to the incredibility, which is a measure of fuzziness of the observations, and obtain the optimum values of the incredibility under various additional noise environments. 2. Preliminaries
Let X = (X, Bx, Fx) be an experiment, where X is a set in the real line ~, Bx is the Borel a-field on X, and Fx belongs to a family of probability distributions A on the measurable space (X, Bx). Suppose that the experimentation from X does not provide exact information, but provides only fuzzy information. For mathematical handling of the fuzzy observations in such circumstances, let us introduce the concepts of the fuzzy information system [2] and sample fuzzy information [2] in Definitions 1 and 2, respectively. Definition 1. A fuzzy information system ~ associated with the experiment X is a fuzzy partition (orthogonal system) of X by means of fuzzy events. Definition 2. A fuzzy random sample of size nfrom ~, denoted by ~n~, is a fuzzy partition of a random sample of size n from X, by means of fuzzy events of the random sample, in such a way that each element called sample fuzzy information in ~")is characterized by means of an n-tuple of fuzzy information, x = (xl, ..., x,), xi ~ ~, i = l,...,n. The probability distribution of a fuzzy event was first introduced in [8], which has been directly extended to the case of the fuzzy random sample as follows. Definition 3. The probability distribution of the fuzzy random sample (~"~is a mapping P from ~"~ to [0, 1] given by
P(x) = fx. pK,(xl) ... I~.(x,)dFx.(Xl, ... ,x,),
(1)
where the integral is the Lebesgue-Stieltjes integral, Fxn(Xl . . . . . x,) is the probability distribution on (X ", Bxo) determined by Fx on (X, Bx), #~, is called the membership function of xi, and X"=Xx
... xX.
n times
Let us introduce a few more definitions for the hypotheses testing with fuzzy observations. The definitions of a fuzzy test, a fuzzy test function, and a power function of the fuzzy test are not quite different from the crisp counterparts. For instance, a rule with which we choose for each sample fuzzy information x in (~n~
S.Y. Kim et al. / Fuzzy Sets and Systems 74 (1995) 195-205
197
between the inferences "accept the null hypothesis" and "accept the alternative hypothesis" with probabilities 1 - ~b(x) and ~b(x), respectively, is called a fuzzy test. In addition, the function ~b from ~'~ to [0, 1], allocating to each x the probability ~b(x) of accepting the alternative hypothesis, is called a fuzzy test function. Definition 4. The power function fl, of the fuzzy test is a mapping from A to [0, 1] and is defined by
fl,(Fx) = ~
dp(x)P(x)
for all Fx e A.
Ke~n~
(2)
Definition 5. Let Ao be the set of distributions in A for which a given null hypothesis is true, and let Foe Ao. Then the value sup FoeAofl,(Fo) is called the size of the fuzzy test. If the size of the fuzzy test is at most equal to ct with ct e [0, 1], then the fuzzy test is said to be a fuzzy test of size ot or a size-orfuzzy test.
3. Fuzzy sign detector for known signals
3.1. Observation model
Let us consider the well-known signal-detection problem which is described as a binary hypotheses testing problem: Ho:
1I/= W/,
(3)
Y/= 0el + IV/,
(4)
versus Hi:
for i = 1 . . . . . n. In the above formulation, e~ is the known signal component, W~ is the purely-additive noise component, and Yi is the observation at the ith sampling instant. The quantity 0 is the amplitude parameter which controls the signal strength. The purely-additive noise components Wi, i = 1, ..., n, are assumed to be independent and identically distributed (i.i.d.) with the common continuous probability density function (p.d.f)f. It is assumed that the p.d.f, f is zero-mean, even-symmetric ( f ( y ) = f ( - y)), and unimodal. Note that the observation model is the same as that in most conventional studies on known-signal detection (e.g., [7]). However, some difference in the amount of fuzziness on sample information exists. That is, in this paper the available probability distribution from the experiment is P(x) induced from H 7= l f(Y~). 3.2. Test statistic
The test statistic for the fuzzy signal detection problem based on the fuzzy set theoretic extension of the Neyman-Pearson criterion is defined in terms of the likelihood ratio. The test statistic can easily be derived to be T(x) = ~" gopt(tq),
(5)
i=1
where gopt(Ki) =
In
P(KiIH1) P(x~[ Ho)
= In I#~,(x)f(x - OeDdx I~K,(x)f (x) dx
(6)
In (6), the function #opt(X~) is called the optimum fuzzy nonlinearity and characterizes the detector structure.
