Nonstationary matrix covariances: compact support, long range dependence and quasi-arithmetic constructions

Flexible models for multivariate processes are increasingly important for datasets in the geophysical, environmental, economics and health sciences. Modern datasets involve numerous variables observed at large numbers of space–time locations, with millions of data points being common. We develop a suite of stochastic models for nonstationary multivariate processes. The constructions break into three basic categories—quasi-arithmetic, locally stationary covariances with compact support, and locally stationary covariances with possible long-range dependence. All derived models are nonstationary, and we illustrate the flexibility of select choices through simulation.

[1]  Chunsheng Ma,et al.  Vector random fields with compactly supported covariance matrix functions , 2013 .

[2]  T. Gneiting.  Compactly Supported Correlation Functions , 2002 .

[3]  M. Fuentes.  Spectral methods for nonstationary spatial processes , 2002 .

[4]  Tatiyana V. Apanasovich,et al.  Cross-covariance functions for multivariate random fields based on latent dimensions , 2010 .

[5]  Tilmann Gneiting,et al.  Stochastic Models That Separate Fractal Dimension and the Hurst Effect , 2001, SIAM Rev..

[6]  Emilio Porcu,et al.  Characterization theorems for some classes of covariance functions associated to vector valued random fields , 2011, J. Multivar. Anal..

[7]  Michael Scheuerer,et al.  Covariance Models for Divergence-Free and Curl-Free Random Vector Fields , 2012 .

[8]  Ying Sun,et al.  A Valid Matérn Class of Cross-Covariance Functions for Multivariate Random Fields With Any Number of Components , 2012 .

[9]  Harald Cramer,et al.  On the Theory of Stationary Random Processes , 1940 .

[10]  R. Bhatia.  Positive Definite Matrices , 2007 .

[11]  Emilio Porcu,et al.  Classes of compactly supported covariance functions for multivariate random fields , 2015, Stochastic Environmental Research and Risk Assessment.

[12]  René L. Schilling,et al.  Bernstein Functions: Theory and Applications , 2010 .

[13]  Ü Kuran,et al.  POTENTIAL THEORY ON LOCALLY COMPACT ABELIAN GROUPS , 1977 .

[14]  Jorge Mateu,et al.  Quasi-arithmetic means of covariance functions with potential applications to space-time data , 2006, J. Multivar. Anal..

[15]  Mitio Nagumo.  Über eine Klasse der Mittelwerte , 1930 .

[16]  Holger Wendland,et al.  Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree , 1995, Adv. Comput. Math..

[17]  Alan E. Gelfand,et al.  Multivariate Spatial Modeling for Geostatistical Data Using Convolved Covariance Functions , 2007 .

[18]  Christopher J Paciorek,et al.  Spatial modelling using a new class of nonstationary covariance functions , 2006, Environmetrics.

[19]  T. Gneiting.  Nonseparable, Stationary Covariance Functions for Space–Time Data , 2002 .

[20]  J. Mateu,et al.  On the non-reducibility of non-stationary correlation functions to stationary ones under a class of mean-operator transformations , 2010 .

[21]  Emilio Porcu,et al.  From Schoenberg to Pick–Nevanlinna: Toward a complete picture of the variogram class , 2008, 0812.2936.

[22]  Ronald P. Barry,et al.  Constructing and fitting models for cokriging and multivariable spatial prediction , 1998 .

[23]  T. Gneiting,et al.  Matérn Cross-Covariance Functions for Multivariate Random Fields , 2010 .

[24]  M. Goulard,et al.  Linear coregionalization model: Tools for estimation and choice of cross-variogram matrix , 1992 .

[25]  Douglas W. Nychka,et al.  Covariance Tapering for Likelihood-Based Estimation in Large Spatial Data Sets , 2008 .

[26]  K. Furutsu,et al.  On the theory of radio wave propagation over inhomogenous earth , 1963 .

[27]  Emilio Porcu,et al.  Radial basis functions with compact support for multivariate geostatistics , 2013, Stochastic Environmental Research and Risk Assessment.

[28]  Douglas W. Nychka,et al.  Nonstationary modeling for multivariate spatial processes , 2012, J. Multivar. Anal..

[29]  M. Stein.  Nonstationary spatial covariance functions , 2005 .

[30]  Balaji Rajagopalan,et al.  Daily minimum and maximum temperature simulation over complex terrain , 2012, 1210.1814.

[31]  Martin Schlather,et al.  Some covariance models based on normal scale mixtures , 2011 .

[32]  V. Mandrekar,et al.  Fixed-domain asymptotic properties of tapered maximum likelihood estimators , 2009, 0909.0359.

[33]  Debashis Paul,et al.  A GENERALIZED CONVOLUTION MODEL FOR MULTIVARIATE NONSTATIONARY SPATIAL PROCESSES , 2010 .

[34]  David Higdon,et al.  A process-convolution approach to modelling temperatures in the North Atlantic Ocean , 1998, Environmental and Ecological Statistics.

[35]  Richard Askey.  Radial Characteristics Functions. , 1973 .

[36]  J. Mateu,et al.  ARCHIMEDEAN SPECTRAL DENSITIES FOR NONSTATIONARY SPACE-TIME GEOSTATISTICS , 2009 .

[37]  Montserrat Fuentes,et al.  A new class of nonstationaryspatial models , 2001 .

[38]  D. Nychka,et al.  Covariance Tapering for Interpolation of Large Spatial Datasets , 2006 .

[39]  F. J. Narcowich,et al.  Generalized Hermite interpolation via matrix-valued conditionally positive definite functions , 1994 .

[40]  S. Cohn,et al.  Ooce Note Series on Global Modeling and Data Assimilation Construction of Correlation Functions in Two and Three Dimensions and Convolution Covariance Functions , 2022 .

[41]  Marc G. Genton,et al.  Spatially varying cross-correlation coefficients in the presence of nugget effects , 2013 .

[42]  Calyampudi R. Rao,et al.  Multivariate environmental statistics , 1995 .