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We introduce a new class of spatial Cox processes driven by a Hilbert--valued random log--intensity. We adopt a parametric framework in the spectral domain, to estimate its spatial functional correlation structure. Specifically, we consider a spectral functional, based on the periodogram operator, inspired on Whittle estimation methodology. Strong-consistency of the parametric estimator is proved in the linear case. We illustrate this property in a simulation study under a Gaussian first order Spatial Autoregressive Hilbertian scenario for the log--intensity model. Our method is applied to the spatial functional prediction of respiratory disease mortality in the Spanish Iberian Peninsula, in the period 1980--2015., Submitted (25 pages, 18 figures and seven tables)

A spatial curve dynamical model framework is adopted for functional prediction of counts in a spatiotemporal log-Gaussian Cox process model. Our spatial functional estimation approach handles both wavelet-based heterogeneity analysis in time, and spectral analysis in space. Specifically, model fitting is achieved by minimising the information divergence or relative entropy between the multiscale model underlying the data, and the corresponding candidates in the spatial spectral domain. A simulation study is carried out within the family of log-Gaussian Spatial Autoregressive $$\ell ^{2}$$ -valued processes (SAR $$\ell ^{2}$$ processes) to illustrate the asymptotic properties of the proposed spatial functional estimators. We apply our modelling strategy to spatiotemporal prediction of respiratory disease mortality.

This work adopts a Banach-valued time series framework for component-wise estimation and prediction, from temporal correlated functional data, in presence of exogenous variables. The strong-consistency of the proposed functional estimator and associated plug-in predictor is formulated. The simulation study undertaken illustrates their large-sample size properties. Air pollutants PM10 curve forecasting, in the Haute-Normandie region (France), is addressed by implementation of the functional time series approach presented.

The restriction to the sphere of a homogeneous and isotropic random field defines a spherical isotropic random field. This paper derives central and noncentral limit results for the first Minkowski functional subordinated to homogeneous and isotropic Gaussian and chi-squared random fields, restricted to the sphere in R3. Both scenarios are motivated by their interesting applications in the analysis of the Cosmic Microwave Background (CMB) radiation.

New results on strong-consistency in the trace operator norm are obtained, in the parameter estimation of an autoregressive Hilbertian process of order one (ARH(1) process). Additionally, a strongly-consistent diagonal componentwise estimator of the autocorrelation operator is derived, based on its empirical singular value decomposition.

The Karhunen-Lo`eve expansion and the Fredholm determinant formula are used, to derive an asymptotic Rosenblatt-type distribution of a sequence of integrals of quadratic functions of Gaussian stationary random fields on R d displaying long-range dependence. This distribution reduces to the usual Rosenblatt distribution when d = 1. Several properties of this new distribution are obtained. Specifically, its series representation, in terms of independent chi-squared random variables, is established. Its L´evy-Khintchine representation, and membership to the Thorin subclass of self-decomposable distributions are obtained as well. The existence and boundedness of its probability density then follow as a direct consequence.

Fractional (in time and in space) evolution equations defined on Dirichlet regular bounded open domains, driven by fractional integrated in time Gaussian spatiotemporal white noise, are considered here. Sufficient conditions for the definition of a weak-sense Gaussian solution, in the mean-square sense, are derived. The temporal, spatial and spatiotemporal Holder continuity, in the mean-square sense, of the formulated solution is obtained, under suitable conditions, from the asymptotic properties of the Mittag-Leffler function, and the asymptotic order of the eigenvalues of a fractional polynomial of the Dirichlet negative Laplacian operator on such bounded open domains.

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 Fernandez-Pascual (Stoch Environ Res Risk Assess 30:523–557, 2016).

This paper studies the asymptotic properties of a plug-in predictor, based on the formulation of a componentwise estimator of the autocorrelation operator, for a special class of standard autoregressive Hilbertian processes of order one (ARH(1) processes). In the Gaussian case, double asymptotic functional plug-in prediction intervals are derived. Some numerical examples are considered for illustration.

