Distributed detection and Uniformly Most Powerful tests

Logical fusion rules for merging structured news reports

In this paper we study a wireless passive sensor network. The sensors are deployed to estimate the true values of multiple active target signals. The sensors forward their observation to a fusion center, which processes the observation of each sensor by a set of fusion rules. One achievement of this paper is proposing an unbiased estimator with minimized variance of errors. To do so, we optimize both the power allocation to the sensor nodes and the fusion rules. The optimal solution to the fusion rules are attained analytically, while the power allocation is solved sub-optimally. Finally, our results are reinforced by numerical simulations.

In this paper, we study some relationships between the detection and estimation theories for a binary composite hypothesis test <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> against <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and a related estimation problem. We start with a one-dimensional (1D) space for the unknown parameter space and one-sided hypothesis problems and then extend out results into more general cases. For one-sided tests, we show that the uniformly most powerful (UMP) test is achieved by comparing the minimum variance and unbiased estimator (MVUE) of the unknown parameter with a threshold. Thus for the case where the UMP test does not exist, the MVUE of the unknown parameter does not exist either. Therefore for such cases, a good estimator of the unknown parameter is deemed as a good decision statistic for the test. For a more general class of composite testing with multiple unknown parameters, we prove that the MVUE of a separating function (SF) can serve as the optimal decision statistic for the UMP unbiased test where the SF is continuous, differentiable, positive for all parameters under <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and is negative for the parameters under <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> . We then prove that the UMP unbiased statistic is equal to the MVUE of an SF. In many problems with multiple unknown parameters, the UMP test does not exist. For such cases, we show that if one detector between two detectors has a better receiver operating characteristic (ROC) curve, then using its decision statistic we can estimate the SF more ε-accurately, in probability. For example, the SF is the signal-to-noise ratio (SNR) in some problems. These results motivate us to introduce new suboptimal SF-estimator tests (SFETs) which are easy to derive for many problems. Finally, we provide some practical examples to study the relationship between the decision statistic of a test and the estimator of its corresponding SF.

Uniformly most powerful (UMP) tests are known to exist in one-parameter exponential families when the hypothesis H0 and the alternative hypothesis H1 are given by

The Uniformly Most Powerful (UMP) test, which is one with the highest probability of detection over all unknown parameters in a composite hypothesis test, does not exist in the most practical problems. It is common to derive the suboptimal Generalize Likelihood Ratio (GLR) detector, that is shown to perform close to optimal invariant tests in some problems. The Uniformly Most Powerful Invariant (UMPI) test is the one that is used for the performance evaluation of the GLR detector. However, the GLR or UMPI detectors are not simply derivable in many problems. In this paper, we suggest that the problem be transformed to a new space. We derive the conditions of the transformation, such that the detectors for the original problem and the transformed ones become equivalent. In addition, using this method, we discuss two practical problems (one in radar and other in communication) and derive the GLR detectors. We also find the UMPI detectors in these problems to verify the performance of the derived GLR detector.

This letter is focused on the classic problem of testing samples drawn from independent Bernoulli probability mass functions, when the success probability under the alternative hypothesis is not known. The goal is to provide a systematic taxonomy of the viable detectors (designed according to theoretically-founded criteria) which can be used for the specific instance of the problem. Both One-Sided (OS) and Two-Sided (TS) tests are considered, with reference to: ( $i$ ) identical success probability (a homogeneous scenario) or ( $ii$ ) different success probabilities (a non-homogeneous scenario) for the observed samples. As a result of the study, a complete summary (in tabular form) of the relevant statistics for the problem is provided, along with a discussion on the existence of the Uniformly Most Powerful (UMP) test. Finally, when the Likelihood Ratio Test (LRT) is not UMP, existence of the UMP detector after reduction by invariance is investigated.

SUMMARY The use of major genes is a component of livestock improvement. This requires identification of the genotypes of reproducers, in particular progeny-tested sires after mating with homozygous females. Two test statistics derived from the nonnested test of Cox (1961, Proceedings of the Fouirth Berkeley Svmposium on Mathematical Statistics and Probability 1, 105-123; 1962, Journal of the Royal Statistical Society, Series B 24, 406-424) are proposed. For both tests, asymptotic normality is proved algebraically and the speed of convergence is studied through simulations. In the cases where a uniformly most powerful (UMP) test exists, the power of one of the tests is close to the power of that UMP test. Finally, the tests are applied to a population of poultry carrying the dwarf gene.

