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出信号处理的新算法，特别是对隐藏在噪声信号中的信息的估计。这将建立在申请者以前的贡献基础上。我们将朝三个方向努力。第一个部分将开发新的隐半马尔可夫模型估计算法，扩展了我们以前对隐马尔可夫模型的研究。第二种方法将考虑基于事件的过滤问题，当仅当隐藏信号变化超过指定量或当信号过程跨越某些级别时，才接收到观察到的过程。第三章将用倒向随机微分方程讨论带噪声观测的马尔可夫链的最优控制。*除了信号处理、语音处理和其他领域，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，当离散序列建模时，例如在离散时间或在生物应用中，如果隐藏过程是马尔可夫链，则其在任何状态下的(随机)占用时间都是几何分布的随机变量。在许多应用领域，包括排队论、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占用时间。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，而是调制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找到的结果。在数字世界中，必须对连续时间信号进行采样。传统上，它们是在时间上均匀采样的。基于事件的采样这一术语可以指传统的均匀时间步长采样(有时称为Riemann采样)，但它通常指的是响应于先验定义的事件而进行的采样，例如当信号变化超过指定量时(按增量发送)，或者当信号超过指定的水平时(勒贝格采样)。这项工作的目标将是获得新的可实施的过滤器。基于事件抽样的理论比较复杂，但它有许多实际的好处，包括更便宜的传感器，更少的通信成本和更少的数据需要处理。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，其中倒向随机常微分方程组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.