Optimal detection and error exponents for hidden semi-Markov models

Bajović, Dragana and He, Kanghang and Stanković, Lina and Vukobratović, Dejan and Stanković, Vladimir (2018) Optimal detection and error exponents for hidden semi-Markov models. IEEE Journal on Selected Topics in Signal Processing, 12 (5). pp. 1077-1092. ISSN 1932-4553

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    Abstract

    We study detection of random signals corrupted by noise that over time switch their values (states) between a finite set of possible values, where the switchings occur at unknown points in time. We model such signals as hidden semi-Markov signals (HSMS), which generalize classical Markov chains by introducing explicit (possibly non-geometric) distribution for the time spent in each state. Assuming two possible signal states and Gaussian noise, we derive optimal likelihood ratio test and show that it has a computationally tractable form of a matrix product, with the number of matrices involved in the product being the number of process observations. The product matrices are independent and identically distributed, constructed by a simple measurement modulation of the sparse semi-Markov model transition matrix that we define in the paper. Using this result, we show that the Neyman-Pearson error exponent is equal to the top Lyapunov exponent for the corresponding random matrices. Using theory of large deviations, we derive a lower bound on the error exponent. Finally, we show that this bound is tight by means of numerical simulations.