Estimating uncertainty in density surface models

Abstract

Providing uncertainty estimates for predictions derived from species distribution models is essential for management but there is little guidance on potential sources of uncertainty in predictions and how best to combine these. Here we show where uncertainty can arise in density surface models (a multi-stage spatial modelling approach for distance sampling data), focussing on cetacean density modelling. We propose an extensible, modular, hybrid analytical-simulation approach to encapsulate these sources. We provide example analyses of fin whales Balaenoptera physalus in the California Current Ecosystem.

Keywords: Density surface models; Distance sampling; Environmental uncertainty; Model uncertainty; Spatial modelling; Species distribution modelling; Uncertainty quantification.

Figure 1. Flow diagram showing our process for capturing uncertainty from multiple model components.

In this case we have a GAM model of spatial counts, a detection function and environmental variability to include in the model. Once detection function uncertainty is included in the GAM, some number (n_b) posterior samples of model parameters (${\hat{\beta }}_b$) can be generated using the GAM-plus-other-components mean ($\hat{\beta }$) and covariance matrix (${\mathbf{V}}_{\hat{\beta },\hat{\lambda },\hat{\theta }}$ here). These samples are used to generate potential density surfaces exploring both model parameter and environmental space. Finally, summaries can be calculated from these predictions.

Figure 2. Raw data and results from applying our procedure to the fin whale data.

Top row, left: locations of segment centroids (black dots) for the fin whale survey with observation locations (orange dots) and underlying bathymetry. Top row, middle: predicted density for fin whales in the California Current Ecosystem. Top row, right: estimated standard error for the predictions, using the procedure we outline. Black dots give locations of observations of fin whales in the centre and right plots. Bottom row: confidence surfaces derived from a log-normal approximation; left is lower 2.5%, right is upper 97.5%.

Figure 3. Time series of monthly abundance estimates for fin whales in the California Current Ecosystem.

Dots indicate mean abundance estimates over the days/iterations and lines 95% intervals based on a log-normal approximation.

Figure 4. Comparison of published yearly estimates of fin whale abundance in the California Current Ecosystem.

Dots show abundance estimates and lines their 95% intervals.