Advances in geo-spatial technologies have created data-rich environments which provide extraordinary opportunities to understand the complexity of large and spatially indexed data in ecology and the natural sciences. Our current application concerns analysis of soil nutrients data collected at La Selva Biological Station, Costa Rica, where inferential interest lies in capturing the spatially varying relationships among the nutrients. The objective here is to interpolate not just the nutrients across space, but also associations among the nutrients that are posited to vary spatially. This requires spatially varying cross-covariance models. Fully process-based specifications using matrix-variate processes are theoretically attractive but computationally prohibitive. Here we develop fully process-based low-rank but non-degenerate spatially varying cross-covariance processes that can effectively yield interpolate cross-covariances at arbitrary locations. We show how a particular low-rank process, the predictive process, which has been widely used to model large geostatistical datasets, can be effectively deployed to model non-degenerate cross-covariance processes. We produce substantive inferential tools such as maps of nonstationary cross-covariances that constitute the premise of further mechanistic modeling and have hitherto not been easily available for environmental scientists and ecologists.
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Apanasovich, T. V., and Genton, M. G. (2010), “Cross-Covariance Functions for Multivariate Random Fields Based on Latent Dimensions,” Biometrika, 97, 15–30.
Banerjee, S., Carlin, B. P., and Gelfand, A. E. (2004), Hierarchical Modeling and Analysis for Spatial Data, Boca Raton: Chapman and Hall/CRC Press.
Banerjee, S., and Johnson, G. A. (2006), “Coregionalized Single- and Multi-Resolution Spatially-Varying Growth Curve Modelling With Application to Weed Growth,” Biometrics, 61, 617–625
Banerjee, S., Gelfand, A. E., Finley, A. O., and Sang, H. (2008), “Gaussian Predictive Process Models for Large Spatial Datasets,” Journal of the Royal Statistical Society, Series B, 70, 825–848.
Banerjee, S., Finley, A. O., Waldmann, P., and Ericcson, T. (2010), “Hierarchical Spatial Process Models for Multiple Traits in Large Genetic Trials,” Journal of the American Statistical Association, 105, 506–521.
Cressie, N. (1993), Statistics for Spatial Data (2nd ed.), New York: Wiley.
Cressie, N., and Johannesson, G. (2008), “Fixed Rank Kriging for Very Large Spatial Data Sets,” Journal of the Royal Statistical Society, Series B, 70, 209–226.
Cressie, N. A. C., and Wikle, C. K. (2011), Statistics for Spatio-Temporal Data, New York: Wiley.
Daniels, M. J., and Kass, R. E. (1999), “Nonconjugate Bayesian Estimation of Covariance Matrices and Its Use in Hierarchical Models,” Journal of the American Statistical Association, 94, 1254–1263.
Diez, J. M., and Pulliam, H. R. (2007), “Hierarchical Analysis of Species Distributions and Abundance Across Environmental Gradients,” Ecology, 88, 3144–3152.
Finley, A. O., Banerjee, S., and McRoberts, R. E. (2009), “Hierarchical Spatial Models for Predicting Tree Species Assemblages Across Large Domains,” Annals of Applied Statistics, 3, 1052–1079.
Finley, A. O., Banerjee, S., Ek, A. R., and McRoberts, R. E. (2008), “Bayesian Multivariate Process Modeling for Prediction of Forest Attributes,” Journal of Agricultural, Biological, and Environmental Statistics, 13, 60–83.
Finley, A. O., Sang, H., Banerjee, S., and Gelfand, A. E. (2009), “Improving the Performance of Predictive Process Modeling for Large Datasets,” Computational Statistics & Data Analysis, 53, 2873–2884.
Finzi, A. C., van Breemen, N., and Canham, C. D. (1998), “Canopy Tree-Soil Interactions Within Temperate Forests: Species Effects on pH and Base Cations,” Ecological Applications, 8, 447–454.
Gelfand, A. E., and Banerjee, S. (2010), “Multivariate Spatial Process Models,” in Handbook of Spatial Statistics, eds. A. E. Gelfand, P. Diggle, P. Guttorp, and M. Fuentes, Boca Raton: Taylor and Francis/CRC, pp. 495–516.
Gelfand, A. E., and Ghosh, S. K. (1998), “Model Choice: A Minimum Posterior Predictive Loss Approach,” Biometrika, 85, 1–11.
Gelfand, A. E., Schmidt, A. M., Banerjee, S., and Sirmans, C. F. (2004), “Nonstationary Multivariate Process Modeling Through Spatially Varying Coregionalization” (with discussion), Test, 13, 263–312.
Gelman, A., and Rubin, D. (1992), “Inference From Iterative Simulation Using Multiple Sequences,” Statistical Science, 7, 457–511.
Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (2004), Bayesian Data Analysis (2nd ed.), Boca Raton: Chapman and Hall/CRC Press.
Gneiting, T., and Guttorp, P. (2010), “Continuous-Parameter Stochastic Process Theory,” in Handbook of Spatial Statistics, eds. A. E. Gelfand, P. Diggle, P. Guttorp, and M. Fuentes, Boca Raton: Taylor and Francis/CRC, pp. 17–28.
Gneiting, T., Kleiber, W., and Schlather, M. (2010), “Matérn Cross-Covariance Functions for Multivariate Random Fields,” Journal of the American Statistical Association, 105, 1167–1177.
