When mechanism matters: Bayesian forecasting using models of ecological diffusion.

Ecological diffusion is a theory that can be used to understand and forecast spatio-temporal processes such as dispersal, invasion, and the spread of disease. Hierarchical Bayesian modelling provides a framework to make statistical inference and probabilistic forecasts, using mechanistic ecological models. To illustrate, we show how hierarchical Bayesian models of ecological diffusion can be implemented for large data sets that are distributed densely across space and time. The hierarchical Bayesian approach is used to understand and forecast the growth and geographic spread in the prevalence of chronic wasting disease in white-tailed deer (Odocoileus virginianus). We compare statistical inference and forecasts from our hierarchical Bayesian model to phenomenological regression-based methods that are commonly used to analyse spatial occurrence data. The mechanistic statistical model based on ecological diffusion led to important ecological insights, obviated a commonly ignored type of collinearity, and was the most accurate method for forecasting.

[1]  Noel A Cressie,et al.  Statistics for Spatio-Temporal Data , 2011 .

[2]  Mevin B. Hooten,et al.  Homogenization of Large-Scale Movement Models in Ecology with Application to the Spread of Chronic Wasting Disease in Mule Deer , 2012 .

[3]  R. Hilborn,et al.  The Ecological Detective: Confronting Models with Data , 1997 .

[4]  Mevin B Hooten,et al.  When can the cause of a population decline be determined? , 2016, Ecology letters.

[5]  M. Hooten,et al.  Statistical Agent-Based Models for Discrete Spatio-Temporal Systems , 2010 .

[6]  M. Girolami,et al.  Bayesian Solution Uncertainty Quantification for Differential Equations , 2013 .

[7]  M. Hooten,et al.  Hierarchical Species Distribution Models , 2016 .

[8]  J. Fieberg,et al.  Establishing the link between habitat selection and animal population dynamics , 2015 .

[9]  L. Waller,et al.  Mountains, valleys, and rivers: The transmission of raccoon rabies over a heterogeneous landscape , 2008, Journal of agricultural, biological, and environmental statistics.

[10]  Y. Freund,et al.  Discussion of the Paper \additive Logistic Regression: a Statistical View of Boosting" By , 2000 .

[11]  S. Wood.  Generalized Additive Models: An Introduction with R , 2006 .

[12]  Mevin B Hooten,et al.  The basis function approach for modeling autocorrelation in ecological data. , 2016, Ecology.

[13]  Melvin M. Varughese,et al.  DiffusionRgqd : An R Package for Performing Inference and Analysis on Time-Inhomogeneous Quadratic Diffusion Processes , 2016 .

[14]  J. G. Skellam.  Random dispersal in theoretical populations , 1951, Biometrika.

[15]  Mark A. Ditmer,et al.  Understanding the causes and consequences of animal movement: a cautionary note on fitting and interpreting regression models with time‐dependent covariates , 2012 .

[16]  Mevin B. Hooten,et al.  A hierarchical Bayesian non-linear spatio-temporal model for the spread of invasive species with application to the Eurasian Collared-Dove , 2008, Environmental and Ecological Statistics.

[17]  M. Hooten,et al.  Dynamic spatio-temporal models for spatial data , 2017 .

[18]  Richard Field,et al.  Coefficient shifts in geographical ecology: an empirical evaluation of spatial and non-spatial regression , 2009 .

[19]  J. Elith,et al.  Species Distribution Models: Ecological Explanation and Prediction Across Space and Time , 2009 .

[20]  Christopher K. Wikle,et al.  Hierarchical Bayesian Models for Predicting The Spread of Ecological Processes , 2003 .

[21]  S. Carpenter,et al.  Ecological forecasts: an emerging imperative. , 2001, Science.

[22]  T. Gneiting.  Making and Evaluating Point Forecasts , 2009, 0912.0902.

[23]  Hugo Fort,et al.  Catastrophic shifts in ecosystems: spatial early warnings and management procedures (Inspired in the physics of phase transitions) , 2010 .

[24]  S. Levin,et al.  Diffusion and Ecological Problems: Modern Perspectives , 2013 .

[25]  Richard T. Conant,et al.  BEST PRACTICES IN PREDICTION FOR DECISION‐MAKING: LESSONS FROM THE ATMOSPHERIC AND EARTH SCIENCES , 2003 .

[26]  Yanbing Zheng,et al.  Hierarchical dynamic modeling of outbreaks of mountain pine beetle using partial differential equations , 2010 .

[27]  Ingolf Kühn,et al.  Incorporating spatial autocorrelation may invert observed patterns , 2006 .

[28]  Trevor Hastie,et al.  Generalized linear and generalized additive models in studies of species distributions: setting the scene , 2002 .

[29]  R. Macarthur.  The Problem of Pattern and Scale in Ecology: The Robert H. MacArthur Award Lecture , 2005 .

[30]  S. Wood,et al.  Generalized additive models for large data sets , 2015 .

[31]  David L Smith,et al.  Predicting the spatial dynamics of rabies epidemics on heterogeneous landscapes , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[32]  Suming Jin,et al.  Completion of the 2011 National Land Cover Database for the Conterminous United States – Representing a Decade of Land Cover Change Information , 2015 .

[33]  A. Arab,et al.  Hierarchical spatio-temporal models for environmental processes , 2007 .

