Advertisement

A Fused Lasso Approach to Nonstationary Spatial Covariance Estimation

  • Ryan J. ParkerEmail author
  • Brian J. Reich
  • Jo Eidsvik
Article

Abstract

Spatial data are increasing in size and complexity due to technological advances. For an analysis of a large and diverse spatial domain, simplifying assumptions such as stationarity are questionable and standard computational algorithms are inadequate. In this paper, we propose a computationally efficient method to estimate a nonstationary covariance function. We partition the spatial domain into a fine grid of subregions and assign each subregion its own set of spatial covariance parameters. This introduces a large number of parameters and to stabilize the procedure we impose a penalty to spatially smooth the estimates. By penalizing the absolute difference between parameters for adjacent subregions, the solution can be identical for adjacent subregions and thus the method identifies stationary subdomains. To apply the method to large datasets, we use a block composite likelihood which is natural in this setting because it also operates on a partition of the spatial domain. The method is applied to tropospheric ozone in the US, and we find that the spatial covariance on the west coast differs from the rest of the country.

Keywords

Spatial statistics Nonstationary covariance Regularization Penalized likelihood 

Supplementary material

13253_2016_251_MOESM1_ESM.pdf (90 kb)
Supplementary material 1 (pdf 89 KB)

References

  1. Appel, K., Pouliot, G., Simon, H., Sarwar, G., Pye, H., Napelenok, S., Akhtar, F., and Roselle, S. (2013), “Evaluation of dust and trace metal estimates from the Community Multiscale Air Quality (CMAQ) model version 5.0,” Geoscientific Model Development, 6, 883–899.Google Scholar
  2. Banerjee, S., Gelfand, A. E., Finley, A. O., and Sang, H. (2008), “Gaussian predictive process models for large spatial data sets,” Journal of the Royal Statistical Society: Series B, 70, 825–848.Google Scholar
  3. Besag, J., York, J., and Mollié, A. (1991), “Bayesian image restoration, with two applications in spatial statistics,” Annals of the Institute of Statistical Mathematics, 43, 1–20.Google Scholar
  4. Bien, J. and Tibshirani, R. J. (2011), “Sparse estimation of a covariance matrix,” Biometrika, 98, 807–820.Google Scholar
  5. Bornn, L., Shaddick, G., and Zidek, J. V. (2012), “Modeling nonstationary processes through dimension expansion,” Journal of the American Statistical Association, 107, 281–289.Google Scholar
  6. Chang, Y.-M., Hsu, N.-J., and Huang, H.-C. (2010), “Semiparametric estimation and selection for nonstationary spatial covariance functions,” Journal of Computational and Graphical Statistics, 19, 117–139.Google Scholar
  7. Cressie, N. and Johannesson, G. (2008), “Fixed rank kriging for very large spatial data sets,” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 70, 209–226.Google Scholar
  8. Eidsvik, J., Shaby, B. A., Reich, B. J., Wheeler, M., and Niemi, J. (2014), “Estimation and prediction in spatial models with block composite likelihoods,” Journal of Computational and Graphical Statistics, 23, 295–315.Google Scholar
  9. 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.Google Scholar
  10. Friedman, J. and Popescu, B. E. (2003), “Gradient directed regularization for linear regression and classification”, Tech. rep., Statistics Department, Stanford University.Google Scholar
  11. Fuentes, M. (2002), “Spectral methods for nonstationary spatial processes,” Biometrika, 89, 197–210.Google Scholar
  12. —— (2007), “Approximate likelihood for large irregularly spaced spatial data,” Journal of the American Statistical Association, 102, 321–331.Google Scholar