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S. E Kim et al. / Fuzzy Sets and Systems 74 (1995) 195-205
As a special case of the fuzzy signal detector, let us consider the situation where the available informations 9reater than ½0e~ with the membership are: K_ = "approximately less than ~~ .ve~ and. x + = "'approximately . . . functions x_ and x+ having the following properties: (1) The membership function /~K (x) is monotonically decreasing with l i m x ~ / ~ ( x ) = 0 and limx~-o~#~ ( x ) = 1. (2) The membership function p~.(x) is monotonically increasing with limx~o~/~÷(x)= 1 and limx~_~ pK+(x) = 0. (3) For any x, #~ (x) + / ~ . ( x ) = 1 a n d / ~ (x) = gK+(- x). Typical examples of the membership functions are shown in Fig. 1. In Fig. l(a), the parameter A, which we call the incredibility, is a measure of fuzziness of observations. When the value of A is large, we cannot distinguish between x_ and x +. We consider small values of A assuming that the fuzziness of observations is relatively small. It should be noted that the membership function with A = 0 is for the crisp sign detection problem. The membership functions in Fig. 1(b) are more general than those of Fig. 1(a). In Fig. 1(b), other measures of fuzziness (e.g., [4]) should be used for the fuzziness of observations. The membership functions in Fig. 1(a) are relatively easy to handle mathematically and many cases produces good results.
3.3. Some properties of fuzzy sign nonlinearity In this section, we present several properties of the fuzzy sign nonlinearity. We will denote by 9(x) the fuzzy sign (FS) nonlinearity. From (6) the FS nonlinearity g(x) is given as follows:
g(xi) = [ l n ( f PK (x)f(x - Oei)dx / f l~K ( x ) f ( x ) d x )
for xi = x - ,
(7)
ln(f,~+(x)f(x-Oe,)dx/f~K+(x)f(x)dx)
for x , = x + .
The FS nonlinearity (7) has the following properties. Property 1. 9(x = r~_ ) < 0 and g(x = x+ ) > O. Property 2. g(x = x _ ) = - g ( x = x+).
In other words, the FS nonlinearity is an odd-symmetric function of x. In addition, as a function of 0, the FS nonlinearity has the following properties. Property 3. 9(x = ~:- ) is a decreasing function of O, while O(x = ~c+) is an increasin9 function of O. Property 4. For the membership functions of Fig. 1(a), g(x = ~c_ ) is an increasino function of A, while 9(x = x+ )
is a decreasin9 function of A. For the statistical average and variance of the FS nonlinearity we have Properties 5 and 6. Property 5. E [9(x) l Ho] = - E [ 9 ( x ) I H1] and E[T(x)I Ho] = - E [ T [(x)l Hi ], where E [ ' ] denotes the
expectation. Property 6. VEg(x) l H o ] = V[g(x)lH1]
variance.
and V [ T ( x ) I H o ] = V[T(x)IHx], where V [ ' ] denotes the
S.Y. Kim et al. / Fuzzy Sets and Systems 74 (1995) 195-205
0ei 2
A
0ei -~-
199
0e i T+A
(a) ~t_(x)
~t.(x)
0.5 x
0e 2
(b) Fig. 1. Membership functions of x_ and x+.
