Development of Modal Regression

本项目旨在为经济学、社会学、医学和生物学中常见的异常数据（如偏斜、截断、异构或带有异常值的噪声数据）开发一套新的统计回归模型。这种新的回归方法被称为模态回归，它在给定协变量的情况下找到因变量的条件最可能值（模态），而不是传统回归模型所关注的均值/分位数。作为现有回归工具的补充，模态回归可以揭示可能被条件均值或分位数遗漏的有趣的新数据结构。此外，模态回归具有抗离群值和测量误差的能力，在数据偏倚时可以提供更短的预测间隔，如经济学中的工资、价格和支出，社会学中的教会规模和症状指数。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的数据，当只有在因变量具有下限或上限时才观察到数据时，例如在某个范围内测量的经济指数，就会出现这种情况。这项工作将使那些想要分析经济学、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倾斜或截断数据的科学家和研究人员受益。该项目将为研究生提供培训机会。为实现新的模态回归而开发的软件将公开提供。与现有回归模型平行，研究者将通过对给定协变量x的因变量Y的条件模式进行一些模型假设，为独立和相关（时间序列或空间）数据开发各种参数和非参数模态回归模型。新方法避免了给定x的Y的条件密度的非参数估计，这在x的维数较大时是困难的。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。系统地研究了所得估计器的收敛速率和抽样性质。对于高维数据，研究者将考虑一种新的特征选择工具和模态回归的变量选择方法。此外，研究者将开发一种新的充分降维方法来降低协变量的维数用于模态回归。此外，研究者将开发一个多模态回归曲线存在的异构/混合数据的模态聚类工具。模态聚类方法可以作为混合回归模型的一种替代工具来揭示聚类/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。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.