State-space models' dirty little secrets: even simple linear Gaussian models can have estimation problems

Abstract

State-space models (SSMs) are increasingly used in ecology to model time-series such as animal movement paths and population dynamics. This type of hierarchical model is often structured to account for two levels of variability: biological stochasticity and measurement error. SSMs are flexible. They can model linear and nonlinear processes using a variety of statistical distributions. Recent ecological SSMs are often complex, with a large number of parameters to estimate. Through a simulation study, we show that even simple linear Gaussian SSMs can suffer from parameter- and state-estimation problems. We demonstrate that these problems occur primarily when measurement error is larger than biological stochasticity, the condition that often drives ecologists to use SSMs. Using an animal movement example, we show how these estimation problems can affect ecological inference. Biased parameter estimates of a SSM describing the movement of polar bears (Ursus maritimus) result in overestimating their energy expenditure. We suggest potential solutions, but show that it often remains difficult to estimate parameters. While SSMs are powerful tools, they can give misleading results and we urge ecologists to assess whether the parameters can be estimated accurately before drawing ecological conclusions from their results.

Figure 1. Changes in parameter estimates and state RMSEs associated with varying the measurement error to process stochasticity ratios (σ_ε/σ_η) in the simulations.

(A–C) The boxplots represent the distribution of the parameter estimates (, , ) and the pink circles represent the true (simulated) values. (D) The grey boxplots represent the distribution of the RMSE of the model fitted using the estimated parameter values, while the pink boxplots represent the RMSE when the model is fitted using the true parameter values.

Figure 2. Log likelihood profiles for problematic simulations.

In the first three columns, the curve represents the log likelihood when the focal parameter is fixed (the other parameters are optimise to maximise the log likelihood). The dash lines are the true parameter values (i.e., value used for the simulation), the full lines are the maximum likelihood estimates and the grey bands represent the 95% CI. The last column shows the time-series. The black lines represent the observations, y_t, the red lines the simulated true states, x_t, and the grey dashed lines the estimated states, .

Figure 3. Polar bear movement, parameter estimates of the polar bear sea ice model, and estimates of the total voluntary bear displacement.

(A) Locations of the 15 polar bears used in the analysis, with colours representing different individuals. The map was created in R using the Northern Hemisphere azimuthal equal-area EASE-Grid projection developed for polar sea ice data. (B) Estimated total voluntary displacement over 342 days. (C–H) Parameter estimates of the polar bear sea ice models. The different colours in panels (B,C) represent the three individuals for which either or .