Estimation and filtering of hidden semi Markov models, event based filters and stochastic control.

这项工作的中心目标将是推导新的信号处理算法，特别是对隐藏在噪声信号中的信息的估计。这将以申请人以前的贡献为基础。将追求三个方向。第一部分将开发新的隐半马尔可夫模型估计算法，扩展我们之前在隐马尔可夫模型上的工作。第二个将考虑基于事件的滤波问题，当仅当隐藏信号变化超过指定量或当信号过程越过一定水平时才接收到观察到的过程。第三章将讨论用倒向随机微分方程对有噪声观测马尔可夫链的最优控制。***除了信号处理、语音处理等领域，隐马尔可夫模型的一个重要应用是基因组和蛋白质测序。然而，当建模离散序列时，例如在离散时间或生物应用中，如果隐藏过程是一个马尔可夫链，它在任何状态下的（随机）占用时间是一个几何分布的随机变量。在许多应用领域，包括排队理论、可靠性和维护、生存分析、性能评估、生物学、DNA分析和基因组测序，似乎应该考虑更一般的职业时间。这导致我们考虑半马尔可夫模型。我们将考虑“隐藏的”半马尔可夫模型，即半马尔可夫链不是直接观察到的，而是调节第二个观察到的过程的情况。我们将为这些模型发展在我们的书和以前的出版物中发现的结果。在数字世界中，必须对连续时间信号进行采样。传统上，它们在时间上是均匀采样的。术语“基于事件的采样”可以指传统的均匀时间步进采样（有时称为黎曼采样），但它通常意味着采样是为了响应先验定义的事件，例如当信号变化超过指定量时（发送-on-delta），或者当它越过指定水平时（勒贝格采样）。这项工作的目标将是获得新的可实现的过滤器。基于事件的采样理论更复杂，但有许多实际的好处，包括更便宜的传感器，更低的通信成本和更少的数据处理。***研究的第三条线将使用我们最近在倒向随机微分方程上的结果来研究部分观察到的随机控制问题。伴随过程用倒向随机微分方程来描述；对于部分观测问题，这是一个倒向随机偏微分方程。首先，我们考虑部分观测马尔可夫链的控制问题，其中一个后向随机常微分方程系统将给出确定最优控制的准则。

The central objective of the work will be the derivation of new algorithms for signal processing, in particular the estimation of information hidden in noisy signals. This will build on previous contributions by the applicant. Three directions will be pursued. The first will develop new estimation algorithms for hidden semi Markov models, extending our previous work on hidden Markov models. The second will consider event based filtering problems when the observed process is received only when the hidden signal changes by more than a specified amount or when the signal process crosses certain levels. The third will discuss the optimal control of a noisily observed Markov chain using backward stochastic differential equations.***In addition to signal processing, speech processing and other areas, an important application of hidden Markov models has been to genome and protein sequencing. However, when modelling discrete sequences, for example in discrete time or in biological applications, if the hidden process is a Markov chain its (random) occupation time in any state is a geometrically distributed random variable. In many areas of applications, including queuing theory, reliability and maintenance, survival analysis, performance evaluation, biology, DNA analysis, and genome sequencing, it seems more general occupation times should be considered. This leads us to consider semi-Markov models. We shall consider 'hidden' semi-Markov models, that is situations where the semi-Markov chain is not observed directly but modulates a second, observed, process.  We shall develop for these models the results found in our book and previous publications.  ***In the digital world, continuous-time signals must be sampled. Traditionally, they are sampled uniformly in time. The term 'event-based sampling' can refer to the traditional uniform time step sampling, (sometimes called Riemann sampling), but it usually means that samples are taken in response to a priori defined events, such as when the signal changes by more than a specified amount, (send-on-delta), or when it crosses specified levels, (Lebesgue sampling). The objectives of the work will be to obtain new implementable filters. The theory of event-based sampling is more involved but there are many practical benefits including cheaper sensors, reduced communication costs and less data to process.  ***The third line of research will use our recent results on backward stochastic differential equations to investigate partially observed stochastic control problems. The adjoint process is described by a backward stochastic differential equation; for partially observed problems this is a backward stochastic partial differential equation. Initially we consider this problem for the control of a partially observed Markov chain where a system of backward stochastic ordinary differential equations will give criteria which determine an optimal control.