zbMATH — the first resource for mathematics

Asymptotic theory of cepstral random fields. (English) Zbl 1294.62044
Summary: Random fields play a central role in the analysis of spatially correlated data and, as a result, have a significant impact on a broad array of scientific applications. This paper studies the cepstral random field model, providing recursive formulas that connect the spatial cepstral coefficients to an equivalent moving-average random field, which facilitates easy computation of the autocovariance matrix. We also provide a comprehensive treatment of the asymptotic theory for two-dimensional random field models: we establish asymptotic results for Bayesian, maximum likelihood and quasi-maximum likelihood estimation of random field parameters and regression parameters. The theoretical results are presented generally and are of independent interest, pertaining to a wide class of random field models. The results for the cepstral model facilitate model-building: because the cepstral coefficients are unconstrained in practice, numerical optimization is greatly simplified, and we are always guaranteed a positive definite covariance matrix. We show that inference for individual coefficients is possible, and one can refine models in a disciplined manner. Our results are illustrated through simulation and the analysis of straw yield data in an agricultural field experiment.

62F12 Asymptotic properties of parametric estimators
62M40 Random fields; image analysis
62M30 Inference from spatial processes
62F15 Bayesian inference
spBayes; R
Full Text: DOI Euclid arXiv
[1] Bandyopadhyay, S. and Lahiri, S. N. (2009). Asymptotic properties of discrete Fourier transforms for spatial data. Sankhyā 71 221-259. · Zbl 1193.62164
[2] Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2004). Hierarchical Modeling and Analysis for Spatial Data 101 . Chapman & Hall, Boca Raton, FL. · Zbl 1053.62105
[3] Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B Stat. Methodol. 36 192-236. · Zbl 0327.60067
[4] Besag, J. and Green, P. J. (1993). Spatial statistics and Bayesian computation. J. R. Stat. Soc. Ser. B Stat. Methodol. 55 25-37. · Zbl 0800.62572
[5] Besag, J. E. (1972). On the correlation structure of some two-dimensional stationary processes. Biometrika 59 43-48. · Zbl 0246.62093
[6] Bloomfield, P. (1973). An exponential model for the spectrum of a scalar time series. Biometrika 60 217-226. · Zbl 0261.62074
[7] Bochner, S. (1955). Harmonic Analysis and the Theory of Probability . Univ. California Press, Berkeley. · Zbl 0068.11702
[8] Brillinger, D. R. (2001). Time Series : Data Analysis and Theory. Classics in Applied Mathematics 36 . SIAM, Philadelphia, PA. · Zbl 0983.62056
[9] Chan, G. and Wood, A. T. A. (1999). Simulation of stationary Gaussian vector fields. Statist. Comput. 9 265-268.
[10] Cliff, A. D. and Ord, J. K. (1981). Spatial Processes : Models & Applications . Pion, London. · Zbl 0598.62120
[11] Cressie, N. and Wikle, C. K. (2011). Statistics for Spatio-Temporal Data . Wiley, Hoboken, NJ. · Zbl 1273.62017
[12] Cressie, N. A. C. (1993). Statistics for Spatial Data . Wiley, New York. · Zbl 0825.62477
[13] Dahlhaus, R. and Künsch, H. (1987). Edge effects and efficient parameter estimation for stationary random fields. Biometrika 74 877-882. · Zbl 0633.62094
[14] Fuentes, M. (2002). Spectral methods for nonstationary spatial processes. Biometrika 89 197-210. · Zbl 0997.62073
[15] Fuentes, M., Guttorp, P. and Sampson, P. D. (2007). Using transforms to analyze space-time processes. In Statistical Methods for Spatio-Temporal Systems 77-150. · Zbl 1121.62077
[16] Fuentes, M. and Reich, B. (2010). Spectral domain. In Handbook of Spatial Statistics (A. Gelfand, P. Diggle, M. Fuentes and P. Guttorp, eds.) 57-77. CRC Press, Boca Raton, FL.
[17] Geweke, J. (2005). Contemporary Bayesian Econometrics and Statistics . Wiley, Hoboken, NJ. · Zbl 1093.62107
[18] Guyon, X. (1982). Parameter estimation for a stationary process on a \(d\)-dimensional lattice. Biometrika 69 95-105. · Zbl 0485.62107
[19] Hurvich, C. M. (2002). Multistep forecasting of long memory series using fractional exponential models. International Journal of Forecasting 18 167-179.
