Modeling and Optimal Inference in change-point models with ultra-high dimensional data

关于多变量或张量设置中的变点的建模和推断出现在统计应用的许多领域，例如金融市场，神经成像，计量经济学，社交网络分析和反恐。本文的主要工作是发展随机微分方程、分数阶随机微分方程和常微分方程的数值计算方法。它由三个主要部分组成：鲁棒的推理，在高维张量回归与变点，通过常微分方程的建模和推理的动态系统的变化，改进的推理的模型和模型。 在第一部分中，我将研究具有未知多个变点的多变量系统中的推理问题，其中漂移参数可能满足某些限制。我将研究类似的推理问题的背景下，festival。在这两种情况下，我将考虑漂移系数不一定是线性的一般情况。因此，该方法将适用于由广义Cox-Ingersoll-Ross过程或Ornstein-Uhlenbeck过程生成的数据集的特殊情况。上述问题也将在状态转换的背景下进行研究。除了处理这些模型的复杂性所带来的变化点的存在和不确定性的先验知识的参数，我计划制定渐近结果的情况下，尺寸的估计本身是随机的。 在第二部分中，我将考虑当张量参数被怀疑满足某些限制时，具有多个变点的张量回归模型中的推理问题。我也将在高维数据的背景下研究类似的问题。与最近文献中的类似模型相比，我将放宽误差项的条件，使其不需要独立同分布。依赖结构至多是混合结构。我也将考虑的情况下，误差项是一个长记忆平稳过程。 在第3部分中，我将考虑动态系统的随机版本的建模，例如通常由常微分方程建模的生态或生物医学系统中出现的那些。 我将首先获得ODE轨迹的初步非参数估计，然后使用它们来构造估计函数，以推断ODE的参数。然后，这些推理工具将用于检测此类系统中的多个变点。同样在这种情况下，我将放松常用的强假设的独立性的错误，以较弱的混合依赖。例如，在生态系统中，所提出的模型预计将考虑一些现实的因素，如动物的适应，迁移和/或隐藏策略。

Modeling and inference about change-points in multivariate or tensor setups appear in many areas of statistical applications such as financial markets, neuroimaging, econometrics, social network analysis and counter-terrorism. This proposal is concerned with developing such methodologies in stochastic differential equations (SDE), fractional stochastic differential equations (fSDE), ordinary differential equations (ODEs). It is composed of three main parts: robust inference in SDE and fSDE models subject to changes, improved inference in high dimensional tensor regression with change-points, and modeling and inference in dynamical systems via ODEs. In Part 1, I will study inference problems in some multivariate SDE with unknown multiple change-points for which the drift parameters may satisfy some restrictions. I will study similar inference problems in context of  fSDE. In both cases, I will consider the general case where the drift coefficient is not necessarily linear. Thus, the methods will be applicable to special cases where the datasets are generated by the generalized Cox-Ingersoll-Ross processes or Ornstein-Uhlenbeck processes. The above problems will also be studied in the context of regime switching. In addition to dealing with complexities brought into these models by the presence of change-points and by the uncertainty in the prior knowledge about the parameters, I plan to develop asymptotic results for the situation where the dimensions of the estimators themselves are random. In Part 2, I will consider inference problems in tensor regression models with multiple change-points when the tensor parameter is suspected to satisfy some restriction. I will also study similar problems in the context of high dimensional data. As compared to similar models in recent literature, I will relax the conditions on the error term so that it does not need to be independent and identically distributed. The dependence structure will be at most that of mixingale. I will also consider the case where the error term is a long memory stationary process.  In Part 3, I will consider the modeling of stochastic versions of dynamical systems such as those appearing in ecological or biomedical systems which are commonly modeled by ODEs.  I will first obtain preliminary nonparametric estimators of the trajectory of the ODEs and then use them to construct estimating functions to make inferences about the parameters of the ODEs. These inferential tools will then be used to detect multiple change-points in such systems. Also in this case, I will relax the commonly used strong assumptions of independence of the errors to the weaker mixingale dependence. For instance, in ecological systems, the proposed models are expected to account for some realistic factors such as animal adaptation, migration and/or hiding strategies.