Date of Award


Degree Type


Degree Name

Doctor of Philosophy


A variety of physical and biological phenomena can be modeled by filtered Poisson processes, that is, by Poisson-driven stationary real-valued processes{dollar}{dollar}X(t) = \int\sbsp{lcub}0{rcub}{lcub}t{rcub}\ h(s,\tau;u)N(ds),\ t\ \in\ {lcub}\cal R{rcub},{dollar}{dollar}where {dollar}\tau{dollar} denotes an arbitrary occurrence point and u the associated mark of a marked Poisson process. In these models, that output or response process can be expressed as a sum of the individual responses to each marked occurrence point.;The present study is concerned with specific filtered Poisson process models of two biochemical spectroscopy imaging techniques, Scanning Fluorescence Correlation Spectroscopy (S-FCS) and its extension, Image Correlation Spectroscopy (ICS). These techniques are used in studying certain aspects of cell membrane physiology, specifically the rate or extent of membrane receptor protein clustering or aggregation. This phenomenon has been observed directly by electron microscopy. These measurements constitute an important step towards understanding how such receptors influence or interact with other cell membrane components.;The data obtained from S-FCS and ICS are measurements of fluorescence intensity as a function of position. The intensity process is modeled as{dollar}{dollar}X({lcub}\bf t{rcub}) = \int\sb{lcub}s\sb{lcub}\bf t{rcub}{rcub}\ f\sb{lcub}{lcub}\bf t{rcub}{rcub}(x,y)Y(x,y)N(dxdy) + \gamma\epsilon({lcub}\bf t{rcub}), {lcub}\bf t{rcub}\ \in\ {lcub}\cal Z{rcub}\sp{lcub}d{rcub}{dollar}{dollar}where d = 1 (S-FCS) or d = 2 (ICS),{dollar}{dollar}f\sb{lcub}{lcub}\bf t{rcub}{rcub}(x,y) = \exp(- {lcub}1\over w\sp2{rcub}((x - t\sb1\delta)\sp2 + (y - t\sb2\delta)\sp2)),{dollar}{dollar} enspace{dollar}S\sb{lcub}{lcub}\bf t{rcub}{rcub}{dollar} is a disc centred at {dollar}(t\sb1\delta, t\sb2\delta){dollar} of radius {dollar}{lcub}3w\over \surd 2{rcub}, Y\sb{lcub}(x,y){rcub}{dollar} is the nonnegative integer-valued random mark associated with the occurrence point (x,y) and {dollar}N(\cdot){dollar} is the spatial Poisson counting process. The term {dollar}\gamma\epsilon({lcub}\bf t{rcub}){dollar} is a Gaussian (background) noise process which represents experimental recording error. The observed data constitute a realization of a finite-range (or {dollar}m-{dollar}) dependent stationary process (S-FCS) or random field (ICS).;Certain second-order moment functionals of {dollar}X({lcub}\bf t{rcub}){dollar} contain important model parameters related to protein cluster sizes and densities. Specifically, in the absence of additive noise ({dollar}\gamma{dollar} = 0), the quantity{dollar}{dollar}R \alpha\ {lcub}1\over \bar N{rcub}{dollar}{dollar}where R = {dollar}Var(X({lcub}\bf t{rcub}))\over \mu\sbsp{lcub}X{rcub}{lcub}2{rcub}{dollar} and N is the average number of fluorescent protein clusters in the observed volume. The ratio R is a function of two other parameters, {dollar}\theta\sb1{dollar} = {dollar}\lambda{lcub}\mu\sb{lcub}2Y{rcub}{rcub}{dollar} and {dollar}\theta\sb1{dollar} = w where {dollar}\lambda{dollar} is the Poisson intensity parameter and {dollar}\mu\sb{lcub}2Y{rcub}{dollar} is the second moment of the marking distribution. The main purpose of this study is the consistent interval estimation of these parameters. Nonlinear least squares methods are proposed involving sample autocovariances (S-FCS) and periodograms (ICS). An asymptotic log likelihood procedure is also discussed for ICS.;The estimators are shown to obey central limit theorems as a consequence of the finite-range or mixing dependence of {dollar}\{lcub}X({lcub}\bf t{rcub})\{rcub}{dollar}. Thus, one is able to obtain approximate confidence intervals for important model parameters such as R provided consistent estimators of limiting variances are obtained. Much of this work is concerned with this variance estimation problem and an effective method is proposed.;An extensive Monte Carlo simulation study is undertaken to investigate the asymptotics of the least squares estimator {dollar}\ \theta{dollar} and the ratio estimator {dollar}\ R{dollar} and to implement a variance estimation method. Due to the computationally intensive nature of the ICS calculations, these are performed on a MasPar MP-2 massively parallel (2K) computer. Finally, several data sets are analyzed for both S-FCS and ICS.



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