In this section, we discuss the relationship between optimal parallel Γn system and the optimal Ξn system through the following propositions S(n), T (n), and V(n).
Proposition 4.1. S(n) : For the parallel n-sensor system Γnand for arbitrary prior P (H1) = r, the optimal error probability is achieved by identical sensors and k-out-of-n fusion rules.
Proposition 4.2. T (n) : For the n-sensor Ξn system, if the broadcast sensor has m ≥ 1 ballots in the voting fusion, then the optimal fusion rules are k-out-of-n fusion rules.
Proposition 4.3. V(n) : For the n-sensor Ξn system with a fixed k-out-of-n fusion rule and for arbitrary prior P (H1) = r, the optimal error probability is achieved by identical sensors, i.e., the (n − 1) ordinary sensors ignore the received decision of the broadcast sensor and use local decision rules that are the same as the broadcast sensor one, and that are based on their own observations only.
Lemma 4.6. If the proposition S(2) holds, then the proposition T (3) holds.
Proof. As discussed in the preceding section, the optimal fusion rule for the Ξn system is k-out-of-(n − 1 + m) fusion rules for some m ≥ 0; hence, it suffices to show that for m ≥ 1, the optimal m = 1.
Assume that S(2) holds. Let us consider the case of m = 1 and 1 ≤ k ≤ 3. These are obviously k-out-of-3 fusion rules. Now we consider the case of m = 2 and 1 ≤ k ≤ 4, for which the fusion rule can be expressed as
u0 =
½ 1, if u1+ u2+ 2u3 ≥ k 0, if u1+ u2+ 2u3 < k.
Observe from the above discussion that for the case of m = 2 and k = 1 and the case of m = 2 and k = 4, the fusion rules are equivalent to the 1-out-of-3 fusion rule and the
3-out-of-3 fusion rule, respectively. For the case of m = 2 and k = 2, we have
Thus, the minimum error probability corresponding to the 2-out-of-3 fusion rule is no greater than that for the case of m = 2 and k = 2.
Thus, the minimum error probability of the 2-out-of-3 fusion rule is no greater than that for the case of m = 2 and k = 3.
Note that the fusion rule for the case of m = 3 and k ≤ 2 is equivalent to the fusion rule for the case of m = 2 and k ≤ 2. For the case of m = 3 and k = 3, the fusion rule is u0 = u3,
and its minimum error probability is equal to Pe,min(1) , i.e., the minimum error probability of the single sensor; hence, it can not be the optimal fusion rule. The fusion rule for the case of m = 3 and k = 4, 5 is equivalent to the fusion rule for the case of m = 2 and k = 3, 4.
Finally, it is easy to see that the analysis for the case of m ≥ 3 is identical to the analysis for the case of m = 3.
The lemma is then substantiated since we have shown that for all m > 0 fusion rules, there are some k-out-of-n fusion rules that have error probabilities no greater than the original ones.
Lemma 4.7. For n ≥ 3, if
1. conditions (4.7), (4.8) and (4.9) are satisfied with θ0 = θ1 = η for 1 < k < n;
2. either PD(λ)λ
PF (λ)
is monotonic, or ˜Pe,n(n)> min
λ1,...,λn
(Pe(n));
3. either 1−PD(λ)λ
1−PF (λ)
is monotonic, or ˜Pe,1(n)> min
λ1,...,λn
(Pe(n)),
then proposition V(n) holds.
Proof. The proof is straightforward; hence, we omit it.
Lemma 4.8. If propositions S(2) and V(3) hold, then proposition S(3) holds.
Proof. For a fixed fusion rule, we have from the formula of the error probability of the parallel
system prob-ability of the parallel 2-sensor system with the LLRT thresholds λ1and λ2 and the fusion rule Υ0(u1, u2), respectively. PD<1>(λ1, λ2) and PF<1>(λ1, λ2) are defined similarly. Note that in the above derivation, the second inequality comes from proposition S(2), the third inequality follows from proposition T (3), and the last equality comes from proposition V(3).
