Finite multivariate density mixtures: applications and new approaches

多变量数据集的聚类分析和分类是现代统计和数据科学中的两个最重要的任务。通过利用基于Copula的密度混合，该项目将引入新的方法来对表现出复杂的相关性属性的多变量数据进行聚类和分类。建议的方法将提供更可靠的聚类解决方案，这些解决方案可用于例如开发新的治疗药物和更精确地量化基因相互作用。这个项目将让本科生和研究生参与这项跨学科研究的计算方面的工作。该项目的特别重点将是加强对统计科学的更广泛参与以及来自代表性不足群体的受训者的参与。许多学生将通过总部设在普渡大学的全国数学科学博士研究联盟(俗称数学联盟)招收。多元混合模型在许多统计领域都有广泛的应用。它们在多变量高维数据的聚类和分类中特别有用。大多数基于模型的多变量数据聚类技术都是基于多变量正态模型及其直接推广。然而，这种方法有很大的局限性，因为在大多数情况下，很难对非椭圆形状的簇进行建模。即使在可能的情况下，这些模型在组合包括连续和离散随机变量的混合类型的多变量数据方面的能力往往也是有限的。该项目将开发一种替代现有的基于模型的多变量数据聚类方法。这一替代方案的基础是使用基于Copula的密度混合物，这将允许对各种依赖结构进行建模。此外，拟议的解决方案还将允许对包含连续和离散观测的数据集进行原则性分组。这个项目的直接结果将是开发出一系列计算效率高的算法。这样的算法将在最小的假设下提供可靠的对象聚类。建议的方法将广泛适用于转录数据中的基因和其他生物实体的聚类以及文本文档的聚类，其中维度的数量可能等于词汇表的大小。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Cluster analysis and classification of multivariate datasets are two of the most important tasks in modern statistical and data sciences. By leveraging copula-based density mixtures, this project will introduce new methods for clustering and classification of multivariate data which exhibit sophisticated dependence properties. The proposed methods will provide more reliable clustering solutions that could be in turn used, for example, for development of new therapeutic drugs and more precise quantification of gene interactions. This project will involve students, both at the undergraduate and graduate levels, to work on the computational aspects of this interdisciplinary research. The particular focus of the project will be on bolstering broader participation in statistical sciences and involvement of mentees from the underrepresented groups. Many of the students will be recruited through the National Alliance for Doctoral Studies in the Mathematical Sciences (commonly called simply “Math Alliance”) that is headquartered at Purdue University.Multivariate mixture models are widely applicable in many areas of statistics. They are particularly useful in clustering and classification of multivariate high-dimensional data. The majority of model-based clustering techniques for multivariate data are based on multivariate normal models and their direct generalizations. However, this approach is quite restrictive since, in most cases, it is difficult to model clusters of non-elliptical shapes. Even where it is possible, these models tend to be limited in their capabilities to cluster multivariate data of mixed types that include both continuous and discrete random variables. This project will develop an alternative to the existing model-based clustering methods for multivariate data. This alternative is based on using copula-based density mixtures which will allow for modeling a vast variety of dependence structures. Furthermore, the proposed solution will also allow for the principled clustering of datasets containing both continuous and discrete observations. The immediate outcome of this project will be the development of a family of computationally efficient algorithms. Such algorithms will provide reliable clustering of objects under the minimal assumptions. The proposed methodology will be widely applicable to such areas as clustering of genes and other biological entities in transcriptomics data as well as clustering of text documents, where the number of dimensions may be equal to the size of the vocabulary.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.