Multi-hypothesis Testing Research Articles

A class of sequential multi-hypothesis tests

In this article, we deal with sequential testing of multiple hypotheses. In the general scheme of construction of optimal tests based on the backward induction, we propose a modification which provides a simplified (generally speaking, and suboptimal) version of the optimal test, for any particular criterion of optimization. We call this DBC version (the one with Dropped Backward Control) of the optimal test. In particular, for the case of two simple hypotheses, dropping backward control in the Bayesian test produces the classical sequential probability ratio test (SPRT). Similarly, dropping backward control in the modified Kiefer-Weiss solutions produces Lorden’s 2-SPRTs. In the case of more than two hypotheses, we obtain in this way new classes of sequential multi-hypothesis tests, and investigate their properties. The efficiency of the DBC-tests is evaluated with respect to the optimal Bayesian multi-hypothesis test and with respect to the matrix sequential probability ratio test (MSPRT) by Armitage. In a multihypothesis variant of the Kiefer-Weiss problem for binomial proportions the performance of the DBC-test is numerically compared with that of the exact solution. In a model of normal observations with a linear trend, the performance of the DBC-test is numerically compared with that of the MSPRT. Some other numerical examples are presented. In all the cases the proposed tests exhibit a very high efficiency with respect to the optimal tests (more than 99.3% when sampling from Bernoulli populations) and/or with respect to the MSPRT (even outperforming the latter in some scenarios).

A numerical approach to sequential multi-hypothesis testing for Bernoulli model

In this article, we deal with the problem of sequential testing of multiple hypotheses. The main goal is minimizing the expected sample size (ESS) under restrictions on the error probabilities. We take, as a criterion of minimization, a weighted sum of the ESSs evaluated at some points of interest in the parameter space, aiming at its minimization under restrictions on the error probabilities. We use a variant of the method of Lagrange multipliers based on the minimization of an auxiliary objective function (called Lagrangian) combining the objective function with the restrictions, taken with some constants called multipliers. Subsequently, the multipliers are used to make the solution comply with the restrictions. We develop a computer-oriented method of minimization of the Lagrangian function that provides, depending on the specific choice of the parameter points, optimal tests in different concrete settings, as in Bayesian, Kiefer-Weiss, and other settings. To exemplify the proposed methods for the particular case of sampling from a Bernoulli population, we develop a set of computer algorithms for designing sequential tests that minimize the Lagrangian function and for the numerical evaluation of test characteristics like the error probabilities and the ESS. We implement the algorithms in the R programming language. The program code is available in a public GitHub repository. For the Bernoulli model, we made a series of computer evaluations related to the optimality of sequential multi-hypothesis tests, in a particular case of three hypotheses. A numerical comparison with the matrix sequential probability ratio test is carried out. A method of solution of the multi-hypothesis Kiefer-Weiss problem is proposed and is applied for a particular case of three hypotheses in the Bernoulli model.

Sequential Multi-Hypothesis Testing in Multi-Armed Bandit Problems: An Approach for Asymptotic Optimality

We consider a multi-hypothesis testing problem involving a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -armed bandit. Each arm’s signal follows a distribution from a vector exponential family. The actual parameters of the arms are unknown to the decision maker. The decision maker incurs a delay cost for delay until a decision and a switching cost whenever he switches from one arm to another. His goal is to minimise the overall cost until a decision is reached on the true hypothesis. Of interest are policies that satisfy a given constraint on the probability of false detection. This is a sequential decision making problem where the decision maker gets only a limited view of the true state of nature at each stage, but can control his view by choosing the arm to observe at each stage. An information-theoretic lower bound on the total cost (expected time for a reliable decision plus total switching cost) is first identified, and a variation on a sequential policy based on the generalised likelihood ratio statistic is then studied. Due to the vector exponential family assumption, the signal processing at each stage is simple; the associated conjugate prior distribution on the unknown model parameters enables easy updates of the posterior distribution. The proposed policy, with a suitable threshold for stopping, is shown to satisfy the given constraint on the probability of false detection. Under a continuous selection assumption, the policy is also shown to be asymptotically optimal in terms of the total cost among all policies that satisfy the constraint on the probability of false detection.

