Spatial statistics, narrowly defined, is the study of stochastic processes in Euclidean spaces, like the locations of trees in a forest or patches of land and water in a satellite image. Broadly defined, it covers any stochastic process with dependence more complicated than a time series, like the inheritance of genetic traits over generations of a family tree or like the network of social interactions in a community.
Some of my work in this area deals with two endangered species: the Asian wild horse and the California condor. Both are extinct in the wild and are now bred in captivity so that wild populations can be established again when suitable habitat can be provided and the population size is large enough. Good management of the breeding program, avoiding inbreeding and loss of rare genes, requires statistical genetics, because factors such as the relatedness of wild-caught founders of the population and the fraction of each founder's genes that have passed down to the current population cannot be directly observed, only estimated.
These problems, like many others in spatial statistics, involve massive calculation because there are no exact formulas for the things we want to calculate. The best computational methods for such problems, the so-called "bootstrap" and "Monte Carlo" methods, simulate the stochastic process of interest in the computer, and do statistics on the simulations to get the desired answers.
Some of my work in this area deals with maximum likelihood estimation in hard problems where the likelihood cannot be calculated exactly and where there are inequality constraints on the parameters. This work allows us to use likelihood methods for problems that were considered unapproachable just a few years ago and has interesting connections with time series, Markov chains, asymptotics, and optimization theory.
- constrained maximum likelihood
- Markov chain Monte Carlo
- Monte Carlo likelihood
- spatial statistics