Bayesian calibration of mathematical models: Optimization of model structure and examination of the role of process error covariance

Abstract The integration of Bayesian inference techniques with mathematical modeling offers a promising means to improve ecological forecasts and management actions over space and time, while accounting for the uncertainty underlying model predictions. In this study, we address two important questions related to the ramifications of the statistical assumptions typically made about the model structural error and the prospect of Bayesian calibration to guide the optimization of model complexity. Regarding the former issue, we examine statistical formulations that whether postulate conditional independence or explicitly accommodate the covariance among the error terms for various model endpoints. Our analysis evaluates the differences in the posterior parameter patterns and predictive performance of a limiting nutrient (phosphate)–phytoplankton–zooplankton–detritus (particulate phosphorus) model calibrated with three alternative statistical configurations. The lessons learned from this exercise are combined with those from a second comparative analysis that aims to optimize model structure. In particular, we selected three formulas of the zooplankton mortality term (linear, hyperbolic, sigmoidal) and examine their capacity to determine the posterior parameterization as well as the reproduction of the observed ecosystem patterns. Our analysis suggests that the statistical characterization of the model error as well as the mathematical representation of specific ecological processes can be influential to the inference drawn by a modeling exercise. Our findings could be useful when selecting the most suitable statistical framework for model calibration and/or making informative decisions about model structure optimization. In the absence of adequate prior knowledge, we also advocate the use of Bayesian model averaging for obtaining weighted averages of the forecasts from different model structures and/or statistical descriptions of the process error terms.

[1]  Weitao Zhang,et al.  Addressing equifinality and uncertainty in eutrophication models , 2008 .

[2]  James O. Berger,et al.  A Framework for Validation of Computer Models , 2007, Technometrics.

[3]  Demetris Koutsoyiannis,et al.  One decade of multi-objective calibration approaches in hydrological modelling: a review , 2010 .

[4]  P. Reichert,et al.  Linking statistical bias description to multiobjective model calibration , 2012 .

[5]  A. O'Hagan,et al.  Bayesian calibration of computer models , 2001 .

[6]  T. Mayer,et al.  Inorganic Contaminants in Suspended Solids from Hamilton Harbour , 1990 .

[7]  Barbara A. Adams-Vanharn,et al.  Evaluation of the current state of mechanistic aquatic biogeochemical modeling: citation analysis and future perspectives. , 2006, Environmental science & technology.

[8]  Sylvia Richardson,et al.  Markov Chain Monte Carlo in Practice , 1997 .

[9]  Dominik Endres,et al.  A new metric for probability distributions , 2003, IEEE Transactions on Information Theory.

[10]  Craig A. Stow,et al.  Eutrophication risk assessment using Bayesian calibration of process-based models : application to a mesotrophic lake , 2007 .

[11]  Mevin B. Hooten,et al.  Assessing First-Order Emulator Inference for Physical Parameters in Nonlinear Mechanistic Models , 2011 .

[12]  C. Wikle.  Hierarchical Models in Environmental Science , 2003 .

[13]  Andrew M. Edwards,et al.  The role of higher predation in plankton population models , 2000 .

[14]  Andrew W. Roddam,et al.  Measurement Error in Nonlinear Models: a Modern Perspective , 2008 .

[15]  R. Strawderman,et al.  Bayesian estimation of input parameters of a nitrogen cycle model applied to a forested reference watershed, Hubbard Brook Watershed Six , 2005 .

[16]  Gerard McMahon,et al.  On the application of multilevel modeling in environmental and ecological studies. , 2010, Ecology.

[17]  H. Malchow.  Nonequilibrium structures in plankton dynamics , 1994 .

[18]  L. M. Berliner,et al.  Hierarchical Bayesian space-time models , 1998, Environmental and Ecological Statistics.

[19]  Murray N. Charlton.  The Hamilton Harbour remedial action plan: eutrophication , 2001 .

[20]  Fred A. Johnson,et al.  BAYESIAN INFERENCE AND DECISION THEORY—A FRAMEWORK FOR DECISION MAKING IN NATURAL RESOURCE MANAGEMENT , 2003 .

[21]  Henrik Madsen,et al.  Comparison of different automated strategies for calibration of rainfall-runoff models , 2002 .