Decoder Ties Do Not Affect the Error Exponent of the Memoryless Binary Symmetric Channel

The generalized Poor-Verdú error lower bound established by Chang <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> (2020) for multihypothesis testing is studied in the classical channel coding context. It is proved that for any sequence of block codes sent over the memoryless binary symmetric channel (BSC), the minimum probability of error (under maximum likelihood decoding) has a relative deviation from the generalized bound that grows at most linearly in blocklength. This result directly implies that for arbitrary codes used over the BSC, decoder ties can only affect the subexponential behavior of the minimum probability of error.

Sequential controlled sensing for composite multihypothesis testing

The problem of multihypothesis testing with controlled sensing of observations is considered. The distribution of observations collected under each control is assumed to follow a single-parameter exponential family distribution. The goal is to design a policy to find the true hypothesis with minimum expected delay while ensuring that the probability of error is below a given constraint. The decision-maker can reduce the delay by intelligently choosing the control for observation collection in each time slot. A policy for this problem is derived that satisfies given constraints on the error probability, and it is shown that this policy is asymptotically optimal in the sense that it asymptotically achieves an information-theoretic lower bound on the expected delay. Numerical results are provided that illustrate an application of the policy to medical diagnostic inference.

A framework to analyze decision strategies for multi-band spectrum sensing in cognitive radios

Spectrum sensing is an essential aspect of cognitive radios that allows secondary users (SUs) to opportunistically use licensed frequency bands while the primary user is idle. In this paper, we assume the available spectrum is divided into multiple bands, and SUs search over these bands to detect the empty bands, which is known as multi-band spectrum sensing. Most of the recent work focus on spectrum sensing in a given band which can be modeled as a binary hypothesis problem and has performance metrics such as receiver operating characteristics (ROC) curves. However, this framework is not appropriate for multi-band spectrum sensing problem, which can be modeled as a multi-hypothesis testing problem. The contribution of this paper is three-fold: (i) a novel approach to evaluate the sensing performance of the multi-band spectrum-sensing using localization ROCs; (ii) optimal spectrum-sensing strategies that maximize the area under the localization ROC curves that can be used as performance benchmarks for any other multi-band spectrum sensing technique; (iii) a framework to analyze the energy expenditure of multi-band spectrum sensing.

Detection of Spectrum Misuse Behavior in Satellite-Terrestrial Spectrum Sensing Based on Multi-Hypothesis Tests

As the increasing of both users’ demand and the number of communication satellites, spectrum resource is becoming increasingly scarce. Cognitive radio (CR) can effectively alleviate the scarcity of spectrum resource. By using CR technology in the satellite and terrestrial integrated network, the satellite can share the spectrum resource with the terrestrial communication equipment to improve the spectrum utilization. However, due to the satellite communication’s inherent openness and the increasing intelligent level of devices, it makes the satellite communication vulnerable to spectrum misuse, which seriously threatens the reliability and efficiency of the satellite communication system. This paper studies the detection of spectrum misuse in the satellite-terrestrial spectrum sharing system. First, we model the problem as a ternary hypothesis test problem by using generalized multi-hypothesis Neyman-Pearson (GMNP). Then, we utilize a two-step method to solve the complex formula problem in the multiple authorized users scenario. Reasonable decision thresholds are derived by means of the generalized likelihood ratio test (GLRT) and maximum a posterior (MAP) criterion, respectively. Finally, the performance of detection is evaluated and analyzed comprehensively to verify the feasibility of the study.

Suboptimal Low Complexity Joint Multi-Target Detection and Localization for Non-Coherent MIMO Radar With Widely Separated Antennas

In this article, the problem of simultaneously detecting and localizing multiple targets in homogeneous noise environment is considered for non-coherent multiple-input multiple-output (MIMO) radar with widely separated antennas. By assuming that the a prior knowledge of target number is available, an optimal solution to this problem is presented first. It is essentially a maximum-likelihood (ML) estimator searching the parameters of interest in a high-dimensional state space. However, the complexity of this solution increases exponentially with the number $G$ of targets. Besides, if the number of targets is unknown, a multi-hypothesis testing strategy to verify all the possible hypotheses on target number is required, which further complicates this method. In order to devise computationally feasible methods for practical applications, we split the high-dimensional maximization into $G$ disjoint sub-optimization problems by sequentially detecting targets and then clearing their interference for the subsequent detection of remaining targets. In this way, we further propose two fast and robust suboptimal solutions which allow to trade performance for a much lower implementation complexity. In addition, the multi-hypothesis testing is no longer required when target number is unknown. Simulation results show that the proposed algorithms can correctly detect and accurately localize multiple targets even when targets lie in the same range bins. Experimental data recorded by three small radars are also provided to demonstrate the efficacy of the proposed algorithms.

