Mass-Balanced Multiscale Models on Trees

In many multiscale problems, the relationship between the different levels of the multiscale process is deterministic. For example, the total volume of petroleum in a given region at a coarser level is the sum of the volumes of petroleum in the corresponding finer regions. This type of property is known as balance of mass. In this chapter, we revise the tree modeling approach of Kolaczyk and Huang (2001) for the analysis of multiscale processes subject to mass balance. Their approach is strongly based on the approach of Allan S. Willsky and his coauthors, presented in Chapter 7, but generalized to cases where the number of descendants is not constant and to include the cases of Gaussian and Poisson observations. The case of Poisson observations is a generalization of the work of Kolaczyk (1999) for multiscale Poisson processes on dyadic trees. As in Willsky’s tree model, the multiscale model of Kolaczyk and Huang induces rich covariance structures whose analytic expressions were derived by Louie and Kolaczyk (2004).