Electronic Journal of Statistics

Towards a complete picture of stationary covariance functions on spheres cross time

Philip White and Emilio Porcu

Full-text: Open access

Abstract

With the advent of wide-spread global and continental-scale spatiotemporal datasets, increased attention has been given to covariance functions on spheres over time. This paper provides results for stationary covariance functions of random fields defined over $d$-dimensional spheres cross time. Specifically, we provide a bridge between the characterization in Berg and Porcu (2017) for covariance functions on spheres cross time and Gneiting’s lemma (Gneiting, 2002) that deals with planar surfaces.

We then prove that there is a valid class of covariance functions similar in form to the Gneiting class of space-time covariance functions (Gneiting, 2002) that replaces the squared Euclidean distance with the great circle distance. Notably, the provided class is shown to be positive definite on every $d$-dimensional sphere cross time, while the Gneiting class is positive definite over $\mathbb{R} ^{d}\times \mathbb{R} $ for fixed $d$ only.

In this context, we illustrate the value of our adapted Gneiting class by comparing examples from this class to currently established nonseparable covariance classes using out-of-sample predictive criteria. These comparisons are carried out on two climate reanalysis datasets from the National Centers for Environmental Prediction and National Center for Atmospheric Research. For these datasets, we show that examples from our covariance class have better predictive performance than competing models.

Article information

Source
Electron. J. Statist., Volume 13, Number 2 (2019), 2566-2594.

Dates
Received: August 2018
First available in Project Euclid: 2 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1564732820

Digital Object Identifier
doi:10.1214/19-EJS1593

Keywords
Bayesian statistics covariance functions global data great circle distance spatiotemporal statistics sphere

Rights
Creative Commons Attribution 4.0 International License.

Citation

White, Philip; Porcu, Emilio. Towards a complete picture of stationary covariance functions on spheres cross time. Electron. J. Statist. 13 (2019), no. 2, 2566--2594. doi:10.1214/19-EJS1593. https://projecteuclid.org/euclid.ejs/1564732820