Multi‐hypothesis test for close targets detection in co‐located MIMO radar

To distinguish/detect multiple close targets is of great significance in many areas, and the typical tool is the GLRT detection. However, there will be distinguishing difficulty and mutual interference when present targets are very close. Furthermore, it is difficult for binary decision theory to tell where the targets locate exactly as it can only make a decision that whether there is one or two targets present. As it cannot utilise the prior information of target number, it also suffers a poor detection performance. To overcome those disadvantages, the authors bring in multi-hypothesis test (MHT) to the multi-target model within co-located MIMO radar. They also apply the sequential probability ratio test (SPRT) to locate the multiple range-located targets efficiently, whose angles are known as a prior. The simulations show that authors’ SPRT-based detection is more fast and constant comparing to the existing GLRT-based detection taken within cell-by-cell.

Near earth space object detection using parallax as multi-hypothesis test criterion.

The US Strategic Command (USSTRATCOM) operated Space Surveillance Network (SSN) is tasked with Space Situational Awareness (SSA) for the U.S. military. This system is made up of Electro-Optic sensors, such as the Ground-based Electro-Optical Deep Space Surveillance (GEODSS) and RADAR based sensors, such as the Space Fence Gaps. They remain in the tracking of Resident Space Objects (RSO's) in Geosynchronous Orbits (GEO), due to limitations of SST and GEODSS system implementation. This research explores a reliable, ground-based technique used to quickly determine an RSO's altitude from a single or limited set of observations. Implementation of such sensors into the SSN would mitigate GEO SSA performance gaps. The research entails a method used to distinguish between the point spread function (PSF) observed by a star and the PSF observed from an RSO by using Multi-Hypothesis Testing with parallax as a test criterion. Parallax is the effect that an observed object will appear to shift when viewed from different positions. This effect is explored by generating PSFs from telescope observations of space objects at different baselines. The research has shown the PSF of an RSO can be distinguished from that of a star using single, simultaneous observations from reference and parallax sensing telescopes. This report validates these techniques with both simulations and experimental data from the SST and Naval Observatory sensors.

Multi-Target Detection and Adaptive Waveform Design for Cognitive MIMO Radar

Multi-target detection is the main function of radar. Cognitive radar, which can adaptively investigate the radar scene and determine the next action based on previous measurements, has better performance. In this paper, a multi-target detection method and an adaptive waveform design algorithm for cognitive MIMO radar are proposed. In this method, the multi-target detection is modeled as a multi-hypothesis testing. The multi-hypothesis testing is investigated according to sequentially received data. Along with the multi-target detection method, an adaptive waveform design algorithm based on information theory is proposed to improve the efficiency of the multi-hypothesis testing. We adopt semi-definite relaxation technique and semi-definite programming to tackle the nonconvex design problem. Numerical examples demonstrate that the proposed multi-target detection method has better performance than the classical target detection method, and the proposed adaptive waveform design algorithm can significantly improve the performance of the proposed multi-target detection method.

WFMHT Method of Orbit Maneuver Detection Based on Space-Based Bearing-Only Measurement

The method of orbit maneuver detection of space targets is investigated using the space-based bearing-only measurement, which aims to acquire a real-time or nearly real-time awareness of orbit maneuver in the space situation awareness. First, the model for estimating real-time motion of a space target is presented, which only uses the space-based bearing-only measurements. The innovation characteristics of the normal orbit and orbit maneuver are analyzed and compared. Second, based on the hypothesis test methods of the distribution characteristic of the stochastic sequence, the WFMHT (i.e., weighted fusion of multi hypothesis tests) method with the innovation is put forward to detect the orbit maneuver. Furthermore, the criterion of determining the weight coefficients is studied. Finally, the method is validated by numeric simulations. The results show that the highest gained success rate is up to 36% with the WFMHT method than the prevalent Chi2 method. With the WFMHT method, the detection system achieves a strengthened robustness with greatly shortened detection window. The research will be beneficial to construction of our space situation awareness system.

