Sequential joint signal detection and signal-to-noise ratio estimation

The sequential analysis of the problem of joint signal detection and signal-to-noise ratio (SNR) estimation for a linear Gaussian observation model is considered. The problem is posed as an optimization setup where the goal is to minimize the number of samples required to achieve the desired (i) type I and type II error probabilities and (ii) mean squared error performance. This optimization problem is reduced to a more tractable formulation by transforming the observed signal and noise sequences to a single sequence of Bernoulli random variables; joint detection and estimation is then performed on the Bernoulli sequence. This transformation renders the problem easily solvable, and results in a computationally simpler sufficient statistic compared to the one based on the (untransformed) observation sequences. Experimental results demonstrate the advantages of the proposed method, making it feasible for applications having strict constraints on data storage and computation.Comment: 5 pages, Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 201

Asymptotically Optimal Procedures for Sequential Joint Detection and Estimation

We investigate the problem of jointly testing multiple hypotheses and estimating a random parameter of the underlying distribution in a sequential setup. The aim is to jointly infer the true hypothesis and the true parameter while using on average as few samples as possible and keeping the detection and estimation errors below predefined levels. Based on mild assumptions on the underlying model, we propose an asymptotically optimal procedure, i.e., a procedure that becomes optimal when the tolerated detection and estimation error levels tend to zero. The implementation of the resulting asymptotically optimal stopping rule is computationally cheap and, hence, applicable for high-dimensional data. We further propose a projected quasi-Newton method to optimally choose the coefficients that parameterize the instantaneous cost function such that the constraints are fulfilled with equality. The proposed theory is validated by numerical examples.Comment: 13 pages, 3 figures, 1 table, 2 pages supplementing material. Under review in the IEEE Transactions on Signal Processin

Design and Analysis of Optimal and Minimax Robust Sequential Hypothesis Tests

In this dissertation a framework for the design and analysis of optimal and minimax robust sequential hypothesis tests is developed. It provides a coherent theory as well as algorithms for the implementation of optimal and minimax robust sequential tests in practice. After introducing some fundamental concepts of sequential analysis and optimal stopping theory, the optimal sequential test for stochastic processes with Markovian representations is derived. This is done by formulating the sequential testing problem as an optimal stopping problem whose cost function is given by a weighted sum of the expected run-length and the error probabilities of the test. Based on this formulation, a cost minimizing testing policy can be obtained by solving a nonlinear integral equation. It is then shown that the partial generalized derivatives of the optimal cost function are, up to a constant scaling factor, identical to the error probabilities of the cost minimizing test. This relation is used to formulate the problem of designing optimal sequential tests under constraints on the error probabilities as a problem of solving an integral equation under constraints on the partial derivatives of its solution function. Finally, it is shown that the latter problem can be solved by means of standard linear programming techniques without the need to calculate the partial derivatives explicitly. Numerical examples are given to illustrate this procedure. The second half of the dissertation is concerned with the design of minimax robust sequential hypothesis tests. First, the minimax principle and a general model for distributional uncertainties is introduced. Subsequently, sufficient conditions are derived for distributions to be least favorable with respect to the expected run-length and error probabilities of a sequential test. Combining the results on optimal sequential tests and least favorable distributions yields a sufficient condition for a sequential test to be minimax optimal under general distributional uncertainties. The cost function of the minimax optimal test is further identified as a convex statistical similarity measure and the least favorable distributions as the distributions that are most similar with respect to this measure. In order to obtain more specific results, the density band model is introduced as an example for a nonparametric uncertainty model. The corresponding least favorable distributions are stated in an implicit form, based on which a simple algorithm for their numerical calculation is derived. Finally, the minimax robust sequential test under density band uncertainties is discussed and shown to admit the characteristic minimax property of a maximally flat performance profile over its state space. A numerical example for a minimax optimal sequential test completes the dissertation