This dissertation deals with point process models and their applications to imaging and messenger RNA (mRNA) transcription. We address three problems. The first problem arises in two-photon laser scanning microscopy. We model the process by which photons are counted by a detector which suffers from a dead period upon registration of a photon. In this model, we assume that there are a Poisson (α) number of excited molecules, with exponentially distributed waiting times for the emissions of photons. We derive the exact distribution of all observed counts, rather than grouped counts which were used earlier. We use it to get improved estimates of the Poisson intensity, which leads to images with higher signal-to-noise ratio. This improvement is because grouping of count data results in loss of information. We illustrate this improvement on imaging data of paper fibers. Next, we study two variants of this model: the first uses a finite time horizon and the second considers gamma waiting times for the emissions.
The second problem concerns the Conway-Maxwell-Poisson distribution for count data. This family has been proposed as a generalization of the Poisson for handling overdispersion and underdisperson. Because the normalizing constant of this family is hard to compute, good approximations for it are needed. We provide a statistical approach to derive an existing approximation more simply. However, this approximation does not perform well across all the parameter ranges. Therefore, we introduce correction terms to improve its performance. For other parts of the parameter space, we use the geometric and Bernoulli distributions, with correction terms based on Taylor expansions. Using numerical examples, we show that our approximations are much better than earlier proposed methods.
In the last problem, we present a new application for Conway-Maxwell-Poisson family.
We use the generalized linear model setting of this family to study mRNA counts. We then compare its performance with the existing methods used for modeling mRNAs, such as the negative binomial. This empirical model can be a good modeling tool for dispersed mRNA count data when a biophysically based model is not available
