Statistical inference in finite mixture of regressions and mixture-of-experts models in high-dimensional spaces, and varying coefficient finite mixture of regression models

导言：近年来，我们见证了从生物、医学到工程、社会科学和计量经济学等不同科学研究领域的大规模数据的兴起，俗称大数据。在分析这类数据时，一个常见的统计问题是将感兴趣的响应变量建模为大量特征的小子集的函数。这称为特征选择问题。除了噪声累积和虚假相关性之外，高维数据中未观察到的异质性使特征选择问题变得更加困难。有限混合回归(FMR)和专家混合(MOE)是捕捉数据异质性的强有力的统计模型。该方案的第一部分重点研究了FMR/MOE中的特征选择、估计和选择后推理问题。第二部分是变系数有限混合回归(VC-FMR)模型，其中回归系数随时间等指标变量的光滑函数而变化。例如，在市场细分研究中，消费者对产品的偏好往往会随着时间的推移而在不同的细分市场上发生变化。VC-FMR模型为建模这类涉及异质功能数据的现象提供了一种自然的工具。然而，这些相对较新的模型的方法学和计算工具在很大程度上还没有被探索。 目标：我的研究计划的一个重点是开发合理的统计方法和计算高效的算法，用于高维FMR/MOE中的估计、特征选择以及选择后推理，如假设检验和可信区间。本文研究的另一个重点是VC-FMR中的估计和特征选择。我的较长期目标是关注复杂的时间序列数据，以及高维、异质和相关的数据。 方法：研究高维FMR/MOE模型中同时进行参数估计和特征选择的正则化技术LASSO/SCAD。将研究用于数值计算的坐标下降型期望最大化(EM)算法。基于样本分裂技术，对稀疏FMR/MOE中参数的假设检验和置信度区间等选择后推理进行了探索，并将基于正则化局部核似然的方法用于VC-FMR模型中的函数参数估计和特征选择。 影响：我建议的研究计划将解决高维FMR/MOE模型以及VC-FMR模型中尚未解决的统计问题，并为更广泛的统计受众感兴趣的实际问题提供解决方案。然后，建议的方法可以立即用于解决生物、工程、健康科学和市场研究等领域的科学问题。

INTRODUCTION: In recent years, we have witnessed the rise of large scale data, colloquially referred to as big data, in different fields of scientific research ranging from biology and medicine to engineering, the social sciences and econometrics. A common statistical problem of interest in the analysis of such data is to model a response variable of interest as a function of a small subset of a large number of features. This is referred to as a feature selection problem. In addition to noise accumulation and spurious correlation, unobserved heterogeneity in high-dimensional data makes the feature selection problem even harder. Finite mixture of regressions (FMR) and mixture-of-experts (MOE) are powerful statistical models for capturing heterogeneity in data. The first part of this proposal focuses on feature selection, estimation and post-selection inference problems in FMR/MOE. The second part concerns varying coefficient finite mixture of regression (VC-FMR) models in which regression coefficients change as smooth functions of an index variable such as time. For example, in market segmentation research, consumer preferences for products often change over time and across different market segments. VC-FMR models provide a natural tool for modeling such phenomena which involves heterogeneous functional data. However, methodological and computational tools for these relatively new models are largely unexplored.  OBJECTIVES: An emphasis of my research program is on developing sound statistical methodology and computationally efficient algorithms for estimation, feature selection, and also post-selection inference such as hypothesis testing and confidence intervals in FMR/MOE in high dimensions. Another focal point of my research concerns estimation and feature selection in VC-FMR. My longer term objectives focus on complex time series data and high-dimensional heterogeneous and dependent data.  METHODS: I will study the regularization techniques LASSO/SCAD for simultaneous parameter estimation and feature selection in FMR/MOE models in high dimensions. Coordinate descent-type expectation-maximization (EM) algorithms will be investigated for numerical computations. Post-selection inference such as hypothesis testing and confidence intervals for parameters in sparse FMR/MOE will be explored based on sample splitting techniques. Regularized local kernel likelihood-based methods will be used for functional parameter estimation and feature selection in VC-FMR models.  IMPACT: My proposed research program will address unresolved statistical issues in FMR/MOE models in high dimensions as well as in VC-FMR models, and offer solutions to practical problems of interest to a broader statistical audience. The proposed methods could then immediately be used to solve scientific problems in areas such as biology, engineering, the health sciences, and marketing research.