Binary Hypothesis Testing Problem Research Articles

Locally Most Powerful Invariant Tests for the Properness of Quaternion Gaussian Vectors

Previous works have addressed the second-order statistical characterization of quaternion random vectors, introducing different properness definitions, and presenting the generalized likelihood ratio tests (GLRTs) for determining the kind of quaternion properness. This paper considers the more challenging problem of deriving the locally most powerful invariant tests (LMPITs), which can be obtained, even without an explicit expression for the maximal invariants, thanks to the Wijsman's theorem. Specifically, we consider three binary hypothesis testing problems involving the two main kinds of quaternion properness, and show that the LMPIT statistics are given by the Frobenius norm of three previously defined sample coherence matrices. The proposed detectors exhibit interesting connections with the problem of testing for the properness of a complex vector, and with the problems of testing for the sphericity of a four-dimensional real (or two-dimensional complex proper) vector. Additionally, some numerical examples show that in general, the proposed LMPITs outperform their GLRT counterparts, and in some cases the performance gap is very noticeable.

Energy-Efficient Detection in Wireless Sensor Networks Using Likelihood Ratio and Channel State Information

In this paper we investigate transmission scheduling by Medium Access Control (MAC) for energy-efficient detection using Wireless Sensor Networks (WSN). We consider the binary hypothesis testing problem. The decision is made by an access point and is based on received data from sensors that transmit through a fading channel. We study the significance of exploiting both Channel-State Information (CSI) and Likelihood-Ratio Information (LRI) to design an adequate MAC protocol that minimizes the total transmission energy required for optimal detection. We formulate the access problem as a history-dependent decision process. The optimal solution is mathematically intractable and suffers from exponential complexity as a function of model size. Hence, we propose an approximate solution using the Markov property to reduce complexity and make the problem mathematically tractable. We designed the LRI and CSI Based Access (LCBA) protocol based on this solution. The LCBA protocol trades off between LRI and CSI to reduce the total transmission energy. Simulation results show a significant performance gain of LCBA over existing approaches.

Joint Design of Spectrum Sensing and Channel Access in Cognitive Radio Networks

Spectrum sensing is an essential functionality of cognitive radio networks. However, the effect of errors in the spectrum sensing process on the performance of the multiple access layer of both primary and secondary networks has not gained much attention. This paper aims at bridging the gap between the study of spectrum sensing and the multiple access of cognitive radio networks. To achieve this goal we pose and answer the question how the spectrum sensing errors affects the performance of cognitive radio networks from a multiple access protocol design point of view. The negative effects of the spectrum sensing errors on the throughput of both primary and secondary networks are characterized through queuing theory analysis of both networks. To alleviate these negative effects a novel joint design of the spectrum sensing and channel access mechanisms is proposed. This design is based on the observation that, in a binary hypothesis testing problem, the value of the test statistics could be used as a confidence measure for the test outcome. Therefore, this value will be used to define different channel access probabilities for secondary users. Results reveal a significant performance improvement in the maximum stable throughput of both primary and secondary networks by virtue of the proposed technique.

Channel optimization for binary hypothesis testing

AbstractIn this paper we consider the classical binary hypothesis testing problem where the iid samples are obtained through a channel. Our goal is to study the relationship between the channel capacity and the goodness of the estimation measured by the Chernoff information in order to get an upper bound on the estimation performances as well as some insight on the structure of the optimal channel.

On the Invariance, Coincidence, and Statistical Equivalence of the GLRT, Rao Test, and Wald Test

Three common techniques to discriminate between alternatives in a binary hypothesis testing problem are: the generalized likelihood ratio test (GLRT), the Rao test, and the Wald test. In this paper, we investigate some characteristics of the corresponding decision statistics and provide their expressions for some problems of particular interest in statistical signal processing. First of all, we focus on the invariance of the Rao and Wald tests with respect to transformations leaving the testing problem unaltered. Then, we introduce necessary and sufficient conditions in order for their decision statistics to coincide with twice the logarithm of the GLRT statistic. Finally, we present some detection problems, usually encountered in practical signal processing applications, where the decision variables of the three quoted tests are equivalent, namely related by strictly monotonic transformations.