[34]  Catherine A Calder,et al.  Accounting for uncertainty in ecological analysis: the strengths and limitations of hierarchical statistical modeling. , 2009, Ecological applications : a publication of the Ecological Society of America.

[35]  Mevin B Hooten,et al.  Models for Bounded Systems with Continuous Dynamics , 2009, Biometrics.

[36]  Niklaus E. Zimmermann,et al.  MULTISCALE ANALYSIS OF ACTIVE SEED DISPERSAL CONTRIBUTES TO RESOLVING REID'S PARADOX , 2004 .

[37]  J Elith,et al.  A working guide to boosted regression trees. , 2008, The Journal of animal ecology.

[38]  M. Hooten,et al.  A general science-based framework for dynamical spatio-temporal models , 2010 .

[39]  J. Andrew Royle,et al.  Hierarchical Spatiotemporal Matrix Models for Characterizing Invasions , 2007, Biometrics.

[40]  Adrian E. Raftery,et al.  Use and communication of probabilistic forecasts , 2014, Stat. Anal. Data Min..

[41]  Mevin B. Hooten,et al.  A guide to Bayesian model selection for ecologists , 2015 .

[42]  Paul R Moorcroft,et al.  Mechanistic home range models and resource selection analysis: a reconciliation and unification. , 2006, Ecology.

[43]  R. Tibshirani,et al.  Generalized Additive Models , 1991 .

[44]  Owen L. Petchey,et al.  The ecological forecast horizon, and examples of its uses and determinants , 2015, bioRxiv.

[45]  R. Veit,et al.  Partial Differential Equations in Ecology: Spatial Interactions and Population Dynamics , 1994 .

[46]  A. Raftery,et al.  Strictly Proper Scoring Rules, Prediction, and Estimation , 2007 .

[47]  James S. Clark,et al.  Why environmental scientists are becoming Bayesians , 2004 .

[48]  Harold Hotelling,et al.  Differential Equations Subject to Error, and Population Estimates , 1927 .

[49]  Xiao-Li Meng,et al.  POSTERIOR PREDICTIVE ASSESSMENT OF MODEL FITNESS VIA REALIZED DISCREPANCIES , 1996 .

[50]  Peter Buhlmann,et al.  BOOSTING ALGORITHMS: REGULARIZATION, PREDICTION AND MODEL FITTING , 2007, 0804.2752.

[51]  Mevin B. Hooten,et al.  Bayesian Models: A Statistical Primer for Ecologists , 2015 .

[52]  S. Wood,et al.  GAMs with integrated model selection using penalized regression splines and applications to environmental modelling , 2002 .

[53]  Mevin B. Hooten,et al.  Computationally Efficient Statistical Differential Equation Modeling Using Homogenization , 2013 .

[54]  B. Bentz,et al.  Phenology and density-dependent dispersal predict patterns of mountain pine beetle (Dendroctonus ponderosae) impact , 2014 .

[55]  Mevin B. Hooten,et al.  Restricted spatial regression in practice: geostatistical models, confounding, and robustness under model misspecification , 2015 .

[56]  S. Levin.  The problem of pattern and scale in ecology , 1992 .

[57]  Sabine Himmel,et al.  Partial Differential Equations For Scientists And Engineers , 2016 .

[58]  Mevin B. Hooten,et al.  The Bayesian Group Lasso for Confounded Spatial Data , 2017 .

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

[60]  C. Allen,et al.  Adaptive invasive species distribution models: a framework for modeling incipient invasions , 2015, Biological Invasions.

[61]  J. Hodges,et al.  Adding Spatially-Correlated Errors Can Mess Up the Fixed Effect You Love , 2010 .

[62]  Mevin B Hooten,et al.  Homogenization, sex, and differential motility predict spread of chronic wasting disease in mule deer in southern Utah , 2013, Journal of Mathematical Biology.

[63]  Robert M. May,et al.  Stability and Complexity in Model Ecosystems , 2019, IEEE Transactions on Systems, Man, and Cybernetics.

[64]  S. Chandrasekhar.  Stochastic problems in Physics and Astronomy , 1943 .

[65]  C. Patlak.  Random walk with persistence and external bias , 1953 .

[66]  Mevin B Hooten,et al.  Dynamic occupancy models for explicit colonization processes. , 2016, Ecology.

[67]  Mollie E. Brooks,et al.  Generalized linear mixed models: a practical guide for ecology and evolution. , 2009, Trends in ecology & evolution.

[68]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[69]  Simon N. Wood,et al.  Generalized Additive Models for Gigadata: Modeling the U.K. Black Smoke Network Daily Data , 2017 .

[70]  Caz M Taylor,et al.  The spatial spread of invasions: new developments in theory and evidence , 2004 .

[71]  Mevin B Hooten,et al.  An integrated data model to estimate spatiotemporal occupancy, abundance, and colonization dynamics. , 2017, Ecology.

[72]  P. Turchin.  Quantitative analysis of movement : measuring and modeling population redistribution in animals and plants , 1998 .

[73]  E. Williams,et al.  CHRONIC WASTING DISEASE OF CAPTIVE MULE DEER: A SPONGIFORM ENCEPHALOPATHY1 , 1980, Journal of wildlife diseases.