A New Algorithm for Nonparametric Sequential Detection

Transcription

1 A New Algorithm for Nonparametric Sequential Detection Shouvik Ganguly, K. R. Sahasranand and Vinod Sharma Department of Electrical Communication Engineering Indian Institute of Science, Bangalore, India {sanandkr, ABSTRACT We consider nonparametric sequential hypothesis testing problem when the distribution under the null hypothesis is fully known but the alternate hypothesis corresponds to a general family of distributions. We propose a simple algorithm to address the problem. Its performance is analysed and asymptotic properties are proved. The simulated and analysed performance of the algorithm is compared with an earlier algorithm addressing the same problem with similar assumptions. Finally, we provide a justification for our model motivated by a Cognitive Radio scenario and modify the algorithm for optimizing performance when information about the prior probabilities of occurrence of the two hypotheses is available. Key words: Non-parametric Hypothesis Testing, Asymptotic Analysis. I. INTRODUCTION Presently there is a scarcity of spectrum due to the proliferation of wireless services. However, it has been observed that much of the licensed spectrum remains underutilised for most of the time. Cognitive Radios (CRs) are proposed as a solution to this problem []. These are designed to exploit the unutilised spectrum for their communication, without causing interference to the primary users. This is achieved through spectrum sensing by the CRs, to gain knowledge about spectrum usage by the primary users. The problem of spectrum sensing by CRs can be interpreted as a detection problem. This problem has been recently studied actively in the context of sensor networks ([], [3]) and cognitive radios ([4], [5], [6]). Detection problems can be classified as fixed sample size or sequential ([], [3]). In a fixed sample framework, the decision has to be made based on a fixed number of samples, and a likelihood ratio test turns out This work was partially supported by a grant from ANRC /4/$3.00 c 04 IEEE to be optimal for a simple binary hypothesis problem. In a sequential framework, samples are taken until some conditions are fulfilled, and once the process of taking samples has stopped, a decision is arrived at. It is known that in case of a single node, Sequential Probability Ratio Test (SPRT) outperforms other sequential or fixed sample size detectors in the simple binary hypothesis problem ([7]). But optimal solutions in the decentralized setup are not available ([8]). [9], [0] and [] have also studied the spectrum sensing problem in Cognitive Radio. See also other references in the surveys [], [0] and []. For nonparametric sequential setup, [] has provided separate algorithms for different problems like changes in mean, variance, etc. Two tests, t-test and rank tests for testing a parameter are the most prominent sequential tests [3]. A more relevant nonparametric test useful in our generality is the sequential version of Kolmogorov-Smirnoff test [4]. References [5] and [6] have studied the single node as well as distributed decentralised detection problem in a sequential framework, with a noisy reporting MAC. Algorithms in [5] require the probability distributions involved from a parametric family. The approach in [6] is non-parametric in the sense that it assumes very little knowledge of one of the distributions. It was shown in [6] that the algorithm KT-SLRT developed in that paper performs better than Hoeffding test (which is asymptotically optimal for discrete alphabet) and non-parametric detectors formed by approximating the unknown density by kernel density estimators and differential entropy estimators (these were compared via some examples with the estimators provided in [6]). KT-SLRT was also compared with sequential Kolmogorov-Smirnoff test and was found to perform better (not reported in [6]). In this paper, we present a simpler algorithm (which requires no quantization, decision about the range to be considered or universal source codes as in [6]) to address the single-node version of the problem studied in [6].

