Topics in multifractal measures, nonparametrics and biostatistics

Attila Frigyesi

Centre for Mathematical Sciences
Mathematical Statistics
Lund University,
2004

ISBN 91-628-5993-5
LUNFMS--1014-2004

Abstract:

This thesis consists of four papers. The first two papers, which comprise the main part of the thesis, deal with an unexpected connection between kernel density estimators and dimension spectra for multifractal measures. The third paper presents a fully automated expert system for the diagnosis of pulmonary embolism from ventilation/perfusion scintigraphy. The final paper concerns statistical properties of the parameters of the operational model of pharmacological agonism, a widely applied model for dose-response curves in pharmacology.

In the first paper kernel density estimators for singular distributions are studied. The density estimator f is a function of the sample size and the bandwidth. It was found that the integral of H(f), where H is a suitable "magnifying" functional, diverges as the sample increases to infinity and the bandwidth goes to 0.

In the second paper it is shown that, for a particular choice of H, the velocity with which the integral of H(f) diverges depends on the q:th generalized Hentschel-Procaccia dimension of the measure from which the sample is drawn. This gives a new way to estimate dimension spectra for multifractal measures. An alternative kernel-based method that gives the correlation integral as a special case is also studied, which enables the estimation of the correlation dimension.

The classic way of estimating generalized fractal dimensions with the aid of grids gives the generalized Rényi dimension. For q>-1 this is proved to be equivalent to the generalized Hentschel-Procaccia dimension. For q<-1 the Rényi dimension may depend on the choice of grid and thus be different from the uniquely defined Hentschel-Procaccia dimension. Examples of such measures are given.

Key words:

singular distribution functions, absolute continuity, kernel density estimates, multifractal measures, dimension spectrum, fractal dimension estimation, generalized dimensions, Rényi dimension, Hentschel-Procaccia dimension, correlation dimension, box counting, automated method, pulmonary embolism, V/Q-scan, ventilation/perfusion scintigraphy, dose-response curves, concentration-response curves, operational model of pharmacological agonism, confidence interval, mixed effects