Distributed Detection of a Nuclear Radioactive Source using Fusion of Correlated Decisions
|
|
- Juliet Nelson
- 5 years ago
- Views:
Transcription
1 Distributed Detection of a uclear Radioactive Source using Fusion of Correlated Decisions Ashok Sundaresan and Pramod K. Varshney Department of Electrical Engineering and Computer Science Syracuse University Syracuse, U.S.A. asundare,varshney@syr.edu ageswara S.V. Rao Computer Science and Mathematics Division Oak Ridge ational Laboratory Oak Ridge, U.S.A. raons@ornl.gov Abstract A distributed detection method is developed for the detection of a nuclear radioactive source using a small number of radiation counters. Local one bit decisions are made at each sensor over a period of time and a fusion center makes the global decision. A novel test for the fusion of correlated decisions is derived using the theory of copulas and optimal sensor thresholds are obtained using the ormal copula function. The performance of the derived fusion rule is compared with that of the Chair-Varshney rule. An increase in detection performance is observed. A method to estimate the correlation between the sensor observations using only the vector of sensor decisions is also proposed. I. ITRODUCTIO Detection of radiation from nuclear materials has become an important task due to the increasing threats from potential terrorist activities. One possible scenario is the dispersion of radioactive material using a conventional explosive device, namely the so called dirty bomb, in a densely populated area. The radioactive materials used in creating a dirty bomb are usually isotopes like Cs-137 that are widely used in industries and in hospitals for medical purposes and can be obtained with considerable ease. The radioactive materials for clandestine activities will need to be transported to the destination place. The task is to detect the low level radiations from the vehicles carrying these sources before they reach their destination. We propose a system comprising of a network of radiation counters operating collaboratively to detect the presence of a radioactive source. Such a network of sensors could be deployed at suitable places along the road side or at places like weigh stations, inspection stations, etc. Detection of radioactive sources using sensor networks has received some attention off-late. In [1], the authors examine the increase in signal-to-noise ratio obtained by a simple combination of data from networked sensors compared to a single sensor. The costs and benefits of using a network of radiation detectors for radioactive source detection are analyzed and evaluated in [2]. In [3], the authors propose a Bayesian methodology by assuming independence of sensor observations. While this work is rigorous, the independence property is not satisfied in practice since the sensor measurements are correlated based on the relative sensor locations and with respect to the source. In this work, we propose a novel distributed framework for radiation source detection and exploit the correlation of sensor observations for improved detection performance. Due to the nature of the problem, the radioactive sensors used are expensive high precision devices requiring more battery power unlike low cost, low power sensors used in most sensor network applications [4]. This makes deployment of a large number of sensors infeasible. In this work, we assume the presence of a few sensors (typically ranging from 1-5 monitoring a region for the possible presence of a radiation source, and propose a distributed bandwidth-constrained scheme for radioactive source detection. The sensors provide radiation counts based on the intensity of the radiation, and a decision regarding the presence or absence of a radioactive source may be made based on a local threshold. The radiation counts typically follow a Poisson distribution with the parameter proportional to inverse square distance [5]. Thus, sensors that are closer to the source register higher counts compared to farther ones, and such a relationship makes the measurements correlated, and hence non-independent. But the individual sensor measurements that correspond to the margins of the joint measurement distribution are fairly well known for this problem [6]. However, in general the correlations are not a priori known and must be explicitly accounted for in combining the information from multiple sensors. The radiation source or target to be detected is assumed to be stationary over a period of time. A sequence of binary decisions (over a length of time are made at the individual sensors and sent to a fusion center which combines them to make a final decision on the presence or absence of the radioactive source. As noted previously, if a radioactive source is indeed present, the sensor decisions at any instant of time would be correlated since all sensors observe a common random phenomenon. Fusion of correlated decisions has been studied in [7], [8]. These approaches assumed a complete knowledge of the joint distribution of the sensor observations. Such methods are feasible in special cases such as when sensor observations are realizations of multivariate Gaussian random variables. We present a novel approach using copulas to fuse correlated decisions and obtain optimal thresholds for sensor quantization. Using the copula theory, joint distribution functions can be constructed from the marginal distributions even when the observations are correlated non-gaussian random
2 variables. Hence the fusion strategy described in this paper is particularly attractive in practical cases where the underlying distributions are non-gaussian. The rest of the paper is arranged as follows. The problem is described in Section II. The design of the optimal fusion rule and individual sensor tests is considered in Section III. Experimental results are shown in Section IV. Some concluding remarks are drawn in Section V. II. PROBLEM FORMULATIO The problem is formulated as a binary hypothesis testing problem with the H 0 hypothesis indicating the absence of any radioactive source and the H 1 hypothesis indicating the presence of a radioactive source. The observations received by the sensors under both hypotheses are as follows. H 0 : z ij = b ij + w ij H 1 : z ij = c ij + b ij + w ij i = 1, 2; j = 1,..., i = 1, 2; j = 1,..., where b ij, c ij and w ij are the background radiation count, source radiation count and measurement noise respectively, at sensor i located at (x i, y i during the j th time interval. The background radiation count received during the time interval (0, t] is assumed to be Poisson distributed with known rate λ b. The source radiation count at sensor i located at (x i, y i is assumed to be Poisson distributed with rate λ ci. We assume an isotropic behavior of radiation in the presence of the source so that the rate λ ci is a function of the source intensity A, and distance of the i th sensor from the source given by λ ci = (1 A (x 0 x i 2 + (y 0 y i 2 (2 where (x 0, y 0 represent the source coordinates. The measurement noise w ij is Gaussian distributed with a known variance w. 2 The background radiation count b ij and measurement noise w ij are assumed to be spatially and temporally independent. This implies that under the H 0 hypothesis, sensor observations are independent over space and time. Under the H 1 