Guhaniyogi, R., Finley, A. O., Banerjee, S., and Gelfand, A. E. (2011), “Adaptive Gaussian Predictive Process Models for Large Spatial Datasets,” Environmetrics, 22, 997–1007.
Harville, D. A. (1997), Matrix Algebra From a Statistician’s Perspective, New York: Springer.
Henderson, H. V., and Searle, S. R. (1981), “On Deriving the Inverse of a Sum of Matrices,” SIAM Review, 23, 53–60.
Hodges, J. S., and Reich, B. J. (2010), “Adding Spatially-Correlated Errors Can Mess up the Fixed Effect You Love,” American Statistician, 64, 335–344.
Holste, E. K., Kobe, R. K., and Vriesendorp, C. F. (2011), “Seedling Growth Responses to Soil Nutrients in a Wet Tropical Forest Understory,” Ecology, 92, 1828–1838.
Houlton, B. Z., Wang, Y. P., Vitousek, P. M., and Field, C. B. (2008), “A Unifying Framework for Dinitrogen Fixation in the Terrestrial Biosphere,” Nature, 454, 327–331.
Kang, E. L., and Cressie, N. (2011), “Bayesian Inference for the Spatial Random Effects Model,” Journal of the American Statistical Association, 106, 972–983.
Kobe, R. K., and Vriesendorp, C. F. (2009), “Size of Sampling Unit Strongly Influences Detection of Seedling Limitation in a Wet Tropical Forest,” Ecology Letters, 12, 220–228.
Majumdar, A., Paul, D., and Bautista, D. (2010), “A Generalized Convolution Model for Multivariate Nonstationary Spatial Processes,” Statistica Sinica, 20, 675–695.
McCarthy-Neumann, S., and Kobe, R. K. (2010), “Conspecific Plant-Soil Feedbacks Reduce Survivorship and Growth of Tropical Tree Seedlings,” Journal of Ecology, 98, 396–407.
Ovaskainen, O., Hottola, J., and Siitonen, J. (2010), “Modeling Species Co-occurrence by Multivariate Logistic Regression Generates New Hypotheses on Fungal Interactions,” Ecology, 9, 2414–2521.
Paciorek, C. J. (2010), “The Importance of Scale for Spatial-Confounding Bias and Precision of Spatial Regression Estimators,” Statistical Science, 107–125.
Pourahmadi, M. (1999), “Joint Mean-Covariance Model With Applications to Longitudinal Data: Unconstrained Parameterisation,” Biometrika, 86, 677–690.
Rao, C. R. (1973), Linear Statistical Inference and Its Applications (2nd ed.), New York: Wiley.
Robert, C. P., and Casella, G. (2010), An Introduction to Monte Carlo Methods With R, New York: Springer.
Roberts, G. O., and Rosenthal, J. S. (2009), “Examples of Adaptive MCMC,” Journal of Computational and Graphical Statistics, 18, 349–367.
Royle, J. A., and Berliner, L. M. (1999), “A Hierarchical Approach to Multivariate Spatial Modeling and Prediction,” Journal of Agricultural, Biological, and Environmental Statistics, 4, 29–56.
Sang, H., Jun, M., and Huang, J. Z. (2011), “Covariance Approximation for Large Multivariate Spatial Data Sets With an Application to Multiple Climate Model Errors,” Annals of Applied Statistics, 4, 2519–2548.
Stein, M. L. (1999), Interpolation of Spatial Data: Some Theory of Kriging, New York: Springer.
— (2008), “A Modeling Approach for Large Spatial Datasets,” Journal of the Korean Statistical Society, 37, 3–10.
Townsend, A. R., Asner, G. P., and Cleveland, C. C. (2008), “The Biogeochemical Heterogeneity of Tropical Soils,” Trends in Ecology & Evolution, 23, 424–431.
Wackernagel, H. (2006), Multivariate Geostatistics: An Introduction With Applications (3rd ed.), New York: Springer.
Waddle, J. H., Dorazio, R. M., Walls, S. C., Rice, K. G., Beauchamp, J., Schuman, M. J., and Mazzotti, F. J. (2010), “A New Parameterization for Estimating Co-occurrence of Interacting Species,” Ecological Applications, 20, 1467–1475.
Walker, T. W., and Syers, J. K. (1976), “The Fate of Phosphorus During Pedogenesis,” Geoderma, 15, 1–19.
Wardle, D. A., Walker, L. R., and Bardgett, R. D. (2004), “Ecosystem Properties and Forest Decline in Contrasting Long-Term Chronosequences,” Science, 305, 509–512.
Yaglom, A. M. (1987), Correlation Theory of Stationary and Related Random Functions, Vol. I, New York: Springer.
Zhang, H. (2007), “Maximum-Likelihood Estimation for Multivariate Spatial Linear Coregionalization Models,” Environmetrics, 18, 125–139.
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Guhaniyogi, R., Finley, A.O., Banerjee, S. et al. Modeling Complex Spatial Dependencies: Low-Rank Spatially Varying Cross-Covariances With Application to Soil Nutrient Data. JABES 18, 274–298 (2013). https://doi.org/10.1007/s13253-013-0140-3
- Gaussian spatial process
- Predictive process
- Tropical soil nutrients