Hence, we have min
λ1,λ2,λ3
(Pe(3)) = min
λ1=λ2=λ3
(Pe(3)), and the lemma is proved.
Theorem 4.3. If proposition S(2) holds, and if for n = 3 the conditions in Lemma (4.7) hold, then proposition S(3) holds.
Proof. The theorem can be easily obtained from the above lemmas.
Theorem 4.4. If propositions S(2), T (l) and V (l) hold for n ≥ l ≥ 3, then proposition S(l) holds for n ≥ l ≥ 1.
Proof. The theorem can be easily obtained from the above lemmas.
4.7 The Parallel Three-sensor System
Now we turn to the classification of exponential sources problem. Although we only discuss the optimal performance of the parallel three-sensor system, similar approach can be applied for the analysis of the system with more than three sensors.
For the classification problem of exponential sources, we can rewrite (4.8) as log
Combining (4.16) and (4.9), we obtain Q(n−1)D,k (θ0) − Q(n−1)D,k−1(θ1) rewrite the above equation as
Jk(n)(ξ) = Jk(n)(1), (4.19)
Some examples of Jk(n)(ξ)s are listed here for reference.
Taking the derivative of Jk(n)(ξ) with respect to ξ, we can show numerically that for a 6= b, Jk(n)(ξ) is either monotonically increasing or monotonically decreasing for 0 < ξ < 1 and 1 < k < n; hence, for 1 < k < n, (4.19) is satisfied if, and only if, a = b, i.e., θ0 = θ1. In other words, conditions (4.7), (4.8) and (4.9) are satisfied, only if θ0 = θ1. Moreover, from Lemma 4.1, (4.7) and (4.8), we learn that θ0 = θ1 results in θ0 = θ1 = η, i.e., conditions (4.7), (4.8) and (4.9) are satisfied, only when θ0 = θ1 = η for 1 < k < n.
In the above discussion, we know numerically that the first condition in the Lemma 4.7 holds. In addition, the third condition holds since
λ
is monotonic. As for the second condition in Lemma 4.7, since PD(λ)λ
PF (λ)
Hence,
η,θmin0,θ1
( ˜Pe,n(n)) = min
λ1
(Pe(1)) > min
λ1,...,λn
(Pe(n)),
Now, from Lemma 4.7, we notice that proposition V(n) holds. Moreover, if proposition T (l) holds for n ≥ l ≥ 3, then proposition S(l) also holds for n ≥ l ≥ 1, i.e., the optimal performance is achieved by the parallel systems with identical sensors.
The above arguments are, however, built partly based on numerical results. Nonetheless, for the relatively small numbers of sensors, we can show the same results analytically. In the following, we show the optimality of identical sensors on the classification of exponential sources for the parallel three-sensor system. Firstly, we will show the monotonicity of J2(3)(ξ).
One can show the monotonicity of other Jk(n)(ξ) for relatively small n in the same way.
Lemma 4.9. J2(3)(ξ) is a monotonic function for a 6= b.
Proof. Denote the ratio between a and b as ρ = ab. Then J2(3)(ξ) can be expressed as J2(3)(ξ) = a2ξ− bξ
bξ(1 − bξ)+ 1 = 1 − a−ξ − ρξ a−ξ − ρ−ξ. Taking the derivative of the above formula with respect to ξ, we have
J20(3)(ξ) = Ω(ξ) (a−ξ − ρ−ξ)2, where
Ω(ξ) = (−a−ξlog(a) + ρ−ξlog(ρ))(a−ξ− ρξ) − (a−ξ − ρ−ξ)(−a−ξlog(a) − ρξlog(ρ)).
Now taking the derivative of Ω(ξ) with respect to ξ, we have
Ω0(ξ) = ((log(ρ))2− (log(a))2)a−ξ(ρξ− ρ−ξ).
Thus, either Ω0(ξ) > 0 for ξ > 0 or Ω0(ξ) < 0 for ξ > 0. Moreover, Ω(0) = 0; hence, either Ω(ξ) > 0 for ξ > 0 or Ω(ξ) < 0 for ξ > 0. Thus, we have confirmed that J20(3)(ξ) is
either positive for all ξ > 0 or negative for all ξ > 0, i.e., J2(3)(ξ) is a monotonic function for 1 > ξ > 0.