A linear multiple regression model in function spaces is formulated, under temporal correlated errors. This formulation involves kernel regressors. A generalized least-squared regression parameter estimator is derived. Its asymptotic normality and strong consistency is obtained, under suitable conditions. The correlation analysis is based on a componentwise estimator of the residual autocorrelation operator. When the dependence structure of the functional error term is unknown, a plug-in generalized least-squared regression parameter estimator is formulated. Its strong-consistency is proved as well. A simulation study is undertaken to illustrate the performance of the presented approach, under different regularity conditions. An application to financial panel data is also considered., This paper has been submitted to TEST Journal, and now the reviewing process status is on the submitted first revised version on June, 2018

This work derives new results on strong consistent estimation and prediction for autoregressive processes of order 1 in a separable Banach space B. The consistency results are obtained for the componentwise estimator of the autocorrelation operator in the norm of the space $\mathcal{L}(B)$ of bounded linear operators on B. The strong consistency of the associated plug-in predictor then follows in the $B$-norm. A Gelfand triple is defined through the Hilbert space constructed in Kuelbs' Lemma \cite{Kuelbs70}. A Hilbert--Schmidt embedding introduces the Reproducing Kernel Hilbert space (RKHS), generated by the autocovariance operator, into the Hilbert space conforming the Rigged Hilbert space structure. This paper extends the work of \cite{Bosq00} and \cite{LabbasMourid02}., Comment: 37 pages (Supplementary Material has been included) with 6 figures. Manuscript accepted, in press

This paper presents new results on functional analysis of variance for fixed effect models with correlated Hilbert-valued Gaussian error components. The geometry of the reproducing kernel Hilbert space of the error term is considered in the computation of the total sum of squares, the residual sum of squares, and the sum of squares due to the regression. Under suitable linear transformation of the correlated functional data, the distributional characteristics of these statistics, their moment generating and characteristic functions, are derived. Fixed effect linear hypothesis testing is finally formulated in the Hilbert-valued multivariate Gaussian context considered.

Potential theory and Dirichlet’s priciple constitute the basic elements of the well-known classical theory of Markov processes and Dirichlet forms. This paper presents new classes of fractional spatiotemporal covariance models, based on the theory of non-local Dirichlet forms, characterizing the fundamental solution, Green kernel, of Dirichlet boundary value problems for fractional pseudodifferential operators. The elements of the associated Gaussian random field family have compactly supported non-separable spatiotemporal covariance kernels admitting a parametric representation. Indeed, such covariance kernels are not self-similar but can display local self-similarity, interpolating regular and fractal local behavior in space and time. The associated local fractional exponents are estimated from the empirical log-wavelet variogram. Numerical examples are simulated for illustrating the properties of the space–time covariance model class introduced.

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.

The article introduces spatial long-range dependent models based on the fractional difference operators associated with the Gegenbauer polynomials. The results on consistency and asymptotic normality of a class of minimum contrast estimators of long-range dependence parameters of the models are obtained. A methodology to verify assumptions for consistency and asymptotic normality of minimum contrast estimators is developed. Numerical results are presented to confirm the theoretical findings.

This paper presents new results on prediction of linear processes in function spaces. The autoregressive Hilbertian process framework of order one (ARH(1) process framework) is adopted. A componentwise estimator of the autocorrelation operator is formulated, from the moment-based estimation of its diagonal coefficients, with respect to the orthogonal eigenvectors of the auto-covariance operator, which are assumed to be known. Mean-square convergence to the theoretical autocorrelation operator, in the space of Hilbert-Schmidt operators, is proved. Consistency then follows in that space. For the associated ARH(1) plug-in predictor, mean absolute convergence to the corresponding conditional expectation, in the considered Hilbert space, is obtained. Hence, consistency in that space also holds. A simulation study is undertaken to illustrate the finite-large sample behavior of the formulated componentwise estimator and predictor. The performance of the presented approach is compared with alternative approaches in the previous and current ARH(1) framework literature, including the case of unknown eigenvectors., 50 pages (with 4 figures)

New results on functional prediction of the Ornstein-Uhlenbeck process in an autoregressive Hilbert-valued and Banach-valued frameworks are derived. Specifically, consistency of the maximum likelihood estimator of the autocorrelation operator, and of the associated plug-in predictor is obtained in both frameworks., 30 pages with 10 figures. Supplementary material (8 pages) is also included

In this paper we present novel results on the asymptotic be-havior of the so-called Ibragimov minimum contrast estimates. The case of tapered data for various models of Gaussian random fields is investigated. MSC 2010 subject classifications: Primary 62F12, 62M30; secondary 60G60.

