Stochastic Environmental Research and Risk Assessment, 2018
Structural characteristics of random field excursion sets defined by threshold exceedances provid... more Structural characteristics of random field excursion sets defined by threshold exceedances provide meaningful indicators for the description of extremal behaviour in the spatiotemporal dynamics of environmental systems, and for risk assessment. In this paper a conditional approach for analysis at global and regional scales is introduced, performed by implementation of risk measures under proper model-based integration of available knowledge. Specifically, quantile-based measures, such as Value-at-Risk and Average Value-at-Risk, are applied based on the empirical distributions derived from conditional simulation for different threshold exceedance indicators, allowing the construction of meaningful dynamic risk maps. Significant aspects of the application of this methodology, regarding the nature and the properties (e.g. local variability, dependence range, marginal distributions) of the underlying random field, as well as in relation to the increasing value of the reference threshold, are discussed and illustrated based on simulation under a variety of scenarios.
Physical space-time metrics are used in environmental modeling to define "distance" between point... more Physical space-time metrics are used in environmental modeling to define "distance" between points in the space-time domain of a physical attribute (contaminant concentration, exposure, temperature etc.). Assessing a space-time metric is often a considerably more complicated affair than assessing a purely spatial metric. This is because the physical space-time metric suggests a certain concept of distance that blends space and time to make space-time, but at the same time, it views time as a dissimilar quantity. In this work, the determination of space-time metrics takes advantage of the strong links between the physical characteristics of the real-world attribute and the geometrical features of the composite space-time domain within which the attribute occurs. Via physical law an explicit connection is established between attribute's space-time dependence structure (represented by the covariance function) and attribute's domain geometry (expressed by the metric coefficients). The derived physical geometry equation can be solved for the metric coefficients. The solution depends not only on the form of the physical law, but also on the boundary/initial conditions and the randomness sources. The proposed approach turns metric coefficients into physically meaningful parameters, allowing better understanding of the space-time characteristics than the ad hoc and arbitrary metric selection in purely technical terms.
In this article, the current situation of the teaching of Statistics in Spain, mainly in Secondar... more In this article, the current situation of the teaching of Statistics in Spain, mainly in Secondary and High School Education, is assessed. The analysis, made by the Spanish Society of Statistics and Operations Re-search (SEIO), was transmitted to the Spanish Ministry of Education, Culture and Sports, in the context of the discussion and implementation of the new Education law called LOMCE.
We consider a fractional-order differential equation involving fractal activity time to represent... more We consider a fractional-order differential equation involving fractal activity time to represent the stochastic behaviour of a log-price process of an underlying asset. The log-price process is defined in terms of fractional integration of the fractional derivative of Brownian motion on fractal time. A stable solution to the extrapolation and filtering problems associated is obtained in terms of covariance vaguelette functions (Angulo and Ruiz-Medina 1999). A simulation study is carried out to illustrate the methodology presented.
In this paper, we apply the theory of pseudodifferential operators and Sobolev spaces to characte... more In this paper, we apply the theory of pseudodifferential operators and Sobolev spaces to characterize fractional and multifractional probability densities. In the fractional case, local regularity properties of the probability density function are given in terms of fractional moment conditions satisfied by the characteristic function. Conversely, the parameter defining the order of the fractional Sobolev space where the characteristic function
The use of general descriptive names, registered names, trademarks, etc. in this publication does... more The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Recent Advances in Stochastic Modeling and Data Analysis, 2007
... Applications in 2D Lognormal Diffusions* Ramon Gutierrez1, Conception Roldan2, Ramon Gutierre... more ... Applications in 2D Lognormal Diffusions* Ramon Gutierrez1, Conception Roldan2, Ramon Gutierrez-Sanchez1, and Jose M. Angulo1 1 ... the null hypothesis pi with 0*=(0,... To), we have that 1/l" np ... [Gutierrez et al., 2005] R. Gutierrez, C. Roldan, R. Gutierrez-Sanchez, and JM ...
