轨道数据的聚类分析

The analysis of trajectory data, that is, longitudinal data with a group-based approach, is attractive because of long tradition of group-based theorizing. Examples include theories of personality development, drug use,learning, language and conceptual development, depression, eating disorders, alcoholism, anxiety and the development of prosocial behaviors such as conscience and of antisocial behaviors such as conduct disorder and delinquency. There are several problems with the existing methods. Firstly, although assumptions are different, the existing methods model the trajectories based on Gaussian distribution. As a result, for nonnormal data a multiple trajectory solution may be identified even when only one real group exits. In addition, the growth curve in existing methods is specified by a polynomial function. The misspecifications of the growth curve would lead to biased estimation and classification. Finally, how many groups should be included is often unknown a prior. Few work has been devoted to the selection of the number of groups, predominantly using the AIC and BIC criterion. However, the properties of these methods are elusive, making it difficult to evaluate them. In the project, we propose penalized transformation approaches for trajectory groups based on single, multiple and mixed trajectory groups,respectively. We allow the distribution of the response, the growth curve, and the number of the groups to be unspecified. We simultaneously estimate them using a penalty pseudo-likelihood and estimating equation. The large sample properties of the proposed estimators will be established so we can make inferences for the parameters and the functions. The proposed methods will be applied to real medical and economicsl data.

轨道数据是以整个纵向一连串观察为单位的数据，广泛出现在医学、心理学、经济学等。本项目研究轨道数据的聚类，用于探索个体发展演变的可能方式及其风险因素。现有方法有如下几个问题：1.只考虑正态的轨道数据。对非正态数据，即使只有一个类别，也可能给出多类别结果；2.要求生长曲线已知，这在实际中很难提前知道，错误的生长曲线会导致错误分类；3. 用AIC或BIC准则简单判断聚类个数，理论性质不清楚，无法做进一步的统计推断。本项目分别针对单轨道数据、多重轨道数据及混合多重轨道数据，研究一般数据框架（包括正态、非正态）下，当生长曲线及聚类数都未知时轨道数据的分类。我们通过未知转换将数据正态化并用非参数方法模拟生长曲线。我们进一步提出判罚拟似然及估计方程方法同时估计这些未知函数、未知参数及聚类数并研究估计的大样本性质。研究结果将用于分析实际的医学及经济学数据。

在轨道及相关数据分析的三个方面取得重要进展，发展了一套新的半参数估计及理论，可有效地估计结构复杂的非参数及半参数模型；针对复杂的半参数及非参数模型下的高维轨道及相关数据，给出了计算可行的正则化方法，并确立了理论性质，包括sure screening性质及Oracle性质；对于不完全观测及有缺失的轨道及相关数据分析，通过迭代及核估计算法，发展了一系列理论有效，计算简单的估计方法处理缺失及轨道数据带来的相关性问题。.在项目设立期间，共发表学术论文19篇，国际统计学的前四刊物2篇；计量经济学顶级刊物4篇；国际统计学重要刊物Statistica Sinica(4篇), Scandinavian Journal of Statistics(2篇)等。成功申请到国家自然科学基金重点项目1项，国家自然科学基金海外及港澳学者合作研究基金2项，组建校级重点研究基地1项；申请专利项目1个。.在该项目部分支持下，项目负责人林华珍教授于2017年获教育部长江学者特聘教授及百千万人才国际级人选、2019年荣获首批“四川省教书育人名师”。项目主要参与人周岭2017年度获钟家庆数学奖，2019年度入选中组部“青年千人计划”及四川省千人计划。