Proofs of Properties 1-6 can be found in Appendix A. The FS nonlinearity which constitutes the test statistic of the fuzzy sign detector compensates for the effects of the noise. The observation in Property 3 is quite natural since 0 is directly related to the signal to noise ratio: that is, the confidence of the information increases as 0 increases. In Property 4 we can also make a similar observation. 3.4. Performance characteristics
Here we will consider the performance characteristics of the fuzzy sign detector with the membership functions in Fig. 1(a). Let us investigate performance characteristics of the fuzzy sign detector and compare them to those of the crisp sign detector. The proofs of these theorems are given in Appendix B.
Theorem 1. Let tPa be {xl the subset of the sample space in which the fuzzy sign detector with the incredibility A accepts H1 } and ~Po be {xl the subset of the sample space in which the crisp sign detector accepts H1 }. Then ~eo w_ ~pa, w_ ~pa2for any A1 < A2.
Corollary 1. For A ~ < A 2 and a preassigned size at, we have El'~bcrisp(~C)I Ho']/> El'q~fuz~y(t¢)I Ho] 14=a, 1> El-~bfu~y(~C)I Ho-] la=a2 and
E[~bcri~p(X)I HI] i> E[q~fu~y(X)]H~] I~=~,/> E[Oroz~y(,c)l H , ] I~=~,
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where (~crisp(/~) and (])fuzzy(X) a r e the crisp and fuzzy sign test functions, respectively, and E l . ] means the expectation with respect to the noise p.d.f, including additional noise. Corollary 1 follows immediately from Theorem 1. Thus for a preassigned size, which may not be equal to the false alarm probability, the false alarm and detection probabilities of the crisp sign detector are greater than those of the fuzzy sign detector in practical noise environments. Corollary 1 suggests that the difference between a preassigned size and the false alarm probability resulting from the effect of practical noise including additional noise can be diminished by employing the fuzzy set theoretic approach. In other words, since the incredibility decreases the false alarm probability and additional noise increases the false alarm probability, we can find a value of the incredibility for which the false alarm probability equals to the preassigned ~ in practical noise environments including additional noise. As examples, the false alarm and detection probabilities of the crisp and fuzzy sign detectors in various cases are given in Tables 1-4 when additional noise has normal distributions. It should be remarked that the crisp and fuzzy sign detectors in Tables 1-4 are designed for a preassigned size under the assumption that no additional noise exists. We assumed that the desired size ~ = 0.002, e~ = 1, i = 1, ..., 20, the sample size n = 20, and W~ ~N(0, 1), that is, the noise component is normally distributed with zero mean and unit variance. From Tables 1-4, we see that the crisp sign detector provides better performance than the fuzzy sign detector when no additional noise exists. The performance of the fuzzy sign detector, however, is closer to the expected performance than that of the crisp sign detector when additional noise exists. That is, the false alarm Table 1 Comparison of the false alarm probabilities and detection probabilities of the fuzzy sign detector and the crisp sign detector when 0 = 1 Fuzzy sign detector Additional Noise
Crisp sign detector A=0.5
No Noise N(0, 1) N(0,2) N(0,3)
A=I.0
A=2.0
Pfa
Pd
Pfa
lPd
-'°fa
Pd
Pfa
Pd
0.0020 0.0093 0.0170 0.0238