Facial expression recognition using best tree RD-LGP encoded features and HMM

Hand and face gestures enable deaf people to communicate in their daily lives rather than speaking. This paper describes a hidden Markov model (HMM)-based framework for face sign recognition and detection. The observation vectors used to characterize the states of the HMM are obtained using the best tree local gradient pattern (LGP) encoded features. Each face gesture is modeled as a five-state HMM. The problem of facial expression classification is posed as a composite seven-classes multi-hypothesis Bayesian test. The likelihood ratio test showed that the overall recognition rate for the proposed model is higher than the HMM-local binary pattern descriptor by 6.4%. The overall recognition rate is enhanced by 8.6% using the discrete wavelet packet best tree decomposition filter as a pre-processing noise removal tool. In addition, the overall recognition rate ranges from 84.3%, for the seven classes Bayesian test, to 100%, for lower number of classes depending upon the type of the face gesture. The proposed face expression algorithm reduces significantly the computational complexity of previous HMM-based face expression recognition systems, and still preserve the recognition rate.

Probabilistic Analysis of Light Curves

To date, most light curve analyses take place using time and frequency domain analyses that generally lack in their ability to quantify the information content of the data. In this paper, we examine the information content of light curves using information theoretic and functional data approaches to characterize physical and dynamic attributes of space objects from non-resolved photometric observations. The information content is examined in a probabilistic context where a set of simulated light curves for a diverse set of object shapes, sizes and dynamics are used to demonstrate the application and value of Functional Data Analysis, data clustering and information theory. The results confirm the value of these approaches by correctly categorizing independent sets of light curve measurements and quantifying the likelihood of a given light curve being associated with a specific object. These analytical tools can also be applied to better understand how well a given set of observations characterize an object and, hence, guide the necessity of future observations. In this paper, we also comment on how the proposed ideas can enable the use of light curves, treated as a functional observation, for joint object tracking and characterization within a multi-hypothesis testing framework such as Finite Set Statistics.

Performance Analysis of Sequential Detection of Primary User Number Based on Multihypothesis Sequential Probability Ratio Test

In cognitive radio networks, a priori information on the number of primary users (PUs) is helpful to estimate more specific parameters, such as carrier frequency, direction of arrival, and location. We propose a sequential canonical correlation technique (S-CCT) method to estimate the number of active PUs quickly and accurately. The proposed method is based on the classical CCT and multi-hypothesis sequential probability ratio test. We also deduce the asymptotic expressions of average sample number and detection probability. Simulation results show that our proposed S-CCT method can achieve better performance with fewer samples than the CCT.

Constructive minimax classification of discrete observations with arbitrary loss function

This paper develops a multihypothesis testing framework for calculating numerically the optimal minimax test with discrete observations and an arbitrary loss function. Discrete observations are common in data processing and make tractable the calculation of the minimax test. Each hypothesis is both associated to a parameter defining the distribution of the observations and to an action which describes the decision to take when the hypothesis is true. The loss function measures the gap between the parameters and the actions. The minimax test minimizes the maximum classification risk. It is the solution of a finite linear programming problem which gives the worst case classification risk and the worst case prior distribution. The minimax test equalizes the classification risks whose prior probabilities are strictly positive. The minimax framework is applied to vector channel decoding which consists in classifying some codewords transmitted on a binary asymmetric channel. The Hamming metric is used to measure the number of differences between the emitted codeword and the decoded one.