Error exponents in hypothesis testing for correlated states on a spin chain

We study various error exponents in a binary hypothesis testing problem and extend recent results on the quantum Chernoff and Hoeffding bounds for product states to a setting when both the null hypothesis and the alternative hypothesis can be correlated states on a spin chain. Our results apply to states satisfying a certain factorization property; typical examples are the global Gibbs states of translation-invariant finite-range interactions as well as certain finitely correlated states.

A Fast Multiple-Source Detection and Localization Array Signal Processing Algorithm Using the Spatial Filtering and ML Approach

We propose a computationally efficient algorithm for detection of multiple signals that also gives a rough estimation of their direction of arrivals (DOAs). The narrowband received signals from a uniform linear array are first filtered by a set of orthogonal filters, e.g., by a fast Fourier transformation, in order to separate the sources into multiple spatial intervals. This transformation converts the complicated multihypothesis problem of source detection and localization into multiple binary hypothesis testing problems. For an additive white Gaussian noise (AWGN) environment, the maximum-likelihood (ML) solution of these interrelated tests requires substantially less computational complexity than that of the multihypothesis problem. For each spatial interval, a binary test detects the presence of a single source and thus gives a rough localization. We employ generalized-likelihood ratio (GLR) tests as the detection criterion, assuming that the number of sources, their power, and the noise variance are all unknown. We also show that the optimal uniformly most powerful invariant (UMPI) detector does not exist. However, we derive a UMPI detector that uses some extra information and as a result provides an upper bound performance for evaluation of any invariant detector. Simulations illustrate that the proposed noniterative GLR test performs efficiently for various number of observed data snapshots and signal-to-noise-ratios (SNR)s, and its performance is comparable to the upper bound performance. We used the proposed algorithm for the initialization of the iterative implementation of the standard ML localization. This combination is a high-resolution localization algorithm with a low computational complexity

A Simple Algorithm for Defect Detection From a Few Radiographies

The paper concerns the radiographic nondestructive testing of well-manufactured objects. The detection of anomalies is addressed from the statistical point of view as a binary hypothesis testing problem with nonlinear nuisance parameters. A new simple and numerically stable detection scheme is proposed as an alternative to the conventional generalized likelihood ratio test which becomes untractable in the non-linear case. This original decision rule detects the anomalies with a loss of a negligible part of optimality with respect to an optimal test designed for the “closest” hypothesis testing problem with linear nuisance parameters. The inspection of nuclear fuel rods is discussed to illustrate the relevance of the proposed theoretical solution.

Reducing Probability of Decision Error Using Stochastic Resonance

The problem of reducing the probability of decision error of an existing binary receiver that is suboptimal using the ideas of stochastic resonance is solved. The optimal probability density function of the random variable that should be added to the input is found to be a Dirac delta function, and hence, the optimal random variable is a constant. The constant to be added depends upon the decision regions and the probability density functions under the two hypotheses and is illustrated with an example. Also, an approximate procedure for the constant determination is derived for the mean-shifted binary hypothesis testing problem

A Decision Support Framework for Automated Screening of Diabetic Retinopathy

The early signs of diabetic retinopathy (DR) are depicted by microaneurysms among other signs. A prompt diagnosis when the disease is at the early stage can help prevent irreversible damages to the diabetic eye. In this paper, we propose a decision support system (DSS) for automated screening of early signs of diabetic retinopathy. Classification schemes for deducing the presence or absence of DR are developed and tested. The detection rule is based on binary-hypothesis testing problem which simplifies the problem to yes/no decisions. An analysis of the performance of the Bayes optimality criteria applied to DR is also presented. The proposed DSS is evaluated on the real-world data. The results suggest that by biasing the classifier towards DR detection, it is possible to make the classifier achieve good sensitivity.

