Kevin Gross, Anthony R. Ives, and Erik V. Nordheim. 2005. Estimating fluctuating vital rates from time-series data: a case study of aphid biocontrol. Ecology 86:740–752.

Appendix A. Model factorization. See Appendix H for references cited.

Let X_t denote the state of the system at time t, and let Y_t denote data collected at time t, t=1,...,T.  Let superscripts denote a sequence (i.e., X^T = X₁, …, X_T).  Let _t denote time-varying vital rates, let  denote process-model parameters that are constant over time and for all cutting cycles, and let denote parameters in the process model.  Finally, is the parameter that sets the smoothness constraint.  For a single cutting cycle, the joint probability distribution of all parameters and data factors as

A directed acyclic graph for this model is shown in Fig. A1.

FIG. A1.  Directed acyclic graph for the model. The series X1, …, XT  is the trajectory of the hidden state, and the series Y1, …, YT is the data. Solid lines denote stochastic relationships, dotted lines denote deterministic relationships, boxes indicate observed data, and circles indicate model quantities (after Spiegelhalter et al. 1996).