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

隐半马尔可夫模型的估计和过滤、基于事件的过滤器和随机控制。

• 批准号: RGPIN-2015-06084
• 负责人: Elliott, Robert
• 金额: $ 2.19万
• 依托单位: University of Calgary
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=650105
• 关键词: Estimation; filtering; hidden; semi; Markov

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.

工作的核心目的是推导用于信号处理的新算法，特别是噪声信号中隐藏的信息的估计。这将基于适用的先前贡献。将追求三个指示。第一个将为隐藏的半马尔可夫模型开发新的估计算法，从而扩展了我们先前在隐藏的马尔可夫模型上的工作。仅当隐藏信号变化超过指定量或信号过程越过某些级别时，第二个将考虑基于事件的过滤问题。第三将讨论使用向后随机微分方程对噪声观察到的马尔可夫链的最佳控制。但是，当对离散序列进行建模时，例如在离散时间或生物应用中，如果隐藏过程是马尔可夫链，则在任何状态下占据（随机）占用时间是几何分布的随机变量。在许多领域，包括排队理论，可靠性和维护，生存分析，绩效评估，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。这使我们考虑了半马尔可夫模型。我们将考虑“隐藏”的半马尔科夫模型，这是未直接观察到半马尔可夫链但观察到的一秒钟的过程的情况。我们将为这些模型开发书籍和以前出版物中发现的结果。 ***在数字世界中，必须采样连续的时间信号。传统上，它们在及时均匀地进行了采样。术语“基于事件的采样”可以是指传统的统一时间步长采样（有时称为Riemann采样），但这通常意味着取样以响应先验定义的事件，例如当信号变化超过指定量时（send-on-on-on-on-on-n-delta），或当它交叉指定的级别（lebesgue samplemplats sampling）。工作的目标将是获得新的可实施过滤器。基于事件的抽样理论更多地参与其中，但是有许多实际收益，包括廉价传感器，降低的通信成本和更少的处理数据。 ***第三线研究将使用我们的最新结果对向后的随机微分方程进行研究，以研究部分观察到的随机控制问题。伴随过程由向后的随机微分方程描述。对于部分观察到的问题，这是一个向后的随机部分差分方程。最初，我们认为控制了部分观察到的马尔可夫链的问题，在该链中，后向随机的普通微分方程系统将提供确定最佳控制的标准。