One-step Prediction of Financial Time Series

This paper examines the one-step prediction of financial time series from a binary decision theory perspective. Under the assumption that the decision statistic of the binary hypothesis testing problem is a Gaussian random variable, bounds for the forecasting efficiency of the hypothesis testing problem are derived. When the criterion for forecasting performance is the total return over the investment period, an optimisation problem is formulated to compute an optimally weighted decision statistic for the binary hypothesis testing problem. Numerical results are illustrated using weekly time series of excess return between two US dollar bond portfolios having six months duration difference. In particular, it is shown that, on average, a 27 basis point excess return per annum is possible against a given benchmark by carrying out active duration management.

On the influence of a detection step on lower bounds for deterministic parameter estimation

A wide variety of actual processing requires a detection step, whose main effect is to restrict the set of observations available for parameter estimation. Therefore, as a contribution to the theoretical formulation of the joint detection and estimation problem, we address the derivation of lower bounds for deterministic parameters conditioned by a binary hypothesis testing problem. The main result is the introduction of a general scheme-detailed in the particular case of CRB-enabling the derivation of conditional deterministic MSE lower bounds. To prove that it is meaningful, we also show, with the help of a fundamental application, that the problem of lower bound tightness at low SNR may arise from an incorrect lower bound formulation that does not take into account the true nature of the problem under investigation: a joint detection-estimation problem.

Randomized data selection in detection with applications to distributed signal processing

Performing robust detection with resource limitations such as low-power requirements or limited communication bandwidth is becoming increasingly important in contexts involving distributed signal processing. One way to address these constraints consists of reducing the amount of data used by the detection algorithms. Intelligent data selection in detection can be highly dependent on a priori information about the signal and noise. In this paper, we explore detection strategies based on randomized data selection and analyze the resulting algorithms' performance. Randomized data selection is a viable approach in the absence of reliable and detailed a priori information, and it provides a reasonable lower bound on signal processing performance as more a priori information is incorporated. The randomized selection procedure has the added benefits of simple implementation in a distributed environment and limited communication overhead. As an example of detection algorithms based upon randomized selection, we analyze a binary hypothesis testing problem, and determine several useful properties of detectors derived from the likelihood ratio test. Additionally, we suggest an adaptive detector that accounts for fluctuations in the selected data subset. The advantages and disadvantages of this approach in distributed sensor networks applications are also discussed.

Optimal bi-level quantization of i.i.d. sensor observations for binary hypothesis testing

We consider the problem of binary hypothesis testing using binary decisions from independent and identically distributed (i.i.d). sensors. Identical likelihood-ratio quantizers with threshold /spl lambda/ are used at the sensors to obtain sensor decisions. Under this condition, the optimal fusion rule is known to be a k-out-of-n rule with threshold k. For the Bayesian detection problem, we show that given k, the probability of error is a quasi-convex function of /spl lambda/ and has a single minimum that is achieved by the unique optimal /spl lambda//sub opt/. Except for the trivial situation where one hypothesis is always decided, we obtain a sufficient and necessary condition on /spl lambda//sub opt/, and show that /spl lambda//sub opt/ can be efficiently obtained via the SECANT algorithm. The overall optimal solution is obtained by optimizing every pair of (k, /spl lambda/). For the Neyman-Pearson detection problem, we show that the use of the Lagrange multiplier method is justified for a given fixed k since the objective function is a quasi-convex function of /spl lambda/. We further show that the receiver operating characteristic (ROC) for a fixed k is concave downward.

Thresholding method for dimensionality reduction in recognition systems

Often recognition systems must be designed with a relatively small amount of training data. Plug-in test statistics suffer from large estimation errors, often causing the performance to degrade as the measurement vector dimension increases. Choosing a better test statistic or applying a method of dimensionality reduction are two possible solutions to this problem. In this paper, we consider a recognition problem where the data for each population are assumed to have the same parametric distribution but differ in their unknown parameters. The collected vectors of data as well as their components are assumed to be independent. The system is designed to implement a plug-in log-likelihood ratio test with maximum-likelihood (ML) estimates of the unknown parameters instead of the true parameters. Because a small amount of data is available to estimate the parameters, the performance of such a system is strongly degraded relative to the performance with known parameters. To improve the performance of the system we define a thresholding function that, when incorporated into the plug-in log-likelihood ratio, significantly decreases the probability of error for binary and multiple hypothesis testing problems for the exponential class of populations. We analyze the modified test statistic and present the results of Monte Carlo simulation. Special attention is paid to the complex Gaussian model with zero mean and unknown variances.