Theorem 4.5. For the classification of exponential sources in the parallel three-sensor sys-tem, the optimal local decision rules are identical for all sensors.
Proof. From Theorem 4.3, proposition S(2) holds as shown in Section 4.3. Also, from the above discussion, the conditions in Lemma 4.7 are satisfied. Thus, the theorem is valid.
4.8 Problems with Similar ROCs
As long as the hypothesis testing problem has similar ROCs as those discussed for expo-nential hypothesis sources, their performances should be able to be evaluated in similar manner. Here, we illustrate two such examples. Classification problems of this sort might be encountered in survival analysis and failure time analysis.
The first example considers the following hypothesis testing problem:
H1 : Xi = min(Zi,1, · · · , Zi,β)
versus
H0 : Xi = min(Zi,1, · · · , Zi,γ)
for i = 1, 2, . . . , n and β < γ, where Xi is the observation of i-th sensor, and {Zi,j} are independent and identically distributed random variables with the associated PDF wZ(z) and CDF WZ(z). Thus, Xi has CDF FX(xi) = 1 − Pr(min(Zi,1, · · · , Zi,β) > xi) = 1 − (1 − WZ(xi))β and PDF fX(xi) = β(1 − WZ(xi))β−1wZ(xi) when H1 is true, and has CDF GX(xi) = 1 − Pr(min(Zi,1, · · · , Zi,γ) > xi) = 1 − (1 − WZ(xi))γ and PDF gX(xi) = γ(1 − WZ(xi))γ−1wZ(xi) when H0 is true.
From the discussion in the previous sections, we know that each sensor should apply local likelihood ratio tests as the local decision rules, and the local likelihood ratio test threshold for i-th sensor is given by
λi = fX(xi)
gX(xi) = ξ(1 − WZ(xi))β−γ,
where ξ = βγ. The detection probability PD and the false alarm probability PF for i-th sensor are therefore
PD(λi) = µ ξ
λi
¶β˜
and
PF(λi) = µξ
λi
¶γ˜ ,
where ˜β = γ−ββ and ˜γ = γ−βγ . We also have
PD(PF(i)) = PF(i)ξ and
λi = dPD(PF(i))
dPF(i) = ξPD(PF(i)) PF(i) ,
where by abusing the notations, PF(i) and PD(PF(i)) are the false alarm probability and the detection probability for the i-th sensor, respectively. Hence, the ROC of this classification problem is of the same form as the aforementioned classification of exponential sources problem. The discussion for the classification of exponential sources can accordingly be well-fit to this problem.
The second example is an analogue of the first example. Consider the following hypothesis testing problem:
H1 : Xi = max(Zi,1, · · · , Zi,β) versus
H0 : Xi = max(Zi,1, · · · , Zi,γ)
for i = 1, 2, . . . , n and β < γ, where Xi is the observation of i-th sensor, {Zi,j} are inde-pendent and identically distributed random variables with the associated PDF wZ(z) and CDF WZ(z). Thus, Xi has CDF FX(xi) = Pr(max(Zi,1, · · · , Zi,β) ≤ xi) = WZ(xi)β and PDF fX(xi) = βWZ(xi)β−1wZ(xi) when H1is true, and has CDF GX(xi) = Pr(max(Zi,1, · · · , Zi,γ) ≤ xi) = WZ(xi)γ and PDF gX(xi) = γWZ(xi)γ−1wZ(xi) when H0 is true.