0.7530 0.5672 0.4761 0.4220
0.0016 0.0077 0.0144 0.0203
0.7301 0.5405 0.4496 0.3963
8.6E-4 0.0048 0.0086 0.0123
0.6390 0.4424 0.3564 0.2764
1.7E-4 0.0011 0.0023 0.0035
0.4525 0.2701 0.2024 0.1673
Table 2 Comparison of the false alarm probabilities and detection probabilities of the fuzzy sign detector and the crisp sign detector when 0 = 2 Fuzzy sign detector Additional Noise
Crisp sign detector d = 0.5
No Noise N(0,1) N (0, 2) N (0, 3)
A = 1.0
A = 2.0
Pfa
Pd
Pfa
Pd
Pfa
Pd
Pfa
Pd
0.0020 0.0318 0.0812 0.1308
0.9999 0.9993 0.9971 0.9933
0.0013 0.0222 0.0593 0.0983
0.9999 0.9984 0.9937 0.9870
3.7E-4 0.0096 0.0299 0.0542
0.9999 0.9966 0.9876 0.9756
9.0E-6 6.2E-4 0.0029 0.0066
0.9980 0.9682 0.9187 0.8692
S.Y. Kim et al. / Fuzzy Sets and Systems 74 (1995) 195-205
201
Table 3 Comparison of the false alarm probabilities and detection probabilities of the fuzzy sign detector and the crisp sign detector when 0 = 3 Fuzzy sign detector Additional Noise
Crisp sign detector A = 0.5
No Noise N(0, 1) N(0,2) N(0,3)
A = 1.0
A = 2.0
Pfa
Pd
Pf~
Pd
Pfa
Pd
Pfa
Pd
0.0020 0.0638 0.1852 0.3035
0.9999 0.9999 0.9999 0.9999
0.0011 0.0453 0.1431 0.2458
0.9999 0.9999 0.9999 0.9999
2.0E-4 0.0172 0.0714 0.1416
0.9999 0.9999 0.9999 0.9999
9.5E-7 4.9E-4 0.0042 0.0126
0.9999 0.9999 0.9997 0.9988
Table 4 Comparison of the false alarm probabilities and detection probabilities of the fuzzy sign detector and the crisp sign detector when 0 = 5 Fuzzy sign detector Additional Noise
Crisp sign detector A=0.5
No Noise N(0, 1) N(0,2) N(0,3)
A=I.0
Pfa
Pd
Pfa
Pd
Pfa
Pd
Pfa
Pd
0.0020 0.0770 0.2528 0.4314
0.9999 0.9999 0.9999 0.9999
0.0011 0.0587 0.2184 0.3941
0.9999 0.9999 0.9999 0.9999
1.7E-4 0.0284 0.1397 0.2863
0.9999 0.9999 0.9999 0.9999
1.0E-7 6.5E-4 0.0114 0.0449
0.9999 0.9999 0.9999 0.9999
Table 5 Relationship between the optimum values of the incredibility and additional noises Optimum value of the incredibility Additional noise
No Noise N(0, 0.2) N(0, 0.4) N(0, 0.6) N(0, 0.8) N(0, 1.0) N(0, 1.2) N(0,1.4) N(0,1.6) N(0,1.8) N(0, 2.0) N(0, 2.2) N(0, 2.4) N(0, 2.6) N(0, 2.8) N 10, 3.0)
A=2.0
0=1
0=2
0=3
0.000 0.344 0.679 0.996 1.293 1.572 1.835 2.088 2.333 2.572 2.807 3.041 3.274 3.508 3.743 3.980
0.000 0.344 0.682 1.005 1.311 1.600 1.873 2.136 2.392 2.624 2.861 3.095 3.328 3.561 3.794 4.028
0.000 0.344 0.687 1.021 1.341 1.644 1.920 2.201 2.459 2.706 2.947 3.183 3.416 3.647 3.878 4.109
S.Y. Kim et al. / Fuzzy Sets and Systems 74 (1995) 195-205
202
probabilities of the fuzzy sign detector are more similar to the preassigned ones than those of the crisp sign detector under additional noise environments. It should be remarked that the numerical results in Tables 1-4 are obtained with an arbitrarily chosen values of the incredibility; in other words, we may partially alleviate the effect of additional noise by just assigning an appropriate value to the incredibility without estimating the statistical characteristics of additional noise.
Theorem 2. For any zero-mean symmetric additional noise there exists a fuzzy sign detector whose false alarm probability equals to the preassigned size. Theorem 2 implies that there exists an optimum fuzzy sign detector of which performance is the same as that of the crisp sign detector with exact information on additional noise. When additional noise is normally distributed with zero-mean, the values of the incredibility of optimum fuzzy sign detectors whose false alarm probabilities equal to the preassigned size are given in Table 5.