Fractional-order pseudodifferential equations are considered to represent ocean climate variability when anomalous diffusion processes affect heat transfer in ocean surface. The driven process of these equations is assumed to be a regular spatiotemporal Gaussian random field representing normal conditions in the ocean. Linear regression in the log-wavelet domain is applied for the estimation of the parameters characterizing the pseudodifferential equation defining the anomalous diffusion process. The non-parametric framework is adopted in the estimation of the probability distribution of the driven spatiotemporal random field. Finally, ocean surface temperature values are approximated by plug-in least-square estimation from the computed parameter estimates, the estimated distribution of the driven process, and the integral version of the fractional-order pseudodifferential equation. The ability of the approach presented to process strong spatial-correlated ocean surface temperature curve data is illustrated with a real-data example, where sample information from weather stations in Hawaii ocean is analyzed.

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.

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.

This paper deals with the estimation of hidden periodicities in a non-linear regression model with stationary noise displaying cyclical dependence. Consistency and asymptotic normality are established for the least-squares estimates.

A minimum contrast criterion for consistently estimating the spatial fractal dimension in the wavelet domain is presented in [M. D. Ruiz--Medina and R. M. Crujeiras, Comm. Statist. Theory Methods., 40 (2011), pp. 3599--3613] for a class of semiparametric fractal Gaussian random fields, which includes fractional Brownian motion and Riesz--Bessel motion as particular cases. In this work, asymptotic normality of this minimum contrast estimator is obtained using central limit results for multiple stochastic integrals (see [N. Nualart and G. Peccati, Ann. Probab., 33 (2005), pp. 91--115]).

This paper derives the weak-sense Gaussian solution to a family of\ud fractional-in-time and multifractional-in-space stochastic partial differential\ud equations, driven by fractional-integrated-in-time spatiotemporal white\ud noise. Some fundamental results on the theory of pseudodifferential operators\ud of variable order, and on the Mittag-Leffler function are applied to\ud obtain the temporal, spatial and spatiotemporal H¨older continuity, in the\ud mean-square sense, of the derived solution.

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.

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.

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.

In this article, the estimation of spatiotemporal long-range dependence is formulated in the spectral wavelet domain. Sample information is provided by functional spectral data. Their high local singularity at the origin is captured by the wavelet transform. Weak consistency of the spectral wavelet estimators proposed is derived. Two functional estimation algorithms are implemented. A simulation study is developed to illustrate the efficiency of the computational methods derived. An approximation to the empirical convergence rate of the spectral wavelet periodogram is computed in some simulated examples.

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.

This article addresses the problem of defining a general scaling setting in which Gaussian and non-Gaussian limit distributions of linear random fields can be obtained. The linear random fields considered are defined by the convolution of a Green kernel, satisfying suitable scaling conditions, with a non-linear transformation of a Gaussian centered homogeneous random field. The results derived cover the weak-dependence and strong-dependence cases for such Gaussian random fields. Extension to more general random initial conditions defined, for example, in terms of non-linear transformations of χ2-random fields, is also discussed. For an example, we consider the random fractional diffusion equation. The vectorial version of the limit theorems derived is also formulated, including the limit distribution of the parabolically rescaled solution to the Burgers equation in the cases of weakly and strongly dependent initial potentials.

The autoregressive Hilbertian process framework has been introduced in Bosq (2000). This book provides the nonparametric estimation of the autocorrelation and covariance operators of the autoregressive Hilbertian processes. The asymptotic properties of these estimators are also provided. The maximum likelihood approach still remains unexplored. This paper obtains the asymptotic distribution of the maximum likelihood (ML) estimators of the auto-covariance operator of the Hilbert-valued innovation process, and of the autocorrelation operator of a Gaussian autoregressive Hilbertian process of order one. A real data example is analyzed in the financial context for illustration of the performance of the projection maximum likelihood estimation methodology in the context of missing data.

In this article, we study the effect of the geometry of a domain with variable local dimension on the regularity/singularity of the restriction of a multifractional random field on such a domain. The theories of reproducing kernel Hilbert spaces (RKHS) and generalized random fields are applied. Fractional Sobolev spaces of variable order are considered as RKHSs of random fields satisfying certain elliptic multifractional pseudodifferential equations. The multifractal spectra of these random fields are trivial due to the regularity assumptions on the variable order of the fractional derivatives. In this article, we introduce a family of RKHSs defined by isomorphic identification with the trace on a compact heterogeneous fractal domain of a fractional Sobolev space of variable order. The local regularity/singularity order of functions in these spaces, which depends on the variable order of the fractional Sobolev space considered and on the local dimension of the domain, is derived. We also study the spectra...

