Set-indexed and multiparameter stochastic processes

由多维时间参数或更广泛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，并且近年来一直是许多研究活动的焦点。然而，从完全有序的索引集合(时间)到偏序的索引集合(多维时间或集合)带来了许多挑战。特别是，描述过程的动态属性(即过程如何演变)变得复杂得多。在这项研究中，我们将考虑对仅部分可观测的集合指标过程的分析，即该过程可能在某些(随机)集合上被遮蔽。在这种情况下，我们说进程是按集合过滤的。我们将研究过滤数据的生存分析，以及优先类型的检验。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，但它的行为在随机集上发生变化，而随机集是无法观察到的。在这种情况下，我们希望找到一种检测集合的最佳方式。所有这些技术都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.