hypothesis, the sensors observe a spatio-temporal phenomenon giving rise to spatial and temporal correlation. The overall problem is solved in a distributed fashion. It consists of determining individual sensor thresholds to form sensor decisions and the fusion test to declare the global decision using the vector of sensor decisions. In this work, we assume temporal independence and while designing the system, focus on exploiting only spatial correlation between the sensors for improved detection performance. Also, in this paper, the problem is solved for a known signal case, i.e., values of source intensity A and source coordinates (x 0, y 0 are assumed to be known. A. Decision Fusion III. SYSTEM DESIG For the sake of simplicity, in this paper we will assume that two sensors are observing the common phenomenon over time intervals each of length (0, t] (see Figure 1. Let us Fig. 1. A two sensor distributed detection scheme assume that τ 1 and τ 2 are individual sensor thresholds used for making the one-bit decisions. Then the sensor decisions, at any time interval 1 i, are quantized versions of sensor observations defined as { 0 if < z 1i τ 1, u 1i = Q(z 1i = (3 1 if τ 1 z 1i < { 0 if < z 2i < τ 2, u 2i = Q(z 2i = (4 1 if τ 2 < z 2i < Also let P r(u 1i = 1 H 1 = p 1, P r(u 1i = 1 H 0 = q 1 P r(u 2i = 1 H 1 = p 2, P r(u 2i = 1 H 0 = q 2 If f(z 1i H 1 and f(z 1i H 0 are the conditional density functions (under H 1 and H 0 respectively of the i th observation received at sensor 1 (z 1i, then it can be readily seen that p 1 = q 1 = τ 1 τ 1 f(z 1i H 1 dz 1i f(z 1i H 0 dz 1i We can define p 2 and q 2 in a similar manner. Under H 0, the observation received by any sensor during a particular time interval i, 1 i is given by z i = b i + w i where the sensor subscript has been omitted for notational convenience. It is obvious that z i follows the hierarchical
3 distribution [9] z i (b i, 2 w b i P oisson(λ b Hence under H 0, the marginal probability density function (pdf of z i is given by f(z i = f(b = k b, z i = = k b =0 f(z i b = k b P (b = k b k b =0 k b =0 [ ] 1 exp (z i k b 2 exp( λ b λ k b b 2π 2 w 2w 2 k b! From the above equation it is obvious that f(z i is an infinite sum of scaled Gaussian densities. Hence f(z i under H 0 is a Gaussian mixture distribution with the following mean and variance. E(z i = E(E(z i b = E(b = λ b var(z i = E(var(z i b + var(e(z i b = 2 w + λ b ote that the Gaussian mixture has its components centered around the Poisson counts and weighted by the Poisson count probabilities. Components centered around count values close to the Poisson rate λ b are more heavily weighted. In this paper, we approximate the Gaussian mixture distribution by a Gaussian distribution with the same mean and the variance. Similar approximations have been employed in the literature [10]. Thus, f(z i H 0 (λ b, w 2 + λ b Similarly, under the H 1 hypothesis, f(z i H 1 (λ b + λ c, 2 w + λ b + λ c where λ c is a function of the sensor s position relative to the source and hence may be different for sensor 1 and sensor 2. Once the pdfs of individual sensor s observations under both hypotheses are known, we can readily evaluate ( τ1 λ b λ c1 p 1 = Q (5 2 w + λ b + λ c1 ( τ2 λ b λ c2 p 2 = Q 2 w + λ b + λ c2 ( τ1 λ b q 1 = Q 2 w + λ b ( τ2 λ b q 2 = Q 2 w + λ b where, Q(. is the complementary cumulative distribution function of the standard ormal. Let u 1 and u 2 be the vector of sensor decisions, then the optimal test at the fusion center is the likelihood ratio test (LRT given by [11] (6 (7 (8 Λ(u = P (u 1, u 2 H 1 P (u 1, u 2 H 0 H 1 H 0 γ (9 Assuming temporal independence of sensor decisions, the optimal fusion statistic becomes T (u = P (u 1i, u 2i H 1 P (u (10 1i, u 2i H 0 Let P (u 1i = 0, u 2i = 0 H 1 = P 00, P (u 1i = 0, u 2i = 1 H 1 = P 01 P (u 1i = 1, u 2i = 0 H 1 = P 10, P (u 1i = 1, u 2i = 1 H 1 = P 11 P (u 1i = 0, u 2i = 0 H 0 = Q 00, P (u 1i = 0, u 2i = 1 H 0 = Q 01 P (u 1i = 1, u 2i = 0 H 0 = Q 10, P (u 1i = 1, u 2i = 1 H 0 = Q 11 Then the joint probability mass function (pmf of u 1 and u 2 under H 1 and H 0 is given by P (u 1i, u 2i H 1 = P (1 u1i(1 u2i 00 P (1 u1iu2i 01 P u1i(1 u2i 10 P u1iu2i 11 (11 P (u 1i, u 2i H 0 = Q (1 u1i(1 u2i 00 Q (1 u1iu2i 01 Q u1i(1 u2i 10 Q u1iu2i 11 (12 Using Eq. (11 and Eq. (12 in Eq. (10, taking log on both sides and simplifying, we get log Λ(u = C 1 where, u 1i + C 2 u 2i + C 3 u 1i u 2i (13 C 1 = log P 10Q 00 P 00 Q 10 C 2 = log P 01Q 00 P 00 Q 01 C 3 = log P 00P 11 Q 01 Q 10 P 01 P 10 Q 00 Q 11 It is known that u 1i, u 2i and u 3i = u 1i u 2i are each Bernoulli random variables. The success probabilities of u 1i and u 2i are p 1 and p 2 respectively under H 1 and q 1 and q 2 respectively under H 0. Let the success probability of u 3i be p 3 = P 11 under H 1 and q 3 = Q 11 under H 0. Under the assumption of time independence, u 1i, u 2i and u 3i are each Binomial distributed. Using Laplace-DeMoivre approximation [12], the optimal fusion test statistic is Gaussian distributed under both hypotheses. Let µ 1 and 2 1 be the mean and the variance of log Λ(u under H 1 and µ 0 and 2 0 be the mean and the variance of log Λ(u under H 0. Then it can be shown that µ 0 = [C 1 q 1 + C 2 q 2 + C 3 q 3 ] ( = [C 2 1q 1 (1 q 1 + C 2 2q 2 (1 q 2 + C 2 3q 3 (1 q 3 ] (15 µ 1 = [C 1 p 1 + C 2 p 2 + C 3 p 3 ] ( = [C 2 1q 1 (1 q 1 + C 2 2q 2 (1 q 2 + C 2 3q 3 (1 q 3 ] (17
4 The system probability of false alarm (P F A and system probability of detection (P D are now given by ( γ µ 1 P D = Q 1 ( γ µ 0 P F A = Q 0 (18 (19 where, γ is the threshold for the fusion test. Under the eyman-pearson criterion, γ can be obtained by constraining P F A = α as below γ = 0 Q 1 (P F A + µ 0 (20 For performing the test at the fusion center, we require the quantities P 00, P 01, P 10, P 11, Q 00, Q 01, Q 10, Q 11 that completely specify the joint conditional pmfs of the sensor decisions u 1i and u 2i under both hypotheses. Because of the independence of sensor observations and hence sensor decisions under H 0, we get Q 00 = (1 q 1 (1 q 2 Q 01 = (1 q 1 q 2 Q 10 = q 1 (1 q 2 Q 11 = q 1 q 2 However, under H 1, the observations are correlated and the joint pmf of the sensor decisions cannot be evaluated in a straightforward manner. Under H 1, the probabilities P 00, P 01, P 10, P 11 need to be calculated as follows. P 00 = τ 1 τ 2 z 1i= z 2i= P 01 = P 10 = P 11 = τ 1 z 1i= z 1i=τ 1 z 2i= z 1i=τ 1 f(z 1i, z 2i H 1 dz 1i dz 2i (21 z 2i=τ 2 f(z 1i, z 2i H 1 dz 1i dz 2i (22 τ 2 f(z 1i, z 2i H 1 dz 1i dz 2i (23 z 2i=τ 2 f(z 1i, z 2i H 1 dz 1i dz 2i (24 otice that the joint distribution of the sensor observations under H 1 (f(z 1i, z 2i H 1 is required to calculate the probabilities P 00, P 01, P 10, P 11. It is known that the marginals f(z 1i H 1 and f(z 2i H 1 are Gaussian. However, a conclusion about the joint density of z 1i and z 2i under H 1 cannot be made directly since z 1i and z 2i do not originate from a bi-variate Gaussian distribution. Here we employ the copula theory to construct the joint distribution of z 1i and z 2i under the H 1 hypothesis. B. Copula Theory Recently a lot of progress has been made in the study of copulas and their applications in statistics. Copulas are basically functions that join or couple multivariate distribution functions to their one-dimensional marginal distribution functions [13]. Another definition of copulas states that they are joint distribution functions of uniform distributed random variables. The role of copulas in relating multivariate distribution functions and their univariate marginals is explained by Sklar s theorem [13], [14], which is stated as follows. Sklar s Theorem: Consider an m-dimensional continuous distribution function F with continuous marginal distribution functions F 1,..., F m. Then there exists a unique copula C, such that for all x 1,..., x m in [, ] F (x 1, x 2,..., x m = C(F 1 (x 1, F 2 (x 2,..., F m (x m (25 Conversely, consider a copula C and univariate cdfs F 1,..., F m, then F as defined in Eq.