4. Concluding remarks In this paper, we reconsidered the signal detection theory based on the fuzzy set theory, As a special case, the fuzzy sign detector was obtained, and some properties of the fuzzy sign non-linearity were described. We showed that the fuzzy sign detector has lower sensitivity of performance to small deviation of assumed noise statistic. The incredibility, which is a measure of fuzziness of the observations, has been shown to be able to control the false alarm probability when additional noise exists. The same procedure can be applicable to the random signal detection problem which is now under investigation. It is shown that the performance characteristics of the fuzzy signal detectors are strongly related to the incredibility of the available information. It seems to be an important problem to choose the incredibility for a given noise characteristic, which is also under investigation. An adaptive fuzzy signal detection scheme seems to be possible and is also under investigation.
Appendix A. Proofs of properties Proof of Property 1. Under the assumption that 0 > 0, since P(x_ I Ho) > P(x_ I H~) and P(x+ I Ho) < P(x+ IH1), we have g(x_ ) < 0 and g(x+ ) > O. [] Proof of Property 2. Since P ( x - I H o ) = P ( x + I H 1 ) and P ( x + l H o ) = P ( x - I H 1 ) , we have g ( r _ ) = -g(x+). [] Proof of Property 3. Since P(x_ I Ho) is an increasing function of 0 and P(x_ I H t ) is a decreasing function of 0, it is easy to see that g(~:_ ) is a decreasing function of 0, and that g(x +) is an increasing function of 0. [] Proof of Property 4. First we show that P(~c+ [HI) is a decreasing function of A and P(x_ I H1) is an increasing function of A. It is clear to see that, for A 1 > 0, P(tc+ {H,)14=o - P(x+ {H,){a=~,
S.Y. Kim et al. / Fuzzy Sets and Systems 74 (1995) 195-205
203
__~l)f(y_~)dy_f_° ~(l+ Y'~rf JJV--Oe,'J~dy _ ~ ) f ( y --~-)dy Oe,\ - Jo [a' -2\ l f l --~)f(Y +Oe,\__~)dy
>
O, where we used a change of variables. That is, P(x+ IH~) is maximum at A = 0. Now for A~ < A2: P(K+ I H1)Ia=A, - P(x+ I H~)la=a2
e(x+ I H1)ld=d, } - TOe,\ )dy
{P(K+ I nl)l~ = 0 - P(r+ I H,)ld=a2} -- {P(x+ I Hx)la=o 4:1(1
>
>
-
lYl~f(y-~)dy-f]'
~(1-1~)f(y
- N + a,/ \
O.
Similarly, it can be shown that P(x_ I H~) is an increasing function of A. It thus follows that a decreasing function of A and g(x_) is an increasing function of A. []
g(x+) is
Proof of Property 5. We have
E[g(x)l H~] = g(x+)P(x+I H~) + g(x_)P(x_ I H1) = g(x+) {P(x+ I H ~ ) - P(x_ Inx)} = g ( x + ) { P ( x - I H o ) - P(x+ I Ho)} -- - {g(~c+)P(~+ I Ho) +
g(~c_)P(x_I Ho)}
= - E[g(K) I Ho]. Similarly, we can show that E[T(K)I Ho] = -
E[T(x)I H~].
[]
Proof of Property 6. We have E [ g 2 ( x ) l H 1 ] = g2(x+)e(x+ IH1) + g 2 ( x - ) P ( x -
H1)
= g2(x+){P(x+ IH1) + P(x_ IH~) = g2(t¢+)
= g2(x+)P(x+ IHo) + g 2 ( x - ) P ( x = E[g2(x)l
Ho)
Ho].