Export citation

References

  • Alegria, A. and Porcu, E. (2017). The Dimple Problem related to Space-Time Modeling under the Lagrangian Framework., Journal of Multivariate Analysis 162 110-121.
  • Alegria, A., Porcu, E., Furrer, R. and Mateu, J. (2017). Covariance Functions for Multivariate Gaussian Fields Evolving Temporally over Planet Earth., Technical Report, University Federico Santa Maria.
  • Apanasovich, T. and Genton, M. (2010). Cross-Covariance Functions for Multivariate Random Fields Based on Latent Dimensions., Biometrika 97 15-30.
  • Banerjee, S. (2005). On Geodetic Distance Computations in Spatial Modeling., Biometrics 61 617–625.
  • Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2014)., Hierarchical Modeling and Analysis for Spatial Data. Crc Press.
  • 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 (Statistical Methodology) 70 825–848.
  • Barbosa, V. S. and Menegatto, V. A. (2017). Strict Positive Definiteness on Products of Compact Two–Point Homogeneous Spaces., Integral Transforms and Special Functions 28 56–73.
  • Bateman, H. (1954)., Tables of Integral Transforms, Volume I. McGraw-Hill Book Company, New York.
  • Berg, C. (2008). Stieltjes-Pick-Bernstein-Schoenberg and Their Connection to Complete Monotonicity. In, Quantitative Methods for Current Environmental Issues 15–45.
  • Berg, C. and Porcu, E. (2017). From Schoenberg Coefficients to Schoenberg Functions., Constructive Approximation 45 217–241.
  • Brown, T. A. (1974). Admissible Scoring Systems for Continuous Distributions Technical Report No. P-5235, The Rand Corporation, Santa Monica, California.
  • Castruccio, S. and Stein, M. L. (2013). Global Space-Time Models for Climate Ensembles., Ann. Appl. Statist. 7 1593–1611.
  • 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.
  • Cressie, N. and Wikle, C. K. (2015)., Statistics for Spatio-Temporal Data. John Wiley & Sons.
  • Dai, F. and Xu, Y. (2013)., Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer.
  • Datta, A., Banerjee, S., Finley, A. O. and Gelfand, A. E. (2016a). Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets., Journal of the American Statistical Association 111 800-812.
  • Datta, A., Banerjee, S., Finley, A. O., Hamm, N. A. and Schaap, M. (2016b). Nonseparable Dynamic Nearest Neighbor Gaussian Process Models for Large Spatio-Temporal Data with an Application to Particulate Matter Analysis., The Annals of Applied Statistics 10 1286–1316.
  • Folland, C. K., Rayner, N. A., Brown, S., Smith, T., Shen, S., Parker, D., Macadam, I., Jones, P., Jones, R. N. and Nicholls, N. (2001). Global Temperature Change and its Uncertainties since 1861., Geophysical Research Letters 28 2621–2624.
  • Furrer, R., Genton, M. G. and Nychka, D. (2006). Covariance Tapering for Interpolation of Large Spatial Datasets., Journal of Computational and Graphical Statistics 15 502–523.
  • Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A. and Rubin, D. B. (2014)., Bayesian Data Analysis 2. CRC press Boca Raton, FL.
  • Gneiting, T. (2002). Nonseparable, Stationary Covariance Functions for Space-Time Data., Journal of the American Statistical Association 97 590–600.
  • Gneiting, T. (2013). Strictly and Non-Strictly Positive Definite Functions on Spheres., Bernoulli 19 1327–1349.
  • Gneiting, T. and Raftery, A. E. (2007). Strictly Proper Scoring Rules, Prediction, and Estimation., Journal of the American Statistical Association 102 359–378.
  • Gneiting, T. and Schlather, M. (2004). Stochastic Models That Separate Fractal Dimension and the Hurst Effect., SIAM Rev. 46 269-282.
  • Gradshteyn, I. S. and Ryzhik, I. M. (2007)., Tables of Integrals, Series, and Products, seventh ed. Academic Press, Amsterdam.
  • Gramacy, R. B. and Apley, D. W. (2015). Local Gaussian Process Approximation for Large Computer Experiments., Journal of Computational and Graphical Statistics 24 561–578.
  • Guella, J. C., Menegatto, V. A. and Peron, A. P. (2016a). An Extension of a Theorem of Schoenberg to a Product of Spheres., Banach Journal of Mathematical Analysis 10 671–685.
  • Guella, J. C., Menegatto, V. A. and Peron, A. P. (2016b). Strictly Positive Definite Kernels on a Product of Spheres II., SIGMA 12.
  • Guella, J. C., Menegatto, V. A. and Peron, A. P. (2017). Strictly Positive Definite Kernels on a Product of Circles., Positivity 21 329–342.
  • Hansen, J., Sato, M., Ruedy, R., Lo, K., Lea, D. W. and Medina-Elizade, M. (2006). Global Temperature Change., Proceedings of the National Academy of Sciences 103 14288–14293.
  • Hansen, J., Ruedy, R., Sato, M. and Lo, K. (2010). Global Surface Temperature Change., Reviews of Geophysics 48.
  • Heaton, M. J., Datta, A., Finley, A. O., Furrer, R., Guinness, J., Guhaniyogi, R., Gerber, F., Gramacy, R. B., Hammerling, D., Katzfuss, M. et al. (2018). A Case Study Competition among Methods for Analyzing Large Spatial Data., Journal of Agricultural, Biological and Environmental Statistics 1–28.
  • Heine, V. (1955). Models for Two-Dimensional Stationary Stochastic Processes., Biometrika 42 170–178.