(25 is a multivariate cdf with marginals F 1,..., F m. As a direct consequence of the above theorem,we obtain by differentiating both sides of Eq.(25, ( m f(x 1,..., x m = f(x i c(f 1 (x 1,..., F m (x m (26 where, c is termed as the copula density given by c(k = m (C(k 1,..., k m k 1,..., k m (27 where, k i = F i (x i. Thus, we can construct a joint density function with specified marginal densities by employing Eq.(26. The choice of a copula function to construct the joint density is an important consideration here. Various families of copula functions exist in the literature [13], [14]. However, it is not very clear as to which copula function should be used in which case. It is conjectured by many authors that the use of different copula functions may exhibit different dependence behavior among the random variables. In this work, we make use of the ormal (or Gaussian copula to construct the joint density function of the sensor observations under the H 1 hypothesis. The use of other copula functions is under investigation. The copula density for a ormal copula can be obtained easily from Eq.(26 as below. c(φ(x 1,..., φ(x m = Φ(x 1,..., x m φ(x 1,..., φ(x m (28 where, Φ is the multivariate ormal density function and φ is the univariate normal density function. The ormal copula incorporates the dependencies among the random variables in a manner exactly similar to the way a multivariate ormal distribution does by using the covariance matrix. Hence, to use the ormal copula in constructing a joint distribution, the linear correlation coefficients between the random variables are needed. In some cases, the correlation between the random
5 variables may be available in advance or evaluated from a correlation model. Otherwise, they need to be estimated from the data. This will be elucidated in more detail in Section IV. Making use of the ormal copula density(see Eq.(28 to construct the joint distribution of the sensor observations under H 1, we get f(z 1i, z 2i H 1 =f(z 1i H 1 f(z 2i H 1 c g (F Z1i H 1 (z 1i, F Z2i H 1 (z 2i (29 where, c g (u, v is the ormal copula density evaluated from Eq.(28. On simplifying Eq.(29, f(z 1i, z 2i H 1 is found to be nothing but the bi-variate Gaussian density as expected. It is important to note here that the use of a different copula function would not have resulted in the bi-variate Gaussian density but might have still served our purpose of determining the joint probabilities. Using the determined joint density function of the observations under H 1, expressions for the probabilities P 00, P 01, P 10, P 11 can be obtained by using Eqs.(21-(24. C. Optimal Threshold for Local Sensors In Section III-A, we assumed that τ 1 and τ 2 are individual sensor thresholds. It can be seen that P D and P F A given by Eq.(18 and Eq.(19 are functions of τ 1 and τ 2. Constraining P F A = α, P D can be written as P D (τ 1, τ 2 = Q( 0(τ 1, τ 2 Q 1 (α + µ 0 (τ 1, τ 2 µ 1 (τ 1, τ 2 1 (τ 1, τ 2 (30 The sensor thresholds are chosen to maximize P D at a particular value of P F A. Hence the optimal sensor thresholds are given by (τ 1, τ 2 = arg max τ 1,τ 2 P D (τ 1, τ 2 (31 For the results shown in this paper, a search algorithm is used to perform the above optimization. IV. EXPERIMETAL IVESTIGATIOS As mentioned previously, in this work we present results for the case when the count rate of the source at each sensor (λ ci is known. The count rate is determined by using the following source parameter values: A = 10, (x 0, y 0 = (10, 10. It is assumed that both sensors are located equidistant from the source at a distance 4 units from the source resulting in equal values of λ ci (λ c1 = λ c2 = λ c = A mean background radiation with count λ B = 10 and measurement noise with variance 2 w = 10 is considered. It is assumed that the sensors are observing the phenomena over = 100 time intervals each of length one second. A. Known Correlation Case In this case, we assume that the correlation between the random variables z 1i and z 2i is known. Spatial correlation functions modeling the correlation between two sensors as a function of the distance between them exist in literature [15] and may be used in this case. Starting from bi-variate ormal, bi-variate Poisson random variables with mean equal to λ c are generated (using the probability integral transform and then applying the inverse Poisson distribution function with varying values of correlation (ρ s. Due to the effect of background radiation and measurement noise, the correlation between the sensor observations (ρ z at any time instant i is reduced and is given by ρ s λ c ρ z = λ c + λ b + w 2 (32 For each value of ρ s and consequently ρ z, the expression for P D as a function of τ 1 and τ 2 by constraining the value of P F A is obtained from Eq.(30 and the same is maximized w.r.t τ 1 and τ 2 to obtain the optimal sensor thresholds. The maximum value of P D is noted. The same is repeated for various values of P F A and the ROC is generated. The experiment is repeated for different values of ρ s and the results are plotted as shown in Figure 2. For comparison purposes, we also evaluated the detection performance of the system that assumes independence of sensor decisions under the H 1 hypothesis. Under the conditional independence assumption, the term C 3 in Eq.(13 becomes zero and the optimal test statistic reduces to log Λ 2 (u = C 1 u 1i + C 2 u 2i (33 which is nothing but the Chair-Varshney test statistic [16]. Also, the joint probabilities of the sensor decisions under the H 1 hypothesis are now given by, P 00 = (1 p 1 (1 p 2 P 01 = (1 p 1 p 2 P 10 = p 1 (1 p 2 P 11 = p 1 p 2 Using the Laplace-DeMoivre approximation [12], the Chair- Varshney test statistic is also Gaussian distributed (assuming time independence whose mean and variance under either hypothesis can be calculated from Eqs.(14-(17 by substituting C 3 = 0. P D and P F A for the Chair-Varshney statistic, as functions of τ 1 and τ 2, are obtained and the detection performance of the Chair-Varshney statistic is also evaluated in the same manner as described afore (see Figure 2. From Figure 2, it is clear that the detection performance is improved by taking correlation into account. The proposed approach is able to do much better than the Chair-Varshney fusion rule even at low values of correlation. The increase in P D is noticeable especially at lower values of P F A which is desirable. As expected, the detection performance increases with increase in signal correlation. B. Unknown Correlation Case In this case, the correlation between z 1 and z 2 needs to be estimated first before the pdf of the test statistic given by Eq.(13 can be evaluated under either hypotheses. A two step procedure is adopted here and is described as follows.
