Current Forestry Reports

, Volume 3, Issue 4, pp 269–280 | Cite as

Bayesian Methods for Quantifying and Reducing Uncertainty and Error in Forest Models

  • Marcel van OijenEmail author
Modelling Productivity and Function (M Battaglia, Section Editor)
Part of the following topical collections:
  1. Topical Collection on Modelling Productivity and Function


Purpose of review

Forest models are tools for analysis and prediction of productivity and other services. Model outputs can only be useful if possible errors in inputs and model structure are recognized. However, errors cannot be quantified directly, making uncertainty inevitable. In this paper, we aim to clarify terminological confusion around the concepts of error and uncertainty and review current methods for addressing uncertainty in forest modelling.

Recent findings

Modellers increasingly recognize that all uncertainties—in data, model inputs and model structure—can be represented using probability distributions. This has stimulated the use of Bayesian methods for quantifying and reducing uncertainty and error in models of forests and other vegetation. The Achilles’ heel of Bayesian methods has always been their computational demand, but solutions are being found.


We conclude that future work will likely include (1) more use of Bayesian methods, (2) more use of hierarchical modelling, (3) replacement of model spin-up by Bayesian calibration, (4) more use of ensemble modelling and Bayesian model averaging, (5) new ways to account for model structural error in calibration, (6) better software for Bayesian calibration of complex models, (7) faster Markov chain Monte Carlo algorithms, (8) more use of model emulators, (9) novel uncertainty visualization techniques, (10) more use of graphical modelling and (11) more use of risk analysis.


Calibration Discrepancy Ensemble modelling Hierarchical modelling Risk analysis Visualization 



I thank Peter Levy (CEH-Edinburgh) for his comments on this paper.

Compliance with Ethical Standards

Conflict of Interest

Dr. Van Oijen states that there are no conflicts of interests to declare.

Human and Animal Rights

This article does not contain any studies with human or animal subjects performed by any of the authors.


Papers of particular interest, published recently, have been highlighted as: • Of importance •• Of major importance