[20] Kedem, B. and Fokianos, K. (2002). Regression Models for Time Series Analysis . Wiley, Hoboken, NJ. · Zbl 1011.62089
[21] Kizilkaya, A. (2007). On the parameter estimation of 2-D moving average random fields. IEEE Transactions on Circuits and Systems II : Express Briefs 54 989-993.
[22] Kizilkaya, A. and Kayran, A. H. (2005). ARMA-cepstrum recursion algorithm for the estimation of the MA parameters of 2-D ARMA models. Multidimens. Syst. Signal Process. 16 397-415. · Zbl 1079.62086
[23] Li, H., Calder, C. A. and Cressie, N. (2007). Beyond Moran’s I: Testing for spatial dependence based on the spatial autoregressive model. Geographical Analysis 39 357-375.
[24] Mardia, K. V. and Marshall, R. J. (1984). Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika 71 135-146. · Zbl 0542.62079
[25] McElroy, T. and Holan, S. (2009). A local spectral approach for assessing time series model misspecification. J. Multivariate Anal. 100 604-621. · Zbl 1169.62076
[26] McElroy, T. S. and Findley, D. F. (2010). Selection between models through multi-step-ahead forecasting. J. Statist. Plann. Inference 140 3655-3675. · Zbl 1404.62089
[27] McElroy, T. S. and Holan, S. H. (2014). Supplement to “Asymptotic theory of cepstral random fields.” . · Zbl 1294.62044
[28] Moran, P. A. P. (1950). Notes on continuous stochastic phenomena. Biometrika 37 17-23. · Zbl 0041.45702
[29] Noh, J. and Solo, V. (2007). A true spatio-temporal test statistic for activation detection in fMRI by parametric cepstrum. In IEEE International Conference on Acoustics , Speech and Signal Processing , 2007. ICASSP 2007. 1 I-321. IEEE, Honolulu, HI.
[30] Pierce, D. A. (1971). Least squares estimation in the regression model with autoregressive-moving average errors. Biometrika 58 299-312. · Zbl 0226.62066
[31] Politis, D. N. and Romano, J. P. (1995). Bias-corrected nonparametric spectral estimation. J. Time Series Anal. 16 67-103. · Zbl 0811.62088
[32] Politis, D. N. and Romano, J. P. (1996). On flat-top kernel spectral density estimators for homogeneous random fields. J. Statist. Plann. Inference 51 41-53. · Zbl 0847.62080
[33] Pourahmadi, M. (1984). Taylor expansion of \(\operatornameexp(\sum^\infty_k=0a_kz^k)\) and some applications. Amer. Math. Monthly 91 303-307. · Zbl 0555.30002
[34] R Development Core Team (2012). R : A Language and Environment for Statistical Computing . R foundation for statistical computing, Vienna, Austria.
[35] Rosenblatt, M. (1985). Stationary Sequences and Random Fields . Birkhäuser, Boston, MA. · Zbl 0597.62095
[36] Rosenblatt, M. (2000). Gaussian and Non-Gaussian Linear Time Series and Random Fields . Springer, New York. · Zbl 0933.62082
[37] Rue, H. and Held, L. (2010). Discrete spatial variation. In Handbook of Spatial Statistics (A. Gelfand, P. Diggle, M. Fuentes and P. Guttorp, eds.). Chapman & Hall, London.
[38] Sandgren, N. and Stoica, P. (2006). On nonparametric estimation of 2-D smooth spectra. IEEE Signal Processing Letters 13 632-635.
[39] Solo, V. (1986). Modeling of two-dimensional random fields by parametric cepstrum. IEEE Trans. Inform. Theory 32 743-750. · Zbl 0622.62090
[40] Stein, M. L. (1999). Interpolation of Spatial Data : Some Theory for Kriging . Springer, New York. · Zbl 0924.62100
[41] Taniguchi, M. and Kakizawa, Y. (2000). Asymptotic Theory of Statistical Inference for Time Series . Springer, New York. · Zbl 0955.62088
[42] Tonellato, S. F. (2007). Random field priors for spectral density functions. J. Statist. Plann. Inference 137 3164-3176. · Zbl 1114.62097
[43] Whittle, P. (1954). On stationary processes in the plane. Biometrika 41 434-449. · Zbl 0058.35601
[44] Wood, A. T. A. and Chan, G. (1994). Simulation of stationary Gaussian processes in \([0,1]^d\). J. Comput. Graph. Statist. 3 409-432.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.