Since E 2 [g(x)l Ho] = E 2 [g(x)l HI ] from Property 5, we have V [g(x)l Ho] = can show that V [ T ( x ) I H o ] = V[T(x)IH1]. []
V[g(x)IHI]. Similarly, we
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S.Y. Kim et al. / Fuzzy Sets and Systems 74 (1995) 195 205
Appendix B. Proofs of theorems Proof of Theorem 1. Let u+ and u_ be the n u m b e r of K+'s and x ' s
in the observation x, respectively, F = {r j} be the ordered sample space with elements arranged in the decreasing order of corresponding value of the FS test statistic or equivalently in the decreasing order of the value (u + - u_ ), and zj be t h e j t h element o f F . Since P(x+ I Ho) is an increasing function of A, P(x÷ I Ho) + P(~:- I Ho) = 1, and 0 ~< P ( x t Ho) ~< ½, we see that P ( x _ I H o ) P ( x + I Ho) is an increasing function of A. N o w we see that P(xl Ho) can be expressed as follows: P ( K t H o ) = f i P(~clIHo)
i=1
= t l q U+ i = l { P ( x - I H o ) P ( x + l H o ) } F l i = ~U U, P ( x - I H o ) ,
for u_ > u + ,
HT=~{P(x-IHo)P(~+IHo)}FIT--+~" P ( x + l H o ) ,
(B.1)
for u_ ~
F r o m (B. 1) we can easily see that P(x[ Ho) is an increasing function of A when u ~< u + and P(xl Ho) can be an increasing or a decreasing function of A when u_ > u+. N o w let F+ and F_ be the subset of the sample space in which P(x[ Ho) is an increasing function of A and the subset of the sample space in which P(xl Ho) is a decreasing function of A, respectively. Then we can see that F+ is the same as {zi[ 1 ~ j ~< # ( F ÷ )} and F_ is the same as {r~ I # (F + ) + 1 ~
Proof of Theorem 2. Since the nonlinearities of the crisp and fuzzy sign detectors take a form of the likelihood ratio, we can implement the crisp sign detector with the performance preassigned if we k n o w the exact probabilities of K+ and K for A = 0 under Ho. U n d e r the assumption that the noise p.d.f, is even symmetric a b o u t the origin, we have P(x+ [Ho) + P ( x _ l H o ) = 1, P ( x _ I H 1 ) + P ( x + I H 1 ) = 1, and P(~c I H o ) = P ( x + I H 1 ) for any A ~>0. Therefore if we k n o w the value of P ( x IHo), we can also find the values of P(K+ IHo), P(~c [H1), and P(x+ IH1). Consequently if P(K [ Ho) obtained without considering additional noise for A = A 1 is the same as the exact probability of x for A = 0 under Ho when the information on additional noise is available, the decision rule of the fuzzy sign detector with A = A 1 is the same as that of the o p t i m u m crisp sign detector under additional noise. In addition P(K_ I Ho) is a continuous and decreasing function of A, and the exact probability of x_ for A = 0 under Ho when additional noise exists is lower than P ( x _ I Ho)l A=o. Then we see that there exists
Table 6 P0c] Ho) and yj P(rjl Ho) when A = 0.5 and A = 1 P(~c]Ho)
All x~'s are h-+ Three x~'sare ~+ and one h-~is hTwo xi's are x+ and two xi's are x One xl is x+ and three xi's are • All xi's are ~:
1 4 6 4 1
)~ j P(rjl Ho)
A=0.5
A=I
A=0.5
A=I
8.059E-4 3.977E-3 1.963E-2 9.687E-2 4.780E-1
1.453E-3 5.988E-3 2.468E-2 1.018E-I 4.195E-1
8.059E-4 1.671E-2 1.345E-1 5.220E-1 1
1.453E-3 2.541E-2 1.735E-1 5.807E-1 1
S.Y. Kim et al. / Fuzzy Sets and Systems 74 (1995) 195-205
205
a value of A # 0 for which the probability P(x_ IHo) obtained without considering additional noise is equal to the exact probability of x_ for A = 0 under H0, from which it follows that there exists an optimum value of A for a preassigned performance when zero-mean additional noise exists. []
Acknowledgement The authors wish to thank the anonymous reviewers for their invaluable comments and suggestions. This research was supported in part by the Non-Directed Research Fund, Korea Research Foundation, in 1993, for which the authors would also like to express their thanks.
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