6 Fig. 2. Detection Performance for Known Correlation Case First the sensor thresholds are obtained by assuming the sensor decisions to be independent under H 1 and using the Chair-Varshney test statistic as detailed in the previous section. The dependence between the sensor observations is evaluated (estimated from the decision vectors (u 1 and u 2 using a nonparametric rank correlation measure, Kendall s τ [13]. Given a vector (a, b of observations from a bi-variate random vector (A, B, Kendall s τ is defined ([13] as the ratio of the difference in the number of concordant pairs (c and discordant pairs (d to the total number of pairs of observations, i.e., k τ = c d c + d (34 (a i, b i and (a j, b j are said to be concordant if (a i a j (b i b j > 0 and discordant if (a i a j (b i b j < 0. An interesting property of the Kendall s τ correlation measure is that it remains invariant under non-decreasing transformations of the original data. Because of this property and from Eq.(3 and Eq.(4 we can infer that Kendall s τ between u 1 and u 2 is equal to that between z 1 and z 2. Once Kendall s τ (k τ between z 1 and z 2 is known, the linear correlation coefficient between z 1 and z 2 is given by [14] ( πkτ ρ(z 1, z 2 = sin (35 2 In the second step, using the correlation estimated in the first step the pdf of the optimal test statistic (see Eq.(13 is obtained under both hypotheses. The same procedure described in Section IV-A is then carried out to determine optimum sensor thresholds that maximize P D at a given value of P F A. The results of our simulation are shown in Table 1. The results are shown for the case when A = 10, λ c1 = λ c2 = 0.625, λ B = 10 and w 2 = 10. The correlation between the source signal received at the two sensors ρ s = 0.9. A degradation in performance is noticeable compared to the known signal case. This is expected since the correlation is being estimated from a sequence of quantized data. However, the use of correlation still has increased the detection performance compared to the Chair-Varshney test especially for low P F a values. ote that the length of time over which decisions are obtained plays an important role here. The longer the observation time better will the estimate of the correlation between the sensor observations. V. COCLUSIO A distributed scheme using a network of two sensors for detection of a nuclear radioactive source was developed. A new fusion test taking correlation of sensor decisions was developed and used to determine optimal sensor quantization thresholds. The performance of the proposed scheme was compared to the Chair-Varshney test which assumes conditional independence of sensor decisions. The proposed scheme is able to achieve a better detection performance than the Chair- Varshney fusion rule. Our future work will consist of investigating methods to estimate the covariance matrix of the ormal copula function by utilizing training data obtained by generating a set of measurements under both hypotheses. In our case, these measurements can be generated in two different ways: a Poisson and Gaussian distributions can be used to generate c ij, b ij and w ij, which will be added to provide z ij as in Eq. (1, or b the intensity level λ ci can be computed based on the source parameters, and radiation sensors can be simultaneously subjected to radiation levels λ ci and λ b in a controlled laboratory environment. The output of the sensors can taken as random samples of (z 1, z 2. In future work, we will also consider the use of other copula functions to construct the joint density of sensor observations. Also, a more general detection problem in the event of unknown source location parameters will be considered.
7 ACKOWLEDGEMET This material is based on work supported by UT Battelle, LLC Subcontract number , with funding originating from Department of Energy Contract umber DE-AC05-00OR REFERECES [1] R.J. emzek, J.S. Dreicer, D.C. Torney and T.T. Warnock, Distributed sensor networks for detection of mobile radioactive sourcesr, IEEE Trans. on uclear Science, vol. 15, no. 4, pp , Aug [2] D.L. Stephens,Jr. and A.J. Peurrung, Detection of moving radioactive sources using sensor networks, IEEE Trans. on uclear Science, vol. 51, no. 5, pp , Oct [3] S.M. Brennan, A.M. Mielke and D.C. Torney, Radioactive source detection by sensor networks, IEEE Trans. on uclear Science, vol. 52, no. 3, pp , June [4] G.F. Knoll, Radiation Detection and Measurement. John-Wiley, [5] R.E. Lapp and H.L. Andrews, uclear Radiation Physics. Prentice- Hall, ew Jersey, [6] D. Mihalas and B.W. Mihalas, Foundations of Radiation Hydrodynamics. Courier Dover Publications, [7] E. Drakopoulos and C.C. Lee, Optimum multisensor fusion of correlated local decisions, IEEE Trans. on Aerospace and Elect. Syst., vol. 27, no. 4, pp , July [8] M. Kam, Q. Zhu and W.S. Gray, Optimum data fusion of correlated local decisions in multiple sensor detection systems, IEEE Trans. on Aerospace and Elect. Syst., vol. 28, pp , July [9] G. Casella and R.L. Berger, Statistical Inference. Duxbury Press, Belmont, CA, [10] Y. Bar-Shalom and X.R. Li, Multitarget-Multisensor Tracking: Princliples and Techniques. YBS, Storrs, CT, [11] P.K. Varshney, Distributed Detection and Data Fusion. Springer-Verlag, ew York, [12] A. Papoulis, Probability, Random Variables and Stochastic Processes. McGraw-Hill, ew York, [13] R.B. elsen, An Introduction to Copulas. Springer-Verlag, ew York, [14] T. Schmidt, Coping with Copulas. Risk Books, [15] M.C. Vuran, O.B. Akan, I.F. Akyildiz, Spatio-temporal correlation: theory and applications for wireless sensor networks, Computer etworks, vol. 45, Issue 3, pp , June [16] Z. Chair and P.K. Varshney, Optimal data fusion in multiple sensor detection systems, IEEE Trans. on Aerospace and Elect. Syst., vol. 22, pp , Jan 1986.
Pramod K. Varshney. EECS Department, Syracuse University This research was sponsored by ARO grant W911NF
Pramod K. Varshney EECS Department, Syracuse University varshney@syr.edu This research was sponsored by ARO grant W911NF-09-1-0244 2 Overview of Distributed Inference U i s may be 1. Local decisions 2.
More informationCensoring for Type-Based Multiple Access Scheme in Wireless Sensor Networks
Censoring for Type-Based Multiple Access Scheme in Wireless Sensor Networks Mohammed Karmoose Electrical Engineering Department Alexandria University Alexandria 1544, Egypt Email: mhkarmoose@ieeeorg Karim
More informationBAYESIAN DESIGN OF DECENTRALIZED HYPOTHESIS TESTING UNDER COMMUNICATION CONSTRAINTS. Alla Tarighati, and Joakim Jaldén
204 IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP) BAYESIA DESIG OF DECETRALIZED HYPOTHESIS TESTIG UDER COMMUICATIO COSTRAITS Alla Tarighati, and Joakim Jaldén ACCESS
More informationThreshold Considerations in Distributed Detection in a Network of Sensors.
Threshold Considerations in Distributed Detection in a Network of Sensors. Gene T. Whipps 1,2, Emre Ertin 2, and Randolph L. Moses 2 1 US Army Research Laboratory, Adelphi, MD 20783 2 Department of Electrical
More informationCooperative Spectrum Sensing for Cognitive Radios under Bandwidth Constraints
Cooperative Spectrum Sensing for Cognitive Radios under Bandwidth Constraints Chunhua Sun, Wei Zhang, and haled Ben Letaief, Fellow, IEEE Department of Electronic and Computer Engineering The Hong ong
More informationبسم الله الرحمن الرحيم
بسم الله الرحمن الرحيم Reliability Improvement of Distributed Detection in Clustered Wireless Sensor Networks 1 RELIABILITY IMPROVEMENT OF DISTRIBUTED DETECTION IN CLUSTERED WIRELESS SENSOR NETWORKS PH.D.