  1. 1.
    Mäkelä A, del Río M, Hynynen J, Hawkins MJ, Reyer C, Soares P, et al. Using stand-scale forest models for estimating indicators of sustainable forest management. For Ecol Manag. 2012;285:164–78.CrossRefGoogle Scholar
  2. 2.
    Hickler T, Rammig A, Werner C. Modelling CO2 impacts on forest productivity. Current Forestry Reports. 2015;1:69–80.CrossRefGoogle Scholar
  3. 3.
    • Reyer C. Forest productivity under environmental change—a review of stand-scale modeling studies. Current Forestry Reports. 2015;1:53–68. This paper is useful for forest modellers aiming to provide global assessments of the impacts of environmental change: it identifies those forest types and parts of the globe for which data are at present lacking. CrossRefGoogle Scholar
  4. 4.
    Fontes L, Bontemps J-D, Bugmann H, Van Oijen M, Gracia C, Kramer K, et al. Models for supporting forest management in a changing environment. Forest Systems. 2010;3:8–29.CrossRefGoogle Scholar
  5. 5.
    Hartig F, Dyke J, Hickler T, Higgins SI, O’Hara RB, Scheiter S, et al. Connecting dynamic vegetation models to data—an inverse perspective. J Biogeogr. 2012;39:2240–52.CrossRefGoogle Scholar
  6. 6.
    Jaynes ET. Probability theory: The logic of science. Cambridge: Cambridge University Press; 2003.Google Scholar
  7. 7.
    Sivia D, Skilling J. Data analysis: a Bayesian tutorial. 2nd ed. Oxford: Oxford University Press, U.S.A; 2006.Google Scholar
  8. 8.
    Ogle K, Barber JJ. Bayesian data in plant physiological and ecosystem ecology. In: Lüttge U, Beyschlag W, Murata J, editors. Progress in botany. Springer: Berlin; 2008. p. 281–311.CrossRefGoogle Scholar
  9. 9.
    Van Oijen M, Cameron D, Butterbach-Bahl K, Farahbakhshazad N, Jansson P-E, Kiese R, et al. A Bayesian framework for model calibration, comparison and analysis: application to four models for the biogeochemistry of a Norway spruce forest. Agric For Meteorol. 2011;151:1609–21.CrossRefGoogle Scholar
  10. 10.
    • Van Oijen M, Cameron D, Levy PE, Preston R. Correcting errors from spatial upscaling of nonlinear greenhouse gas flux models. Environ Model Softw. 2017;94:157–65. This paper reviews and revives the powerful yet neglected idea that errors caused by spatial aggregation of the inputs of nonlinear models can be estimated well using a Taylor-expansion of the model, together with information on spatial variability of the inputs. CrossRefGoogle Scholar
  11. 11.
    Cameron DR, Van Oijen M, Werner C, Butterbach-Bahl K, Grote R, Haas E, et al. Environmental change impacts on the C- and N-cycle of European forests: a model comparison study. Biogeosciences. 2013;10:1751–73.CrossRefGoogle Scholar
  12. 12.
    Van Oijen M, Ågren GI, Chertov O, Kellomäki S, Komarov A, Mobbs D, et al. Evaluation of past and future changes in European forest growth by means of four process-based models. In: Kahle H-P, Karjalainen T, Schuck A, Ågren GI, Kellomäki S, Mellert K, Prietzel J, Rehfuess K-E, Spiecker H, editors. Causes and consequences of forest growth trends in Europe: results of the recognition project. Leiden: Brill; 2008. p. 183–99.Google Scholar
  13. 13.
    Van Oijen M, Reyer C, Bohn F, Cameron D, Deckmyn G, Flechsig M, et al. Bayesian calibration, comparison and averaging of six forest models, using data from Scots pine stands across Europe. For Ecol Manag. 2013;289:255–68.CrossRefGoogle Scholar
  14. 14.
    Levy PE, Wendler R, Van Oijen M, Cannell MG, Millard P. The effect of nitrogen enrichment on the carbon sink in coniferous forests: uncertainty and sensitivity analyses of three ecosystem models. Water, Air, Soil Pollution: Focus. 2005;4:67–74.CrossRefGoogle Scholar
  15. 15.
    Van Oijen M, Thomson A. Toward Bayesian uncertainty quantification for forestry models used in the United Kingdom Greenhouse Gas Inventory for land use, land use change, and forestry. Clim Chang. 2010;103:55–67.CrossRefGoogle Scholar
  16. 16.
    Hyvönen R, Ågren GI, Linder S, Persson T, Cotrufo MF, Ekblad A, et al. The likely impact of elevated [CO2], nitrogen deposition, increased temperature and management on carbon sequestration in temperate and boreal forest ecosystems: a literature review. New Phytol. 2007;173:463–80.CrossRefGoogle Scholar
  17. 17.
    Rollinson CR, Liu Y, Raiho A, Moore DJP, McLachlan J, Bishop DA, et al. Emergent climate and CO2 sensitivities of net primary productivity in ecosystem models do not agree with empirical data in temperate forests of eastern North America. Glob Chang Biol. 2017;23:2755–67.CrossRefGoogle Scholar
