Towards a complete picture of stationary covariance functions on spheres cross time. (English) Zbl 1426.62285

White, Philip; Porcu, Emilio

Electron. J. Stat. 13, No. 2, 2566-2594 (2019).

Summary: 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 [C. Berg and E. Porcu, Constr. Approx. 45, No. 2, 217–241 (2017; Zbl 1362.43004)] for covariance functions on spheres cross time and T. Gneiting’s lemma [J. Am. Stat. Assoc. 97, No. 458, 590–600 (2002; Zbl 1073.62593)] 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, loc. cit.] 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.

MSC:

62M40	Random fields; image analysis
62M30	Inference from spatial processes
60G60	Random fields
62P12	Applications of statistics to environmental and related topics

Keywords:

Bayesian statistics; covariance functions; global data; great circle distance; spatiotemporal statistics; sphere; random fields

Citations:

Zbl 1362.43004; Zbl 1073.62593

Software:

GPvecchia; FRK; spBayes

References:

[1]	Alegria, A. and Porcu, E. (2017). The Dimple Problem related to Space-Time Modeling under the Lagrangian Framework., Journal of Multivariate Analysis162 110-121. · Zbl 1374.60093
[2]	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.
[3]	Apanasovich, T. and Genton, M. (2010). Cross-Covariance Functions for Multivariate Random Fields Based on Latent Dimensions., Biometrika97 15-30. · Zbl 1183.62164
[4]	Banerjee, S. (2005). On Geodetic Distance Computations in Spatial Modeling., Biometrics61 617-625.
[5]	Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2014)., Hierarchical Modeling and Analysis for Spatial Data. Crc Press. · Zbl 1053.62105
[6]	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. · Zbl 05563371
[7]	Barbosa, V. S. and Menegatto, V. A. (2017). Strict Positive Definiteness on Products of Compact Two-Point Homogeneous Spaces., Integral Transforms and Special Functions28 56-73. · Zbl 1361.42027
[8]	Bateman, H. (1954)., Tables of Integral Transforms, Volume I. McGraw-Hill Book Company, New York.
[9]	Berg, C. (2008). Stieltjes-Pick-Bernstein-Schoenberg and Their Connection to Complete Monotonicity. In, Quantitative Methods for Current Environmental Issues 15-45.
[10]	Berg, C. and Porcu, E. (2017). From Schoenberg Coefficients to Schoenberg Functions., Constructive Approximation45 217-241. · Zbl 1362.43004
[11]	Castruccio, S. and Stein, M. L. (2013). Global Space-Time Models for Climate Ensembles., Ann. Appl. Statist.7 1593-1611. · Zbl 1454.62436
[12]	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. · Zbl 05563351
[13]	Cressie, N. and Wikle, C. K. (2015)., Statistics for Spatio-Temporal Data. John Wiley & Sons. · Zbl 1273.62017
[14]	Dai, F. and Xu, Y. (2013)., Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer. · Zbl 1275.42001
[15]	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 Association111 800-812.
[16]	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 Statistics10 1286-1316. · Zbl 1391.62269
[17]	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 Letters28 2621-2624.
[18]	Furrer, R., Genton, M. G. and Nychka, D. (2006). Covariance Tapering for Interpolation of Large Spatial Datasets., Journal of Computational and Graphical Statistics15 502-523.
[19]	Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A. and Rubin, D. B. (2014)., Bayesian Data Analysis2. CRC press Boca Raton, FL. · Zbl 1279.62004
[20]	Gneiting, T. (2002). Nonseparable, Stationary Covariance Functions for Space-Time Data., Journal of the American Statistical Association97 590-600. · Zbl 1073.62593
[21]	Gneiting, T. (2013). Strictly and Non-Strictly Positive Definite Functions on Spheres., Bernoulli19 1327-1349. · Zbl 1283.62200
[22]	Gneiting, T. and Raftery, A. E. (2007). Strictly Proper Scoring Rules, Prediction, and Estimation., Journal of the American Statistical Association102 359-378. · Zbl 1284.62093
[23]	Gneiting, T. and Schlather, M. (2004). Stochastic Models That Separate Fractal Dimension and the Hurst Effect., SIAM Rev.46 269-282. · Zbl 1062.60053
[24]	Gradshteyn, I. S. and Ryzhik, I. M. (2007)., Tables of Integrals, Series, and Products, seventh ed. Academic Press, Amsterdam. · Zbl 1208.65001
[25]	Gramacy, R. B. and Apley, D. W. (2015). Local Gaussian Process Approximation for Large Computer Experiments., Journal of Computational and Graphical Statistics24 561-578.
[26]	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 Analysis10 671-685. · Zbl 1348.43008
[27]	Guella, J. C., Menegatto, V. A. and Peron, A. P. (2016b). Strictly Positive Definite Kernels on a Product of Spheres II., SIGMA12. · Zbl 1351.33013