More informationOptimal Mean-Square Noise Benefits in Quantizer-Array Linear Estimation Ashok Patel and Bart Kosko
IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 12, DECEMBER 2010 1005 Optimal Mean-Square Noise Benefits in Quantizer-Array Linear Estimation Ashok Patel and Bart Kosko Abstract A new theorem shows that
More informationFalse Discovery Rate Based Distributed Detection in the Presence of Byzantines
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS () 1 False Discovery Rate Based Distributed Detection in the Presence of Byzantines Aditya Vempaty*, Student Member, IEEE, Priyadip Ray, Member, IEEE,
More informationFusion of Decisions Transmitted Over Fading Channels in Wireless Sensor Networks
Fusion of Decisions Transmitted Over Fading Channels in Wireless Sensor Networks Biao Chen, Ruixiang Jiang, Teerasit Kasetkasem, and Pramod K. Varshney Syracuse University, Department of EECS, Syracuse,
More informationChapter 2. Binary and M-ary Hypothesis Testing 2.1 Introduction (Levy 2.1)
Chapter 2. Binary and M-ary Hypothesis Testing 2.1 Introduction (Levy 2.1) Detection problems can usually be casted as binary or M-ary hypothesis testing problems. Applications: This chapter: Simple hypothesis
More informationDetection theory 101 ELEC-E5410 Signal Processing for Communications
Detection theory 101 ELEC-E5410 Signal Processing for Communications Binary hypothesis testing Null hypothesis H 0 : e.g. noise only Alternative hypothesis H 1 : signal + noise p(x;h 0 ) γ p(x;h 1 ) Trade-off
More informationQUANTIZATION FOR DISTRIBUTED ESTIMATION IN LARGE SCALE SENSOR NETWORKS
QUANTIZATION FOR DISTRIBUTED ESTIMATION IN LARGE SCALE SENSOR NETWORKS Parvathinathan Venkitasubramaniam, Gökhan Mergen, Lang Tong and Ananthram Swami ABSTRACT We study the problem of quantization for
More informationProbability and Stochastic Processes
Probability and Stochastic Processes A Friendly Introduction Electrical and Computer Engineers Third Edition Roy D. Yates Rutgers, The State University of New Jersey David J. Goodman New York University
More informationDistributed estimation in sensor networks
in sensor networks A. Benavoli Dpt. di Sistemi e Informatica Università di Firenze, Italy. e-mail: benavoli@dsi.unifi.it Outline 1 An introduction to 2 3 An introduction to An introduction to In recent
More informationEUSIPCO
EUSIPCO 3 569736677 FULLY ISTRIBUTE SIGNAL ETECTION: APPLICATION TO COGNITIVE RAIO Franc Iutzeler Philippe Ciblat Telecom ParisTech, 46 rue Barrault 753 Paris, France email: firstnamelastname@telecom-paristechfr
More informationSTONY BROOK UNIVERSITY. CEAS Technical Report 829
1 STONY BROOK UNIVERSITY CEAS Technical Report 829 Variable and Multiple Target Tracking by Particle Filtering and Maximum Likelihood Monte Carlo Method Jaechan Lim January 4, 2006 2 Abstract In most applications
More informationDistributed Binary Quantizers for Communication Constrained Large-scale Sensor Networks
Distributed Binary Quantizers for Communication Constrained Large-scale Sensor Networks Ying Lin and Biao Chen Dept. of EECS Syracuse University Syracuse, NY 13244, U.S.A. ylin20 {bichen}@ecs.syr.edu Peter
More informationDistributed Detection and Estimation in Wireless Sensor Networks: Resource Allocation, Fusion Rules, and Network Security
Distributed Detection and Estimation in Wireless Sensor Networks: Resource Allocation, Fusion Rules, and Network Security Edmond Nurellari The University of Leeds, UK School of Electronic and Electrical
More informationMINIMUM EXPECTED RISK PROBABILITY ESTIMATES FOR NONPARAMETRIC NEIGHBORHOOD CLASSIFIERS. Maya Gupta, Luca Cazzanti, and Santosh Srivastava
MINIMUM EXPECTED RISK PROBABILITY ESTIMATES FOR NONPARAMETRIC NEIGHBORHOOD CLASSIFIERS Maya Gupta, Luca Cazzanti, and Santosh Srivastava University of Washington Dept. of Electrical Engineering Seattle,
More informationSliding Window Test vs. Single Time Test for Track-to-Track Association
Sliding Window Test vs. Single Time Test for Track-to-Track Association Xin Tian Dept. of Electrical and Computer Engineering University of Connecticut Storrs, CT 06269-257, U.S.A. Email: xin.tian@engr.uconn.edu
More informationSIGNAL STRENGTH LOCALIZATION BOUNDS IN AD HOC & SENSOR NETWORKS WHEN TRANSMIT POWERS ARE RANDOM. Neal Patwari and Alfred O.
SIGNAL STRENGTH LOCALIZATION BOUNDS IN AD HOC & SENSOR NETWORKS WHEN TRANSMIT POWERS ARE RANDOM Neal Patwari and Alfred O. Hero III Department of Electrical Engineering & Computer Science University of
More informationDependence. Practitioner Course: Portfolio Optimization. John Dodson. September 10, Dependence. John Dodson. Outline.
Practitioner Course: Portfolio Optimization September 10, 2008 Before we define dependence, it is useful to define Random variables X and Y are independent iff For all x, y. In particular, F (X,Y ) (x,
More informationMultivariate Distribution Models
Multivariate Distribution Models Model Description While the probability distribution for an individual random variable is called marginal, the probability distribution for multiple random variables is
More informationSIGNAL STRENGTH LOCALIZATION BOUNDS IN AD HOC & SENSOR NETWORKS WHEN TRANSMIT POWERS ARE RANDOM. Neal Patwari and Alfred O.
SIGNAL STRENGTH LOCALIZATION BOUNDS IN AD HOC & SENSOR NETWORKS WHEN TRANSMIT POWERS ARE RANDOM Neal Patwari and Alfred O. Hero III Department of Electrical Engineering & Computer Science University of
More informationNovel spectrum sensing schemes for Cognitive Radio Networks
Novel spectrum sensing schemes for Cognitive Radio Networks Cantabria University Santander, May, 2015 Supélec, SCEE Rennes, France 1 The Advanced Signal Processing Group http://gtas.unican.es The Advanced
More informationg(.) 1/ N 1/ N Decision Decision Device u u u u CP
Distributed Weak Signal Detection and Asymptotic Relative Eciency in Dependent Noise Hakan Delic Signal and Image Processing Laboratory (BUSI) Department of Electrical and Electronics Engineering Bogazici
More informationChapter 5 continued. Chapter 5 sections
Chapter 5 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions
More informationDependence. MFM Practitioner Module: Risk & Asset Allocation. John Dodson. September 11, Dependence. John Dodson. Outline.
