Compound Markov counting processes and their applications to modeling infinitesimally over-dispersed systems

We propose an infinitesimal dispersion index for Markov counting processes. We show that, under standard moment existence conditions, a process is infinitesimally (over-) equi-dispersed if, and only if, it is simple (compound), i.e. it increases in jumps of one (or more) unit(s), even though infinitesimally equi-dispersed processes might be under-, equi- or over-dispersed using previously studied indices. Compound processes arise, for example, when introducing continuous-time white noise to the rates of simple processes resulting in Lévy-driven SDEs. We construct multivariate infinitesimally over dispersed compartment models and queuing networks, suitable for applications where moment constraints inherent to simple processes do not hold.Continuous time, Counting Markov process, Birth-death process, Environmental stochasticity, Infinitesimal over-dispersion, Simultaneous events

Statistical Inference for Partially Observed Markov Processes via the R Package pomp

Partially observed Markov process (POMP) models, also known as hidden Markov models or state space models, are ubiquitous tools for time series analysis. The R package pomp provides a very flexible framework for Monte Carlo statistical investigations using nonlinear, non-Gaussian POMP models. A range of modern statistical methods for POMP models have been implemented in this framework including sequential Monte Carlo, iterated filtering, particle Markov chain Monte Carlo, approximate Bayesian computation, maximum synthetic likelihood estimation, nonlinear forecasting, and trajectory matching. In this paper, we demonstrate the application of these methodologies using some simple toy problems. We also illustrate the specification of more complex POMP models, using a nonlinear epidemiological model with a discrete population, seasonality, and extra-demographic stochasticity. We discuss the specification of user-defined models and the development of additional methods within the programming environment provided by pomp.Comment: In press at the Journal of Statistical Software. A version of this paper is provided at the pomp package website: http://kingaa.github.io/pom

Systemic Infinitesimal Over-dispersion on General Stochastic Graphical Models

Stochastic models of interacting populations have crucial roles in scientific fields such as epidemiology and ecology, yet the standard approach to extending an ordinary differential equation model to a Markov chain does not have sufficient flexibility in the mean-variance relationship to match data (e.g. \cite{bjornstad2001noisy}). A previous theory on time-homogeneous dynamics over a single arrow by \cite{breto2011compound} showed how gamma white noise could be used to construct certain over-dispersed Markov chains, leading to widely used models (e.g. \cite{breto2009time,he2010plug}). In this paper, we define systemic infinitesimal over-dispersion, developing theory and methodology for general time-inhomogeneous stochastic graphical models. Our approach, based on Dirichlet noise, leads to a new class of Markov models over general direct graphs. It is compatible with modern likelihood-based inference methodologies (e.g. \cite{ionides2006inference,ionides2015inference,king2008inapparent}) and therefore we can assess how well the new models fit data. We demonstrate our methodology on a widely analyzed measles dataset, adding Dirichlet noise to a classical SEIR (Susceptible-Exposed-Infected-Recovered) model. We find that the proposed methodology has higher log-likelihood than the gamma white noise approach, and the resulting parameter estimations provide new insights into the over-dispersion of this biological system.Comment: 47 page

An iterated block particle filter for inference on coupled dynamic systems with shared and unit-specific parameters

We consider inference for a collection of partially observed, stochastic, interacting, nonlinear dynamic processes. Each process is identified with a label called its unit, and our primary motivation arises in biological metapopulation systems where a unit corresponds to a spatially distinct sub-population. Metapopulation systems are characterized by strong dependence through time within a single unit and relatively weak interactions between units, and these properties make block particle filters an effective tool for simulation-based likelihood evaluation. Iterated filtering algorithms can facilitate likelihood maximization for simulation-based filters. We introduce an iterated block particle filter applicable when parameters are unit-specific or shared between units. We demonstrate this algorithm by performing inference on a coupled epidemiological model describing spatiotemporal measles case report data for twenty towns

Macroeconomic effects on mortality revealed by panel analysis with nonlinear trends

Many investigations have used panel methods to study the relationships between fluctuations in economic activity and mortality. A broad consensus has emerged on the overall procyclical nature of mortality: perhaps counter-intuitively, mortality typically rises above its trend during expansions. This consensus has been tarnished by inconsistent reports on the specific age groups and mortality causes involved. We show that these inconsistencies result, in part, from the trend specifications used in previous panel models. Standard econometric panel analysis involves fitting regression models using ordinary least squares, employing standard errors which are robust to temporal autocorrelation. The model specifications include a fixed effect, and possibly a linear trend, for each time series in the panel. We propose alternative methodology based on nonlinear detrending. Applying our methodology on data for the 50 US states from 1980 to 2006, we obtain more precise and consistent results than previous studies. We find procyclical mortality in all age groups. We find clear procyclical mortality due to respiratory disease and traffic injuries. Predominantly procyclical cardiovascular disease mortality and countercyclical suicide are subject to substantial state-to-state variation. Neither cancer nor homicide have significant macroeconomic association.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS624 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org