Statistical inverse problems for population processes

Population processes are stochastic processes that record the dynamics of the number of individuals in a population, and have many different applications in a broad range of areas. Population processes are often modelled as Markov processes, and have the important feature that transitions correspond either to an increase or a decrease in the population size. These two types of transitions are often referred to as births and deaths. A specific class of population processes is the class of birth-death processes, where transitions can only increase or decrease the population by one at a time. In many real-life situations the dynamics of a population is affected by exogenous, often unobservable, factors. Therefore, this thesis considers population processes of which the parameters are affected by an underlying stochastic process, referred to as the background process. The aim is to find reliable inference techniques to estimate the parameters, including those related to the background process, from discrete-time observations of the population size. The statistical inference is complicated severely by the fact that a substantial part of the process is unobserved. First, the underlying background process is not observed. Second, only the population size is observed, which is the net effect of all the transitions in the dynamics of the population. Last, the population size is observed in discrete time, hence the transitions in between two consecutive observations are not observed. In this thesis we show a collection of techniques to overcome these complications for a variety of population processes. The aspects in which the models differ, ask for specific inference techniques. For a certain class of Markov-modulated population processes, we show how the well-known EM algorithm can be used to estimate the model parameters. In these models, the background process is a finite, continuous-time Markov chain and the parameters of the population process switch between distinct values at the jump times of this Markov chain. An algorithm is presented that iteratively maximizes the likelihood function and at the same time updates the parameter estimates. A generalization of the conventional birth-death process, involving a background process, is the quasi birth-death process. We use the Erlangization technique to evaluate the likelihood function for this kind of processes, which can then be maximized numerically to obtain maximum likelihood estimates. A specific model in the class of quasi birth-death processes is a birth-death process of which the births follow a hypoexponential distribution with L phases and are controlled by an on/off mechanism. We call this the on/off-seq-L process, and use it to model the dynamics of populations of mRNA molecules in single living cells. Numerical complications related to the likelihood maximization are analyzed and solutions are presented. Based on real-life data, we illustrate the estimation method, and perform a model selection procedure on the number of phases and on the on/off mechanism. Last, we consider a class of discrete-time multivariate population processes under Markov-modulation. In these models, the population process is defined on a network with finitely many nodes. In addition to the births and deaths that can occur at each of the nodes, the individuals follow a probabilistic route through the network. We introduce the saddlepoint technique and show how it can be used to evaluate the likelihood function based on observations of the network population vector. The likelihood function can again be maximized numerically to obtain maximum likelihood estimates. Throughout the thesis, the accuracy of the inference methods is investigated by extensive simulation studies.