MFM Practitioner Module: Risk & Asset Allocation September 11, 2013 Before we define dependence, it is useful to define Random variables X and Y are independent iff For all x, y. In particular, F (X,Y
More informationSemi-parametric predictive inference for bivariate data using copulas
Semi-parametric predictive inference for bivariate data using copulas Tahani Coolen-Maturi a, Frank P.A. Coolen b,, Noryanti Muhammad b a Durham University Business School, Durham University, Durham, DH1
More informationDecision Fusion With Unknown Sensor Detection Probability
208 IEEE SIGNAL PROCESSING LETTERS, VOL. 21, NO. 2, FEBRUARY 2014 Decision Fusion With Unknown Sensor Detection Probability D. Ciuonzo, Student Member, IEEE, P.SalvoRossi, Senior Member, IEEE Abstract
More informationRobust Binary Quantizers for Distributed Detection
1 Robust Binary Quantizers for Distributed Detection Ying Lin, Biao Chen, and Bruce Suter Abstract We consider robust signal processing techniques for inference-centric distributed sensor networks operating
More informationIntroduction to Probability and Statistics (Continued)
Introduction to Probability and Statistics (Continued) Prof. icholas Zabaras Center for Informatics and Computational Science https://cics.nd.edu/ University of otre Dame otre Dame, Indiana, USA Email:
More informationEstimation, Detection, and Identification CMU 18752
Estimation, Detection, and Identification CMU 18752 Graduate Course on the CMU/Portugal ECE PhD Program Spring 2008/2009 Instructor: Prof. Paulo Jorge Oliveira pjcro @ isr.ist.utl.pt Phone: +351 21 8418053
More informationSequential Procedure for Testing Hypothesis about Mean of Latent Gaussian Process
Applied Mathematical Sciences, Vol. 4, 2010, no. 62, 3083-3093 Sequential Procedure for Testing Hypothesis about Mean of Latent Gaussian Process Julia Bondarenko Helmut-Schmidt University Hamburg University
More informationEECS564 Estimation, Filtering, and Detection Exam 2 Week of April 20, 2015
EECS564 Estimation, Filtering, and Detection Exam Week of April 0, 015 This is an open book takehome exam. You have 48 hours to complete the exam. All work on the exam should be your own. problems have
More informationTarget Localization in Wireless Sensor Networks with Quantized Data in the Presence of Byzantine Attacks
Target Localization in Wireless Sensor Networks with Quantized Data in the Presence of Byzantine Attacks Keshav Agrawal, Aditya Vempaty, Hao Chen and Pramod K. Varshney Electrical Engineering Department,
More informationCopulas. Mathematisches Seminar (Prof. Dr. D. Filipovic) Di Uhr in E
Copulas Mathematisches Seminar (Prof. Dr. D. Filipovic) Di. 14-16 Uhr in E41 A Short Introduction 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 The above picture shows a scatterplot (500 points) from a pair
More informationSensor Tasking and Control
Sensor Tasking and Control Sensing Networking Leonidas Guibas Stanford University Computation CS428 Sensor systems are about sensing, after all... System State Continuous and Discrete Variables The quantities
More informationOptimal Sensor Rules and Unified Fusion Rules for Multisensor Multi-hypothesis Network Decision Systems with Fading Channels
Optimal Sensor Rules and Unified Fusion Rules for Multisensor Multi-hypothesis Network Decision Systems with Fading Channels Qing an Ren Yunmin Zhu Dept. of Mathematics Sichuan University Sichuan, China
More informationImputation Algorithm Using Copulas
Metodološki zvezki, Vol. 3, No. 1, 2006, 109-120 Imputation Algorithm Using Copulas Ene Käärik 1 Abstract In this paper the author demonstrates how the copulas approach can be used to find algorithms for
More informationDetection Performance Limits for Distributed Sensor Networks in the Presence of Nonideal Channels
1 Detection Performance imits for Distributed Sensor Networks in the Presence of Nonideal Channels Qi Cheng, Biao Chen and Pramod K Varshney Abstract Existing studies on the classical distributed detection
More informationOn prediction and density estimation Peter McCullagh University of Chicago December 2004
On prediction and density estimation Peter McCullagh University of Chicago December 2004 Summary Having observed the initial segment of a random sequence, subsequent values may be predicted by calculating
More informationThe Instability of Correlations: Measurement and the Implications for Market Risk
The Instability of Correlations: Measurement and the Implications for Market Risk Prof. Massimo Guidolin 20254 Advanced Quantitative Methods for Asset Pricing and Structuring Winter/Spring 2018 Threshold
More informationProceedings of the 2016 Winter Simulation Conference T. M. K. Roeder, P. I. Frazier, R. Szechtman, E. Zhou, T. Huschka, and S. E. Chick, eds.
Proceedings of the 2016 Winter Simulation Conference T. M. K. Roeder, P. I. Frazier, R. Szechtman, E. Zhou, T. Huschka, and S. E. Chick, eds. A SIMULATION-BASED COMPARISON OF MAXIMUM ENTROPY AND COPULA
More informationGeneralizing Stochastic Resonance by the Transformation Method
Generalizing Stochastic Resonance by the Transformation Method Steven Kay Dept. of Electrical, Computer, and Biomedical Engineering University of Rhode Island Kingston, RI 0288 40-874-5804 (voice) 40-782-6422
More informationDigital Transmission Methods S
Digital ransmission ethods S-7.5 Second Exercise Session Hypothesis esting Decision aking Gram-Schmidt method Detection.K.K. Communication Laboratory 5//6 Konstantinos.koufos@tkk.fi Exercise We assume
More informationMaximum Likelihood Estimation
Connexions module: m11446 1 Maximum Likelihood Estimation Clayton Scott Robert Nowak This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License Abstract
More informationThe Bayes classifier
The Bayes classifier Consider where is a random vector in is a random variable (depending on ) Let be a classifier with probability of error/risk given by The Bayes classifier (denoted ) is the optimal
More informationEmulating Nuclear Emissions with a Pulsed Laser
1 Emulating Nuclear Emissions with a Pulsed Laser Benjamin J. Hockman, Jianxin Sun and Herbert G. Tanner Abstract The paper presents an approach to emulate the Poisson process observed by a sensor when
More informationDetection theory. H 0 : x[n] = w[n]
Detection Theory Detection theory A the last topic of the course, we will briefly consider detection theory. The methods are based on estimation theory and attempt to answer questions such as Is a signal
More informationUniversity of Siena. Multimedia Security. Watermark extraction. Mauro Barni University of Siena. M. Barni, University of Siena
Multimedia Security Mauro Barni University of Siena : summary Optimum decoding/detection Additive SS watermarks Decoding/detection of QIM watermarks The dilemma of de-synchronization attacks Geometric
More informationOn the Optimality of Likelihood Ratio Test for Prospect Theory Based Binary Hypothesis Testing
1 On the Optimality of Likelihood Ratio Test for Prospect Theory Based Binary Hypothesis Testing Sinan Gezici, Senior Member, IEEE, and Pramod K. Varshney, Life Fellow, IEEE Abstract In this letter, the
More informationAn Invariance Property of the Generalized Likelihood Ratio Test
352 IEEE SIGNAL PROCESSING LETTERS, VOL. 10, NO. 12, DECEMBER 2003 An Invariance Property of the Generalized Likelihood Ratio Test Steven M. Kay, Fellow, IEEE, and Joseph R. Gabriel, Member, IEEE Abstract
More informationTechnical Report 1004 Dept. of Biostatistics. Some Exact and Approximations for the Distribution of the Realized False Discovery Rate
Technical Report 14 Dept. of Biostatistics Some Exact and Approximations for the Distribution of the Realized False Discovery Rate David Gold ab, Jeffrey C. Miecznikowski ab1 a Department of Biostatistics,
More informationMultiple Bits Distributed Moving Horizon State Estimation for Wireless Sensor Networks. Ji an Luo
Multiple Bits Distributed Moving Horizon State Estimation for Wireless Sensor Networks Ji an Luo 2008.6.6 Outline Background Problem Statement Main Results Simulation Study Conclusion Background Wireless
More informationProbability and Information Theory. Sargur N. Srihari
Probability and Information Theory Sargur N. srihari@cedar.buffalo.edu 1 Topics in Probability and Information Theory Overview 1. Why Probability? 2. Random Variables 3. Probability Distributions 4. Marginal
More information2. What are the tradeoffs among different measures of error (e.g. probability of false alarm, probability of miss, etc.)?
