Change-point detection - theory and applications

变化点是指观测或数据遵循两种不同模型的位置或时间：之前和之后。对变点问题的统计研究可以追溯到佩奇(1955)，并在20世纪80年代以来蓬勃发展。这些研究在广泛的领域得到了应用，包括质量控制、金融、环境计量学、医学、遗传学和地理学。这项拟议的研究计划的一个重要部分是开发快速而准确的变点检测方法。*解决多变点问题的一个很有前途的方法是通过对数据序列进行适当的分割将其转化为变量选择问题。因此，建立了这两个问题之间的联系，并提出了通过以下两个关键步骤来处理多变点检测的框架：(1)应用一致变量选择方法的最新进展来检测变点；(2)采用改进的过程来提高变点估计的精度。*这项拟议研究计划的第一部分是继续研究大型和复杂数据集中的变量选择方法，开发使用统计计算软件R实现的快速有效的算法，并建立统计理论来支持所开发的算法。*其次，为了准确地确定变化点的位置，将以加权双部分和的形式对检验统计量的分布函数应用鞍点近似。该近似非常准确，不需要使用模拟或使用布朗桥进行近似。*第三，利用相似图和最小生成树表示，提出了基于图的加权双部分和统计量来测试变化点。这种方法的动机是研究高通量数据，其中每个观测的维度可以大于序列的长度。*这项建议寻求资金，以开展变点检测和变量选择之间的关系的系统研究，开发准确的鞍点近似，并探索在许多领域的应用。在本研究计划中开发的基本统计学理论将显著提高我们对多变点检测的理解，并为在许多科学领域(包括高通量遗传学、环境计量学、地震勘探和金融)收集的高维数据开发新的测试统计方法指明方向。**

A change point refers to a location or time at which observations or data obey two different models: before and after. Statistical studies of change-point problems date back to Page (1955) and have flourished especially since the 1980s. These studies have found applications in a wide range of areas, including quality control, finance, environmetrics, medicine, genetics and geography. An essential part of this proposed research program is to develop fast and accurate methods for change-point detection. ***A promising method for tackling the multiple change-point problem is to convert it to the variable selection problem by segmenting the data sequence properly. Thus, a connection between the two problems is established, and a framework is proposed to tackle multiple change point detection via the following two key steps: (1) apply the recent advances in consistent variable selection methods to detect change points; (2) employ a refined procedure to improve the accuracy of change-point estimation. ******The first part of this proposed research program is to continue the investigation of variable selection methods in large and complex data sets, to develop fast and efficient algorithms implemented using the statistical computing software R, and to establish statistical theory to support the developed algorithms. ******Second, in order to accurately refine the location of the change-point, a saddlepoint approximation will be applied to the distribution function of the test statistic, in the form of a weighted double partial sum. The approximation is very accurate and does not require the use of simulations or approximation using the Brownian bridge. ******Third, graph-based weighted double partial sum statistics for testing change-points will be proposed, utilizing similarity graphs and minimum spanning tree representations. This approach is motivated by the study of high-throughput data, where the dimension of each observation can be larger than the length of the sequence.***This proposal seeks funding to carry out a systematic study of relationships between change-point detection and variable selection, to develop accurate saddlepoint approximations, and to explore applications in many fields. The fundamental statistical theory developed in this research program will improve our understanding of multiple change-point detection significantly and point the way towards developing new test statistics for high-dimensional data collected in many areas of science, including high-throughput genetics, environmetrics, seismic exploration, and finance.**