Dynamic Inverse Prediction and Sensitivity Analysis With High-Dimensional Responses: Application to Climate-Change Vulnerability of Biodiversity

  • James S. Clark
  • David M. Bell
  • Matthew Kwit
  • Amanda Powell
  • Kai Zhu


Sensitivity analysis (SA) of environmental models is inefficient when there are large numbers of inputs and outputs and interactions cannot be directly linked to input variables. Traditional SA is based on coefficients relating the importance of an input to an output response, generating as many as one coefficient for each combination of model input and output. In many environmental models multiple outputs are part of an integrated response that should be considered synthetically, rather than by separate coefficients for each output. For example, there may be interactions between output variables that cannot be defined by standard interaction terms for input variables. We describe dynamic inverse prediction (DIP), a synthetic approach to SA that quantifies how inputs affect the combined (multivariate) output. We distinguish input interactions (specified as a traditional product of input variables) from output interactions (relationships between outputs not directly linked to inputs). Both contribute to traditional SA coefficients and DIP in ways that permit interpretation of unexpected model results.

An application of broad and timely interest, anticipating effects of climate change on biodiversity, illustrates how DIP helps to quantify the important input variables and the role of interactions. Climate affects individual trees in competition with neighboring trees, but interest lies at the scale of species and landscapes. Responses of individuals to climate and competition for resources involve a number of output variables, such as birth rates, growth, and mortality. They are all components of ‘individual health’, and they interact in ways that cannot be linked to observed inputs, through allocation constraints. We show how prior dependence is introduced to aid interpretation of inputs in the context of ecological resource modeling. We further demonstrate that a new approach to multiplicity (multiple-testing) correction can be implemented in such models to filter through the large number of input combinations. DIP provides a synthetic index of important inputs, including climate vulnerability in the context of competition for light and soil moisture, based on the full (multivariate) response. By aggregating in specific ways (over individuals, years, and other input variables) we provide ways to summarize and rank species in terms of their vulnerability to climate change.

This article has supplementary material online.

Key Words

Biodiversity Climate change Forest dynamics Hierarchical models Interactions Model selection Multiple testing Risk analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

13253_2013_139_MOESM1_ESM.pdf (92 kb)
Pairwise correlations between input variables for selected models by species. Input variable names are given in Table 1. The column at right shows the variance inflation factor (VIF). (PDF 92 kB)