Operating Characteristic and Average Sample Number of Binary and Multi-Hypothesis Sequential Probability Ratio Test

The operating characteristic (OC) and average sample number (ASN) of the sequential probability ratio test (SPRT) and multi-hypothesis SPRT (MSPRT) are studied. We consider the case where the observation sequence is independent but not necessarily identically distributed. Also, the thresholds for the test can be time varying. Based on the governing equations for OC and ASN of the SPRT developed in our previous work, a solution for the general case is proposed. The governing equations for OC and ASN of the MSPRT are also obtained. Numerical solutions for MSPRT are developed. Basically, the solutions rely on approximating the original test by truncation, that is, truncating the test at some finite time $K$ . We show that under some mild conditions, the approximation error diminishes as $K$ increases, at the cost of increased computation. Numerical examples are provided to demonstrate our solutions by comparing with Monte Carlo simulations, Simon’s lower bound, and Dragalin’s method (if available) for ASN.

Probabilistic Decision Making with Spikes: From ISI Distributions to Behaviour via Information Gain.

Computational theories of decision making in the brain usually assume that sensory 'evidence' is accumulated supporting a number of hypotheses, and that the first accumulator to reach threshold triggers a decision in favour of its associated hypothesis. However, the evidence is often assumed to occur as a continuous process whose origins are somewhat abstract, with no direct link to the neural signals - action potentials or 'spikes' - that must ultimately form the substrate for decision making in the brain. Here we introduce a new variant of the well-known multi-hypothesis sequential probability ratio test (MSPRT) for decision making whose evidence observations consist of the basic unit of neural signalling - the inter-spike interval (ISI) - and which is based on a new form of the likelihood function. We dub this mechanism s-MSPRT and show its precise form for a range of realistic ISI distributions with positive support. In this way we show that, at the level of spikes, the refractory period may actually facilitate shorter decision times, and that the mechanism is robust against poor choice of the hypothesized data distribution. We show that s-MSPRT performance is related to the Kullback-Leibler divergence (KLD) or information gain between ISI distributions, through which we are able to link neural signalling to psychophysical observation at the behavioural level. Thus, we find the mean information needed for a decision is constant, thereby offering an account of Hick's law (relating decision time to the number of choices). Further, the mean decision time of s-MSPRT shows a power law dependence on the KLD offering an account of Piéron's law (relating reaction time to stimulus intensity). These results show the foundations for a research programme in which spike train analysis can be made the basis for predictions about behavior in multi-alternative choice tasks.

Controlled Sensing for Sequential Multihypothesis Testing with Controlled Markovian Observations and Non-Uniform Control Cost

A new model for controlled sensing for multihypothesis testing is proposed and studied in the sequential setting. This new model, termed a controlled Markovian observation model, exhibits a more complicated memory structure in the controlled observations than existing models. In addition, instead of penalizing just the delay until the final decision time as in standard sequential hypothesis testing problems, a much more general cost structure is considered that entails accumulating the total control cost with respect to an arbitrary control cost function. An asymptotically optimal test is proposed for this new model and is shown to satisfy an optimality condition formulated in terms of decision-making risk. It is shown that the optimal causal control policy for the controlled sensing problem is self-tuning, in the sense of maximizing an inherent “inferential” reward simultaneously under every hypothesis, with the maximal value being the best possible corresponding to the case where the true hypothesis is known at the outset. Another test is also proposed to meet distinctly predefined constraints on the various decision risks nonasymptotically, while retaining asymptotic optimality.

Effects of Phasor Measurement Uncertainty on Power Line Outage Detection

Phasor measurement unit (PMU) technology provides an effective tool to enhance the wide-area monitoring systems (WAMSs) in power grids. Although extensive studies have been conducted to develop several PMU applications in power systems (e.g., state estimation, oscillation detection and control, voltage stability analysis, and line outage detection), the uncertainty aspects of PMUs have not been adequately investigated. This paper focuses on quantifying the impact of PMU uncertainty on power line outage detection and identification, in which a limited number of PMUs installed at a subset of buses are utilized to detect and identify the line outage events. Specifically, the line outage detection problem is formulated as a multi-hypothesis test, and a general Bayesian criterion is used for the detection procedure, in which the PMU uncertainty is analytically characterized. We further apply the minimum detection error criterion for the multi-hypothesis test and derive the expected detection error probability in terms of PMU uncertainty. The framework proposed provides fundamental guidance for quantifying the effects of PMU uncertainty on power line outage detection. Case studies are provided to validate our analysis and show how PMU uncertainty influences power line outage detection.