Rosa M. Espejo

Spatial autoregressive functional plug-in prediction of ocean surface temperature

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 bio-optical database: The World-wide Ocean Optics Database. Cross Validation (C.V.) procedures are applied for the evaluation of the estimation results derived.

Functional time series analysis of spatio–temporal epidemiological data

Spatio–temporal statistical models have been proposed for the analysis of the temporal evolution of the geographical pattern of mortality (or incidence) risks in disease mapping. However, as far as we know, functional approaches based on Hilbert-valued processes have not been used so far in this area. In this paper, the autoregressive Hilbertian process framework is adopted to estimate the functional temporal evolution of mortality relative risk maps. Specifically, the penalized functional estimation of log-relative risk maps is considered to smooth the classical standardized mortality ratio. The reproducing kernel Hilbert space (RKHS) norm is selected for definition of the penalty term. This RKHS-based approach is combined with the Kalman filtering algorithm for the spatio–temporal estimation of risk. Functional confidence intervals are also derived for detecting high risk areas. The proposed methodology is illustrated analyzing breast cancer mortality data in the provinces of Spain during the period 1975–2005. A simulation study is performed to compare the ARH(1) based estimation with the classical spatio–temporal conditional autoregressive approach.

Integration of spatial functional interaction in the extrapolation of ocean surface temperature anomalies due to global warming

Abstract The aim of this paper is to derive spatiotemporal extrapolation maps of ocean surface temperature to investigate two global warming effects: On the one hand, the reduction of daily heat fluxes from the sea into the air at the end of the day and during the night, in tropical regions. On the other hand, the strengthening of ocean current flows, due to the increase of ocean surface minimum daily temperature differences between two connected ocean regions. These maps are constructed from the spatial functional time series framework. Specifically, the spatial functional extrapolation of ocean surface temperature from Hawaii Ocean to the Gulf of Mexico reflects an increase of Hawaii Ocean surface temperature in the last 15 years, caused by the reduction of daily heat fluxes from the sea into the air. Furthermore, for the two connected regions of Indian Ocean, and the eastern coast of Australia, the spatial functional extrapolation results derived show more pronounced differences between ocean surface minimum daily temperatures in the year 2003 than in the years 1995–1997. Thus, a strengthening of the flow of the East Australian Current is appreciated.

Spatial functional normal mixed effect approach for curve classification

This paper proposes a spatial functional formulation of the normal mixed effect model for the statistical classification of spatially dependent Gaussian curves, in both parametric and state space model frameworks. Fixed effect parameters are represented in terms of a functional multiple regression model whose regression operators can change in space. Local spatial homogeneity of these operators is measured in terms of their Hilbert–Schmidt distances, leading to the classification of fixed effect curves in different groups. Assuming that the Gaussian random effect curves obey a spatial autoregressive dynamics of order one [SARH(1) dynamics], a second functional classification criterion is proposed in order to detect local spatially homogeneous patterns in the mean quadratic functional variation of Gaussian random effect curve increments. Finally, the two criteria are combined to detect local spatially homogeneous patterns in the regression operators and in the functional mean quadratic variation, under a state space approach. A real data example in the financial context is analyzed as an illustration.

Moment and Bayesian wavelet regression from spatially correlated functional data

A new wavelet-based estimation methodology, in the context of spatial functional regression, is proposed to discriminate between small-scale and large scale variability of spatially correlated functional data, defined by depth-dependent curves. Specifically, the discrete wavelet transform of the data is computed in space and depth to reduce dimensionality. Moment-based regression estimation is applied for the approximation of the scaling coefficients of the functional response. While its wavelet coefficients are estimated in a Bayesian regression framework. Both regression approaches are implemented from the empirical versions of the scaling and wavelet auto-covariance and cross-covariance operators, characterizing the correlation structure of the spatial functional response. Weather stations in ocean islands display high spatial concentration. The proposed estimation methodology overcomes the difficulties arising in the estimation of ocean temperature field at different depths, from long records of ocean temperature measurements in these stations. Data are collected from The World-Wide Ocean Optics Database. The performance of the presented approach is tested in terms of 10-fold cross-validation, and residual spatial and depth correlation analysis. Additionally, an application to soil sciences, for prediction of electrical conductivity profiles is also considered to compare this approach with previous related ones, in the statistical analysis of spatially correlated curves in depth.

Spatial-depth functional estimation of ocean temperature from non-separable covariance models

Spatial-depth functional regression is applied for the estimation of ocean temperature, with projection onto the eigenvectors of the empirical covariance operator of the functional response (i.e., onto the Empirical Orthogonal Functions in space and depth). Moment-based estimation is performed to approximate the regression operators in the subspace generated by the empirical eigenvectors associated with nonnull eigenvalues. In addition, Bayesian estimation is performed to approximate the regression operators in the subspace generated by the empirical eigenvectors associated with almost null eigenvalues. The cross-validation results obtained, together with the spatial-depth residual correlation analysis carried out on a real data set for the South Atlantic area, to the east of Argentina and the Falkland Islands, represent an improvement on those provided by the wavelet-based approach recently proposed in Fernández-Pascual (Stoch Environ Res Risk Assess 30:523–557, 2016).

Heterogeneous Spatial Dynamical Regression in a Hilbert-Valued Context

This article introduces a Hilbert-valued spatially dynamic regression model. The spatially heterogeneous functional trend is modeled by functional multiple regression, with varying regression operators. The spatial autoregressive Hilbertian model of order one (SARH(1) model, see [37]) is considered to represent the spatial correlation and dynamics displayed by the functional error term. The RKHS theory is applied in the construction of suitable bases for projection and regularization of the associated estimation problems. The performance of the proposed Hilbert-valued modeling and estimation methodology is illustrated with a real-data example, related to financing decisions from firm panel data.

Least-Squares Estimation of Multifractional Random Fields in a Hilbert-Valued Context

This paper derives conditions under which a stable solution to the least-squares linear estimation problem for multifractional random fields can be obtained. The observation model is defined in terms of a multifractional pseudodifferential equation. The weak-sense and strong-sense formulations of this problem are studied through the theory of fractional Sobolev spaces of variable order, and the spectral theory of multifractional pseudodifferential operators and their parametrix. The Theory of Reproducing Kernel Hilbert Spaces is also applied to define a stable solution to the direct and inverse estimation problems. Numerical projection methods are proposed based on the construction of orthogonal bases of these spaces. Indeed, projection into such bases leads to a regularization, removing the ill-posed nature of the estimation problem. A simulation study is developed to illustrate the estimation results derived. Some open research lines in relation to the extension of the derived results to the multifractal process context are also discussed.

Gegenbauer random fields

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