2 Our algorithm has the added advantage of better performance in most cases, as borne out by simulations and analysis. The paper is organized as follows. Section II presents the system model and the algorithm. Section III provides theoretical analysis of the algorithm. Section IV compares our algorithm to KTSLRT in [6]. Section V provides a generalization of our algorithm along with an explanation in the CR setup. Section VI concludes the paper. II. SYSTEM MODEL A node is making observations and needs to settle the following hypothesis testing problem. Observation X k is made at time k. We assume {X k, k 0} are i.i.d. (independent identically distributed). The distribution of the observations is either P 0 or P. The decision to be made is, H 0 : if the probability distribution is P 0 and H : if the probability distribution is P. We assume that P 0 is known, but that P belongs to the family{p : D(P P 0 ) λ and H(P ) H(P 0 )}, where D(P P 0 ) is the divergence E P [log P(X) P ] 0(X) and H(P ) is the entropy, or differential entropy, of the distribution P. Divergence measure is very popular in statistics and is used for testing goodness of fit, Bayesian asymptotics and model selection and information theory [7]. It is fundamentally related to the difficulty in discriminating between two probability measures [8]. Convergence in Divergence implies convergence in Hellinger distance and total variation distance, other commonly used distance measures in statistics. Our motivation for this setup is the Cognitive Radio (CR) system. A CR node has to detect if a channel is free (the primary node is not transmitting) or not. When the channel is free then the observations X k are the receiver noise with distribution P 0. This will often be known (even Gaussian) and hence it is reasonable to assume that P 0 is known (see, however, the generalization in Section V). But when the primary is transmitting, it could be using adaptive modulation and coding, unknown to the secondary node, and even the fading of the wireless channel from the primary transmitter to the local CR node may be time-varying and not known to the receiver. This leads to an unknown distribution P under H. Also, then H(P ) H(P 0 ) will be satisfied. We will elaborate in Section V how this scenario can lead to the above class of distributions specified for H. We have chosen sequential framework for detection because it provides faster decisions on the average, for any probabilities of errors. Our detection algorithm, motivated by SPRT, works as follows. On receiving X k at time k, we compute W k as W k = W k log P 0 (X k ) H(P 0 ) λ, W 0 = 0. () If W k log α, we decide H ; if W k log β, we decide H 0 ; otherwise we wait for the next observation. The constants 0 < α, β <, are specified based on the probabilities of errors P [ Decide H i H j ], i j. Motivation for algorithm () is as follows. {W k } is a random walk with mean drift = λ under H 0 and D(P P 0 ) + H(P ) H(P 0 ) λ λ under H. Thus under H 0, W k will tend to a.s. and under H, to + a.s. In the rest of the paper we analyze the performance of this algorithm. We will show that the present algorithm provides a better performance than [6]. Also, unlike in [6], it is simpler to implement because it does not require a universal source coder and it also does not require quantization of observations. In the following, E i [.] and P i (.) denote the expectation and probability, respectively, under H i, i = 0,. III. PERFORMANCE ANALYSIS In this section we provide performance of the test. Let N inf{n : W n > log α}, N 0 inf{n : W n < log β}, N min(n 0, N ). Lemma 3.. P (N < ) = under H 0 and H. Proof: Under H 0, W n a.s. Therefore, P 0 (N < ) P 0 (N 0 < ) =. The proof is similar under H. We will use the notation that P F A P 0 ( decide H ) and P MD P ( decide H 0 ). Theorem 3.. a) P F A α s where s is a solution of E 0 [e s(h(p0)+log P0(X k)+ λ ) ] =. b) P MD β s where s is a solution of E [e s ( H(P 0) log P 0(X k ) λ ) ] =.

