Population time series: Process variability, observation errors, missing values, lags, and hidden states

Population sample data are complex; inference and prediction require proper accommodation of not only the nonlinear interactions that determine the expected future abundance, but also the stochasticity inherent in data and variable (often unobserved) environmental factors. Moreover, censuses may occur sporadically, and observation errors change with sample methods and effort. The state variable (usually density or abundance) may be hidden from view and known only through highly indirect observational schemes (such as public health records, hunting reports, or fossil/archeological surveys). We extend the basic state-space model for time-series analysis to accommodate these dominant sources of variability that influence population data. Using examples, we show how different types of process error and observation error, unequal sample intervals, and missing values can be accounted for within the flexible framework of Bayesian state-space models. We provide algorithms based on Gibbs sampling that can be used to obtain posterior estimates of population states and of model parameters. For models that can be linearized, results can be used for direct sampling of the posterior, including those with missing values and unequal sample intervals. For nonlinear models, we make use of Metropolis-Hastings within the Gibbs sampling framework. Examples derive from long-term census and population data. We illustrate the extension to discrete state variables with multiple stages using a Time-series Susceptible-Infected-Recovered (TSIR) model for mid 20th-century measles infection in London, where birth rates are assumed known, but susceptibles and infected individuals arise from imperfect reporting.