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Topics in multifractal measures, nonparametrics and biostatistics

Frigyesi, Attila LU (2004)
Abstract (Swedish)
Popular Abstract in Swedish

Denna avhandling består av fyra uppsatser. De två första, som utgör huvuddelen av avhandlingen, behandlar en oväntad koppling mellan kärnskattningar av fördelningar och dimensionsspektra för multifraktala mått. Den tredje uppsatsen presenterar ett helautomatiskt expertsystem för detektering av lungembolier med lungscintigrafi. Den avslutande uppsatsen avhandlar statistiska egenskaper för parametrarna i den så kallade operationella modellen för farmakologisk agonism, en vida använd modell för dos-responskurvor i farmakologi.



I den första uppsatsen studeras kärnskattningar av singulära fördelningar. Den skattade täthetsfunktionen f är en funktion av stickprovsstorleken och... (More)
Popular Abstract in Swedish

Denna avhandling består av fyra uppsatser. De två första, som utgör huvuddelen av avhandlingen, behandlar en oväntad koppling mellan kärnskattningar av fördelningar och dimensionsspektra för multifraktala mått. Den tredje uppsatsen presenterar ett helautomatiskt expertsystem för detektering av lungembolier med lungscintigrafi. Den avslutande uppsatsen avhandlar statistiska egenskaper för parametrarna i den så kallade operationella modellen för farmakologisk agonism, en vida använd modell för dos-responskurvor i farmakologi.



I den första uppsatsen studeras kärnskattningar av singulära fördelningar. Den skattade täthetsfunktionen f är en funktion av stickprovsstorleken och bandbredden. Vi fann att integralen av H(f), där H är en lämplig ”förstorande” funktional divergerar då stickprovsstorleken går mot oändliheten och bandbredden går mot 0.



I den andra uppsatsen visas att, för ett speciellt val av H, så beror hastigheten varmed H(f) divergerar på den q:te generaliserade Hentschel-Procaccia-dimensionen för det mått från vilken stickprovet dras. Detta ger ett nytt sätt att skatta dimensionsspektra för multifraktala mått. En alternativ kärnskattningsbaserad metod studeras också, vilken möjliggör skattning av korrelationsdimensionen.



Det klassiska sättet att skatta generaliserade fraktala dimensioner med rutnät ger den generaliserade Rényidimensionen. För q>-1 visas att denna dimension är lika med Hentschel-Procaccia dimensionen. För q<-1 så kan Rényidimensionen bero på val av rutnät och på så sätt avvika från den entydigt definierade Hentschel-Procaccia-dimensionen. Exempel på sådana mått ges. (Less)
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... (More)
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. (Less)
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author
opponent
  • professor Cutler, Collen D., Univ. of Waterloo, Canada
organization
publishing date
type
Thesis
publication status
published
subject
keywords
confidence interval, mixed effects, Statistics, operations research, programming, actuarial mathematics, Statistik, operationsanalys, programmering, aktuariematematik, operational model of pharmacological agonism, concentration-response curves, dose-response curves, ventilation/perfusion scintigraphy, V/Q-scan, pulmonary embolism, automated method, box counting, correlation dimension, Hentschel-Procaccia dimension, Rényi dimension, generalized dimensions, fractal dimension estimation, dimension spectrum, multifractal measures, kernel density estimates, absolute continuity, singular distribution functions
pages
130 pages
publisher
Centre for Mathematical Sciences, Lund University
defense location
Centre for Mathematical Sciences, MH:C
defense date
2004-04-06 10:15
ISSN
1404-0034
ISBN
91-628-5993-5
language
English
LU publication?
yes
id
92ceb537-9f8f-467e-ad0d-836860280472 (old id 466780)
date added to LUP
2007-09-27 15:07:40
date last changed
2016-09-19 08:44:54
@phdthesis{92ceb537-9f8f-467e-ad0d-836860280472,
  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.<br/><br>
<br/><br>
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.<br/><br>
<br/><br>
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.<br/><br>
<br/><br>
The classic way of estimating generalized fractal dimensions with the aid of grids gives the generalized Rényi dimension. For q&gt;-1 this is proved to be equivalent to the generalized Hentschel-Procaccia dimension. For q&lt;-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.},
  author       = {Frigyesi, Attila},
  isbn         = {91-628-5993-5},
  issn         = {1404-0034},
  keyword      = {confidence interval,mixed effects,Statistics,operations research,programming,actuarial mathematics,Statistik,operationsanalys,programmering,aktuariematematik,operational model of pharmacological agonism,concentration-response curves,dose-response curves,ventilation/perfusion scintigraphy,V/Q-scan,pulmonary embolism,automated method,box counting,correlation dimension,Hentschel-Procaccia dimension,Rényi dimension,generalized dimensions,fractal dimension estimation,dimension spectrum,multifractal measures,kernel density estimates,absolute continuity,singular distribution functions},
  language     = {eng},
  pages        = {130},
  publisher    = {Centre for Mathematical Sciences, Lund University},
  school       = {Lund University},
  title        = {Topics in multifractal measures, nonparametrics and biostatistics},
  year         = {2004},
}