The local likelihood ratio test threshold for i-th sensor is
λi = fX(xi)
gX(xi) = ξWZ(xi)β−γ,
where ξ = βγ. The detection probability PD and the false alarm probability PF for i-th sensor are equal to
PD(λi) = 1 − µξ
λi
¶β˜
and
PF(λi) = 1 − µ ξ
λi
¶˜γ ,
where ˜β = γ−ββ and ˜γ = γ−βγ . By the above setting, we immediately have
1 − PD(PF(i)) = (1 − PF(i))ξ
and
λi = dPD(PF(i))
dPF(i) = ξ1 − PD(PF(i)) 1 − PF(i) ,
where PF(i) and PD(PF(i)) are again the false alarm probability and the detection probability for the i-th sensor, respectively. Hence, the ROC of this classification problem is also a mirror of the ROC of the classification of exponential sources. Consequently, with slight modification, we can have similar results as the classification of exponential sources problem.
4.8.1 Decentralized Classification of Heavy-tailed Sources Prob-lems
The heavy-tailed distributions, specifically the Pareto distribution, are related to the self-similar phenomena in a way that if the packet inter-arrival process is modelled as i.i.d. Pareto random variables, the packet counting process is asymptotically second-order self-similar process with H = (3 − α)/2, where α is the Pareto parameter.
In practical control of the network traffic, one might need to test whether its self-similarity is weak or strong to determine whether the long-range dependence can or cannot be ignored.
To reduce the response time and to alleviate the load of network, a decentralized scheme for the detection of the self-similarity might be useful. As a result, one might need to consider the following binary hypothesis testing problem:
H1 : fX(xi) = β 1 xβ+1i versus
H0 : gX(xi) = γ 1 xγ+1i , or equivalently,
H1 : FX(xi) = 1 − 1 xβi versus
H0 : GX(xi) = 1 − 1 xγi
for i = 1, 2, . . . , n, β < γ and xi ≥ 1, where xi is the observed value of the random variable Xi with the associated PDF fX(xi) and CDF FX(xi). Here, we assume that {Xi} form a set of independent and identically distributed random variables. For a fixed fusion rule, the local likelihood ratio tests are
β xβ+1i
γ xγ+1i
R λi,
or equivalently,
xi R ti
for i = 1, 2, . . . , n, where λi and ti are some constants to be decided, and ti = (λiγβ)γ−β1 . It turns out that PD(λ) and PF(λ) of the above testing problem have the same forms as the classification of exponential sources problem in Section 4.1; hence, the previous result can be applied to the testing problem for the Pareto distributions directly.
4.9 Gaussian Classification Problems
All the previous parts in this chapter discuss mainly on the classification of exponential sources problem (or problems with the same ROC). In this section, we briefly discuss another classification problem that has drawn more attention among researchers, i.e., the classifica-tion of Gaussian sources problem.
Let us introduce some notations first.
Definition 4.2. If X is a Gaussian random variable with mean µ and variance σ2, then it has a probability density function
cX(x; µ, σ) = 1
√2πσe−(x−µ)22σ2 = φ(x−µσ ) σ , and a distribution function
CX(x; µ, σ) = Z x
−∞
cX(x; µ, σ) = Φ
µx − µ σ
¶ ,
where φ(·) and Φ(·) are respectively the probability density function and cumulant distribu-tion funcdistribu-tion of the standard normal distribudistribu-tion, i.e., the Gaussian distribudistribu-tion with µ = 0 and variance σ = 1.
We then concern the following binary hypothesis testing problem for Gaussian distribu-tions:
H1 : P (xi) = c(xi; µ, 1) versus
H0 : P (xi) = c(xi; −µ, 1)
for i = 1, 2, . . . , n and β < γ, where xi is the observation of i-th sensor, and without loss of generality, we assume σ = 1. For a fixed fusion rule, it is known that the optimal local decision rules are local likelihood ratio tests, namely,
√1
2πσe−(x−µ)22σ2
√1
2πσe−(x+µ)22σ2 R λi
or equivalently,
xi R ηi
for i = 1, 2, . . . , n, where λi and ηi are some constants to be decided.
The optimal strategy of the two-sensor system under this setting has been solved in [34], in which they showed analytically that the identical local decision rules are optimal.
