Autoregressive spatial spectral estimates

Abstract

Autoregressive spectral density estimation for stationary random fields on a regular spatial lattice has many advantages relative to kernel based methods. It provides a guaranteed positive-definite estimate even when suitable edge-effect correction is employed, is simple to compute using least squares and necessitates no choice of kernel. We truncate a true half-plane infinite autoregressive representation to estimate the spectral density. The truncation length is allowed to diverge in all dimensions in order to avoid the potential bias which would accrue due to truncation at a fixed lag-length. Consistency and strong consistency of the proposed estimator, both uniform in frequencies, are established. Under suitable conditions the asymptotic distribution of the estimate is shown to be zero-mean normal and independent at fixed distinct frequencies, mirroring the behaviour for time series. A small Monte Carlo experiment examines finite sample performance. We illustrate the technique by applying it to Los Angeles house price data and a novel analysis of voter turnout data in a US presidential election. Technically the key to the results is the covariance structure of stationary random fields defined on regularly spaced lattices. We study this in detail and show the covariance matrix to satisfy a generalization of the Toeplitz property familiar from time series analysis. JEL classifications : C14, C18, C21

Nonparametric prediction with spatial data

“…in 93% of cases. In view of these findings, our simulation study suggests the use of the flexible exponential algorithm proposed in this paper together with the AR spectral estimator of Gupta (2018) in moderate to large sample sizes. Finally, we observe similar and sometimes even better prediction performance, for suitable bandwidths, when performing prediction from a misspecified data generating processing (the cases , q = 1, 2, 3, 4, with n * = 5, model (4.1) τ = 0.05, 0.10).…”

“…The flexible exponential approach requires a nonparametric estimate of f (λ). Two such estimates are available to use: the first one based on the tapered periodogram described in (3.9), which we denote f (λ), and the second based on the autoregressive approach in Gupta (2018). The latter also provides a rival prediction methodology based on a nonparametric algorithm using AR model fitting, extending well established results for d = 1, see Bhansali (1978) and Lewis and Reinsel (1985).…”

“…They most closely match those obtained when (p 1 , p 2 ) = (2, 2), (2, 3) in (4.7). In the latter case we compare the order selection criteria proposed by Gupta (2018), which include the usual FPE and BIC (denoted with a ) as well as corrected version that account for the spatial case (denoted with and ¯). The FPE tends to favour longer lag lengths no matter which version is used, as do BIC and BIC.…”

Nonparametric Prediction With Spatial Data

“…For the smallest sample size x20,20 can outperform x20,20 and x20,20 , but with increasing n * the latter two clearly begin to dominate. An inspection of Tables 1-4 reveals that the use of the flexible exponential algorithm proposed in this paper together with either the tapered periodogram or the AR spectral estimator of Gupta (2018) outperforms autoregressive prediction in moderate to large sample sizes. There is little to choose from between the two best performing algorithms, and a practitioner might choose to use either one.…”

“…The flexible exponential approach requires a nonparametric estimate of f (λ). Two such estimates are available to use: the first one based on the tapered periodogram described in (3.8), which we denote f (λ), and the second based on the autoregressive approach in Gupta (2018). The latter also provides a rival https://doi.org/10.1017/S0266466622000226 Published online by Cambridge University Press prediction methodology based on a nonparametric algorithm using AR model fitting, extending well established results for d = 1, see Bhansali (1978) and Lewis and Reinsel (1985).…”