Development of Modal Regression

该项目旨在为异常数据开发一套新的统计回归模型，例如偏斜，截断，异质或具有离群值的噪声数据，这些数据在经济学，社会学，医学和生物学中常见。新的回归方法称为模态回归（modal regression），它在给定协变量的情况下寻找因变量的条件最可能值（众数），而不是传统回归模型关注的均值/分位数，作为对现有回归工具的补充，模态回归可以揭示条件均值或分位数可能遗漏的有趣的新数据结构。此外，模态回归对离群值和测量误差具有抵抗力，并且可以在数据偏斜时提供更短的预测间隔，例如经济学中的工资，价格和支出以及社会学中的教堂规模和症状指数。此外，与传统的均值或分位数回归不同，模态回归可以直接应用于截断数据，当数据仅在因变量具有下限或上限时才被观察到时，例如在某个范围内测量的经济指数。这项工作将有利于科学家和研究人员，他们希望分析经济学，社会科学，市场营销，医学研究，公共卫生，生物学和农业等领域的扭曲或截断数据。该项目将为研究生提供培训机会。与现有的回归模型平行，研究人员将为独立和相关（时间序列或空间）数据开发各种各样的参数和非参数模态回归模型，方法是在给定协变量x的情况下，对因变量Y的条件模式施加一些模型假设。新方法避免了在给定x的情况下对Y的条件密度进行非参数估计，而当x的维数很大时，这是很困难的。研究者将开发一种模态期望最大化算法来简化模态回归的计算。将系统地研究由此产生的估计量的收敛速度和抽样特性。对于高维数据，研究者将考虑一种新的特征选择工具和变量选择方法进行模态回归。此外，研究者将开发一种新的充分降维方法来降低模态回归协变量的维数。此外，研究人员将开发一个模态聚类工具，用于存在多模态回归曲线的异质/混合数据。模态聚类方法可以作为混合回归模型的替代工具，以揭示聚类/非均匀数据结构，并提供一种自然的方法来估计组件/聚类的数量，这一直是一个具有挑战性的问题。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

This project seeks to develop a set of new statistical regression models for abnormal data, such as skewed, truncated, heterogeneous, or noisy data with outliers, which are commonly seen in economics, sociology, medicine, and biology. The new regression method, named modal regression, finds the conditional most probable value (mode) of a dependent variable given covariates, rather than the mean/quantile that the traditional regression models focus on. As a complement to existing regression tools, the modal regression could reveal interesting new data structure that is possibly missed by the conditional mean or quantiles. In addition, modal regression is resistant to outliers and measurement errors, and can provide shorter prediction intervals when the data are skewed, such as salary, prices, and expenditures in economics and church sizes and symptom indices in sociology. Furthermore, unlike traditional mean or quantile regression, the modal regression can be directly applied to the truncated data, which arises when the data are observed only when the dependent variable has a lower or upper limit, such as an economic index measured within some range. This work will benefit scientists and researchers who want to analyze skewed or truncated data in fields that include economics, social sciences, marketing, medical studies, public health, biology, and agriculture. This project will provide training opportunities for graduate students. Software developed for implementing the new modal regression will be made publicly available.Parallel to existing regression models, the investigator will develop a wide variety of parametric and nonparametric modal regression models for both independent and dependent (time series or spatial) data by imposing some model assumptions on the conditional mode of a dependent variable Y given covariates x. The new method avoids the nonparametric estimation of conditional density of Y given x, which is difficult when the dimension of x is large. The investigator will develop a modal expectation-maximization algorithm to simplify the computation of the modal regression. The convergence rate and sampling properties of the resulting estimators will be systematically studied. For high dimensional data, the investigator will consider a new feature selection tool and variable selection methods for modal regression. In addition, the investigator will develop a new sufficient dimension reduction method to reduce the dimension of covariates for modal regression. Furthermore, the investigator will develop a modal clustering tool for heterogeneous/mixture data where multiple modal regression curves exist. The modal clustering method can serve as an alternative tool for mixture regression models to reveal the clustered/inhomogeneous data structure and provide a natural way to estimate the number of components/clusters, which has long been a challenging problem.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.