  1. Bevin, K., and Kirkby, M. (1979), “A Physically Based, Variable Contributing Area Model of Basin Hydrology,” Hydrological Sciences Bulletin, 24, 43–69. CrossRefGoogle Scholar
  2. Canham, C. D., and Thomas, R. Q. (2010), “Frequency, Not Relative Abundance, of Temperate Tree Species Varies Along Climate Gradients in Eastern North America,” Ecology, 91, 3433–3440. CrossRefGoogle Scholar
  3. Caswell, H. (2000), Matrix Population Models, Sunderland: Sinauer. Google Scholar
  4. Chib, S. (1995), “Marginal Likelihood From the Gibbs Output,” Journal of the American Statistical Association, 90, 1313–1321. MathSciNetCrossRefMATHGoogle Scholar
  5. Clark, J. S., Bell, D. M., Chu, C., Courbaud, B., Dietze, M., Hersh, M., HilleRisLambers, J., Ibáñez, I., LaDeau, S. L., McMahon, S. M., Metcalf, C. J. E., Mohan, J., Moran, E., Pangle, L., Pearson, S., Salk, C., Shen, Z., Valle, D., and Wyckoff, P. (2010), “High Dimensional Coexistence Based on Individual Variation: A Synthesis of Evidence,” Ecological Monographs, 80, 569–608. CrossRefGoogle Scholar
  6. Clark, J. S., Bell, D. M., Hersh, M. H., and Nichols, L. (2011a), “Climate Change Vulnerability of Forest Biodiversity: Climate and Resource Tracking of Demographic Rates,” Global Change Biology, 17, 1834–1849. CrossRefGoogle Scholar
  7. Clark, J. S., Bell, D. M., Hersh, M. H., Kwit, M., Moran, E., Salk, C., Stine, A., Valle, D., and Zhu, K. (2011b), “Individual-Scale Variation, Species-Scale Differences: Inference Needed to Understand Diversity,” Ecology Letters, 14, 1273–1287. CrossRefGoogle Scholar
  8. Clyde, M., and George, E. I. (2004), “Model Uncertainty,” Statistical Science, 19, 81–94. MathSciNetCrossRefMATHGoogle Scholar
  9. Clyde, M., and Ghosh, J. (2010). A Note on the Bias in Estimating Posterior Probabilities in Variable Selection. Biometrika, in press. Google Scholar
  10. Cooper-Ellis, S. M., Foster, D. R., Carlton, G., and Lezberg, A. (1999), “Forest Response to Catastrophic Wind: Results From an Experimental Hurricane,” Ecology, 80, 2683–2696. CrossRefGoogle Scholar
  11. Cressie, N., Calder, C. A., Clark, J. S., Ver Hoef, J. M., and Wikle, C. K. (2009), “Accounting for Uncertainty in Ecological Analysis: The Strengths and Limitations of Hierarchical Statistical Modeling,” Ecological Applications, 19, 553–570. CrossRefGoogle Scholar
  12. de Kroon, H., van Groenendael, J., and Ehrleń, J. (2000), “Elasticities: A Review of Methods and Model Limitations,” Ecology, 81, 607–618. CrossRefGoogle Scholar
  13. Delpierre, N., Soudani, K., François, C., Köstner, B., Pontailler, J.-Y., Nikinmaa, E., Misson, L., Aubinet, M. Bernhofer, C., Granier, A., Grünwald, T., Heinesch, B., Longdoz, B., Ourcival, J.-J., Rambal, S., Vesala, T., and Dufrêne, E. (2008), “Exceptional Carbon Uptake in European Forests During the Warm Spring of 2007: A Data–Model Analysis,” Global Change Biology, 15, 1455–1474. CrossRefGoogle Scholar
  14. Dietze, M., and Clark, J. S. (2008), “Rethinking Gap Dynamics: The Impact of Damaged Trees and Sprouts,” Ecological Monographs, 78, 331–347. CrossRefGoogle Scholar
  15. Ettinger, A. K., Ford, K. R., and HilleRisLambers, J. (2011), “Climate Determines Upper, but Not Lower, Range Limits in Pacific Northwestern Conifers,” Ecology, 92, 1323–1331. CrossRefGoogle Scholar
  16. Fieberg, J., and Jenkins, K. J. (2005), “Assessing Uncertainty in Ecological Systems Using Global Sensitivity Analyses: A Case Example of Simulated Wolf Reintroduction Effects on Elk,” Ecological Modelling, 187, 259–280. CrossRefGoogle Scholar
  17. Frelich, L. E., and Reich, P. B. (2010), “Will Environmental Changes Reinforce the Impact of Global Warming on the Prairie-Forest Border of Central North America?,” Frontiers in Ecology and the Environment, 8, 371–378. CrossRefGoogle Scholar
  18. Gelfand, A. E., and Ghosh, S. K. (1998), “Model Choice: A Minimum Posterior Predictive Loss Approach,” Biometrika, 85, 1–11. MathSciNetCrossRefMATHGoogle Scholar
  19. Gelfand, A. E., Silander, J. A. Jr., Wu, S., Latimer, A., Lewis, P. O., Rebelo, A. G., and Holder, M. (2005), “Explaining Species Distribution Patterns Through Hierarchical Modeling,” Bayesian Analysis, 1 (1), 41–92. MathSciNetCrossRefGoogle Scholar
  20. George, E. I., and Mccullough, R. E. (1993), “Variable Selection Via Gibbs Sampling,” Journal of the American Statistical Association, 88, 881–889. CrossRefGoogle Scholar
  21. — (1997), “Approaches for Bayesian Variable Selection,” Statistica Sinica, 7, 339–373. MATHGoogle Scholar
  22. Gneiting, T., and Raftery, A. E. (2007), “Strictly Proper Scoring Rules, Prediction, and Estimation,” Journal of the American Statistical Association, 102, 359–378. MathSciNetCrossRefMATHGoogle Scholar
