基于数据驱动的复杂数据变点检测方法研究

Change-point detection plays a critical role in the modeling, estimation and prediction of time-series data. Recent years have seen its popularity in statistics and many other fields. Various detection algorithms have been proposed in conjunction with criteria for the determination of the number of change-points, which mainly based on the Bayesian Information Criterion (or its variants) or thresholding methods. Such criteria introduce additional nuisance parameters such as penalties or thresholds, and the consistency of estimator of the number of change-points (and hence estimators of change-points) could be guaranteed under carefully chosen penalty or threshold. However, the performance when dealing with real data is often questioned due to the unknown model and error structure. This project bases on a data-driven strategy and tries to avoid manually selection of involving nuisance parameters while data-adaptively determining the number of change-points and other tuning parameters in the detection algorithms. In this project, we will investigate some popular and important change-point problems for complex data and construct the consistency results of our detection procedure. Finite-sample performance and real data applications will be also considered.

变点检测对时间序列的建模、估计和预测至关重要，近年来在统计学等各个领域引起了广泛的讨论。文献中现有方法多由估计算法和变点个数估计准则两部分构成，其中变点个数的确定多基于贝叶斯信息准则（或其变体）或临界值法。此类准则均引入了惩罚项或临界值等讨厌参数，虽从大样本理论的角度，在适当惩罚项或临界值下，变点个数（及相应变点估计）的相合性能够确保，但在实际应用中其准确性往往受限于未知模型和未知误差结构。该项目拟从数据驱动的角度着手，针对若干热门的、关键的研究课题，特别是在处理复杂数据时，提出如何避免讨厌参数的人为选择而通过数据本身来确定变点个数和变点估计算法所涉及的其他讨厌参数，同时建立相应的相合性理论，并应用于实际数据分析。变点估计算法的数据驱动化研究，丰富和完善了变点检测的方法和理论体系，再者，作为复杂数据分析的必要步骤之一，以期更好地帮助人类生产生活做好推断、决策。

变点结构是数据异质性的一种典型体现，变点检测对复杂数据的建模、估计和预测至关重要。现有变点检测方法大多涉及一些讨厌参数或调节参数，其选取在实际应用中往往较为敏感。本项目旨在提出数据驱动的变点分析方法，我们围绕“保序”样本分割这一想法，探究其在变点估计相合性和不确定性度量上的应用。我们发现该思想不仅可以实现变点个数的相合估计，进一步地，结合文献中最近提出的不依赖于p值的多重检验方法，其可以用于度量变点估计的不确定性，即控制某种“错误”检测比率。我们通过不同的变点模型，论述了相关数据驱动变点分析方法，并建立了其理论性质。