Spatial autoregressive and moving average Hilbertian processes

Stoch Environ Res Risk Assess

Espejo

2012

This paper addresses the problem of spatial functional extrapolation in the framework of spatial autoregressive Hilbertian processes of order one (SARH(1) processes) introduced in Ruiz-Medina (J Muitivar Anal 102:292-305, 2011a). Moment-based estimators of the operators involved in the state equation of these processes are computed by projection into a suitable orthogonal basis. Specifically, the eigenfunction basis diagonalizing the autocovariance operator is considered. An estimation algorithm is designed for the implementation of the resulting SARH(1)-plug-in projection extrapolator from temporal curves irregularly distributed in space. Its performance is illustrated with a real-data example, where the problem of spatial functional extrapolation of ocean surface temperature profiles is addressed. This problem is crucial in the assessment of climate change anomalies. The data are collected from the public oceanographic biooptical database: The World-wide Ocean Optics Database. Cross Validation (C.V.) procedures are applied for the evaluation of the estimation results derived.

Section: Statistical Modeling and Preliminary Resultsmentioning

confidence: 99%

Spatial autoregressive functional plug-in prediction of ocean surface temperature

Stoch Environ Res Risk Assess

Espejo

2012

“…The autocovariance function { , } of the ( − 1)-factor Gegenbauer process is de ned as the inverse Fourier transformation of its spectrum as usual The proof follows straightforward from [26,Proposition 4], keeping in mind equation (5.4), and Theorem 4.2.…”

Section: Gegenbauer Elds With Multiple Singularitiesmentioning

confidence: 99%

“…where , = 1, 2, 3, satisfy the conditions given in [26,Proposition 3]. Speci cally, they satisfy the following conditions: (C1) For = 1, 2, 3, the operator ∈ L(H) is assumed to admit a spectral decomposition in terms of the eigenvalue sequence { : ∈ ℕ}, and the biorthonormal systems of left eigenvectors { : ∈ ℕ} and right eigenvectors { : ∈ ℕ}, de ning dual Riesz bases of H and H * , and satisfying the following equations:…”

Section: Hilbert-valued Spatial Autoregressive Gegenbauer Random Eldsmentioning

confidence: 99%

“…For example, recently, in Bosq [8], the tensorial product of autoregressive Hilbertian processes is investigated, as extension to the functional context of the doubly geometric processes introduced by Martin [21]. Also, the Hilbert-valued formulation of the spatial unilateral dynamics displayed by the spatial autoregressive models of order one studied in [21] is provided in Ruiz-Medina [26]. The structural properties of this class of spatial Hilbert-valued processes are also investigated in this paper.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Gegenbauer random fields

Espejo

Leonenko

Random Operators and Stochastic Equations

2014

This paper introduces spatial long-range dependence time series models, based on the consideration of fractional di erence operators associated with Gegenbauer polynomials. Their structural properties are analyzed. The spatial autoregressive Gegenbauer case is also studied, including the case of factors with multiple singularities. An extension to the Hilbert-valued context is nally formulated leading to the introduction of a new class of spatial functional time series models.

“…(See also [7], [8], [49] and [50], in the functional time series context, and [27] and [28] in the functional nonparametric regression framework). Functional Analysis of Variance (FANOVA) techniques for highdimensional data with a functional background have played a crucial role, within the functional linear model literature as well.…”

mentioning

confidence: 99%

The effect of the spatial domain in FANOVA models with ARH(1) error term

Álvarez-Liébana

Statistics and Its Interface

2017

SummaryFunctional Analysis of Variance (FANOVA) from Hilbert-valued correlated data with spatial rectangular or circular supports is analyzed, when Dirichlet conditions are assumed on the boundary. Specifically, a Hilbertvalued fixed effect model with error term defined from an Autoregressive Hilbertian process of order one (ARH (1) process) is considered, extending the formulation given in [51]. A new statistical test is also derived to contrast the significance of the functional fixed effect parameters. The Dirichlet conditions established at the boundary affect the dependence range of the correlated error term. While the rate of convergence to zero of the eigenvalues of the covariance kernels, characterizing the Gaussian functional error components, directly affects the stability of the generalized least-squares parameter estimation problem. A simulation study and a real-data application related to fMRI analysis are undertaken to illustrate the performance of the parameter estimator and statistical test derived.