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 G. Sensitivity to smoothness constraint. See Appendix H for references cited.

To understand how our results depend on the smoothness parameter , we re-fit the model in two different ways: with successive values of _t completely disengaged (i.e., no smoothness constraint), and with constant vital rates (i.e., the maximum smoothness constraint). The models were identical to the original model in all other respects. Forcing the vital rates to remain constant resulted in model failure, as the model was unable to accommodate the humped dynamics observed in several cutting cycles (Fig. G1). Completely disengaging successive values of _t did not result in any noticeable improvement in model fit relative to the original model. Marginal posterior distributions for _t were more diffuse without a smoothness constraint than with it (Fig. G2). (Marginal posteriors for _t when _t was forced to remain constant were apparently the most precise of all, though the lack of model fit renders the precision meaningless.) The increased posterior variance in _t with a weakened smoothness constraint is an example of a bias-variance trade-off. As the smoothness constraint is weakened, successive vital rates become less tightly correlated, and the marginal variance in each increases. Conversely, strengthening the smoothness constraint tightens the correlation between successive estimates, which decreases their variance but increases the bias.  

This sensitivity analysis indicates that for this model the general biological conclusions are somewhat robust to the choice of a smoothness constraint. However, there is no guarantee that this will hold in general (Knorr-Held 2000). We also tried to estimate the smoothness parameter directly, by placing a hyperprior distribution on the smoothness parameters for each vital rate. In this case, however, the data were always better fit by weakening the smoothness parameter (decreasing ), and the only way to prevent was to choose a hyperprior that penalized smaller values of . The resulting posterior distributions for were essentially determined by the strength of this penalty in the hyperprior. Consequently fitting from the data resulted in an assumption replacement –– an arbitrary assumption about the value of was replaced by an arbitrary assumption about the strength of the penalty against smaller values of in the hyperprior. Because this resulted in little gain, but incurred the cost of an added layer of complexity, we left fixed at its arbitrarily chosen value.

FIG. G1.  90% posterior predictive intervals, and data, for the baseline model (top row), a model without a smoothness constraint on t (middle row), and a model with constant t for each cutting cycle (bottom row). Note that the model with constant t is unable to accommodate the humped aphid dynamics observed in cutting cycles A1, A3, B1, C1, and C3.

FIG. G2.  Marginal highest posterior density intervals for t for cutting cycle A1. Left column: t for the baseline model. Middle column: t for the model without a smoothness constraint. Right column: t for model where t is forced to remain constant over the entire cutting cycle. For the left and middle columns, a line connects posterior medians for each value of t.