Adaptive Regression for Dependent Data by Combining Different Procedures

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

This proposal concerns research and education on adaptive regression when the random errors are dependent. Many procedures have been (and will be) proposed for nonparametric regression based on different assumptions. In applications, a difficulty a user often faces is the choice of the best method for the data at hand. This is specially the case for high-dimensional function estimation, where to overcome the curse of dimensionality, various parsimonious models such as projection pursuit, CART, neural nets, additive models, MARS, etc. are proposed according to different characterizations of the target function. A main interest in this research is to construct adaptive estimators by combining a collection of candidate procedures. The goal for the combined procedure is to perform automatically as well as (or nearly as well as) the best original procedure without knowing which one it is. The random errors will be assumed to be generally dependent, including both short- and long-range cases. The effects of dependence on adaptation capability will be studied. It is anticipated that theoretically proven and computationally feasible algorithms will be proposed to combine regression procedures targeted at various characteristics of the regression function and different dependence structures for the random errors. Function estimation is an important statistical tool that tries to understand accurately the functional relationships between variables based on data and it has applications in many disciplines for successfully addressing scientific questions. In reality, observations are always subject to random noise (error) from different sources. When the random errors are dependent on each other, the dependence may disguise the functional relationship of interest. Long-range dependence refers to a situation where the errors are still highly correlated even when they occur at times or locations that are far away from each other. It is known that such a long-range dependence makes the estimation of the target function much harder. In applications, the degree of dependence between the errors is usually unknown, which makes the function estimation problem even harder. In this proposal, we intend to develop methods that adaptively handle different degrees of dependence among the errors so that the function of interest can be estimated optimally without knowing the dependence structure of the errors. The research results and related work by others on long-range dependent data will be brought to students at various levels in several statistics courses. Collaborations will be conducted with several professors at Iowa State University and their students in atmospheric science, electrical engineering, agronomy and possibly other fields to appropriately address long-range dependence phenomena, which have been encountered often and known to cause problems in data analysis with the existing statistical methods.