Generalized Whittle–Matérn random field as a model of correlated fluctuations

This paper considers a generalization of the Gaussian random field with covariance function of the Whittle–Matern family. Such a random field can be obtained as the solution to the fractional stochastic differential equation with two fractional orders. Asymptotic properties of the covariance functions belonging to this generalized Whittle–Matern family are studied, which are used to deduce the sample path properties of the random field. The Whittle–Matern field has been widely used in modeling geostatistical data such as sea beam data, wind speed, field temperature and soil data. In this paper we show that the generalized Whittle–Matern field provides a more flexible model for wind speed data.

[1]  Sample path properties of fractional Riesz–Bessel field of variable order , 2008 .

[2]  Y. Gliklikh.  The Langevin Equation , 1997 .

[3]  Riemann–Liouville and Weyl fractional oscillator processes , 2006 .

[4]  G. Temple,et al.  GENERALIZED FUNCTIONS, VOL. I , 1965 .

[5]  K. Knight.  Stable Non-Gaussian Random Processes Gennady Samorodnitsky and Murad S. Taqqu Chapman and Hall, 1994 , 1997, Econometric Theory.

[6]  S. Kotz,et al.  Analytic and Asymptotic Properties of Linnik′s Probability Densities .II. , 1995 .

[7]  Fitting negative spatial covariances to geothermal field temperatures in Nea Kessani (Greece) , 2007 .

[8]  Igor M. Sokolov,et al.  Physics of Fractal Operators , 2003 .

[9]  M. Zukovic,et al.  SPARTAN RANDOM PROCESSES IN TIME SERIES MODELING , 2007, 0709.3418.

[10]  Noel A. C. Cressie,et al.  Statistics for Spatial Data: Cressie/Statistics , 1993 .

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

[12]  S. Bochner.  Lectures on Fourier Integrals. (AM-42) , 1959 .

[13]  M. Stein.  Space–Time Covariance Functions , 2005 .

[14]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[15]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[16]  R. A. Silverman,et al.  Wave Propagation in a Turbulent Medium , 1961 .

[17]  D. F. Hays,et al.  Table of Integrals, Series, and Products , 1966 .

[18]  S. Krantz.  Fractal geometry , 1989 .

[19]  Robert Haining,et al.  Statistics for spatial data: by Noel Cressie, 1991, John Wiley & Sons, New York, 900 p., ISBN 0-471-84336-9, US $89.95 , 1993 .

[20]  P. Guttorp,et al.  Studies in the history of probability and statistics XLIX On the Matérn correlation family , 2006 .

[21]  A. Raftery,et al.  Space-time modeling with long-memory dependence: assessing Ireland's wind-power resource. Technical report , 1987 .

[22]  Katarzyna Sznajd-Weron,et al.  Anomalous Diffusion From Basics to Applications , 1999 .

[23]  Dionissios T. Hristopulos,et al.  Analytic Properties and Covariance Functions for a New Class of Generalized Gibbs Random Fields , 2006, IEEE Transactions on Information Theory.

[24]  Ming Li,et al.  LOCALLY SELF-SIMILAR FRACTIONAL OSCILLATOR PROCESSES , 2007 .

[25]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[26]  P. Whittle.  ON STATIONARY PROCESSES IN THE PLANE , 1954 .

[27]  D. Applebaum.  Stable non-Gaussian random processes , 1995, The Mathematical Gazette.

[28]  B. Matérn.  Spatial variation : Stochastic models and their application to some problems in forest surveys and other sampling investigations , 1960 .

[29]  Y. Almirantis,et al.  Long- and Short-Range Correlations in Genome Organization , 1999 .

[30]  J. R. Wallis,et al.  An Approach to Statistical Spatial-Temporal Modeling of Meteorological Fields , 1994 .

[31]  P. Welch.  The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms , 1967 .

[32]  R. V. Churchill,et al.  Lectures on Fourier Integrals , 1959 .

[33]  S. Samko.  Hypersingular Integrals and Their Applications , 2001 .

