Set-indexed and multiparameter stochastic processes

由多维时间参数索引的随机过程，或者更一般地说，由一类集合索引的随机过程，可以用来模拟许多随机现象，并已成为近年来许多研究活动的焦点。然而，从一个完全有序的索引集（时间）到一个部分有序的索引集（多维时间或集合）的传递带来了许多挑战。特别是，描述过程的动态特性（即过程如何演变）变得复杂得多。优雅的工具，如鞅和马尔可夫过程理论需要扩展到这个更一般的框架。在本研究中，我们将考虑仅部分可观察的集索引过程的分析；也就是说，过程可能在某个（随机）集合上被模糊。在本例中，我们说流程是由集合过滤的。我们将研究筛选数据的生存分析以及优先型试验。我们还将考虑相反的情况，在这种情况下，集索引过程可以完全观察到，但它的行为在一个无法观察到的随机集上发生变化。在这种情况下，我们想要找到一种检测集合的最佳方式。所有这些技术将导致分析地理和生物医学数据集的新方法，其中每个数据点都是多维的。

Random processes indexed by a multidimensional time parameter, or more generally, by a class of sets, may be used to model many stochastic phenomena and have been the focus of much research activity in recent years. However, the passage from a totally ordered index set (time) to a partially ordered one (multidimensional time or sets) introduces many challenges. In particular, describing the dynamical properties of the process (i.e. how the process evolves)becomes far more complicated. Elegant tools such as martingale and Markov process theory need to be extended to this more general framework.In this research, we will be considering the analysis of set-indexed processes that are only partially observable; that is, the process may be obscured on some (random) set. In this case, we say that the process is filtered by the set. We will investigate survival analysis of filtered data and as well precedence-type tests. We will also consider the converse situation, in which the set-indexed process can be totally observed, but its behaviour changes on a random set that cannot be observed. In this case, we want to find an optimum manner of detecting the set.All of these techniques will lead to new methods of analyzing geographic and biomedical data sets in which each data point is multidimensional.