ECE 830 / CS 76 Spring 06 Instructors: R. Willett & R. Nowak Lecture 3: Likelihood ratio tests, Neyman-Pearson detectors, ROC curves, and sufficient statistics Executive summary In the last lecture we
More informationMark your answers ON THE EXAM ITSELF. If you are not sure of your answer you may wish to provide a brief explanation.
CS 189 Spring 2015 Introduction to Machine Learning Midterm You have 80 minutes for the exam. The exam is closed book, closed notes except your one-page crib sheet. No calculators or electronic items.
More informationECE531 Screencast 9.2: N-P Detection with an Infinite Number of Possible Observations
ECE531 Screencast 9.2: N-P Detection with an Infinite Number of Possible Observations D. Richard Brown III Worcester Polytechnic Institute Worcester Polytechnic Institute D. Richard Brown III 1 / 7 Neyman
More informationA Systematic Framework for Composite Hypothesis Testing of Independent Bernoulli Trials
IEEE SIGNAL PROCESSING LETTERS, VOL. 22, NO. 9, SEPTEMBER 2015 1249 A Systematic Framework for Composite Hypothesis Testing of Independent Bernoulli Trials D. Ciuonzo, Member, IEEE, A.DeMaio, Fellow, IEEE,
More informationIntroduction to Statistical Inference
Structural Health Monitoring Using Statistical Pattern Recognition Introduction to Statistical Inference Presented by Charles R. Farrar, Ph.D., P.E. Outline Introduce statistical decision making for Structural
More informationIN HYPOTHESIS testing problems, a decision-maker aims
IEEE SIGNAL PROCESSING LETTERS, VOL. 25, NO. 12, DECEMBER 2018 1845 On the Optimality of Likelihood Ratio Test for Prospect Theory-Based Binary Hypothesis Testing Sinan Gezici, Senior Member, IEEE, and
More informationPROBABILITY AND STOCHASTIC PROCESSES A Friendly Introduction for Electrical and Computer Engineers
PROBABILITY AND STOCHASTIC PROCESSES A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates Rutgers, The State University ofnew Jersey David J. Goodman Rutgers, The State University
More informationFusing Heterogeneous Data for Detection Under Non-stationary Dependence
Syracuse University SURFACE Electrical Engineering and Computer Science College of Engineering and Computer Science 202 Fusing Heterogeneous Data for Detection Under on-stationary Dependence Hao He Syracuse
More informationThe Mixture Approach for Simulating New Families of Bivariate Distributions with Specified Correlations
The Mixture Approach for Simulating New Families of Bivariate Distributions with Specified Correlations John R. Michael, Significance, Inc. and William R. Schucany, Southern Methodist University The mixture
More informationMaximum Likelihood Estimation. only training data is available to design a classifier
Introduction to Pattern Recognition [ Part 5 ] Mahdi Vasighi Introduction Bayesian Decision Theory shows that we could design an optimal classifier if we knew: P( i ) : priors p(x i ) : class-conditional
More informationLIKELIHOOD RECEIVER FOR FH-MFSK MOBILE RADIO*
LIKELIHOOD RECEIVER FOR FH-MFSK MOBILE RADIO* Item Type text; Proceedings Authors Viswanathan, R.; S.C. Gupta Publisher International Foundation for Telemetering Journal International Telemetering Conference
More informationRobust Subspace DOA Estimation for Wireless Communications
Robust Subspace DOA Estimation for Wireless Communications Samuli Visuri Hannu Oja ¾ Visa Koivunen Laboratory of Signal Processing Computer Technology Helsinki Univ. of Technology P.O. Box 3, FIN-25 HUT
More informationEEL 851: Biometrics. An Overview of Statistical Pattern Recognition EEL 851 1
EEL 851: Biometrics An Overview of Statistical Pattern Recognition EEL 851 1 Outline Introduction Pattern Feature Noise Example Problem Analysis Segmentation Feature Extraction Classification Design Cycle
More informationOptimal Sensor Placement for Intruder Detection
Optimal Sensor Placement for Intruder Detection 1 Waseem A. Malik, Nuno C. Martins, and Ananthram Swami arxiv:1109.5466v1 [cs.sy] 26 Se011 I. INTRODUCTION The fields of detection, resource allocation,
More informationMonte Carlo Studies. The response in a Monte Carlo study is a random variable.
Monte Carlo Studies The response in a Monte Carlo study is a random variable. The response in a Monte Carlo study has a variance that comes from the variance of the stochastic elements in the data-generating
More informationTwo hours. To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER.