[28]	Guella, J. C., Menegatto, V. A. and Peron, A. P. (2017). Strictly Positive Definite Kernels on a Product of Circles., Positivity21 329-342. · Zbl 1368.42003
[29]	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 Sciences103 14288-14293.
[30]	Hansen, J., Ruedy, R., Sato, M. and Lo, K. (2010). Global Surface Temperature Change., Reviews of Geophysics48.
[31]	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.
[32]	Heine, V. (1955). Models for Two-Dimensional Stationary Stochastic Processes., Biometrika42 170-178. · Zbl 0067.36504
[33]	Held, I. M. and Soden, B. J. (2006). Robust Responses of the Hydrological Cycle to Global Warming., Journal of Climate19 5686-5699.
[34]	Higdon, D. (2002). Space and Space-Time Modeling using Process Convolutions. In, Quantitative Methods for Current Environmental Issues 37-56. Springer. · Zbl 1255.86016
[35]	Jeong, J. and Jun, M. (2015). A Class of Matern-like Covariance Functions for Smooth Processes on a Sphere., Spatial Statistics11 1-18.
[36]	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 Society77 437-471.
[37]	Katzfuss, M. and Guinness, J. (2017). A General Framework for Vecchia Approximations of Gaussian Processes., arXiv preprint arXiv:1708.06302.
[38]	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 Association103 1545-1555. · Zbl 1286.62072
[39]	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. · Zbl 1274.62360
[40]	Mardia, K. V. and Marshall, R. J. (1984). Maximum Likelihood Estimation of Models for Residual Covariance in Spatial Regression., Biometrika71 135-146. · Zbl 0542.62079
[41]	Matheson, J. E. and Winkler, R. L. (1976). Scoring rules for continuous probability distributions., Management Science22 1087-1096. · Zbl 0349.62080
[42]	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., Nature363 234.
[43]	Parmesan, C. and Yohe, G. (2003). A Globally Coherent Fingerprint of Climate Change Impacts across Natural Systems., Nature421 37.
[44]	Porcu, E., Alegría, A. and Furrer, R. (2018). Modeling Temporally Evolving and Spatially Globally Dependent Data., International Statistical ReviewTo Appear.
[45]	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 Association111 888-898.
[46]	Porcu, E., Gregori, P. and Mateu, J. (2006). Nonseparable Stationary Anisotropic Space-Time Covariance Functions., Stochastic Environmental Research and Risk Assessment21 113-122.
[47]	Porcu, E., Mateu, J. and Bevilacqua, M. (2007). Covariance Functions which are Stationary or Nonstationary in Space and Stationary in Time., Statistica Neerlandica61 358-382. · Zbl 1121.62082
[48]	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. · Zbl 1166.62071
[49]	Porcu, E. and Schilling, R. (2011). From Schoenberg to Pick-Nevanlinna: Towards a Complete Picture of the Variogram Class., Bernoulli17 441-455. · Zbl 1280.60032
[50]	Pounds, J. A., Fogden, M. P. and Campbell, J. H. (1999). Biological Response to Climate Change on a Tropical Mountain., Nature398 611.
[51]	Schilling, R., Song, R. and Vondracek, Z. (2012)., Bernstein Functions. Theory and Applications. De Gruyter. · Zbl 1257.33001
[52]	Schlather, M. (2010). Some Covariance Models Based on Normal Scale Mixtures., Bernoulli16 780-797. · Zbl 1220.62119
[53]	Schoenberg, I. J. (1942). Positive Definite Functions on Spheres., Duke Mathematical Journal9 96-108. · Zbl 0063.06808
[54]	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 Statistics11 481-503. · Zbl 1416.62689
[55]	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: Atmospheres109.
[56]	Stein, M. L. (1999)., Statistical Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York. · Zbl 0924.62100
[57]	Stein, M. L. (2008). A Modeling Approach for Large Spatial Datasets., Journal of the Korean Statistical Society37 3-10. · Zbl 1196.62123
[58]	Stein, M. L. (2014). Limitations on Low Rank Approximations for Covariance Matrices of Spatial Data., Spatial Statistics8 1-19.
[59]	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. · Zbl 1062.62094
[60]	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 Physics59 1225-1232.
[61]	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., Nature427 145.
[62]	Vecchia, A. V. (1988). Estimation and Model Identification for Continuous Spatial Processes., Journal of the Royal Statistical Society. Series B (Statistical Methodology) 297-312.
[63]	White, P. and Porcu, E. (2019). Nonseparable Covariance Models on Circles Cross Time: A Study of Mexico City Ozone., Environmetrics e2558.
[64]	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 Climate18 3021-3031.
[65]	Zhang, H. (2004). Inconsistent Estimation and Asymptotically Equal Interpolations in Model-Based Geostatistics., Journal of the American Statistical Association99 250-261. · Zbl 1089.62538