Multiscale generalised linear models for nonparametric function estimation
1 Department of Mathematics and Statistics, Boston University, Boston, Massachusetts 02215, U.S.A. kolaczyk{at}math.bu.edu, 2 Department of Electrical and Computer Engineering, University of Wisconsin, Madison, Wisconsin 53706, U.S.A. nowak{at}engr.wisc.edu
We present a method for extracting information about both the scale and trend of local components of an inhomogeneous function in a nonparametric generalised linear model. Our multiscale framework combines recursive partitions, which allow for the incorporation of scale in a natural manner, with systems of piecewise polynomials supported on the partition intervals, which serve to summarise the smooth trend within each interval. Our estimators are formulated as solutions of complexity-penalised likelihood optimisations, where the penalty seeks to limit the number of intervals used to model the data. The actual calculation of the estimators may be accomplished using standard software routines for generalised linear models, within the context of efficient, tree-based, polynomial-time algorithms. A risk analysis shows that these estimators achieve the same asymptotic rates in the nonparametric generalised linear model as the classical wavelet-based estimators in the Gaussian ‘function plus noise’ model, for suitably defined ranges of Besov spaces. Numerical simulations show that the method tends to perform at least as well as, and often better than, alternative wavelet-based methodologies in the context of finite samples, while applications to gamma-ray burst data in astronomy and packet loss data in computer network tra.c analysis confirm its practical relevance.
Key Words: Astronomy; Computer network traffic; Minimax; Piecewise polynomial; Recursive partitioning; Wavelet
Received January 2003. Revised April 2004.