[34]  Thomas H. Jordan,et al.  Stochastic Modeling of Seafloor Morphology: Inversion of Sea Beam Data for Second-Order Statistics , 1988 .

[35]  R. Adler,et al.  The Geometry of Random Fields , 1982 .

[36]  D. Wilson,et al.  Quasi-Wavelet Model of Von Kármán Spectrum of Turbulent Velocity Fluctuations , 2004 .

[37]  D. Wilson,et al.  Quasi-Wavelet Models of Turbulent Temperature Fluctuations , 2006 .

[38]  L. Teo,et al.  Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure , 2008, 0807.0022.

[39]  S. Jaffard,et al.  Elliptic gaussian random processes , 1997 .

[40]  Marc G. Genton,et al.  Predictive spatio-temporal models for spatially sparse environmental data , 2005 .

[41]  S. C. Lim,et al.  Weyl and Riemann–Liouville multifractional Ornstein–Uhlenbeck processes , 2007 .

[42]  Malgorzata Kotulska,et al.  Natural fluctuations of an electropore show fractional Lévy stable motion. , 2007, Biophysical journal.

[43]  James S. Harris,et al.  Tables of integrals , 1998 .

[44]  Multivariate Geostatistics , 2004 .

[45]  Xiongzhi Chen,et al.  Stat 598 Presentation on Nonseperable, Stationary Covariance Functions for Space-Time Data , 2010 .

[46]  T. Kármán.  Progress in the Statistical Theory of Turbulence , 1948 .

[47]  R. Adler.  The Geometry of Random Fields , 2009 .

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

[49]  Marc-Thorsten Hütt,et al.  Genome Phylogeny Based on Short-Range Correlations in DNA Sequences , 2005, J. Comput. Biol..

[50]  Klaus Mecke,et al.  Statistical Physics and Spatial Statistics , 2000 .

[51]  Covariance factorisation and abstract representation of generalised random fields , 2000, Bulletin of the Australian Mathematical Society.

[52]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[53]  M. Stein,et al.  A Bayesian analysis of kriging , 1993 .

[54]  Samuel Kotz,et al.  Multivariate T-Distributions and Their Applications , 2004 .

[55]  F. Massey.  The Kolmogorov-Smirnov Test for Goodness of Fit , 1951 .

[56]  J. Chilès,et al.  Geostatistics: Modeling Spatial Uncertainty , 1999 .

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

[58]  L. Pitt,et al.  On the sharp Markov property for Gaussian random fields and spectral synthesis in spaces of Bessel potentials , 2003 .

[59]  K. Mardia,et al.  Consistency of Procrustes Estimators , 1997 .

[60]  곽순섭,et al.  Generalized Functions , 2006, Theoretical and Mathematical Physics.

[61]  B. Minasny,et al.  The Matérn function as a general model for soil variograms , 2005 .

[62]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[63]  S. SIAMJ..  SPARTAN GIBBS RANDOM FIELD MODELS FOR GEOSTATISTICAL APPLICATIONS∗ , 2003 .

[64]  John T. Kent,et al.  Estimating the Fractal Dimension of a Locally Self-similar Gaussian Process by using Increments , 1997 .

[65]  Dietrich Stoyan,et al.  Statistical Physics and Spatial Statistics: The Art Of Analyzing And Modeling Spatial Structures And Pattern Formation , 2010 .

[66]  Roger Woodard,et al.  Interpolation of Spatial Data: Some Theory for Kriging , 1999, Technometrics.

[67]  I. P. Shkarofsky,et al.  Generalized turbulence space-correlation and wave-number spectrum-function pairs , 1968 .

[68]  Noel A Cressie,et al.  Statistics for Spatial Data, Revised Edition. , 1994 .

[69]  A. Walden,et al.  Spectral analysis for physical applications : multitaper and conventional univariate techniques , 1996 .

[70]  L. Goldman,et al.  Airway pressure alters wavelet fractal dynamics and short-range dependence of respiratory variability , 2008, Respiratory Physiology & Neurobiology.