University of Washington - Department of Statistics
Advisor: Peter Guttorp
We consider a special case of the two-dimensional stochastic n-compartment or stepping-stone model. This class of models represents a special type of Markov population process in which the state of the process a given time t is represented by (X1(t),X2(t)) = (X11(t), ... ,X1n(t);X21(t), ... , X2n(t)). In our example, all compartments except the nth are assumed completely unobservable, while in the nth we are able to obtain only a sample of each dimension at discrete time intervals. The first compartment is a hidden linear birth-emigration process. In each of the other k (k > 1; k <= n) compartments we assume only migration to the (k + 1)th compartment. In particular, we apply the two-compartment model to the process of hematopoiesis in large animals.
A recursive updating algorithm is used to compute the likelihood of the hidden process for the one compartment model. The other n-1 compartments are inhomogeneous immigration-death processes for which the likelihood cannot be evaluated by exact methods. We can either use an integral approximation in the inhomogeneous immigration-death transition probabilities in conjunction with the updating algorithm, or use a Markov Chain Monte Carlo approach to evaluate the likelihood conditioned on a fully observed process.
A more realistic model for the hematopoietic process includes an upper bound on the total size of the first compartment, creating a competitive process. This makes even moments difficult to compute. We have studied this model extensively through the use of simulations and have obtained some parameter range estimates. Although the inhomogeneous immigration-death equations cannot be solved to obtain explicit transition probabilities for this case, Markov Chain Monte Carlo methods should still be effective in computing the likelihood. Questions of identifiability in the various forms of the model are also raised.