In this article, we study the effect of the geometry of a domain with variable local dimension on... more 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
Journal of Statistical Planning and Inference, 1999
The principle of minimum description length (MDL) provides an approach for selecting the model cl... more The principle of minimum description length (MDL) provides an approach for selecting the model class with the smallest stochastic complexity of the data among a set of model classes. However, when only incomplete data are available the stochastic complexity for the complete data cannot be numerically computed. In this paper, this problem is solved by introducing a notion of expected stochastic complexity for the complete data conditional on the observed data, which can be computed by the EM algorithm. Based on this notion, model selection from incomplete data can also be performed by the MDL principle. A simulation study is presented for illustration of the methodology.
Computational Statistics & Data Analysis, 2007
The autoregressive Hilbertian model of order one (ARH(1)) is considered to represent the dynamics... more The autoregressive Hilbertian model of order one (ARH(1)) is considered to represent the dynamics of a sequence of spatial functional data. Spatiotemporal interaction is defined in terms of the autocorrelation operator. A diagonalization of ARH(1) models is derived based on the functional principal oscillation pattern (POP) decomposition of such an operator. The results are applied to implement the Kalman filter for spatiotemporal prediction from the information provided by the observation of a finite sequence of spatial functional data.
Communications in Statistics - Theory and Methods, 2008
Long-memory and strong spatial dependence are two features which can arise jointly or separately ... more Long-memory and strong spatial dependence are two features which can arise jointly or separately depending on the tail behavior of the temporal and spatial covariance functions of a given spatiotemporal process. Under certain conditions, such a behavior can be related to the variation of temporal and spatial frequencies in a neighborhood of the origin. In particular, a spatiotemporal process displaying long-memory and/or strong spatial dependence can be built, in terms of separable heavy-tail filters, from an input second-order process satisfying suitable regularity and moment conditions. A parameter estimation method based on the marginal spectral densities is implemented to approximate the long-memory and/or strong-spatial-dependence parameters.
Stochastic Environmental Research and Risk Assessment, 2007
This paper introduces two families of space-time random models with multifractal spatial characte... more This paper introduces two families of space-time random models with multifractal spatial characteristics, respectively generated in continuous and discrete time, and both defined on multifractal spatial domains. The definition of the first class is given in terms of a Feller semigroup generated by a pseudodifferential operator of variable order. In the second case, the spatial process at time t + 1 is obtained by applying a variable order blurring operator to the spatial process at time t and adding the innovation given by a spatiotemporal process uncorrelated in time. Spatial multifractal properties of the two classes of space-time processes introduced are analyzed. The implementation of space-time filtering and prediction techniques is also discussed.
We consider estimation and smoothing for irregularly observed two dimensional spatial processes, ... more We consider estimation and smoothing for irregularly observed two dimensional spatial processes, assumed to be generated by certain stochastic partial differential equations. Such processes have been considered by Jones (1989) who gives an example arising in hydrology. Jones’ work, in turn, is descended from the seminal paper by Whittle (1954) where he studied a two-dimensional Laplace equation and data observed over a complete grid. Our objective here is to adapt the approaches considered in the above papers to processes satisfying more general stochastic partial differential equations where such processes are observed over an irregular grid. We consider various Fourier approximations for the covariance matrix, expressed in terms of the discrete Fourier transform of the spectral density, and apply them to problems of approximating the likelihood function and to minimum mean square error smoothing when the spatial processes are observed with error. The approximations are applied to ...
Methodology and Computing in Applied Probability, 2016
The behavior of generalized relative complexity measures is studied for assessment of structural ... more The behavior of generalized relative complexity measures is studied for assessment of structural dependence in a random vector. A related optimality criterion to sampling network design, which provides a flexible extension of mutual information based methods previously introduced, is formulated. Aspects related to practical implementation and conceptual issues regarding the meaning and potential use of this new approach are discussed. Numerical examples are used for illustration.