  • Held, I. M. and Soden, B. J. (2006). Robust Responses of the Hydrological Cycle to Global Warming., Journal of Climate 19 5686–5699.
  • Higdon, D. (2002). Space and Space-Time Modeling using Process Convolutions. In, Quantitative Methods for Current Environmental Issues 37–56. Springer.
  • Jeong, J. and Jun, M. (2015). A Class of Matern-like Covariance Functions for Smooth Processes on a Sphere., Spatial Statistics 11 1–18.
  • Kalnay, E., Kanamitsu, M., Kistler, R., Collins, W., Deaven, D., Gandin, L., Iredell, M., Saha, S., White, G. and Woollen, J. (1996). The NCEP/NCAR 40-Year Reanalysis Project., Bulletin of the American meteorological Society 77 437–471.
  • Katzfuss, M. and Guinness, J. (2017). A General Framework for Vecchia Approximations of Gaussian Processes., arXiv preprint arXiv:1708.06302.
  • 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.
  • Lindgren, F., Rue, H. and Lindstroem, 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.
  • Mardia, K. V. and Marshall, R. J. (1984). Maximum Likelihood Estimation of Models for Residual Covariance in Spatial Regression., Biometrika 71 135–146.
  • Matheson, J. E. and Winkler, R. L. (1976). Scoring rules for continuous probability distributions., Management Science 22 1087–1096.
  • Melillo, J. M., McGuire, A. D., Kicklighter, D. W., Moore, B., Vorosmarty, C. J. and Schloss, A. L. (1993). Global Climate Change and Terrestrial Net Primary Production., Nature 363 234.
  • Parmesan, C. and Yohe, G. (2003). A Globally Coherent Fingerprint of Climate Change Impacts across Natural Systems., Nature 421 37.
  • Porcu, E., Alegría, A. and Furrer, R. (2018). Modeling Temporally Evolving and Spatially Globally Dependent Data., International Statistical Review To Appear.
  • Porcu, E., Bevilacqua, M. and Genton, M. G. (2016). Spatio-Temporal Covariance and Cross-Covariance Functions of the Great Circle Distance on a Sphere., Journal of the American Statistical Association 111 888–898.
  • Porcu, E., Gregori, P. and Mateu, J. (2006). Nonseparable Stationary Anisotropic Space–Time Covariance Functions., Stochastic Environmental Research and Risk Assessment 21 113–122.
  • Porcu, E., Mateu, J. and Bevilacqua, M. (2007). Covariance Functions which are Stationary or Nonstationary in Space and Stationary in Time., Statistica Neerlandica 61 358–382.
  • Porcu, E., Mateu, J. and Christakos, G. (2010). Quasi-Arithmetic Means of Covariance Functions with Potential Applications to Space-Time Data., J. Multivariate Anal. 100 1830–1844.
  • Porcu, E. and Schilling, R. (2011). From Schoenberg to Pick-Nevanlinna: Towards a Complete Picture of the Variogram Class., Bernoulli 17 441–455.
  • Pounds, J. A., Fogden, M. P. and Campbell, J. H. (1999). Biological Response to Climate Change on a Tropical Mountain., Nature 398 611.
  • Schilling, R., Song, R. and Vondracek, Z. (2012)., Bernstein Functions. Theory and Applications. De Gruyter.
  • Schlather, M. (2010). Some Covariance Models Based on Normal Scale Mixtures., Bernoulli 16 780–797.
  • Schoenberg, I. J. (1942). Positive Definite Functions on Spheres., Duke Mathematical Journal 9 96–108.
  • Shirota, S. and Gelfand, A. E. (2017). Space and Circular Time Log Gaussian Cox Processes with Application to Crime Event Data., The Annals of Applied Statistics 11 481–503.
  • Simmons, A., Jones, P., da Costa Bechtold, V., Beljaars, A., Kållberg, P., Saarinen, S., Uppala, S., Viterbo, P. and Wedi, N. (2004). Comparison of trends and low-frequency variability in CRU, ERA-40, and NCEP/NCAR analyses of surface air temperature., Journal of Geophysical Research: Atmospheres 109.
  • Stein, M. L. (1999)., Statistical Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York.
  • Stein, M. L. (2008). A Modeling Approach for Large Spatial Datasets., Journal of the Korean Statistical Society 37 3–10.
  • Stein, M. L. (2014). Limitations on Low Rank Approximations for Covariance Matrices of Spatial Data., Spatial Statistics 8 1–19.
  • 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.
  • Svensmark, H. and Friis-Christensen, E. (1997). Variation of Cosmic Ray Flux and Global Cloud Coverage—a Missing Link in Solar-Climate Relationships., Journal of Atmospheric and Solar-Terrestrial Physics 59 1225–1232.
  • Thomas, C. D., Cameron, A., Green, R. E., Bakkenes, M., Beaumont, L. J., Collingham, Y. C., Erasmus, B. F., De Siqueira, M. F., Grainger, A. and Hannah, L. (2004). Extinction Risk from Climate Change., Nature 427 145.
  • Vecchia, A. V. (1988). Estimation and Model Identification for Continuous Spatial Processes., Journal of the Royal Statistical Society. Series B (Statistical Methodology) 297–312.
  • White, P. and Porcu, E. (2019). Nonseparable Covariance Models on Circles Cross Time: A Study of Mexico City Ozone., Environmetrics e2558.
  • Wylie, D., Jackson, D. L., Menzel, W. P. and Bates, J. J. (2005). Trends in Global Cloud Cover in Two Decades of HIRS Observations., Journal of Climate 18 3021–3031.
  • Zhang, H. (2004). Inconsistent Estimation and Asymptotically Equal Interpolations in Model-Based Geostatistics., Journal of the American Statistical Association 99 250–261.