[22]  Alena Mudroch,et al.  Assessment of sediment and porewater after one year of subaqueous capping of contaminated sediments in Hamilton Harbour, Canada , 1998 .

[23]  Mark E. Borsuk,et al.  A Bayesian network of eutrophication models for synthesis, prediction, and uncertainty analysis , 2004 .

[24]  Weitao Zhang,et al.  Bayesian calibration of mechanistic aquatic biogeochemical models and benefits for environmental management , 2008 .

[25]  George B. Arhonditsis,et al.  USELESS ARITHMETIC ? LESSONS LEARNT FROM AQUATIC BIOGEOCHEMICAL MODELING , 2008 .

[26]  Kenneth H. Reckhow,et al.  Characterization of parameters in mechanistic models: A case study of a PCB fate and transport model , 1997 .

[27]  Mark E. Borsuk,et al.  On Monte Carlo methods for Bayesian inference , 2003 .

[28]  George B. Arhonditsis,et al.  A Bayesian methodological framework for accommodating interannual variability of nutrient loading with the SPARROW model , 2012 .

[29]  F. Johnson,et al.  Uncertainty and the Adaptive Management of Waterfowl Harvests , 1996 .

[30]  George B. Arhonditsis,et al.  Integration of numerical modeling and Bayesian analysis for setting water quality criteria in Hamilton Harbour, Ontario, Canada , 2011, Environ. Model. Softw..

[31]  George B. Arhonditsis,et al.  Eutrophication model for Lake Washington (USA): Part I. Model description and sensitivity analysis , 2005 .

[32]  John D. Hall,et al.  Progress Toward Delisting a Great Lakes Area of Concern: The Role of Integrated Research and Monitoring in the Hamilton Harbour Remedial Action Plan , 2006, Environmental monitoring and assessment.

[33]  George B. Arhonditsis,et al.  A Bayesian hierarchical framework for calibrating aquatic biogeochemical models , 2009 .

[34]  George B. Arhonditsis,et al.  Eutrophication Risk Assessment in Hamilton Harbour: System Analysis and Evaluation of Nutrient Loading Scenarios , 2010 .

[35]  Edward L. Boone,et al.  A hierarchical zero-inflated Poisson regression model for stream fish distribution and abundance , 2012 .

[36]  George B. Arhonditsis,et al.  9.10 – Integration of Bayesian Inference Techniques with Mathematical Modeling , 2011 .

[37]  S. Sorooshian,et al.  A Shuffled Complex Evolution Metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters , 2002 .

[38]  Weitao Zhang,et al.  Predicting the Frequency of Water Quality Standard Violations Using Bayesian Calibration of Eutrophication Models , 2008 .

[39]  Carl J. Walters,et al.  Large‐Scale Management Experiments and Learning by Doing , 1990 .

[40]  Andrew Gelman,et al.  General methods for monitoring convergence of iterative simulations , 1998 .

[41]  Dave Higdon,et al.  Combining Field Data and Computer Simulations for Calibration and Prediction , 2005, SIAM J. Sci. Comput..

[42]  M. Charlton,et al.  Water Quality Trends in Hamilton Harbour: Two Decades of Change in Nutrients and Chlorophyll a , 2009 .

[43]  S. Sorooshian,et al.  Multiple Criteria Global Optimization for Watershed Model Calibration , 2013 .

[44]  A. M. Edwards,et al.  Zooplankton mortality and the dynamical behaviour of plankton population models , 1999, Bulletin of mathematical biology.

[45]  Radford M. Neal,et al.  Suppressing Random Walks in Markov Chain Monte Carlo Using Ordered Overrelaxation , 1995, Learning in Graphical Models.

[46]  George B. Arhonditsis,et al.  A Bayesian synthesis of predictions from different models for setting water quality criteria , 2012 .

[47]  George B. Arhonditsis,et al.  Predicting the response of Hamilton Harbour to the nutrient loading reductions: A modeling analysis , 2011 .

[48]  Carl J. Walters,et al.  Adaptive Management of Renewable Resources , 1986 .

[49]  George B. Arhonditsis,et al.  Structural changes in lake functioning induced from nutrient loading and climate variability , 2009 .