Research Scientist, NMFS
Affiliate faculty, SAFS, University of Washington
National Marine Fisheries Service
Northwest Fisheries Science Center
2745 Montlake Blvd E
Seattle, WA 98112
My research since my graduate student days has been focused on parsimony in population models: what does it mean, how to define it, and how to cross-validate it. What do I mean by 'parsimony in population models'? Basically, I mean how simple of a model can one use for a particular purpose. For example, if you have a spatial question then you need a spatial model, but what if your question is non-spatial then when can you use a non-spatial model to approximate the spatial process?
My initial graduate work was on spatial population dynamics and specifically how spatial interaction fundamentally changes population dynamics and to what extent does it not fundamentally change populations dynamics. If it does fundamentally change dynamics, then how simple can that spatial model be? At what point can a uniform mixing model approximate the properties of a non-uniform mixing process? The papers "Is diffusion too simple?", "Basic epidemiological concepts in a spatial context", and "Running from trouble" were all exploring these ideas. I reviewed some of these ideas in "Partial differential equation models in ecology". Later I taught a graduate course on spatial dynamics which tried to bring all these ideas together. My post-graduate work was focused on an experimental study of metapopulation dynamics using mosquitoes. This was a huge perturbation experiment to test some ideas about how metapopulation dynamics should change in response to variability in the environment.
Since 1999, I have been working with the National Marine Fisheries Service. I currently work on stochastic population dynamics. I am trying formulate hypotheses about parsimonious descriptions for population processes. I think about general canonical forms for the statistical distributions that population monitoring data should come from. I think about it as, 'does there exist a f(data) ~ g(theta)' under broad conditions -- that is under plausible population processes. Finding such an f and proving that it is distributed with the canonical form g, is primarily what concerns me. Why do I think about such esoteric things? Because I think that the process that generates my data on an endangered species is unknowable and unestimateable in my lifetime, and that the data, in and of themselves, will give me little information about the underlying process.
There are a few reasonable approaches to this problem, as I saw it in January 2006. One approach was to be a pure Bayesian, develop a complicated model, fit that to the data, and use some type of model averaging across the posterior. Bayesians argue that since you model average across the posterior this approach does not suffer from the data-mining problem faced in frequentist statistics when the number of parameters greatly outnumbers the number of data points. My understanding of Bayesian statistics is not sufficiently deep to evaluate that argument to my satisfaction. At the same time, I'm alarmed at the degree of data-mining/model-fudging that goes on in typical model-building. Bayesian approaches seem to be an improvement on that. The problem is that a) I have a hard time accepting vague priors especially in complex models with a gazillion parameters and b) we know a lot about population processes, and we should be able to come up with theoretically motivated informative priors.
My 2007 paper, "A statistical approach to quasi-extinction forecasting", was the culmination of two years of pondering on this problem. Having completed that however I was reminded of the quote that opens Bernt Oksendal's book Stochastic differential equations: "We have not succeeded in answering all our problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things."
After 2007, I became interested in multivariate stochastic processes, particularly interacting systems and the statistical properties of those. I had read Ives et al.'s 2003 paper on using multivariate autoregressive models to understand community dynamics and was really taken with the mathematical ideas. I was working on univariate models with process and measurement error at the time and wanted to work on the multivariate version of these. Eric Ward and I submitted a paper on estimation and model selection for these models. It was rejected for not being full-developed (which was true) and in the process of revising, I decided to fully study the statistical properties of AIC and bootstrap-AIC for these models. This required coding up a constrained Expectation-Maximization algorithm because I needed to do 10,000x of bootstraps and Newton-methods were crashing too much.
Later my colleague Mark Scheuerell wanted to use the algorithm for something different, something which my algorithm wasn't set up to do. Our conversation got me thinking and as I waited at a stop-light on my bike ride home, the outline for a general constrained EM algorithm for time-varying multivariate autoregressive state-space models came to me. It took me the next five years to fully develop, code and test the algorithm (and led to the report Holmes 2012 on the constrained EM algorithm). During that time, I, Mark Scheuerell and colleagues at NCEAS got a NOAA CAMEO grant, which among other things, funded the version 1.0 of the MARSS R package (Holmes, Ward, Wills 2012). In 2012 and looking forward for the next couple years, my time is still largely focused on the MARSS package and algorithm. There is one big huddle that I see left, extension to MARMA state-space models, which I know how to do, but alas as always, the distance between the idea in your head and a finished product (be it a publication or stats package) is long and tedious and in the meantime I have become fascinated by estimating interaction matrices for sparsely interacting systems...