However, for the system with more than two sensors, the desired result that identical sensor system is optimal is still absent. Here, we offer an alternative argument that is partly built on numerical results. Firstly, let us examine the conditions in Lemma 4.7. For Gaussian classification problem, it is easy to show analytically that the second and third conditions are satisfied. As for the first condition, we can show numerically that it is valid for relatively small n. Thus, we can conclude from Theorem 4.3 that the optimal three-sensor system still employ identical sensors.
Chapter 5 Conclusions
5.1 Self-Similar Traffic Generators
In the first part of this dissertation, we propose a filter-based generator for the synthesization of self-similar traffics. It can produce long range dependent traffics with adjustable levels of bustiness and correlation, and is parsimonious in the number of model parameters. Precisely, only three input parameters are required, i.e., the self-similar parameter H (which controls the bustiness and autocorrelation of the synthesized traffic), the mean of the traffic λ, and the length of the filter W (which also determines the effective aggregation size in the variance-time analysis). Despite the finite variance-time scales of the self-similar phenomenon in the synthesized traffic, it actually agrees with the measured behavior of true network traffic, i.e., the self-similar nature only lasts beyond a practically manageable range, but disappears as the considered aggregated window is much further extended [4, Fig. 2]. When it is compared with exiting self-similar traffic synthesizers, e.g., the RMD and the Paxson IFFT algorithm, the proposed filter-based synthesizer has the advantages that the synthetic traffic can be generated on the fly, and always produces non-negative valued traffic.
Comparisons of the complexities of self-similar traffic generators are as follows. Given that the length of the synthesized traffic is n, the number of complex multiplications required
for the Paxson IFFT method [19] is about (n/2)(log2n + 2). Our filter-based approach, on the other hand, requires n × W complex multiplications, where W represents the truncation window size. After analytically analyzing our approach based on variance-time test, we conclude that our synthesizer guarantees the generation of a traffic with desired degree of self-similarity beyond the intended range.
5.2 Correlation Approximation to the Mutual Infor-mation of Self-Similar Processes
We discuss the implications between the correlation coefficient (a quantity that only mea-sures the linear dependance) and mutual information (a quantity that can represent the general dependance) in Chapter 3. We focus on the question that given the correlation coefficients of random sources, what is the minimum possible value of mutual information?
Theorem 3.1 then suggests that for weakly correlated random variables, such as two instances of a self-similar process with a long time lag, half the square of the correlation coefficients is a reasonable approximation to the mutual information, provided they are also weakly dependent in a general sense.
5.3 Bayesian Decentralized Detection
Our investigation of the optimal decentralized system has yielded some interesting results.
Firstly, for the classification of exponential sources problem, the optimality of identical sensor system has been proved for n = 2 and n = 3. For n > 3, we have to rely partly on numerical examination. A byproduct is that for the classification of exponential sources problem, the optimal performance of the optimal serial two-sensor system is the same as the optimal parallel sensor system. It is somewhat surprising since it is known that the serial two-sensor system in general has better performance than the parallel two-two-sensor system [32].
For the general classification problem, we propose a set of propositions on the optimality of the identical system. These propositions can be verified without much difficulty. Moreover, we point out that some classification problems encountered in the survival analysis and failure time analysis, as well as the decentralized detection for the self-similarity via the local measurements of the packet inter arrival times, can be manipulated in the same way.
Finally, for the Gaussian classification problem, we conclude the optimality of identical sensors partly numerically.
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Vita
Chien Yao
Date of Birth August 5, 1975 Place of Birth Chung-li, Taiwan
Educations
• Ph.D, Institute of Electronics, National Chiao-Tung University, Taiwan, 2008.
Advisor: Professor Tihao Chiang and Professor Po-Ning Chen
• Field: Self-Similar Traffic Model, Distributed Detection, Information Theory.
• Dissertation: “Synthesization and Decentralized Identification of Self-Similar Pro-cesses”
• M.S., Institute of Electronics, National Chiao-Tung University, Taiwan, 1999.
– Field: Solid State Physics, Surface Science, Plasma Physics.
– Master Thesis: “Effects of Surface Excitations on the Energy Loss of Moving
– Master Thesis: “Effects of Surface Excitations on the Energy Loss of Moving