Statistical Inference for Complex Data

计算机的出现通常会带来非常大的数据集进行分析。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非例行公事。这项建议调查了两个大的领域，即混合分解，以及存在分类和等级变量的回归。混合分解工作涉及的是观测值是分布空间的p维笛卡尔乘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。这样的数据是自然产生的，或者是在将原始数据聚合到更易于管理的大小但保留其固有信息之后产生的。目标是将这些分布划分为相关的类，并估计相关的类分布。建议将经典数据的Copula理论的思想应用于由分布组成的数据。在这个过程中嵌入了对参数估计进行研究的必要性。将探索不同的划分技术，例如动态聚类，将考虑不同的适配度标准，例如，对数似然分类；以及为底层Copula和相关分布及其参数探索的不同估计方法，例如，最大似然、诸如Parzen截断窗口的非参数方法。所得到的方法将广泛适用于在例如气象学、环境科学、社会科学、卫生保健计划等中产生的那些数据集。当分类变量和等级变量存在时，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，迫切需要开发技术来分析这些数据集。到目前为止，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明确但有意义的方式聚合数据。这种聚合将以列表、间隔或分布的形式产生数据，这些数据现在将具有某种形式的内部结构。本研究将针对这两类数据进行研究。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适当的分布混合。另一个涉及分类和等级数据，目的是建立回归关系，解释管理所涉及变量的潜在过程。由此产生的方法将允许分析和解释目前尚不存在方法的当代数据。

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.