Studies on Multidimensional Analysis of Longitudinal Categorical Data

本研究旨在探讨分析纵向分类资料以量化个体变化的方法。我们专注于一个指标矩阵，其行和列分别与时间点和类别上的个人相关联。从这个数据矩阵中，个体的变化不能用现有的量化方法来量化，为了解决这个困难，我们开发了约束和正则化的量化方法。在约束方法中，增长曲线约束被施加到随时间点分配给个体的分数上：分数被约束为时间上的多项式。通过对真实的数据的处理，证明了该方法的有效性。在正则化方法中，将已有方法中的损失函数与惩罚函数相结合，形成惩罚损失函数。这种方法分为两种方法。一种是使用分数的一阶差来定义惩罚，这需要时间点上的个体分数的同质性。另一种是将分数视为时间的自然三次样条函数，并使用样条的二阶导数来定义惩罚，假设个体分数随时间平滑地变化。这两种方法在模拟和真实的数据分析中都取得了令人满意的结果。

The purpose of this project was to study the methods for analyzing longitudinal categorical data to quantify individuals' changes. We focused on an indicator matrix whose rows and columns associated with the individuals over time-points and with categories, respectively. From this data matrix, individual changes cannot be quantified by the existing quantification method, To deal with this difficulty, we developed constrained and regularized methods for quantification.In the constrained method, the growth curve constraint is imposed on the scores to be assigned to the individuals over time-points : the scores is constrained to be a polynomial in time. The usefulness of this method we developed was shown by its application to real data. This method was further extended to simultaneously perform the clustering of individuals.In the regularized method, the loss function in the existing method is combined with a penalty function, to form a penalized loss function. This method is subdivided into two approaches. One is to define the penalty using first order differences of scores, which requires the homogeneity of individual scores over time-points. The other is to treat the scores as natural cubic spline functions of time and to define the penalty using the second order derivative of the splines, assuming individual scores to change smoothly with time. Both methods gave promising results in simulation and real data analysis.