3 Proof: a) Under H 0, from [9], P F A P 0 [max W k > log α] α s. k Similarly, (b) holds under H. As an example, for Gaussian distributions with different variances under the two hypotheses (see Section IV), s and s will be the solutions of the transcendental equations s + e s(λ+) =, and respectively. e s (λ+) σ σ0 s =, Theorem 3.3: a) Under H 0, N log β = λ a.s. Further, if E 0 log P 0 (X) r <, for some r, b) Under H, E 0 N r log β r = ( λ )r. N log α = D(P P 0 ) + H(P ) H(P 0 ) λ Further, if E log P 0 (X) r <, for some r, E N r log α r = ( D(P P 0 ) + H(P ) H(P 0 ) λ Proof: a) From Random walk results, ([0], Chapter 4) β 0 N 0 log β = λ a.s. and in L. Also, since the increments have finite r th moments, the convergence holds in L r, r. Again from random walk results, under H 0, Therefore, under H 0, P 0 [max W k < ] =. k P 0[N = N 0 ] =. β 0 This, together with 0 N N 0 for all α, β implies that N log β = λ, a.s. and in L r. The proof for part (b) is similar. Note: The condition E log P 0 (x) r < is satisfied for all r > 0 for a large class of distributions including all three distributions considered in the next section. If α, β 0, then the thresholds logα + and logβ, and Theorem 3. indicates the rates at which P F A and P MD tend to zero. Also Theorem 3.3 provides the rate at which the number of samples needed tends to under the two hypotheses. The rates of convergence in Theorem 3.3 are the same as in SPRT although the iting values are different. Similarly the decay of P F A and P MD in Theorem 3.3 are similar to those for SPRT with different s and s [3]. IV. SIMULATIONS In this section, we compare the simulated and theoretical performances of the new algorithm with KT- SLRT [6]. In [6], KT-SLRT was shown to perform better than many other algorithms. In the following simulations, E DD 0.5[E 0 (N) + E (N)] and P e 0.5(P F A + P MD ). The observations {X k } are considered with the following distributions: Pareto P, P 0 P(0, ) and P P(3, ). Lognormal ln N, P 0 ln N (0, 3) and P ln N (3, 3). Gaussian N with P 0 N (0, ) and P N (0, 5). Figures and 4 plot E DD and P e for Pareto and lognormal distributions respectively. We see that the a.s. asymptotic values provide a better approximation as the thresholds increase. Figs and 5 compare performance of our algorithm with KT-SLRT, the algorithm described in [6], for Pareto and Normal distributions ) r. respectively. We see that the present algorithm performs much better. There are various reasons for this. One is that KT-SLRT uses a Universal source code for a finite alphabet source. Thus when the distribution of observations is continuous, we need to quantize it into a finite set. This introduces errors. Also, average Universal source code length only asymptotically approaches H(P ). For a finite (especially small as needed in CR) sample size, this also introduces error. Figure 5 also plots the performance of SPRT which is optimal given P 0, P. We see that our algorithm performs quite close to SPRT. In figure 3, we have compared our algorithm with KT-SLRT and also with Rank test [3], SPRT, Sequential t-test (SEQ-T [3]) and Hoeffding test for Binomial distribution case. Hoeffding test is asymptotically optimal for discrete distributions. Our test is performing better than all these tests except SPRT which uses the exact knowledge of the two hypotheses and then of course it is optimal. V. FURTHER GENERALIZATIONS Let us now consider a generalization of the problem, in which P 0 is not exactly known. Specifically, the hypothesis testing problem we consider is: H 0 : P {P 0 : D(P 0 P 0 ) γλ}, for some 0 γ <. ()