  23. Granier, A., Bréda, N., Longdoz, B., Gross, P., and Ngao, J. (2008), “Ten Years of Fluxes and Stand Growth in a Young Beech Forest at Hesse, North-Eastern France,” Annals of Forest Science, 64, 703. Google Scholar
  24. Hall, S. R. (2009), “Stoichiometrically Explicit Food Webs: Feedbacks Between Resource Supply, Elemental Constraints, and Species Diversity,” Annual Review of Ecology, Evolution, and Systematics, 40, 503–528. CrossRefGoogle Scholar
  25. Hillyer, R. A., and Silman, M. R. (2010), “Changes in Species Interactions Across a 3 km Elevation Gradient: Effects on Plant Migration in Response to Climate Change,” Global Change Biology. doi: 10.1111/j.1365-2486.2010.02268.x. Google Scholar
  26. Hoeting, J. A., Madigan, D., Raftery, A. E., and Volinsky, C. T. (1999), “Bayesian Model Averaging: A Tutorial,” Statistical Science, 14, 382–417. MathSciNetCrossRefMATHGoogle Scholar
  27. Huisman, J., and Weissing, F. J. (2001), “Biological Conditions for Oscillations and Chaos Generated by Multispecies Competition,” Ecology, 82, 2682–2695. CrossRefGoogle Scholar
  28. Knops, J. M. H., Koenig, W. D., and Carmen, W. J. (2007), “Negative Correlation Does Not Imply a Tradeoff Between Growth and Reproduction in California Oaks,” Proceedings of the National Academy of Sciences of the United States of America, 104 (2007), 16982–16985. CrossRefGoogle Scholar
  29. Mitchell, T. J., and Beauchamp, J. J. (1988), “Bayesian Variable Selection in Linear Regression,” Journal of the American Statistical Association, 83, 1023–1032. MathSciNetCrossRefMATHGoogle Scholar
  30. Mund, M., Kutsch, W. L., Wirth, C., Kahl, T., Knohl, A., Skomarkova, M. V., and Schulze, E.-D. (2010), “The Influence of Climate and Fructification on the Inter-Annual Variability of Stem Growth and Net Primary Productivity in an Old-Growth, Mixed Beech Forest,” Tree Physiology, 30, 689–704. CrossRefGoogle Scholar
  31. Revilla, T., and Weissing, F. J. (2008), “Nonequilibrium Coexistence in a Competition Model With Nutrient Storage,” Ecology, 89, 865–877. CrossRefGoogle Scholar
  32. Sánchez-Humanes, B., Sork, V. L., and Espelta, J. M. (2011), “Trade-Offs Between Vegetative Growth and Acorn Production in Quercus Lobata During a Mast Year: The Relevance of Crop Size and Hierarchical Level Within the Canopy,” Oecologia, 166, 101–110. CrossRefGoogle Scholar
  33. Schwarz, S. E. (2011), “Feedback and Sensitivity in an Electrical Circuit: An Analog for Climate Models,” Climatic Change, 106, 315–326. CrossRefGoogle Scholar
  34. Scott, J. G., and Berger, J. O. (2010), “Bayes and Empirical-Bayes Multiplicity Adjustment in the Variable-Selection Problem,” The Annals of Statistics, 38, 2587–2619. MathSciNetCrossRefMATHGoogle Scholar
  35. Souza, G. M. I., Ribeiro, R. V., Sato, A. M., and Oliveira, M. S. (2008), “Diurnal and Seasonal Carbon Balance of Four Tropical Tree Species Differing in Successional Status,” Brazilian Journal of Biology, 68, 781–793. CrossRefGoogle Scholar
  36. Tilman, D. (1980), “A Graphical-Mechanistic Approach to Competition and Predation,” The American Naturalist, 116, 362–393. CrossRefGoogle Scholar
  37. Valladares, F., and Pearcy, R. W. (2002), “Drought Can Be More Critical in the Shade than in the Sun: A Field Study of Carbon Gain and Photo-Inhibition in a Californian Shrub During a Dry el Niño Year,” Plant Cell and Environment, 25, 749–759. CrossRefGoogle Scholar
  38. Valladares, J., Zaragoza-Castells, F., Sánchez-Gómez, D., Matesanz, S., Alonso, B., Portsmuth, A., Delgado, A., and Atkin, O. K. (2008), “Is Shade Beneficial for Mediterranean Shrubs Experiencing Periods of Extreme Drought and Late-Winter Frosts?,” Annals of Botany, 102(6), 923–933. CrossRefGoogle Scholar
  39. Welp, L. R., Randerson, J. T., and Liu, H. P. (2007), “The Sensitivity of Carbon Fluxes to Spring Warming and Summer Drought Depends on Plant Functional Type in Boreal Forest Ecosystems,” Agricultural and Forest Meteorology, 147, 172–185. CrossRefGoogle Scholar
  40. Wikle, C. K., Milliff, R. F., Nychka, D., and Berliner, L. M. (2001), “Spatiotemporal Hierarchical Bayesian Modeling: Tropical Ocean Surface Winds,” Journal of the American Statistical Association, 96, 382–397. MathSciNetCrossRefMATHGoogle Scholar
  41. Zhu, K., Woodall, C. W., and Clark, J. S. (2012), “Failure to Migrate: Lack of Tree Range Expansion in Response to Climate Change,” Global Change Biology. doi: 10.1111/j.1365-2486.2011.02571.x. Google Scholar

Copyright information

© International Biometric Society 2013

Authors and Affiliations

  • James S. Clark
    • 1
  • David M. Bell
    • 1
  • Matthew Kwit
    • 1
  • Amanda Powell
    • 1
  • Kai Zhu
    • 1
  1. 1.Nicholas School of the Environment, Department of Biology, and Department of Statistical ScienceDuke UniversityDurhamUSA

Personalised recommendations