A functional classification methodology, based on the Reproducing Kernel Hilbert Space (RKHS) theory, is proposed for discrimination of gene expression profiles. The parameter function involved in the definition of the functional logistic regression is univocally and consistently estimated, from the minimization of the penalized negative log-likelihood over a RKHS generated by a suitable wavelet basis. An iterative descendent method, the gradient method, is applied for solving the corresponding minimization problem, i.e., for computing the functional estimate. Temporal gene expression data involved in the yeast cell cycle are classified with the wavelet-RKHS-based discrimination methodology considered. A simulation study is developed for testing the performance of this statistical classification methodology in comparison with other statistical discrimination procedures.

Spatial Functional Statistics has emerged as a powerful tool in the spatial and spatiotemporal analysis of data arising, for example, from Agriculture, Geology, Soils, Hydrology, Environment, Ecology, Mining, Oceanography, Air Quality, Remote Sensing, Spatial Econometrics, Epidemiology, just to mention a few areas of application. However, big black holes still exist in the development and implementation of new methodologies and approaches in this context. This paper provides an overview of the main references in the field of Spatial Functional Statistics, as well as the description of some key open research problems in this context.

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.

This paper focuses on the problem of functional statistical classification of gene expression curves. A local-wavelet-vaguelette-based functional logistic regression approach is presented. This approach is specially suitable for the classification of non-stationary singular (non-differentiable) curves. The performance of the methodology proposed is illustrated by implementing it for the classification of yeast cell-cycle temporal gene expression profiles. A simulation study is also carried out for comparison with other functional classification methodologies.

The class of spatial autoregressive Hilbertian models (SARH(1) processes) is considered. The projection estimation methodology proposed here is based on the biorthogonal eigenfunction bases diagonalizing the infinite-dimensional parameters involved in the SARH(1) state equation. These bases remove the ill-posed nature of the functional equation system defining the moment-based estimators of such parameters. The performance of the proposed projection estimation methodology, in the SARH(1) context, is illustrated in terms of simulated and real-data examples. In particular, this methodology provides a suitable spatial functional extrapolation of tropical and subtropical weak-dependence ocean surface temperature profiles, in the absence of high spatial concentration of weather stations, removing computational problems associated with matrix determinant close to zero. Copyright © 2011 John Wiley & Sons, Ltd.

The weak-consistency of the wavelet periodogram is established for a class of semiparametric fractal Gaussian random fields with local behavior equivalent to the fractional Brownian motion. This result is derived considering a class of isotropic wavelet functions of Haar type. As a consequence, the consistency of a minimum contrast estimator of the fractal parameter, based on the wavelet periodogram, is obtained.

The functional statistical framework is considered to address the problem of least-squares estimation of the realizations of fractal and long-range dependence Gaussian random signals, from the observation of the corresponding response surface. The statistical methodology applied is based on the functional regression model. The geometrical properties of the separable Hilbert spaces of functions, where the response surface and the signal of interest lie, are considered for removing the ill-posed nature of the estimation problem, due to the non-locality of the integro-pseudodifferential operators involved. Specifically, the local and asymptotic properties of the spectra of fractal and long-range dependence random fields in the Linnik-type, Dagum-type and auxiliary families are analyzed to derive a stable solution to the associated functional estimation problem. Their pseudodifferential representation and Reproducing Kernel Hilbert Space (RKHS) characterization are also derived for describing the geometrical properties of the spaces where the functional random variables involved in the corresponding regression problem can be found.

This paper addresses the introduction and study of structural properties of Hilbert-valued spatial autoregressive processes (SARH(1) processes), and Hilbert-valued spatial moving average processes (SMAH(1) processes), with innovations given by two-parameter (spatial) matingale differences. For inference purposes, the conditions under which the tensorial product of standard autoregressive Hilbertian (ARH(1)) processes (respectively, of standard moving average Hilbertian (MAH(1)) processes) is a standard SARH(1) process (respectively, it is a standard SMAH(1) process) are studied. Examples related to the spatial functional observation of two-parameter Markov and diffusion processes are provided. Some open research lines are described in relation to the formulation of SARMAH processes, as well as General Spatial Linear Processes in Functional Spaces.

This paper proposes a two-stage correlated non-linear shrinkage estimation methodology for spatial random processes. A block hard thresholding design, based on Shannon’s entropy, is formulated in the first stage. The thresholding design is adaptive to each resolution level, because it depends on the empirical distribution function of the mutual information ratios between empirical wavelet blocks and the random variables of interest, at each scale. In the second stage, a global correlated (inter- and intra-scale) shrinkage is applied to approximate the values of interest of the underlying spatial process. Additionally, a simulation study is developed, in the Gaussian context, to analyze the sensitivity, measured by empirical stochastic ordering, of the entropy-based block hard thresholding stage in relation to the parameters characterizing local variability (fractality) and dependence range of the spatial process of interest, the noise level, and the design of the region of interest.