The linear inverse problem of estimating the input random field in a first-kind stochastic integr... more The linear inverse problem of estimating the input random field in a first-kind stochastic integral equation relating two random fields is considered. For a wide class of integral operators, which includes the positive rational functions of a self-adjoint elliptic differential operator on L2(ℝd), the ill-posed nature of the problem disappears when such operators are defined between appropriate fractional Sobolev spaces. In this paper, we exploit this fact to reconstruct the input random field from the orthogonal expansion (i.e. with uncorrelated coefficients) derived for the output random field in terms of wavelet bases, transformed by a linear operator factorizing the output covariance operator. More specifically, conditions under which the direct orthogonal expansion of the output random field coincides with the integral transformation of the orthogonal expansion derived for the input random field, in terms of an orthonormal wavelet basis, are studied.
This paper introduces a fractional heat equation, where the diffusion operator is the composition... more This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is formulated in terms of the Fourier transform of its spatially and temporally homogeneous Green function. The spectral density of the resulting solution is then obtained explicitly. The result implies that the solution of the fractional heat equation may possess spatial long-range dependence asymptotically.
Stochastic Environmental Research and Risk Assessment, 2007
In this paper, a class of spatiotemporal random field models defined as mean-square solutions of ... more In this paper, a class of spatiotemporal random field models defined as mean-square solutions of fractional versions of the stochastic heat equation are considered. Different sampling schemes in space and time are introduced to solve the problem of estimation of fully parameterized spatiotemporal random fields.
Stochastic Environmental Research and Risk Assessment, 2006
A wavelet-based orthogonal decomposition of the solution to stochastic differential/pseudodiffere... more A wavelet-based orthogonal decomposition of the solution to stochastic differential/pseudodifferential equations of parabolic type is derived in the cases of random initial conditions and random forcing. The family of spatiotemporal models considered can represent anomalous diffusion processes when the spatial operator involved is a fractional or multifractional pseudodifferential operator. The results obtained are applied to the generation of the sample
Stochastic Environmental Research and Risk Assessment, 2018
Structural characteristics of random field excursion sets defined by threshold exceedances provid... more Structural characteristics of random field excursion sets defined by threshold exceedances provide meaningful indicators for the description of extremal behaviour in the spatiotemporal dynamics of environmental systems, and for risk assessment. In this paper a conditional approach for analysis at global and regional scales is introduced, performed by implementation of risk measures under proper model-based integration of available knowledge. Specifically, quantile-based measures, such as Value-at-Risk and Average Value-at-Risk, are applied based on the empirical distributions derived from conditional simulation for different threshold exceedance indicators, allowing the construction of meaningful dynamic risk maps. Significant aspects of the application of this methodology, regarding the nature and the properties (e.g. local variability, dependence range, marginal distributions) of the underlying random field, as well as in relation to the increasing value of the reference threshold, are discussed and illustrated based on simulation under a variety of scenarios.
Physical space-time metrics are used in environmental modeling to define "distance" between point... more Physical space-time metrics are used in environmental modeling to define "distance" between points in the space-time domain of a physical attribute (contaminant concentration, exposure, temperature etc.). Assessing a space-time metric is often a considerably more complicated affair than assessing a purely spatial metric. This is because the physical space-time metric suggests a certain concept of distance that blends space and time to make space-time, but at the same time, it views time as a dissimilar quantity. In this work, the determination of space-time metrics takes advantage of the strong links between the physical characteristics of the real-world attribute and the geometrical features of the composite space-time domain within which the attribute occurs. Via physical law an explicit connection is established between attribute's space-time dependence structure (represented by the covariance function) and attribute's domain geometry (expressed by the metric coefficients). The derived physical geometry equation can be solved for the metric coefficients. The solution depends not only on the form of the physical law, but also on the boundary/initial conditions and the randomness sources. The proposed approach turns metric coefficients into physically meaningful parameters, allowing better understanding of the space-time characteristics than the ad hoc and arbitrary metric selection in purely technical terms.