Two hours MATH38181 To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER EXTREME VALUES AND FINANCIAL RISK Examiner: Answer any FOUR
More informationLecture 5: Likelihood ratio tests, Neyman-Pearson detectors, ROC curves, and sufficient statistics. 1 Executive summary
ECE 830 Spring 207 Instructor: R. Willett Lecture 5: Likelihood ratio tests, Neyman-Pearson detectors, ROC curves, and sufficient statistics Executive summary In the last lecture we saw that the likelihood
More informationOn Noise-Enhanced Distributed Inference in the Presence of Byzantines
Syracuse University SURFACE Electrical Engineering and Computer Science College of Engineering and Computer Science 211 On Noise-Enhanced Distributed Inference in the Presence of Byzantines Mukul Gagrani
More informationIf there exists a threshold k 0 such that. then we can take k = k 0 γ =0 and achieve a test of size α. c 2004 by Mark R. Bell,
Recall The Neyman-Pearson Lemma Neyman-Pearson Lemma: Let Θ = {θ 0, θ }, and let F θ0 (x) be the cdf of the random vector X under hypothesis and F θ (x) be its cdf under hypothesis. Assume that the cdfs
More information5682 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 12, DECEMBER /$ IEEE
5682 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 12, DECEMBER 2009 Hyperplane-Based Vector Quantization for Distributed Estimation in Wireless Sensor Networks Jun Fang, Member, IEEE, and Hongbin
More informationWhen is a copula constant? A test for changing relationships
When is a copula constant? A test for changing relationships Fabio Busetti and Andrew Harvey Bank of Italy and University of Cambridge November 2007 usetti and Harvey (Bank of Italy and University of Cambridge)
More informationLatent Variable Models for Binary Data. Suppose that for a given vector of explanatory variables x, the latent
Latent Variable Models for Binary Data Suppose that for a given vector of explanatory variables x, the latent variable, U, has a continuous cumulative distribution function F (u; x) and that the binary
More informationDetection and Estimation Final Project Report: Modeling and the K-Sigma Algorithm for Radiation Detection
Detection and Estimation Final Project Report: Modeling and the K-Sigma Algorithm for Radiation Detection Sijie Xiong, 151004243 Department of Electrical and Computer Engineering Rutgers, the State University
More informationModelling Dropouts by Conditional Distribution, a Copula-Based Approach
The 8th Tartu Conference on MULTIVARIATE STATISTICS, The 6th Conference on MULTIVARIATE DISTRIBUTIONS with Fixed Marginals Modelling Dropouts by Conditional Distribution, a Copula-Based Approach Ene Käärik
More informationChapter 2 Signal Processing at Receivers: Detection Theory
Chapter Signal Processing at Receivers: Detection Theory As an application of the statistical hypothesis testing, signal detection plays a key role in signal processing at receivers of wireless communication
More informationHST.582J / 6.555J / J Biomedical Signal and Image Processing Spring 2007
MIT OpenCourseWare http://ocw.mit.edu HST.582J / 6.555J / 16.456J Biomedical Signal and Image Processing Spring 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationLocalization and Tracking of Radioactive Source Carriers in Person Streams
Localization and Tracking of Radioactive Carriers in Person Streams Monika Wieneke and Wolfgang Koch Dept. Sensor Data and Information Fusion Fraunhofer FKIE, Wachtberg, Germany Email: monika.wieneke@fkie.fraunhofer.de
More informationGaussian Processes. Le Song. Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012
Gaussian Processes Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 01 Pictorial view of embedding distribution Transform the entire distribution to expected features Feature space Feature
More informationEncoder Decoder Design for Event-Triggered Feedback Control over Bandlimited Channels
Encoder Decoder Design for Event-Triggered Feedback Control over Bandlimited Channels LEI BAO, MIKAEL SKOGLUND AND KARL HENRIK JOHANSSON IR-EE- 26: Stockholm 26 Signal Processing School of Electrical Engineering
More informationCompressed Statistical Testing and Application to Radar
Compressed Statistical Testing and Application to Radar Hsieh-Chung Chen, H. T. Kung, and Michael C. Wicks Harvard University, Cambridge, MA, USA University of Dayton, Dayton, OH, USA Abstract We present
More informationEvidence Theory based Cooperative Energy Detection under Noise Uncertainty
Evidence Theory based Cooperative Energy Detection under Noise Uncertainty by Sachin Chaudhari, Prakash Gohain in IEEE GLOBECOM 7 Report No: IIIT/TR/7/- Centre for Communications International Institute
More informationMinimum Feedback Rates for Multi-Carrier Transmission With Correlated Frequency Selective Fading
Minimum Feedback Rates for Multi-Carrier Transmission With Correlated Frequency Selective Fading Yakun Sun and Michael L. Honig Department of ECE orthwestern University Evanston, IL 60208 Abstract We consider
More informationPerformance Comparison of K-Means and Expectation Maximization with Gaussian Mixture Models for Clustering EE6540 Final Project
Performance Comparison of K-Means and Expectation Maximization with Gaussian Mixture Models for Clustering EE6540 Final Project Devin Cornell & Sushruth Sastry May 2015 1 Abstract In this article, we explore
More informationIntroducing the Normal Distribution
Department of Mathematics Ma 3/103 KC Border Introduction to Probability and Statistics Winter 2017 Lecture 10: Introducing the Normal Distribution Relevant textbook passages: Pitman [5]: Sections 1.2,
More informationA GLRT FOR RADAR DETECTION IN THE PRESENCE OF COMPOUND-GAUSSIAN CLUTTER AND ADDITIVE WHITE GAUSSIAN NOISE. James H. Michels. Bin Liu, Biao Chen
A GLRT FOR RADAR DETECTION IN THE PRESENCE OF COMPOUND-GAUSSIAN CLUTTER AND ADDITIVE WHITE GAUSSIAN NOISE Bin Liu, Biao Chen Syracuse University Dept of EECS, Syracuse, NY 3244 email : biliu{bichen}@ecs.syr.edu
More informationECE 3800 Probabilistic Methods of Signal and System Analysis
ECE 3800 Probabilistic Methods of Signal and System Analysis Dr. Bradley J. Bazuin Western Michigan University College of Engineering and Applied Sciences Department of Electrical and Computer Engineering
More informationBAYESIAN DECISION THEORY
Last updated: September 17, 2012 BAYESIAN DECISION THEORY Problems 2 The following problems from the textbook are relevant: 2.1 2.9, 2.11, 2.17 For this week, please at least solve Problem 2.3. We will
More informationChapter 5. Chapter 5 sections
1 / 43 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions
More informationRECENTLY, wireless sensor networks have been the object
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 4, APRIL 2007 1511 Distributed Sequential Bayesian Estimation of a Diffusive Source in Wireless Sensor Networks Tong Zhao, Student Member, IEEE, and
More informationADAPTIVE CLUSTERING ALGORITHM FOR COOPERATIVE SPECTRUM SENSING IN MOBILE ENVIRONMENTS. Jesus Perez and Ignacio Santamaria
ADAPTIVE CLUSTERING ALGORITHM FOR COOPERATIVE SPECTRUM SENSING IN MOBILE ENVIRONMENTS Jesus Perez and Ignacio Santamaria Advanced Signal Processing Group, University of Cantabria, Spain, https://gtas.unican.es/
More informationLecture 7 Introduction to Statistical Decision Theory
Lecture 7 Introduction to Statistical Decision Theory I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw December 20, 2016 1 / 55 I-Hsiang Wang IT Lecture 7
More informationGaussian processes. Chuong B. Do (updated by Honglak Lee) November 22, 2008
Gaussian processes Chuong B Do (updated by Honglak Lee) November 22, 2008 Many of the classical machine learning algorithms that we talked about during the first half of this course fit the following pattern:
More informationIntroduction to Machine Learning
Introduction to Machine Learning Introduction to Probabilistic Methods Varun Chandola Computer Science & Engineering State University of New York at Buffalo Buffalo, NY, USA chandola@buffalo.edu Chandola@UB
More informationHST 583 FUNCTIONAL MAGNETIC RESONANCE IMAGING DATA ANALYSIS AND ACQUISITION A REVIEW OF STATISTICS FOR FMRI DATA ANALYSIS
HST 583 FUNCTIONAL MAGNETIC RESONANCE IMAGING DATA ANALYSIS AND ACQUISITION A REVIEW OF STATISTICS FOR FMRI DATA ANALYSIS EMERY N. BROWN AND CHRIS LONG NEUROSCIENCE STATISTICS RESEARCH LABORATORY DEPARTMENT
More information