  18. 18.
    Van Oijen M, Cannell MGR, Levy PE. Modelling biogeochemical cycles in forests: state of the art and perspectives. In: Andersson F, Birot Y, Päivinen R, editors. Towards the sustainable use of European forests-Forest ecosystem and landscape research: scientific challenges and opportunities. Joensuu: European Forest Institute; 2004. p. 157–69.Google Scholar
  19. 19.
    Clark JS, Iverson L, Woodall CW, Allen CD, Bell DM, Bragg DC, et al. The impacts of increasing drought on forest dynamics, structure, and biodiversity in the United States. Glob Chang Biol. 2016;22:2329–52.CrossRefGoogle Scholar
  20. 20.
    Johnson MO, Galbraith D, Gloor M, De Deurwaerder H, Guimberteau M, Rammig A, et al. Variation in stem mortality rates determines patterns of above-ground biomass in Amazonian forests: implications for dynamic global vegetation models. Glob Chang Biol. 2016;22:3996–4013.CrossRefGoogle Scholar
  21. 21.
    Schlesinger WH, Dietze MC, Jackson RB, Phillips RP, Rhoades CC, Rustad LE, et al. Forest biogeochemistry in response to drought. Glob Chang Biol. 2016;22:2318–28.CrossRefGoogle Scholar
  22. 22.
    •• Medlyn BE, Zaehle S, De Kauwe MG, Walker AP, Dietze MC, Hanson PJ, et al. Using ecosystem experiments to improve vegetation models. Nat Clim Chang. 2015;5:528–34. This very clear paper can be seen as a counterpoint to the present one in that it does not mention probability theory as a tool to reduce uncertainty, but instead focuses on consistency with mechanistic understanding. CrossRefGoogle Scholar
  23. 23.
    Kennedy MC, O’Hagan A. Bayesian calibration of computer models. J Royal Statistical Society: Series B (Statistical Methodology). 2001;63:425–64.CrossRefGoogle Scholar
  24. 24.
    Fu YH, Campioli M, Van Oijen M, Deckmyn G, Janssens IA. Bayesian comparison of six different temperature-based budburst models for four temperate tree species. Ecol Model. 2012;230:92–100.CrossRefGoogle Scholar
  25. 25.
    Reyer CPO, Flechsig M, Lasch-Born P, van Oijen M. Integrating parameter uncertainty of a process-based model in assessments of climate change effects on forest productivity. Clim Chang. 2016;137:395–409.CrossRefGoogle Scholar
  26. 26.
    Sutton MA, Simpson D, Levy PE, Smith RI, Reis S, van Oijen M, et al. Uncertainties in the relationship between atmospheric nitrogen deposition and forest carbon sequestration. Glob Chang Biol. 2008;14:2057–63.CrossRefGoogle Scholar
  27. 27.
    Minunno F, van Oijen M, Cameron D, Cerasoli S, Pereira J, Tomé M. Using a Bayesian framework and global sensitivity analysis to identify strengths and weaknesses of two process-based models differing in representation of autotrophic respiration. Environ Model Softw. 2013;42:99–115.CrossRefGoogle Scholar
  28. 28.
    Bayes T. An essay towards solving a problem in the doctrine of chances. Philos Trans. 1763;53:370–418.CrossRefGoogle Scholar
  29. 29.
    Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian data analysis, third edition. 3rd ed. Boca Raton: Chapman and Hall/CRC; 2013.Google Scholar
  30. 30.
    • McElreath R. Statistical rethinking: A Bayesian course with examples in R and Stan. Boca Raton: CRC Press; 2016. Perhaps the best introduction to Bayesian methods for absolute beginners, explaining both concepts and methods with impressive clarity. Google Scholar
  31. 31.
    Van Oijen M, Rougier J, Smith R. Bayesian calibration of process-based forest models: bridging the gap between models and data. Tree Physiol. 2005;25:915–27.CrossRefGoogle Scholar
  32. 32.
    Levy PE, Cowan N, Van Oijen M, Famulari D, Drewer J, Skiba U. Estimation of cumulative fluxes of nitrous oxide: Uncertainty in temporal upscaling and emission factors. Eur J Soil Sci. 2017;68:400–11.Google Scholar
  33. 33.
    Patenaude G, Milne R, Van Oijen M, Rowland CS, Hill RA. Integrating remote sensing datasets into ecological modelling: a Bayesian approach. Int J Remote Sens. 2008;29:1295–315.CrossRefGoogle Scholar
  34. 34.
    Höglind M, Van Oijen M, Cameron D, Persson T. Process-based simulation of growth and overwintering of grassland using the BASGRA model. Ecol Model. 2016;335:1–15.CrossRefGoogle Scholar
  35. 35.
    Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E. Equation of state calculations by fast computing machines. J Chem Phys. 1953;21:1087–92.CrossRefGoogle Scholar
