Spectral density estimation for random fields via periodic embeddings.

We introduce methods for estimating the spectral density of a random field on a [Formula: see text]-dimensional lattice from incomplete gridded data. Data are iteratively imputed onto an expanded lattice according to a model with a periodic covariance function. The imputations are convenient computationally, in that circulant embedding and preconditioned conjugate gradient methods can produce imputations in [Formula: see text] time and [Formula: see text] memory. However, these so-called periodic imputations are motivated mainly by their ability to produce accurate spectral density estimates. In addition, we introduce a parametric filtering method that is designed to reduce periodogram smoothing bias. The paper contains theoretical results on properties of the imputed-data periodogram and numerical and simulation studies comparing the performance of the proposed methods to existing approaches in a number of scenarios. We present an application to a gridded satellite surface temperature dataset with missing values.

[1]  R. Dahlhaus,et al.  Edge effects and efficient parameter estimation for stationary random fields , 1987 .

[2]  Jun Zhu,et al.  Nonparametric Bayesian inference for the spectral density function of a random field , 2010 .

[3]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[4]  M. Fuentes.  Approximate Likelihood for Large Irregularly Spaced Spatial Data , 2007, Journal of the American Statistical Association.

[5]  Thomas C. M. Lee,et al.  A stabilized bandwidth selection method for kernel smoothing of the periodogram , 2001, Signal Process..

[6]  D. Lieberman,et al.  Fourier analysis , 2004, Journal of cataract and refractive surgery.

[7]  An asymptotic theory for spectral analysis of random fields , 2017 .

[8]  Dorit Hammerling,et al.  A Case Study Competition Among Methods for Analyzing Large Spatial Data , 2017, Journal of Agricultural, Biological and Environmental Statistics.

[9]  Yudi Pawitan,et al.  Nonparametric Spectral Density Estimation Using Penalized Whittle Likelihood , 1994 .

[10]  Shibin Zhang.  Nonparametric Bayesian inference for the spectral density based on irregularly spaced data , 2020, Comput. Stat. Data Anal..

[11]  A. Wood,et al.  Simulation of Stationary Gaussian Processes in [0, 1] d , 1994 .

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

[13]  Thomas C. M. Lee.  A simple span selector for periodogram smoothing , 1997 .

[14]  Douglas W. Nychka,et al.  Methods for Analyzing Large Spatial Data: A Review and Comparison , 2017 .

[15]  W. Wong,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[16]  Yoshihiro Yajima,et al.  Fourier analysis of irregularly spaced data on Rd , 2007 .

[17]  A. V. Vecchia.  Estimation and model identification for continuous spatial processes , 1988 .

[18]  Joseph P. Romano,et al.  BIAS‐CORRECTED NONPARAMETRIC SPECTRAL ESTIMATION , 1995 .

[19]  J. Raz,et al.  A simple generalised crossvalidation method of span selection for periodogram smoothing , 2001 .

[20]  Zhengyuan Zhu,et al.  Nonparametric spectral density estimation with missing observations , 2009, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing.

[21]  Sofia C. Olhede,et al.  The De-Biased Whittle Likelihood for Second-Order Stationary Stochastic Processes , 2016 .

[22]  J. Early,et al.  The debiased Whittle likelihood , 2019, Biometrika.

[23]  C. Heyde.  Smoothed periodogram asymptotics and estimation for processes and fields with possible long-range dependence , 1993 .

[24]  J. Early,et al.  The De-Biased Whittle Likelihood , 2016 .

[25]  Anne Greenbaum,et al.  Iterative methods for solving linear systems , 1997, Frontiers in applied mathematics.

[26]  X. Guyon.  Parameter estimation for a stationary process on a d-dimensional lattice , 1982 .

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

[28]  Joseph Guinness,et al.  Permutation and Grouping Methods for Sharpening Gaussian Process Approximations , 2016, Technometrics.

[29]  Asymptotic Properties of Discrete Fourier Transforms for Spatial Data , 2009 .

[30]  Chae Young Lim,et al.  Properties of spatial cross-periodograms using fixed-domain asymptotics , 2008 .

[31]  M. Fuentes,et al.  Circulant Embedding of Approximate Covariances for Inference From Gaussian Data on Large Lattices , 2017 .

[32]  Daniel J. Nordman,et al.  A frequency domain empirical likelihood method for irregularly spaced spatial data , 2015, 1503.04985.

[33]  G. Wahba.  Automatic Smoothing of the Log Periodogram , 1980 .

[34]  Michael L. Stein,et al.  Fixed-Domain Asymptotics for Spatial Periodograms , 1995 .

[35]  D. Nychka,et al.  A Multiresolution Gaussian Process Model for the Analysis of Large Spatial Datasets , 2015 .

[36]  A. Raftery,et al.  Strictly Proper Scoring Rules, Prediction, and Estimation , 2007 .

[37]  H. Rue,et al.  An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach , 2011 .

[38]  Ulf Grenander,et al.  A Sieve Method for the Spectral Density , 1985 .

[39]  Michael L. Stein,et al.  Bayesian and Maximum Likelihood Estimation for Gaussian Processes on an Incomplete Lattice , 2014, 1402.4281.

[40]  D. Rubin,et al.  Statistical Analysis with Missing Data. , 1989 .