Nonparametric Estimation using Multi-Scale Generalized Linear Models

• Activity Number: 46
• Type: Contributed
• Date/Time: Sunday, August 11, 2002 : 4:00 PM to 5:50 PM
• Sponsor: General Methodology
• Author(s): Eric Kolaczyk*+ and Robert Nowak
• Affiliation(s): Boston University and Rice University
• Keywords: Wavelets ; Piecewise polynomial ; segmentation

Abstract

Wavelet-based methods have proven to be a powerful choice for modeling temporal or spatial data with inhomogeneous structure. However, most such methods are defined for the canonical "signal-plus-noise'' setting, and extensions to more general data types (e.g., rates, proportions, etc.) are rare or non-existent. In this talk, we introduce a multi-scale, likelihood-based modeling framework to address this issue. Our approach is based upon the simple premise that a wavelet basis differs little in its approximation properties from a collection of piecewise polynomials with support restricted to elements of a nested hierarchy of partitions. We describe how models of this nature can be fit within the generalized linear models (GLM) framework using efficient, tree-based algorithms. The result of this process applied to one-dimensional temporal data may be viewed as an adaptively chosen segmentation with local polynomial fits. We discuss the risk properties associated with these estimators, and describe how they can be made to achieve the same near-optimal rates of convergence associated with the original wavelet-shrinkage estimators. Numerical illustrations are provided.