Model-based Classification of Longitudinal and Functional Data

分类的概念渗透到许多科学研究中，几乎出现在人类活动的每个领域，包括数字分类学和市场细分的经典问题，以及机器学习、实验光谱学和生物技术的现代领域。Fisher的线性分类器在方差分析的精神下最大化了组间的分离，它要求每个组具有一个共同的协方差矩阵的多变量正态性。对于异构协方差矩阵，最优分类器不再是线性的，并且在小样本情况下性能较差。虽然有许多启发式和特别的方法来处理不等协方差的情况，但基于模型的方法使用多元正态分布的混合物和协方差矩阵的谱分解对传统的多元数据显示出很大的希望。本研究的目标是利用协方差矩阵的Cholesky分解代替其谱分解，为纵向、功能和多元时间序列数据开发新的灵活的分类方法。对于这类数据，Cholesky分解更为合适，其分量既可以作为一定的回归系数进行统计解释，也可以根据代表数据中各个群体的椭球的体积、形状和方向进行几何解释。本文提出对使用Cholesky分解的计算、统计和经验方面进行研究，并将结果与使用光谱分解得到的结果进行比较。所采用的方法和工具包括：广义线性和混合模型、因子分析、时间序列分析、存在缺失值的混合模型的最大似然和贝叶斯估计、交叉验证和自举。该研究将经典判别分析扩展到纵向和功能数据。它具有很大的实际意义，并将提供对特定分类方法何时可以预期良好工作的见解，并可能导致发展新的分类标准和方法来区分核爆炸和地震。这个问题对监测全面禁试条约至关重要。拟议工作的更广泛影响可以在收集高维和大量多变量数据的环境中看到，例如临床试验、生物技术、环境监测和全球变化、流行病学和金融计量经济学。

The idea of classification permeates many scientific studies and arises in almost every area of human endeavors including the classical problems of numerical taxonomy and market segmentation, and the modern areas of machine learning, experimental spectroscopy and biotechnology. Fisher's linear classifier which maximizes the separation between the groups in the spirit of analysis of variance requires multivariate normality for each group with a common covariance matrix. For heterogeneous covariance matrices, the optimal classifier is no longer linear and has poor performance for small samples. Though there are many heuristic and ad hoc methods to handle the case of unequal covariances, model-based approaches using mixtures of multivariate normal distributions and the spectral decomposition of the covariance matrices have shown great promise for the traditional multivariate data. The goal of this research is to develop new and flexible classification methods for longitudinal, functional and multivariate time series data using the Cholesky decomposition of covariance matrices instead of their spectral decompositions. For such data, the Cholesky decomposition is more suitable and its components enjoy both statistical interpretation as certain regression coefficients and geometric interpretation in terms of volumes, shapes and orientations of ellipsoids representing various groups in the data. It is proposed to study the computational, statistical and empirical aspects of using the Cholesky decomposition and compare the results with those obtained using the spectral decomposition. The methods and tools to be employed include: generalized linear and mixed models, factor analysis, time series analysis, maximum likelihood and Bayesian estimation of mixture models in the presence of missing values, cross-validation and bootstrap. The proposed research will extend classical discriminant analysis to longitudinal and functional data. It is of great practical interest and will provide insight into when a particular classification method can be expected to work well, and may lead to the development of new classification criteria and methods for discriminating between nuclear explosions and earthquakes. A problem which is of critical importance for monitoring a comprehensive test-ban treaty. The broader impact of the proposed work can be seen in settings where high-dimensional and large amounts of multivariate data are collected, such as clinical trials, biotechnology, environmental monitoring and global change, epidemiology and financial econometrics.