In this article, we introduce a class of Markov processes whose transition probability densities are defined by multifractional pseudodifferential evolution equations on compact domains with variable local dimension. The infinitesimal generators of these Markov processes are given by the trace of strongly elliptic pseudodifferential operators of variable order on such domains. The results derived provide a pseudomultifractal version of some existing special classes of multifractional Markov processes. In particular, pseudostable processes are defined on domains with variable local dimension in this framework. In the case where the local dimension of the domain and the local Holder exponents of the transition probability densities are constant, the existing results on fractal versions of Levy processes are recovered.

A general functional class of spatial scalings is introduced, jointly with the logarithmic transformation of the temporal component, to get the convergence to the Gaussian limit distribution of the solution to the heat and Burgers equations with quadratic external potentials, considering weakly dependent Gaussian random initial conditions. The results derived extend the ones obtained in Leonenko and Ruiz-Medina [31] to a more general spatial scaling setting.

Pseudodifferential evolution models have been widely used in the description of biological, geophysical and environmental systems. We consider the case where functional sample information is available from such systems. Specifically, the observation model is defined in terms of a sequence of spatial realizations of the process of interest, solution to a spatiotemporal pseudodifferential equation, affected by additive strong Hilbertian white noise. In this paper, conditions for a stable computation of the solution to the associated functional filtering problem are established in terms of the covariance operator spectra of the process of interest and of the Hilbertian observation noise. In practice, such conditions are referred to the empirical spectra associated with the covariance operator estimators. A simulation study is developed to illustrate the results derived regarding robustness of the functional estimator against functional variability of the data.

In this paper the problem of functional filtering of an autoregressive Hilbertian (ARH) process, affected by additive Hilbertian noise, is addressed when the functional parameters defining the ARH(p) equation are unknown. The maximum-likelihood estimation of such parameters is obtained from the implementation of an expectation-maximization algorithm. Specifically, a finite-dimensional matrix approximation of the state equation is considered from its diagonalization in terms of the spectral decomposition of the functional parameters involved (Principal-Oscillation-Pattern-based diagonalization). The Expectation step and maximization step are then computed from the forward Kalman filtering followed by a backward Kalman smoothing recursion in terms of the Fourier coefficients associated with such a decomposition.

Estimation of the long-range dependence parameter in spatial processes using a semiparametric approach is studied. An extended formulation of the averaged periodogram method proposed in Robinson [1994. Semiparametric analysis of long memory time series. Ann. Statist. 22, 515–539] is derived, considering a certain homogeneous and isotropic behaviour of the spectral distribution in the low frequencies. The weak consistency of the estimator proposed is proved.

In this article, we consider self-similar covariogram models. In particular, these models arise in the study of the mean quadratic variation of fractional Brownian surfaces. For parameter estimation, three methods, respectively based on the integrated periodogram, the variogram, and the wavelet transform, are compared. Suitable implementation of these methods is achieved in the generalized random field framework. Namely, the connected fractality and heavy-tailed parameters are considered in the choice of functional bases for model simulation.

Functional data models provides a suitable framework for the statistical analysis of several environmental phenomena involving continuous time evolution and/or spatial variation. The functional autoregressive model of order p, p ≥ 1, (FAR(p)) extends to the infinite-dimensional space context the classical autoregressive model AR(p) (see, for example, Mourid T (1993) Processus autoregressiifs d’ordre superieur. Acad Sci t.317(Ser. I):1167–1172). In this paper, we derive a multidimensional diagonalization of the functional parameters (operators) involved in the FAR(p), p > 1, formulation. The functional state equation is then transformed into a discrete system of scalar state equations. The decomposition obtained is optimal regarding information on spatiotemporal interaction affecting the evolution of the spatial behavior of the process of interest. For functional prediction and filtering, we implement the Kalman filter equations from the diagonal version derived for FAR(p) models.

For a suitable scaling of the solution to the one-dimensional heat equation with spatial-dependent coefficients and weakly dependent random initial conditions, the convergence to the Gaussian limiting distribution is proved. The scaling proposed and methodology followed allow us to obtain Gaussian scenarios for related equations such as the one-dimensional Burgers equation as well as for the multidimensional formulation of both the heat and Burgers equations. Furthermore, the investigation of non-Gaussian scenarios is opened with a different proposed scaling, proving the convergence of the second-order moments.