4 Fig. 3: Performance Comparison for Binomial Distribution. Fig. : Performance for Pareto Distribution. Top: Detection Delay; Bottom: Error rate. Fig. : Performance Comparison with KT-SLRT for Pareto Distribution. H : P {P : D(P P 0 ) λ and H(P ) > H(P 0), (3) for all P 0 H 0 }. The detection algorithm remains the same except that now we write the test statistic as W k = W k log ˆP 0 (X k ) H( ˆP 0 ) υλ. For good performance we should pick ˆP 0 from the class in () and choose υ carefully. We elaborate on this in the following. Fig. 4: Performance for lognormal Distribution. Top: Detection Delay; Bottom: Error rate. First, we motivate this problem from a practical CR standpoint. In a CR setup, H 0 actually indicates the presence of only noise, while under H, the observatios are signal + noise. Due to electromagnetic interference, the receiver noise can be changing with time ([]). Thus we assume that the noise power P N is bounded as σn,l P N σn,h. Similarly, let the signal power be bounded as σs,l P S σs,h. Now we formulate these constraints in the form of the problem ()-(3), where we should select appropriate P 0, λ and γ. We will compute these assuming we

5 Fig. 5: Performance Comparison for Gaussian Distribution. are iting ourselves to Gaussian distributions but will see that these work well in general. This happens because as the number of observations increases, the distributions of statistics usually converge to Gaussian distribution. This fact is often used in statistics to develop non-parametric tests. Eg., t-test. We take, P 0 N (0, σ 0), with σ 0 determined from the given bounds as follows. Given two Gaussian distributions Q 0 and Q with zero mean and variances σ 0 and σ respectively, D(Q Q 0 ) = ln σ 0 + σ (σ σ0 ). (4) Let f(σ) ln σ 0 σ + (σ σ0 ). We choose σ 0 such that f(σ N,L ) = f(σ N,H ). This can be achieved for some σ 0 (σ N,L, σ N,H ), since f is convex with a minimum at σ 0. This choice ensures that P 0 is at some sort of a centre of the class of distributions under consideration in H 0. We now choose γλ f(σ N,L ) = f(σ N,H ). For the class of distributions considered under H (X = signal + noise, independent of each other), σ N,L + σ S,L E[X ] σ N,H + σ S,H. We take, λ inf f(σ) = f( σ (σn,l +σ S,L,σ N,H +σ S,H ) Next we compute ˆP 0. If the X k has distribution P i for i = 0,, then the drift is D(P 0 ˆP 0 ) + H(P 0) H( ˆP 0 ) υλ under H 0, and D(P ˆP 0 ) + H(P ) H( ˆP 0 ) υλ under H. This drift is an important parameter in determining the algorithm performance and will decide ˆP 0. Let W i be the cost of rejecting H i wrongly, and c be the cost of taking each observation. Then, Bayes risk ([]) for the test is given by R c (δ) = π i [W i P i ( reject H i ) + ce i (N)], where i=0 π i is the prior probability of H i. Hence if α = β, from Theorems 3. and 3.3, R c (δ) α 0 log α = c[ π 0 D(P 0 ˆP 0 ) h(p 0 ) + h( ˆP 0 ) + υλ π + D(P ˆP 0 ) + h(p ) h( ˆP ]. (5) 0 ) υλ Following a minimax approach, we first maximize the above expression with respect to P 0 and P, and then minimize the resulting maximal risk w.r.t. ˆP0 and υ. As noted before, we achieve this optimization iting ourselves to only Gaussian family. The second term in (5) is maximized when D(P ˆP 0 ) + h(p ) is minimized. Let us denote the variance of ˆP 0 by Γ. Now, the variances of all eligible P s are greater than Γ. Hence, D(P ˆP 0 ) + h(p ) is minimized when P has the least possible variance, i.e. σ N,L +σ S,L. Using N (0, σ N,L +σ S,L ) in place of P, the second term in (5) becomes (after simplification), π. ( σ N,L +σ S,L Γ ) υλ Similarly, to maximize the first term in (5), we have to minimize D(P 0 ˆP 0 ) + H(P 0) w.r.t. P 0. After this, the first term becomes π 0. υλ ( σ N,H Γ ) Taking x Γ, y = υλ, a = σ N,H and b = σ N,L+σ S,L, (6) the above expression can be written as a function of x and y in the form π 0 g(x, y) = y + ax + π bx y. Minimizing this w.r.t. y yields, y opt = π0 (bx ) + π (ax ) π0 +. (7) π Together with this, we can choose x ( σ N,H, σ N,L σn,l + σ S,L ). In the following, we demonstrate the advantage of optimizing the above parameters on the examples considered in Section IV. The bounds on the noise and signal power were chosen in each case such that the distributions specified in Section IV satisfy those constraints. Also, the thresholds were chosen the same as before. For the following simulations, we have taken Γ = σ N,L + σ N,H and determined y opt in accordance with (7). For Gaussian, P 0 N (0, ), P N (0, 5). ).

