Bayesian calibration, comparison and averaging of six forest models, using data from Scots pine stands across Europe
Abstract
Forest management requires prediction of forest growth, but there is no general agreement about which models best predict growth, how to quantify model parameters, and how to assess the uncertainty of model predictions. In this paper, we show how Bayesian calibration (BC), Bayesian model comparison (BMC) and Bayesian model averaging (BMA) can help address these issues. We used six models, ranging from simple parameter-sparse models to complex process-based models: 3PG, 4C, ANAFORE, BASFOR, BRIDGING and FORMIND. For each model, the initial degree of uncertainty about parameter values was expressed in a prior probability distribution. Inventory data for Scots pine on tree height and diameter, with estimates of measurement uncertainty, were assembled for twelve sites, from four countries: Austria, Belgium, Estonia and Finland. From each country, we used data from two sites of the National Forest Inventories (NFIs), and one Permanent Sample Plot (PSP). The models were calibrated using the NFI-data and tested against the PSP-data. Calibration was done both per country and for all countries simultaneously, thus yielding country-specific and generic parameter distributions. We assessed model performance by sampling from prior and posterior distributions and comparing the growth predictions of these samples to the observations at the PSPs. We found that BC reduced uncertainties strongly in all but the most complex model. Surprisingly, country-specific BC did not lead to clearly better within-country predictions than generic BC. BMC identified the BRIDGING model, which is of intermediate complexity, as the most plausible model before calibration, with 4C taking its place after calibration. In this BMC, model plausibility was quantified as the relative probability of a model being correct given the information in the PSP-data. We discuss how the method of model initialisation affects model performance. Finally, we show how BMA affords a robust way of predicting forest growth that accounts for both parametric and model structural uncertainty.
What the paper shows and why it matters
(AI-generated)
Forest management needs predictions of stand growth, but there is no agreement on which dynamic growth models predict best, how to set their parameters, or how to judge the uncertainty of their forecasts. The models range from simple, parameter-sparse ones to complex process-based ones, and the paper notes that none has come into general use across Europe, possibly because they are hard to parameterise and their transferability is unclear. Bayesian calibration of parameter-rich process-based models was still rare, and Bayesian model comparison had been applied to four forest biogeochemistry models (van Oijen et al. 2011). The paper asks whether one Bayesian framework can calibrate, rank and combine forest models of very different structure.
The paper argues that Bayesian calibration, model comparison and model averaging, used together, offer a general way to fit, rank and combine forest growth models of very different structure while tracking parameter and structural uncertainty. Calibration narrows prior parameter ranges with data, comparison gives each model a probability of being correct given the observations, and averaging weights forecasts by those probabilities. The study tested this on six models with Scots pine data from twelve sites in four European countries, calibrating on two sites per country and testing on a third. Calibration reduced uncertainty in all but the most complex model, mostly through a few parameters. The comparison ranked BRIDGING, of intermediate complexity, first before calibration and 4C after it; the paper suggests that models started from measured height and diameter had an advantage over those started at planting. Averaging with weights from the comparison before calibration predicted out of sample robustly; the average after calibration was only average, partly in-sample.
The paper concludes that model comparison can identify plausible models and that averaging may be a promising tool for forest management, as it is less prone than picking a single model to underestimate predictive uncertainty. For a manager who wants a single model from these six it recommends 4C, but reads the probabilities as a guide, not as formal model probabilities, since the comparison presumes one model is correct. It notes that rankings might differ with more data or other outputs than height and diameter. Trotsiuk et al. (2020) cite the paper as an earlier large-dataset Bayesian calibration of a forest model.
References
- van Oijen et al. (2011). A Bayesian framework for model calibration, comparison and analysis: Application to four models for the biogeochemistry of a Norway spruce forest. Agricultural and Forest Meteorology 151, 1609–1621. doi:10.1016/j.agrformet.2011.06.017
- Trotsiuk et al. (2020). Assessing the response of forest productivity to climate extremes in Switzerland using model–data fusion. Global Change Biology 26, 2463–2476. doi:10.1111/gcb.15011