机器人监控中具有最大返回时间熵的马尔可夫链
Markov Chains with Maximum Return Time Entropy for Robotic Surveillance
摘要：
基于机器人监控应用的需要，研究了一类具有平稳分布和旅行时间的图拓扑结构的马尔可夫链的返回时间熵最大化问题.返回时间熵是马尔可夫链的第一个返回时间的熵在所有图节点上的加权平均值;这个目标函数是一个函数序列，一般不允许一个封闭形式。本文的特点是理论和计算的贡献。首先，我们得到一个离散时间延迟的线性系的返回时间的概率分布，并建立其收敛性。我们表明，目标函数是连续的一个紧凑的集合，因此承认一个全局极大值，一个唯一的全球最优解是已知的，只有完全图与酉旅行时间。然后，我们建立返回时间熵和著名的马尔可夫链的熵率之间的上界和下界。为了数值计算最优马尔可夫链，我们建立了熵、条件熵和截断熵之间的渐近等价性，并提出了一种计算截断熵梯度的迭代方法.最后，我们将这些结果应用于机器人监视问题。我们的数值结果表明，对于一个模型的理性入侵者在原型图拓扑和测试情况下，最大返回时间熵链的性能优于现有的几个马尔可夫链。
英文摘要：
Motivated by robotic surveillance applications, this paper studies the novel problem of maximizing the return time entropy of a Markov chain, subject to a graph topology with travel times and stationary distribution. The return time entropy is the weighted average, over all graph nodes, of the entropy of the first return times of the Markov chain; this objective function is a function series that does not admit in general a closed form. The paper features theoretical and computational contributions. First, we obtain a discrete-time delayed linear system for the return time probability distribution and establish its convergence properties. We show that the objective function is continuous over a compact set and therefore admits a global maximum; a unique globally-optimal solution is known only for complete graphs with unitary travel times. We then establish upper and lower bounds between the return time entropy and the well-known entropy rate of the Markov chain. To compute the optimal Markov chain numerically, we establish the asymptotic equality between entropy, conditional entropy and truncated entropy, and propose an iteration to compute the gradient of the truncated entropy. Finally, we apply these results to the robotic surveillance problem. Our numerical results show that, for a model of rational intruder over prototypical graph topologies and test cases, the maximum return time entropy chain performs better than several existing Markov chains.
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