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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.

##### MSC:
 62F12 Asymptotic properties of parametric estimators 62M40 Random fields; image analysis 62M30 Inference from spatial processes 62F15 Bayesian inference
spBayes; R
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##### References:
 [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.
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