6 For Lognormal, P 0 log N (0, 3), P log N (3, 3). For Pareto, P 0 P(0, ), P P(3, ). We compare the performances in Figs We see that the optimized version performs noticeably better, even for distributions other than Gaussian. Fig. 6: Optimization for Pareto Distribution Fig. 7: Optimization for Lognormal Distribution Fig. 8: Optimization for Gaussian Distribution VI. CONCLUSIONS AND FUTURE WORK We have developed a new sequential algorithm for detection, where under one of the hypotheses, the distribution can belong to a nonparametric family. This can be useful for spectrum sensing in Cognitive Radios. This algorithm is shown to perform better than a previous algorithm which was known to perform well and is also easier to implement. We have also obtained its performance theoretically. The asymptotics are comparable to SPRT and other known algorithms even though it is in the non-parametric setup. Next we have generalized the setup to the case where under both the hypotheses, the distributions belong to nonparametric families. More recently, we have generalized these results to the decentralized setup, where the observations are taken at multiple, geographically distributed sensor/cr nodes. These results will be reported elsewhere. REFERENCES [] I. F. Akyildiz, B. Lo, and R. Balakrishnan, Cooperative spectrum sensing in cognitive radio networks: A survey, Physical Communication, vol. 4, no., pp. 40 6, 0. [] J. Chamberland and V. Veeravalli, Wireless sensors in distributed detection applications, IEEE Signal Processing Magazine, vol. 4, pp. 6 5, 007. [3] P. Varshney and R. Viswanathan, Distributed detection with multiple sensors, in IEEE, Proceedings of, vol. 85, pp , IEEE, 997. [4] G. Fellouris and G. V. Moustakides, Decentralized sequential hypothesis testing using asynchronous communication, IEEE Transactions on Information Theory, vol. 57, no., pp , 0. [5] T. Banerjee, V. Sharma, V. Kavitha, and A. Jayaprakasam, Generalized analysis of a distributed energy efficient algorithm for change detection, IEEE Transactions on Wireless Communication, vol. 0, pp. 9 0, 0. [6] Y. Mei, Asymptotic optimality theory for decentralized sequential hypothesis testing in sensor networks, IEEE Transactions on Information Theory, vol. 54, no. 5, pp , 008. [7] D. Siegmund, Sequential Analysis: Tests and Confidence Intervals. Springer, 985. [8] V. Veeravalli, T. Basar, and H. V. Poor, Decentralised sequential detection with a fusion centre performing the sequential test, IEEE Transactions on Information Theory, vol. 39, no., pp , 993. [9] W. Ejaz, N. u. Hasan, S. Lee, and H. S. Kim, Intelligent spectrum sensing scheme for cognitive radio networks, EURASIP Journal on Wireless Communications and Networking, vol. 6, no., pp , 03. [0] A. Ghasemi and E. Sousa, Spectrum sensing in cognitive radio networks: Requirements, challenges and design tradeoffs, IEEE Communications Magazine, vol. 46, no. 4, pp. 3 39, 008. [] H. Sun, A. Nallanathan, C.-X. Wang, and C. Yunfei, Wideband spectrum sensing for cognitive radio networks: A survey, arxiv, vol. arxiv: [cs.it], 03. [] N. Mukhopadhyay and B. De Silva, Sequential Methods and Their Applications. CRC: Chapman and Hall, 009. [3] Z. Govindarajulu, Sequential Statistics. World Scientific, 004. [4] J. P. R. E.L. Lehmann, Testing Statistical Hypotheses. 3rd edition, Springer, New York, 005. [5] J. K. Sreedharan and V. Sharma, Novel algorithms for distributed sequential hypothesis testing, in Annual Allerton Conference on Communication, Control, and Computing, pp , 0. [6] J. K. Sreedharan and V. Sharma, Nonparametric distributed sequential detection via universal source coding, in Information Theory and Applications Workshop (ITA), pp. 7, 03. [7] I. Csiszar and P. C. Shields, Information theory and statistics [8] R. Bahadur, Some it theorems in statistics, SIAM, 97. [9] S. Asmussen, Applied Probability and Queues. John Wiley and Sons, 987. [0] A. Gut, Stopped random walks : it theorems and applications. Springer-Verlag, New York, 988. [] A. Sahai, S. Mishra, R. Tandra, and K. Woyach, Cognitive radios for spectrum sharing, IEEE Signal Processing Magazine, vol. 6, no., pp. 07 8, 009. [] E. Lehmann and G. Casella, Theory of Point Estimation. Springer Texts in Statistics, Springer, New York, 003.