Topics in Statistical Modelling and Inference with High-Dimensional, Complex Data

技术正在改变我们的生活。现在以不同的粒度收集大量数据。这类数据往往表现出复杂的结构。能够捕捉这种复杂性的统计模型可以加深我们对潜在数据生成机制的理解，并推动科学和工程领域的相关领域的发展。然而，对这样的数据建模可能是具有挑战性的，特别是当我们有内生测量、离群值、遗漏观测或其他异常情况时。*这一拟议研究计划的目标是解决在对高维、复杂数据建模时遇到的统计问题，特别是对时空、金融、商业智能、基因组数据的兴趣。它将专注于开发稳健的、在计算上可行的、快速的模型选择程序，用于在候选模型集可能不包含真实模型的情况下，针对有噪声的高维或超高维复杂数据。它将开发一种基于两阶段正则化方法的通用模型公式中的多个变点检测方法，例如广义线性模型和函数数据模型，该方法将同时估计所有变化点并执行变量选择。它将处理高维回归模型中的多个变点检测，包括高维线性动态面板数据模型和时空回归模型。利用各自的正则化方法和随机搜索方法进一步发展回归聚类。它将提出利用地理邻域信息的非平稳时空建模。它将通过使用惩罚方法有效地对网格数据进行分组来解决时空逆建模问题。它将为海量的商业智能和基因组数据开发出可行的关联规则挖掘。它将讨论金融数据的统计建模，例如长期隐含波动率和交易数据。对于我们提出的方法，我们将进行理论和方法研究；此外，我们将开发计算算法，以确保我们的方法在计算上是可行和有效的，并提供验证我们的发现的数值研究结果。在拟议研究下取得的进展将对统计建模、推理、计算及其在实践中的应用产生重大影响。

Technology is changing our lives. Data is now collected in large volumes at various granularities. Such data often exhibit complex structures. Statistical models able to capture this complexity can further our understanding of the underlying data-generating mechanism and advance relevant fields in science and engineering. However, modelling such data can be challenging, particularly when we have endogenous measurements, outliers, missing observations, or other anomalies.******The objective of this proposed research program is to tackle statistical problems encountered in modelling high-dimensional, complex data, with particular interest on spatio-temporal, financial, business intelligence, genomic data. It will focus on development of robust, computationally feasible and fast model selection procedures for noisy high- or ultra high-dimensional, complex data in the scenario that the set of candidate models may not contain the true one. It will develop a two-stage regularization method based multiple change-point detection in general model formulations, e.g., generalized linear models and functional data models, which will simultaneously estimate all the change-points and perform variable selection. It will deal with multiple change-point detection in high-dimensional regression models, which include high-dimensional linear dynamic panel data models and spatio-temporal regression models. It will further the development in regression clustering using respective regularization methods and stochastic search. It will propose non-stationary spatio-temporal modelling which leverage of geographical neighbourhood information. It will tackle spatio-temporal inverse modelling problems by effectively grouping grid data using penalized approaches. It will develop the feasible association rule mining for the massive business intelligence, and genomic data. It will address statistical modelling of financial data, e.g., long term implied volatility and trading data. For our proposed methods, we will carry out both theoretical and methodological investigations; in addition, we will develop computational algorithms to ensure that our methods are computationally feasible and efficient, and present results from numerical studies which validate our findings. The advancements achieved under the proposed research will produce significant impact in statistical modelling, inference, computing, and their applications in practice.