Optimal Detection and Error Exponents for Hidden Semi-Markov Models
Optimal Detection and Error Exponents for Hidden Semi-Markov Models
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DOI：
10.1109/jstsp.2018.2851506
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发表时间：
2018-10-01
影响因子：
7.5
通讯作者：
Stankovic, Vladimir
中科院分区：
文献类型：
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作者：
Bajovic, Dragana;He, Kanghang;Stankovic, Vladimir
我们研究了被噪声损坏的随机信号检测，随着时间的流逝，其值（状态）在有限的可能值之间，而开关发生在未知点处。我们对隐藏的半马尔科夫信号等信号进行建模，该信号通过在每个状态中花费的时间引入显式（可能是非几何）分布来概括经典的马尔可夫链。假设有两个可能的信号状态和高斯噪声，我们得出了最佳的似然比测试，并表明它具有矩阵乘积的计算典型形式，涉及该产品的矩阵数量是过程观察的数量。产品矩阵是独立的且分布相同的，是通过对本文中定义的稀疏半马尔科夫模型过渡矩阵进行的简单测量调制而构建的。使用此结果，我们表明Neyman-Pearson误差指数等于相应的随机矩阵的顶部Lyapunov指数。使用大偏差理论，我们在误差指数上得出了一个下限。最后，我们证明了通过数值模拟而紧密的。
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, which generalize classical Markov chains by introducing explicit (possibly nongeometric) 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.