The Annals of Statistics

Asymptotic theory of cepstral random fields

Tucker S. McElroy and Scott H. Holan

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Abstract

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.

Article information

Source
Ann. Statist., Volume 42, Number 1 (2014), 64-86.

Dates
First available in Project Euclid: 15 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1389795745

Digital Object Identifier
doi:10.1214/13-AOS1180

Mathematical Reviews number (MathSciNet)
MR3161461

Zentralblatt MATH identifier
1294.62044

Subjects
Primary: 62F12: Asymptotic properties of estimators 62M30: Spatial processes
Secondary: 62F15: Bayesian inference 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Keywords
Bayesian estimation cepstrum exponential spectral representation lattice data spatial statistics spectral density

Citation

McElroy, Tucker S.; Holan, Scott H. Asymptotic theory of cepstral random fields. Ann. Statist. 42 (2014), no. 1, 64--86. doi:10.1214/13-AOS1180. https://projecteuclid.org/euclid.aos/1389795745


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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.
  • [2] Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2004). Hierarchical Modeling and Analysis for Spatial Data 101. Chapman & Hall, Boca Raton, FL.
  • [3] Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B Stat. Methodol. 36 192–236.
  • [4] Besag, J. and Green, P. J. (1993). Spatial statistics and Bayesian computation. J. R. Stat. Soc. Ser. B Stat. Methodol. 55 25–37.
  • [5] Besag, J. E. (1972). On the correlation structure of some two-dimensional stationary processes. Biometrika 59 43–48.
  • [6] Bloomfield, P. (1973). An exponential model for the spectrum of a scalar time series. Biometrika 60 217–226.
  • [7] Bochner, S. (1955). Harmonic Analysis and the Theory of Probability. Univ. California Press, Berkeley.
  • [8] Brillinger, D. R. (2001). Time Series: Data Analysis and Theory. Classics in Applied Mathematics 36. SIAM, Philadelphia, PA.
  • [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.
  • [11] Cressie, N. and Wikle, C. K. (2011). Statistics for Spatio-Temporal Data. Wiley, Hoboken, NJ.
  • [12] Cressie, N. A. C. (1993). Statistics for Spatial Data. Wiley, New York.
  • [13] Dahlhaus, R. and Künsch, H. (1987). Edge effects and efficient parameter estimation for stationary random fields. Biometrika 74 877–882.
  • [14] Fuentes, M. (2002). Spectral methods for nonstationary spatial processes. Biometrika 89 197–210.
  • [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.
  • [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.
  • [18] Guyon, X. (1982). Parameter estimation for a stationary process on a $d$-dimensional lattice. Biometrika 69 95–105.
  • [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.
  • [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.
  • [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.
  • [25] McElroy, T. and Holan, S. (2009). A local spectral approach for assessing time series model misspecification. J. Multivariate Anal. 100 604–621.
  • [26] McElroy, T. S. and Findley, D. F. (2010). Selection between models through multi-step-ahead forecasting. J. Statist. Plann. Inference 140 3655–3675.
  • [27] McElroy, T. S. and Holan, S. H. (2014). Supplement to “Asymptotic theory of cepstral random fields.” DOI:10.1214/13-AOS1180SUPP.
  • [28] Moran, P. A. P. (1950). Notes on continuous stochastic phenomena. Biometrika 37 17–23.
  • [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.
  • [31] Politis, D. N. and Romano, J. P. (1995). Bias-corrected nonparametric spectral estimation. J. Time Series Anal. 16 67–103.
  • [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.
  • [33] Pourahmadi, M. (1984). Taylor expansion of $\operatornameexp(\sum^\infty_k=0a_kz^k)$ and some applications. Amer. Math. Monthly 91 303–307.
  • [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.
  • [36] Rosenblatt, M. (2000). Gaussian and Non-Gaussian Linear Time Series and Random Fields. Springer, New York.
  • [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.
  • [40] Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York.
  • [41] Taniguchi, M. and Kakizawa, Y. (2000). Asymptotic Theory of Statistical Inference for Time Series. Springer, New York.
  • [42] Tonellato, S. F. (2007). Random field priors for spectral density functions. J. Statist. Plann. Inference 137 3164–3176.
  • [43] Whittle, P. (1954). On stationary processes in the plane. Biometrika 41 434–449.
  • [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.

Supplemental materials

  • Supplementary material: Supplement to asymptotic theory of cepstral random fields. The supplement contains a description of further applications of the cepstral model, analysis of straw yield data, as well as all proofs.