The importance of observation versus process error in analyses of global ungulate populations

Abstract

Population abundance data vary widely in quality and are rarely accurate. The two main components of error in such data are observation and process error. We used Bayesian state space models to estimate the observation and process error in time-series of 55 globally distributed populations of two species, Cervus elaphus (elk/red deer) and Rangifer tarandus (caribou/reindeer). We examined variation among populations and species in the magnitude of estimates of error components and density dependence using generalized linear models. Process error exceeded observation error in 75% of all populations, and on average, both components of error were greater in Rangifer than in Cervus populations. Observation error differed significantly across the different observation methods, and predation and time-series length differentially affected the error components. Comparing the Bayesian model results to traditional models that do not separate error components revealed the potential for misleading inferences about sources of variation in population dynamics.

Figure 1. Location map (generated using ArcGIS) of 27 globally distributed Cervus populations and time series of a subset of eight populations.

For each of the eight populations: the solid blue line is the empirical time-series data, the dashed black lines are the upper and lower Highest Posterior Density (HPD) estimates by a first-order autoregressive state-space model, and the yellow and green bars are observation (O error) and process error (P error) estimated by the state-space models. The secondary right y-axis is for the observation and process error bar graphs, and is the same scale for all figures, enabling direct inter-population comparisons. The populations are (starting with the top-left figure and moving clockwise): Northern Range, Yellowstone, USA; Sikhote-Alin Zapovednik, Russia; Population 4, Norway; Bialowieza Primeval Forest, Poland; Petite Pierre National Reserve, France; Isle of Rum, Scotland; Ya Ha Tinda Cervus herd, Banff National Park, Canada; Point Reyes, USA; Northern Range, Yellowstone, USA.

Figure 2. Location map (generated using ArcGIS) of 28 globally distributed Rangifer populations and time series of a subset of eight populations.

For each of the eight populations: the solid red line is the empirical time-series data, the dashed black lines are the upper and lower Highest Posterior Density (HPD) estimates by a first-order autoregressive state-space model, and the yellow and green bars are observation (O error) and process error (P error) estimated by the state-space models. The secondary right y-axis is for the observation and process error bar graphs, and is the same scale for all figures, enabling direct inter-population comparisons. The populations are (starting with the top-left figure and moving clockwise): Denali National Park, Alaska; Krasnoy, Russia; Tyumen, Russia; Alakyla, Finland; Palojarvi, Finland; Upernavik, Greenland; Manitsoq, Greenland; Nelchina, Alaska, USA.

Figure 3. Mean (±SE) of (a) Bayesian state-space model estimates of observation and process error, and ARIMA model estimates of error, and (b) Bayesian state-space and ARIMA modelling estimates of statistical direct density dependence in the detrended time series of 55 globally distributed Cervus (n = 27) and Rangifer (n = 28) populations.

Figure 4. Mean (±SE) of Bayesian state-space model estimates of observation error found in the detrended time series of 55 globally distributed Cervus (n = 27) and Rangifer (n = 28) populations using different survey methods: aerial counts; ground counts, which included drive counts, line counts, capture-resight counts, road counts and horseback counts; harvest counts; and, snow-track counts.

Figure 5. Generalized linear model (GLM) generated relationships between (a) Bayesian estimates of observation error, (b) Bayesian estimates of process error, and (c) ARIMA estimates of error and the time series length, respectively.

The relationships (95% CI = grey shaded region) were calculated while keeping other explanatory variables constant in the respective GLMs.