Statistical Foundations of Model-Based Variable Clustering

变量聚类问题是遗传学、神经科学、社会学、宏观经济学等众多领域的基石。在神经科学中，它有助于寻找新的功能相连的区域。在遗传学上，它有助于发现功能未被充分发掘或未知的基因。在宏观经济方面，它可以帮助创建新的经济指数。尽管这个问题具有广泛的重要性和潜在的影响，但在文献中还没有得到系统的方法论和理论处理。虽然集群算法很多，而且有很长的历史，但评估它们输入的有效性有点武断。在这个项目中，提出了一种基于模型的概率方法。这将使一个统一的原则性统计变量聚类框架的发展成为可能。具体地说，该项目将引入和研究用于重叠和非重叠变量聚类的潜在变量模型。重点是：(I)引入可识别的潜在变量模型进行聚类。这将为估计提供明确的目标，并将促进对集群的科学解释。(2)开发为这些模型量身定做的多项式时间算法。(3)为集群算法的理论分析建立一个统一的框架，重点是极小极大最优性和高维推理。(4)研究基于模型的聚类算法对下游分析的影响，重点是图形模型、回归和分类。

The problem of variable clustering is a corner stone in a multitude of areas such as genetics, neuroscience, sociology, macroeconomics, to name a few. In neuroscience, it aids in finding new functionally connected areas. In genetics, it helps advance the discovery of genes with under-explored or unknown functions. In macro-economics it can assist with the creation of new economic indices. Despite its wide-spread importance and potential impact, this problem has not received a systematic methodological and theoretical treatment in the literature. Although clustering algorithms abound, and have a very long history, assessing the validity of their input is somewhat arbitrary. A probabilistic, model-based approach is put forward in this project. This will enable the development of a unified framework for principled statistical variable clustering.Specifically, this project will introduce and investigate classes of latent variable models for overlapping and non-overlapping variable clustering. The focal points are: (I) The introduction of identifiable latent variable models for clustering. This will provide well defined targets for estimation, and will facilitate the scientific interpretation of the clusters. (II) The development of polynomial time algorithms tailored to these models. (III) The creation of a unifying framework for the theoretical analysis of clustering algorithms, with emphasis on minimax optimality and high dimensional inference. (IV) The study of the impact of model based clustering algorithms on downstream analyses, with emphasis on graphical models, regression and classification.