High Dimensional Mixture Models

本项目的目的是发展高维数据混合模型的理论、统计方法和计算方法。在理论部分，通过研究这些模型的地形结构及其与其他高维方法（如局部线性回归和层次树）的关系，研究增强了这些模型的潜在统计应用。利用扩散过程的思想，正在构造新的核密度。评估这些模型中可识别性的重要方面的新方法正在开发中。除了这些基本的理论发展，研究者正在创建一套方法来拟合扩散混合模型，并在高维上评估它们的拟合。这种方法发展的一个关键部分是在计算增强方面。统计学界面临着现代科学的巨大挑战，那就是在数据革命之后开发新的科学推理工具。现代数据具有潜在的高维度，并且收集的单位数量庞大。被称为混合模型的概率模型在描述数据样本的异质性方面有着悠久的历史。它们非常灵活，并提供了数据结构关键特性的简洁图像。不幸的是，这一困难领域的有限理论发展阻碍了它们在高维问题中的应用。该研究项目针对该领域仍然存在的一些关键困难，包括将该方法与其他现有方法集成，将该方法扩展到新的数据类型，以及更好地理解该模型的高维结构。从这些发展中产生的方法正在被转化为计算包，以便科学家使用。

The purpose of this project is the development of theory, statistical methodology, and computationalmethods for use in mixture models in high dimensional data. In the theoretical portion, the investigatorenhances the potential statistical applications of these models by examining their topographical structure and their relationship to other high-dimensional methods such as local linear regression andhierarchical trees. New kernel densities are being constructed by the use of the idea ofdiffusion processes. New methods to assess the important aspects of identifiability in these modelsare under development. In addition to these basic theoretical developments, the investigator is creating a set of methods designed to fit diffusion mixture models, and to assess their fit, in high dimensions. A key part of this methodological development is occuring in computational enhancements.The statistics community is faced with a great challenge by modern science, and that is todevelop new tools for scientific inference in the aftermath of the data revolution. Modern data is potentially high in dimension, and massive in the number of collected units. The probability models called mixture models have had a long history of use in describing heterogeneity in data samples. They are extremely flexible, and provide a compact picture of the key features of the data structure. Unfortunately, limited theoretical developments in this difficult area have held back their use in high dimensional problems. This research project targets a number of the key difficulties remaining in this area, including the integration of this methodology with other existing ones, the expansionof this methodology into new data types, and a better understanding of this model's structure invery high dimensions. The methods that arise from these developments are being turned into computational packages so that they can be used by scientists.