In this article, the current situation of the teaching of Statistics in Spain, mainly in Secondar... more In this article, the current situation of the teaching of Statistics in Spain, mainly in Secondary and High School Education, is assessed. The analysis, made by the Spanish Society of Statistics and Operations Re-search (SEIO), was transmitted to the Spanish Ministry of Education, Culture and Sports, in the context of the discussion and implementation of the new Education law called LOMCE.
We consider a fractional-order differential equation involving fractal activity time to represent... more We consider a fractional-order differential equation involving fractal activity time to represent the stochastic behaviour of a log-price process of an underlying asset. The log-price process is defined in terms of fractional integration of the fractional derivative of Brownian motion on fractal time. A stable solution to the extrapolation and filtering problems associated is obtained in terms of covariance vaguelette functions (Angulo and Ruiz-Medina 1999). A simulation study is carried out to illustrate the methodology presented.
In this paper, we apply the theory of pseudodifferential operators and Sobolev spaces to characte... more In this paper, we apply the theory of pseudodifferential operators and Sobolev spaces to characterize fractional and multifractional probability densities. In the fractional case, local regularity properties of the probability density function are given in terms of fractional moment conditions satisfied by the characteristic function. Conversely, the parameter defining the order of the fractional Sobolev space where the characteristic function
The use of general descriptive names, registered names, trademarks, etc. in this publication does... more The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Recent Advances in Stochastic Modeling and Data Analysis, 2007
... Applications in 2D Lognormal Diffusions* Ramon Gutierrez1, Conception Roldan2, Ramon Gutierre... more ... Applications in 2D Lognormal Diffusions* Ramon Gutierrez1, Conception Roldan2, Ramon Gutierrez-Sanchez1, and Jose M. Angulo1 1 ... the null hypothesis pi with 0*=(0,... To), we have that 1/l" np ... [Gutierrez et al., 2005] R. Gutierrez, C. Roldan, R. Gutierrez-Sanchez, and JM ...
In this article, we study the effect of the geometry of a domain with variable local dimension on... more 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
Journal of Statistical Planning and Inference, 1999
The principle of minimum description length (MDL) provides an approach for selecting the model cl... more The principle of minimum description length (MDL) provides an approach for selecting the model class with the smallest stochastic complexity of the data among a set of model classes. However, when only incomplete data are available the stochastic complexity for the complete data cannot be numerically computed. In this paper, this problem is solved by introducing a notion of expected stochastic complexity for the complete data conditional on the observed data, which can be computed by the EM algorithm. Based on this notion, model selection from incomplete data can also be performed by the MDL principle. A simulation study is presented for illustration of the methodology.
Computational Statistics & Data Analysis, 2007
The autoregressive Hilbertian model of order one (ARH(1)) is considered to represent the dynamics... more The autoregressive Hilbertian model of order one (ARH(1)) is considered to represent the dynamics of a sequence of spatial functional data. Spatiotemporal interaction is defined in terms of the autocorrelation operator. A diagonalization of ARH(1) models is derived based on the functional principal oscillation pattern (POP) decomposition of such an operator. The results are applied to implement the Kalman filter for spatiotemporal prediction from the information provided by the observation of a finite sequence of spatial functional data.
Communications in Statistics - Theory and Methods, 2008
Long-memory and strong spatial dependence are two features which can arise jointly or separately ... more Long-memory and strong spatial dependence are two features which can arise jointly or separately depending on the tail behavior of the temporal and spatial covariance functions of a given spatiotemporal process. Under certain conditions, such a behavior can be related to the variation of temporal and spatial frequencies in a neighborhood of the origin. In particular, a spatiotemporal process displaying long-memory and/or strong spatial dependence can be built, in terms of separable heavy-tail filters, from an input second-order process satisfying suitable regularity and moment conditions. A parameter estimation method based on the marginal spectral densities is implemented to approximate the long-memory and/or strong-spatial-dependence parameters.