  13. Furrer, R., Genton, M. G., and Nychka, D. (2006), “Covariance tapering for interpolation of large spatial datasets,” Journal of Computational and Graphical Statistics, 15.Google Scholar
  14. Higdon, D. (1998), “A process-convolution approach to modelling temperatures in the North Atlantic Ocean,” Environmental and Ecological Statistics, 5, 173–190.Google Scholar
  15. Higdon, D., Swall, J., and Kern, J. (1999), “Non-stationary spatial modeling”, in Bayesian Statistics 6, pp. 761–768.Google Scholar
  16. Hsu, N.-J., Chang, Y.-M., and Huang, H.-C. (2012), “A group lasso approach for non-stationary spatial-temporal covariance estimation,” Environmetrics, 23, 12–23.Google Scholar
  17. Kaufman, C. G., Schervish, M. J., and Nychka, D. W. (2008), “Covariance tapering for likelihood-based estimation in large spatial data sets,” Journal of the American Statistical Association, 103, 1545–1555.Google Scholar
  18. Land, S. R. and Friedman, J. H. (1997), “Variable fusion: A new adaptive signal regression method,” Tech. rep., Department of Statistics, Carnegie Mellon University, Pittsburgh.Google Scholar
  19. Lin, Y. and Zhang, H. H. (2006), “Component selection and smoothing in smoothing spline analysis of variance models,” Annals of Statistics, 34, 2272–2297.Google Scholar
  20. Lindgren, F., Rue, H., and Lindström, J. (2011), “An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach,” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73, 423–498.Google Scholar
  21. Neto, J. H. V., Schmidt, A. M., and Guttorp, P. (2014), “Accounting for spatially varying directional effects in spatial covariance structures,” Journal of the Royal Statistical Society: Series C (Applied Statistics), 63, 103–122.Google Scholar
  22. Nychka, D. and Saltzman, N. (1998), “Design of air-quality monitoring networks,” in Case studies in environmental statistics, Springer, pp. 51–76.Google Scholar
  23. Paciorek, C. J. and Schervish, M. J. (2006), “Spatial modelling using a new class of nonstationary covariance functions,” Environmetrics, 17, 483–506.Google Scholar
  24. Reich, B. J., Chang, H. H., and Foley, K. M. (2014), “A spectral method for spatial downscaling,” Biometrics, 70, 932–942.Google Scholar
  25. Reich, B. J., Eidsvik, J., Guindani, M., Nail, A. J., and Schmidt, A. M. (2011), “A class of covariate-dependent spatiotemporal covariance functions,” The annals of applied statistics, 5, 2265.Google Scholar
  26. Sampson, P. D. (2010), ”Constructions for Nonstationary Spatial Processes”, in Handbook of Spatial Statistics, eds. Gelfand, A. E., Diggle, P. J., Fuentes, M., and Guttorp, P., CRC Press, chap. 9.Google Scholar
  27. Sampson, P. D. and Guttorp, P. (1992), “Nonparametric estimation of nonstationary spatial covariance structure,” Journal of the American Statistical Association, 87, 108–119.Google Scholar
  28. Stein, M. L., Chi, Z., and Welty, L. J. (2004), “Approximating likelihoods for large spatial data sets,” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 66, 275–296.Google Scholar
  29. Storlie, C. B., Bondell, H. D., Reich, B. J., and Zhang, H. H. (2011), “Surface estimation, variable selection, and the nonparametric oracle property,” Statistica Sinica, 21, 679.Google Scholar
  30. Tibshirani, R., Saunders, M., Rosset, S., Zhu, J., and Knight, K. (2005), “Sparsity and smoothness via the fused lasso,” Journal of the Royal Statistical Society: Series B, 67, 91–108.Google Scholar
  31. Tibshirani, R. J. and Taylor, J. (2011), “The solution path of the generalized lasso,” Annals of Statistics, 39, 1335–1371.Google Scholar

Copyright information

© International Biometric Society 2016

Authors and Affiliations

  1. 1. SAS InstituteCaryUSA
  2. 2. North Carolina State UniversityRaleighUSA
  3. 3. Norwegian University of Science and TechnologyTrondheimNorway

Personalised recommendations