Sufficient dimension reduction using conditional variance estimation and related concepts

Abstract

In der Regression untersucht man die bedingte Verteilung der Zielvariable gegeben den Prädiktoren, um z.B. Prognosen zu erhalten. Regression ist einer der meist studierten und angewandten Gebiet der Statistik. Die Modellierung von hochdimensionalen Daten, insbesonders bei einem nichtlinearern Zusammenhang, ist herausfordernd falls die Anzahl der Prädiktoren (p) groß ist. Suffiziente Dimensionsreduktion (SDR) ersetzt den hochdimensionalen Prädiktorvektor durch eine niedrigdimensionalere Projektion, ohne Information über die Zielvariable zu verlieren.

Regression concerns modeling the conditional distribution of a target variable, the response, given a set of other variables, the predictors. Regression is the most widely used approach in Statistical applications. As such, it has been extensively studied since the field of Statistics came to existence. Modeling high-dimensional data is challenging, especially when they are nonlinearly related. Sufficient dimension reduction (SDR) considers regressions where the number of predictors (p) is large and replaces the high dimensional predictor by a lower dimensional reduction (function) without loss of information for the response.This thesis develops novel SDR approaches, the conditional variance and ensemble conditional variance estimators, for the identification and estimation of linear sufficient reductions both for the conditional mean and the conditional cumulative distribution function of the response given the multidimensional predictors. The consistency of both estimators is shown. Moreover, a combination of sufficient dimension reduction with neural networks is derived, which leverages the advantages of both in order to predict the response in the presence of abundant predictors and observations.All three proposed estimators are competitive with respect to current state-of-the-art methods in SDR methodology.