Stochastic Environmental Research and Risk Assessment, 2007
This paper introduces two families of space-time random models with multifractal spatial characte... more This paper introduces two families of space-time random models with multifractal spatial characteristics, respectively generated in continuous and discrete time, and both defined on multifractal spatial domains. The definition of the first class is given in terms of a Feller semigroup generated by a pseudodifferential operator of variable order. In the second case, the spatial process at time t + 1 is obtained by applying a variable order blurring operator to the spatial process at time t and adding the innovation given by a spatiotemporal process uncorrelated in time. Spatial multifractal properties of the two classes of space-time processes introduced are analyzed. The implementation of space-time filtering and prediction techniques is also discussed.
We consider estimation and smoothing for irregularly observed two dimensional spatial processes, ... more We consider estimation and smoothing for irregularly observed two dimensional spatial processes, assumed to be generated by certain stochastic partial differential equations. Such processes have been considered by Jones (1989) who gives an example arising in hydrology. Jones’ work, in turn, is descended from the seminal paper by Whittle (1954) where he studied a two-dimensional Laplace equation and data observed over a complete grid. Our objective here is to adapt the approaches considered in the above papers to processes satisfying more general stochastic partial differential equations where such processes are observed over an irregular grid. We consider various Fourier approximations for the covariance matrix, expressed in terms of the discrete Fourier transform of the spectral density, and apply them to problems of approximating the likelihood function and to minimum mean square error smoothing when the spatial processes are observed with error. The approximations are applied to ...
Methodology and Computing in Applied Probability, 2016
The behavior of generalized relative complexity measures is studied for assessment of structural ... more The behavior of generalized relative complexity measures is studied for assessment of structural dependence in a random vector. A related optimality criterion to sampling network design, which provides a flexible extension of mutual information based methods previously introduced, is formulated. Aspects related to practical implementation and conceptual issues regarding the meaning and potential use of this new approach are discussed. Numerical examples are used for illustration.
The linear inverse problem of estimating the input random field in a first-kind stochastic integr... more The linear inverse problem of estimating the input random field in a first-kind stochastic integral equation relating two random fields is considered. For a wide class of integral operators, which includes the positive rational functions of a self-adjoint elliptic differential operator on L2(ℝd), the ill-posed nature of the problem disappears when such operators are defined between appropriate fractional Sobolev spaces. In this paper, we exploit this fact to reconstruct the input random field from the orthogonal expansion (i.e. with uncorrelated coefficients) derived for the output random field in terms of wavelet bases, transformed by a linear operator factorizing the output covariance operator. More specifically, conditions under which the direct orthogonal expansion of the output random field coincides with the integral transformation of the orthogonal expansion derived for the input random field, in terms of an orthonormal wavelet basis, are studied.
This paper introduces a fractional heat equation, where the diffusion operator is the composition... more This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is formulated in terms of the Fourier transform of its spatially and temporally homogeneous Green function. The spectral density of the resulting solution is then obtained explicitly. The result implies that the solution of the fractional heat equation may possess spatial long-range dependence asymptotically.
Stochastic Environmental Research and Risk Assessment, 2007
In this paper, a class of spatiotemporal random field models defined as mean-square solutions of ... more In this paper, a class of spatiotemporal random field models defined as mean-square solutions of fractional versions of the stochastic heat equation are considered. Different sampling schemes in space and time are introduced to solve the problem of estimation of fully parameterized spatiotemporal random fields.
Stochastic Environmental Research and Risk Assessment, 2006
A wavelet-based orthogonal decomposition of the solution to stochastic differential/pseudodiffere... more A wavelet-based orthogonal decomposition of the solution to stochastic differential/pseudodifferential equations of parabolic type is derived in the cases of random initial conditions and random forcing. The family of spatiotemporal models considered can represent anomalous diffusion processes when the spatial operator involved is a fractional or multifractional pseudodifferential operator. The results obtained are applied to the generation of the sample
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Papers by José Angulo