Joint statistical signal detection and estimation. Part I: Theoretical aspects of the problem

It is known that the problem of binary (or M-ary) hypothesis testing can be addressed in terms of the estimation of one (or M−1) discrete parameters which can assume the values 0 and 1. In the same fashion, the problem of composite hypothesis testing and parameter estimation can be seen as the joint estimation of a set of mixed discrete and continuous parameters, which can be modelled as random variables. In this paper, a comparison between serial and joint detection and parameter estimation schemes is carried out. The mathematical formulation is set up for the case of maximum a posteriori detectors and estimators; analytical results are derived for a meaningful case study, which permit to deeply understand the different mechanisms which govern the two schemes. The obtained results show that, although in quite different fashions, both schemes achieve the optimum average performance for a given amount of a priori information.

Detection and classification of spectrally equivalent processes using higher order statistics

This paper addresses the problem of detecting two spectrally equivalent processes, which are modeled by a noisy AR process and an ARMA process. Higher order statistics (HOS) are shown to be efficient tools for detection. Two HOS-based detectors are derived for the binary hypothesis testing problem (i.e., known signal spectrum): The first detector exploits the asymptotic Gaussianity of the sample estimates of the cumulants. The second detector exploits the singularity of a certain matrix based on HOS. The more general composite hypothesis testing problem (i.e., with unknown signal spectrum) is then considered and a detector proposed. The performances of the different detectors are compared in terms of receiver operating characteristic (ROC) curves. Approximate closed-form expressions are derived for the threshold and the ROCs of the three detectors.

A unified framework for evaluating bounds on the Bayesian cost

A unified framework for evaluating bounds on the average cost of an optimum Bayesian receiver with arbitrary cost assignments is presented. The framework is developed based on formulating the binary hypothesis testing problem from a decision-theoretic perspective. This formulation results in a representation for the minimum average cost that is analogous to that for the minimum probability of error. Taking advantage of this analogy, a whole new series of generalized bounds on the minimum average cost is obtained by employing the well-developed theory of bounds of the minimum probability of error problem available in the literature. To demonstrate the applicability of the proposed unified framework, two upper bounds on the minimum cost, that generalize the known Bhattacharyya and Chernoff upper bounds on the minimum probability of error, are derived. The unified framework is also used to obtain a new generalized class of upper and lower bounds in terms of a modified form of the f-divergence. All new bounds derived in the paper are shown to reduce to the probability of error bounds under special cost assignments.

An overview of decision networks and organizations

The paper summarizes recent results on both binary and M-ary distributed hypothesis testing problems with decision makers (DMs) organized in structured decision networks. The general problem of finding an optimal organizational structure and decision strategy for such networks is formulated as a functional optimization problem. A normative model to study the effect of interactions between task structure and organizational design on the performance of hierarchical organizations is presented. A binary signal detection model is considered to illustrate the joint impact of organizational design and of task environment on the organizational decision performance. The concept of a congruent organizational structure (i.e., a structure that achieves centralized performance with minimal communication) is introduced, and a graph decomposition algorithm to synthesize congruent structures is discussed.

Distributed binary hypothesis testing with feedback

The problem of binary hypothesis testing is revisited in the context of distributed detection with feedback. Two basic distributed structures with decision feedback are considered. The first structure is the fusion center network, with decision feedback connections from the fusion center element to each one of the subordinate decisionmakers. The second structure consists of a set of detectors that are fully interconnected via decision feedback. Both structures are optimized in the Neyman-Pearson sense by optimizing each decision-maker individually. Then, the time evolution of the power of the tests is derived. Definite conclusions regarding the gain induced by the feedback process and direct comparisons between the two structures and the optimal centralized scheme are obtained through asymptotic studies (that is, assuming the presence of asymptotically many local detectors). The behavior of these structures is also examined in the presence of variations in the statistical description of the hypotheses. Specific robust designs are proposed and the benefits from robust operations are established. Numerical results provide additional support to the theoretical arguments.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>