CoopMAC has been recently proposed as a possible implementation of cooperation protocols in the medium access control (MAC) layer of a wireless network. However, some nodes may refrain from cooperation for selfish purposes, e.g. in order to save energy, in what is called selfish behavior or misbehavior. This protocol violation worsens other nodes' performance and can be avoided if other nodes detect and punish (e.g. banning from the network) misbehaving nodes. However, fading and interference may prevent nodes from cooperating even if they are willing, therefore it is not trivial to identify misbehaving nodes. In a fading scenario where an automatic repeat request (ARQ) protocol is used, we propose a mechanism that allows to detect misbehaving nodes. Two approaches, either based on the uniformly most powerful (UMP) test or on the sequential probability ratio test (SPRT) are considered. The two techniques are characterized and compared in terms of their average detection delay and resulting network performance.

In this study, the authors study the modulation classification of linearly modulated signals including amplitude shift keying (ASK), phase shift keying (PSK) and quadrature amplitude modulation (QAM) signals. The authors consider an unknown frequency non-selective slowly fading channel with an unknown variance additive white Gaussian noise. The authors treat this classification problem as a multi-hypotheses test which is invariant under the complex scale. In such a case, the authors objective is to find uniformly most powerful (UMP) test in the class of invariant decisions. However, the authors find out that the UMPI test does not exist; instead, they provide a most powerful invariant (MPI) PSK signal classifier for known signal to noise ratio and use it as the upper performance bound for any invariant classifier. The authros also propose a hybrid likelihood ratio test (HLRT) solution which can be employed for the classification of linearly modulated signals, inter-family and intra-family. The authors also explain the efficient implementation of these algorithms in some steps. In order to reduce the computational cost, the authors propose quasi-HLRT classifiers for PSK signals. Some simulation examples are provided that show the power of the proposed algorithms in the classification of linearly modulated signals.

The concept of invariance in hypothesis testing is brought to bear on the problem of detecting signals of known form and unknown energy in Gaussian noise of unknown level. The noise covariance function is assumed to be <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K(t,u) = \sigma^2 \pho(t,u)</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\rho(t,u)</tex> is the known form of the covariance function and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sigma^2</tex> is the unknown level. Classical approaches to signal detection depend on the assumption that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K(t,u)</tex> is known completely. Then, a correlation-type receiver that is the uniformly most powerful (UMP) test of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H_o</tex> (signal absent) versus <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H_1</tex> (signal present) can be derived. When <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sigma^2</tex> is unknown, there exists no UMP test. However, it is shown in this paper that there exists a test of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H_o</tex> versus <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H_1</tex> that is UMP-invariant for a very natural group of transformations on the space of observations. The derived test is found to be independent of knowledge about the noise level <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sigma^2</tex> , since the derived test (receiver) contains an error-free estimate of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sigma^2</tex> . This utopian conclusion is reconciled by noting that the derived receiver can never be physically realized. It is shown that any physically realizable version of the receiver has a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> -distributed test statistic. This permits choice of operating receiver thresholds and evaluation of performance characteristics.

Recently, a design was proposed for the Simultaneous Global Drug Development Program (SGDDP) to assess the impact of ethnic factors on the effect of a new treatment for a targeted ethnic (TE) population. It used weighted Z tests to combine the information collected from the TE and non-TE (NTE) subgroups in the SGDDP based on the fundamental assumption on their shared biological commonality. In this article, we mathematically formulated this assumption as the quantitative interaction between treatment effect and subgroup. We used it to more rigorously describe the hypotheses, and showed the unbiasedness of the weighted Z test. Moreover, to study the loss of efficiency from down weighting the NTE information in this SGDDP design, we compared the power of their test with that of the uniformly most powerful (UMP) test, which we showed was also a weighted Z test. We discussed that the choice of weight should balance the maximization of power when the assumption holds and the minimization of bias otherwise.