  36. 36.
    Van Oijen M. Bayesian Calibration (BC) and Bayesian Model Comparison (BMC) of process-based models: Theory, implementation and guidelines. Edinburgh: Centre for Ecology and Hydrology; 2008. Accessed 4 Sept 2017.
  37. 37.
    •• Hartig F, Minunno F, Paul S, Cameron D, Ott T. Package ‘BayesianTools’. The Comprehensive R Archive Network; 2017. Accessed 4 Sept 2017. This document introduces a major software development: an R-package that facilitates the use of Bayesian calibration by means of MCMC for complex process-based models of forests and other vegetation.
  38. 38.
    Dietze MC, Lebauer DS, Kooper R. On improving the communication between models and data. Plant Cell Environ. 2013;36:1575–85.CrossRefGoogle Scholar
  39. 39.
    Minunno F, van Oijen M, Cameron DR, Pereira JS. Selecting parameters for Bayesian calibration of a process-based model: a methodology based on canonical correlation analysis. SIAM/ASA J Uncertainty Quantification. 2013;1:370–85.CrossRefGoogle Scholar
  40. 40.
    Kobayashi K, Salam MU. Comparing simulated and measured values using mean squared deviation and its components. Agron J. 2000;92:345–52.CrossRefGoogle Scholar
  41. 41.
    Cressie N, Calder CA, Clark JS, Hoef JMV, Wikle CK. Accounting for uncertainty in ecological analysis: the strengths and limitations of hierarchical statistical modeling. Ecol Appl. 2009;19:553–70.CrossRefGoogle Scholar
  42. 42.
    Ogle K. Hierarchical Bayesian statistics: merging experimental and modeling approaches in ecology. Ecol Appl. 2009;19:577–81.CrossRefGoogle Scholar
  43. 43.
    Simpson AH, Richardson SJ, Laughlin DC. Soilclimate interactions explain variation in foliar, stem, root and reproductive traits across temperate forests. Glob Ecol Biogeogr. 2016;25:964–78.CrossRefGoogle Scholar
  44. 44.
    Cressie N, Wikle CK. Statistics for spatio-temporal data. Hoboken: Wiley; 2011.Google Scholar
  45. 45.
    Dietze MC, Wolosin MS, Clark JS. Capturing diversity and interspecific variability in allometries: a hierarchical approach. For Ecol Manag. 2008;256:1939–48.CrossRefGoogle Scholar
  46. 46.
    Rougier J. Probabilistic inference for future climate using an ensemble of climate model evaluations. Clim Chang. 2007;81:247–64.CrossRefGoogle Scholar
  47. 47.
    Chandler RE. Exploiting strength, discounting weakness: Combining information from multiple climate simulators. Phil Trans R Soc A 2013;371:20120388.
  48. 48.
    Kavetski D, Kuczera G, Franks SW. Bayesian analysis of input uncertainty in hydrological modeling: 1. Theory. Water Resour Res 2006;42:W03407.
  49. 49.
    Kass RE, Raftery AE. Bayes factors. J Am Stat Assoc. 1995;90:773–95.CrossRefGoogle Scholar
  50. 50.
    Spiegelhalter D, Pearson M, Short I. Visualizing uncertainty about the future. Science. 2011;333:1393–400.CrossRefGoogle Scholar
  51. 51.
    • Milne AE, Glendining MJ, Lark RM, Perryman SAM, Gordon T, Whitmore AP. Communicating the uncertainty in estimated greenhouse gas emissions from agriculture. J Environ Manag. 2015;160:139–53. A thorough assessment of six different ways of communicating uncertainty, both verbal and visual. CrossRefGoogle Scholar
  52. 52.
    •• Van Oijen M, Balkovič J, Beer C, Cameron DR, Ciais P, Cramer W, et al. Impact of droughts on the carbon cycle in European vegetation: a probabilistic risk analysis using six vegetation models. Biogeosciences. 2014;11:6357–75. This paper shows how risk can be formally decomposed as the product of two terms: hazard probability and ecosystem vulnerability. The method is applied to the results of six vegetation models, and predicts future drought risk to increase mainly in Mediterranean vegetation because of changes in hazard probability. CrossRefGoogle Scholar
  53. 53.
    Smith R, Dick J, Trench H, Van Oijen M. Extending a Bayesian belief network for ecosystem evaluation. In: Proceedings of the 2012 Berlin Conference on the Human Dimensions of Global Environmental Change, 5-6 October 2012, Berlin, Germany. 2012. Accessed 4 Sept 2017.
  54. 54.
    Yeluripati JB, van Oijen M, Wattenbach M, Neftel A, Ammann A, Parton W, et al. Bayesian calibration as a tool for initialising the carbon pools of dynamic soil models. Soil Biol Biochem. 2009;41:2579–83.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Centre for Ecology & HydrologyPenicuikUK

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