Statistical Inference for Complex Data

计算机的出现通常会带来大量的、非常大的数据集来进行分析。如何分析随之而来的数据和/或如何从这些数据中收集有用的信息绝不是例行公事。本文研究了两个广泛的领域，即混合分解和在分类和层次变量存在下的回归。混合分解工作关注的是观测值是分布空间的p维笛卡尔积中的分布函数的数据，而不是经典数据在p维空间中的单点。这些数据是自然产生的，或者是在原始数据聚集到更易于管理的大小后，同时保留其固有信息。目标是将这些分布划分为一致的类，并估计相关的类分布。提出将经典数据的联结理论的思想应用到由分布组成的数据中。在这个过程中嵌入了需要研究的参数估计。将探讨不同的划分技术，如动态聚类；将考虑不同的拟合标准，如对数似然分类；并探索了不同的估计方法，用于潜在的copula和相关的分布及其参数，例如，最大似然，非参数方法，如Parzen的截断窗口。由此产生的方法将广泛适用于气象学、环境科学、社会科学、卫生保健计划等领域产生的数据集。当分类变量和层次变量存在时，也将开发回归方法。这将首先对经典数据执行，然后扩展到区间值数据和直方图（或频率）值数据。随着现代计算机产生非常大的数据集，必须开发分析这些数据集的技术。迄今为止，存在的方法很少。这项研究将开发分析这些数据的新方法。第一步是以某种定义良好但有意义的方式汇总数据。这种聚合将产生列表、间隔或分布形式的数据，这些数据现在将具有某种形式的内部结构。研究将集中于这两类数据。一个是数据现在由分布组成的地方，目标是开发方法来识别描述这些数据的分布的适当混合。另一种方法处理分类法和层次数据，其目标是建立回归关系，以解释控制所涉及变量的潜在过程。由此产生的方法将允许对目前没有方法存在的当代数据进行分析和解释。

The advent of computers is routinely bringing large, very large datasets for analysis. How to analyze theattendant data and/or how to glean useful information from these data is anything but routine. This proposal investigates two broad areas, viz., mixture decomposition, and regression in the presence of taxonomy and hierarchical variables. The mixture decomposition work is concerned with data in which the observation is a distribution function in the p-dimensional Cartesian product of distributions space, rather than the single point in p-dimensional space of classical data. Such data arise naturally, or after aggregation of original data to a more manageable size yet retaining its inherent information. A goal is to partition these distributions into coherent classes and to estimate the relevant class distributions. It is proposed to adapt ideas from copula theory for classical data, to data comprised of distributions. Embedded in this process is the need to study parameter estimation. Different partitioning techniques will be explored such as dynamical clustering, different measures of fit criterion will be considered, e.g., log-likelihood classification; and different estimation methods explored for the underlying copulas and the associated distributions and their parameters, e.g., maximum likelihood, nonparametric methods such as Parzen's truncated window. The resulting methodology will have wide applicability to those datasets generated in, e.g., meteorology, environmental science, social sciences, health-care programs, and the like. Regression methods when taxonomy variables, and when hierarchical variables, are present will also be developed. This will first be executed for classical data, and then extended to interval-valued data and to histogram- (or frequency-) valued data.With modern computers generating very large datasets, it is imperativethat techniques be developed for analysing such datasets. To date veryfew methods exist. The research will develop new methodologies foranalysing these data. A first step is to aggregate the data in somewell-defined but meaningful way. This aggregation will thence producedata in the form of lists, intervals, or distributions, and these datawill now have some form of internal structure. The research will focuson such data of two types. One will be where the data now consist ofdistributions, and where the goal is to develop methods to identify theappropraite mixture of distributions that describe these data. Anotherdeals with taxonomy and hierarical data, with the goal of establishingregression relationships that will explain the underlying processgoverning the variables involved. The resulting methodologies willallow analysis and interpretation of contemporary datastes wherecurrently no methods exist.