We present the problem of polarization diversity detection in compound-Gaussian clutter with unknown covariance matrix. To this end we assume that a set of secondary data, free of signal components and with the same covariance structure of the cell under test, is available. Due to the lack of a uniformly most powerful (UMP) detector we resort to a design procedure based upon the Rao and the Wald tests. Specifically we first derive the Rao and the Wald tests assuming that the covariance matrix is known, and then we substitute into the derived decision rules a suitable estimate of the clutter covariance. Interestingly, the newly proposed detectors share the constant false alarm rate (CFAR) property with respect to the texture statistical characterization. Moreover simulation results have shown that the Wald test based detector ensures a performance level higher than the Rao test. We have also conducted a further performance analysis, in the presence of real clutter data and in comparison with the previously proposed generalized likelihood ratio test (GLRT) based receivers, which highlights that, in general, the Wald test receiver outperforms its counterparts. Finally, since the newly proposed decision rules as well as the previously designed GLRTs do not ensure the CFAR property with respect to the clutter covariance matrix, we have developed a sensitivity analysis on the probability of false alarm (P/sub fa/), based on simulated clutter with covariance matrix estimated from real radar data. The results have shown that (P/sub fa/) is only slightly affected by variations in the clutter correlation properties and hence the CFARness is substantially achieved.

Process capability indices are important measures in statistical science and index Cpk is the most commonly used index in managerial decisions. In statistical inferences for Cpk, testing problems are relatively less explored than point and interval estimating problems. Tests for Cpk are often based on confidence intervals obtained from central limit theorems. Unfortunately, most tests are lack of theoretical justifications. In this article, we first propose a test for Cpk that is asymptotically equivalent to the likelihood ratio test. The above results then lead to an asymptotically UMP test for Cpk. In practice, UMP test provides a wise way to reduce the risk of making wrong decisions and save resources.

We consider a binary parallel decentralized detection system. It employs n local detectors (LDs) to observe a binary phenomenon in order to accept one of two hypotheses about the phenomenon (hypothesis H 0 or hypothesis H 1 ). Each LD uses a local decision rule to compress its local observations into a binary local decision. The Data Fusion Center (DFC) uses a fusion rule to combine all binary local decisions into a global binary decision. The combination of the global fusion rule and all local decision rules in the system is referred to as a decision strategy. In many scenarios, the system uses a single (deterministic) decision strategy — the DFC used a single deterministic fusion rule and each LD uses a single deterministic decision rule. A randomized strategy involves a randomized or pseudo — randomized choice between several deterministic decision strategies. In some instances, such randomized strategy is needed to ensure that the Receiver Operator Characteristic (ROC) curve of the architecture is convex. We are interested in scenarios that require that the randomized strategy be coordinated between the DFC and the LDs, maintaining synchronization between the selection scheme of local decision rules and global fusion rules. Circumstances that require such coordinated randomized selection are labeled “dependent randomization.” We present systems that use dependent randomization, and also discuss their response when synchronization is occasionally lost.

Since there has been an increasing interest in the areas of Internet of Things (IoT) and artificial intelligence that often deals with a large number of sensors, this chapter investigates the decision fusion problem for large-scale sensor networks. Due to unavoidable transmission channel interference, we consider sensor networks with nonideal channels that are prone to errors. When the fusion rule is fixed, we present the necessary condition for the optimal sensor rules that minimize the Monte Carlo cost function. For the K-out-of-L fusion rule chosen very often in practice, we analytically derive the optimal sensor rules. For general fusion rules, a Monte Carlo Gauss-Seidel optimization algorithm is developed to search for the optimal sensor rules. The complexity of the new algorithm is of the order of OLN compared with OLNL of the previous algorithm that was based on Riemann sum approximation, where L is the number of sensors and N is the number of samples. Thus, the proposed method allows us to design the decision fusion rule for large-scale sensor networks. Moreover, the algorithm is generalized to simultaneously search for the optimal sensor rules and the optimal fusion rule. Finally, numerical examples show the effectiveness of the new algorithms for large-scale sensor networks with nonideal channels.