F. J. Alonso

The Kriged Kalman filter

In recent years there has been growing interest in spatial-temporal modelling, partly due to the potential of large scale data in pollution and global climate monitoring to answer important environmental questions. We consider the Kriged Kalman filter (KKF), a powerful modelling strategy which combines the two wellestablished approaches of (a) Kriging, in the field of spatial statistics, and (b) the Kalman filter, in general state space formulations of multivariate time series analysis. We give a brief introduction to the model and describe its various properties, and highlight that the model allows prediction in time as well as in space, simultaneously. Some special cases of the time series model are considered. We give some procedures to implement the model, also illustrated through a practical example. The paper concludes with a discussion.

A state-space model approach to optimum spatial sampling design based on entropy

We consider the spatial sampling design problem for a random field X. This random field is in general assumed not to be directly observable, but sample information from a related variable Y is available. Our purpose in this paper is to present a state-space model approach to network design based on Shannons definition of entropy, and describe its main points with regard to some of the most common practical problems in spatial sampling design. For applications, an adaptation of Ko et al.s (1995) algorithm for maximum entropy sampling in this context is provided. We illustrate the methodology using piezometric data from the Velez aquifer (Malaga, Spain).

A study on sampling design for optimal prediction of space–time stochastic processes

Abstract. Optimal selection of sampling strategies is considered for the prediction of spatio-temporal processes in a state-space-model framework. General conditions are assumed in relation to the basic elements of the problem: modelling space-time interaction, formulating prediction objectives, defining the type and structure of sampling configurations, and formulating optimality criteria. An empirical study, involving a diversity of cases selected within two different examples, is carried out with the aim of illustrating some aspects of interest inherent to the problem considered, with special emphasis on highlighting the important effect of the space-time interaction structure on the ratios of information associated with different possible sampling configurations.

Semi-parametric statistical approaches for space-time process prediction

The problem of estimation and prediction of a spatial-temporal stochastic process, observed at regular times and irregularly in space, is considered. A mixed formulation involving a non- parametric component, accounting for a deterministic trend and the effect of exogenous variables, and a parametric component representing the purely spatio-temporal random variation is proposed. Correspondingly, a two-step procedure, first addressing the estimation of the non- parametric component, and then the estimation of the parametric component is developed from the residual series obtained, with spatial-temporal prediction being performed in terms of suitable spatial interpolation of the temporal variation structure. The proposed model formula-tion, together with the estimation and prediction procedure, are applied using a Gaussian ARMA structure for temporal modelling to space-time forecasting from real data of air pollution concentration levels in the region surrounding a power station in northwest Spain.

Estimation and smoothing from incomplete data for a class of lattice processes

Multidimensional discrete-parameter processes with factorable covariance structure are of great importance for applications and approximations to certain continuous parameter processes. In practical situations, usually only incomplete data are available, so state-space schemes are normally used for modelling and prediction. In this work we describe maximum-likelihood estimation and smoothing for doubly geometric lattice processes using incomplete data. The procedure proposed is based on an application of the EM algorithm, and is inspired by its use in time-series analysis. Minimum mean-square-error prediction is also described. Extension to more general models is commented on. Some examples using simulated data are provided.

A study on sensitivity of spatial sampling designs to a priori discretization schemes

In a previous paper (Environ. Ecol. Stat. 5 (1998) 29.) we presented an entropy-based approach to spatial sampling design in a state-space model framework. We now address the problem of sensitivity of optimal designs with respect to the configuration of the set of potential observation sites considered, as well as to the model specifications. The latter involve both the spatial dependence structure of the variable of interest and its relationship with the observable variable. To analyze several aspects related to this problem, we have developed an extensive empirical study, from which we conclude the critical influence that the a priori selection of candidate observation sites can have on the final sampling designs for different situations. 2004 Elsevier Ltd. All rights reserved.

Spectral-Marginal-Based Estimation of Spatiotemporal Long-Range Dependence

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.

Application of em-type algorithms to spatial data

We consider the problem of maximum-likelihood estimation and smoothing for lattice processes using incomplete data. In a previous paper (Alonso et al. 1996) the authors developed a methodology based on an application of the EM algorithm on a state-space framework for this problem. Now, the procedure is extended using new versions of EM-type algorithms (ECM and MCECM). This has computational advantages, especially when there are many parameters to estimate. The problem of estimating the asymptotic covariance matrix for the parameter estimators is also considered (supplemented EM-type algorithms). The steps are described through an application considering the underlying state model to have an AR(l)xAR(l) structure and extension to more general models is commented on. As an example, we apply the method to the data presented by Kempton and Howes (1981).

Parameter Estimation of Self-Similar Spatial Covariogram Models

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.

Discrete-time Markovian arrival processes to model multi-state complex systems with loss of units and an indeterminate variable number of repairpersons

Abstract In this study, three discrete-time multi-state complex systems subject to multiple events are modeled, in a well structured form, as Markovian arrival processes with marked arrivals. The systems, ranked by the number of events affecting the online unit, have multiple and variable repairpersons, and the online unit are partitioned into performance stages. The first system is subject only to internal failures. The second, additionally, considers external shocks, which can produce any of three consequences; extreme failure, degradation of the internal performance of the online unit or cumulative damage. Failure may be repairable or non-repairable. The repair facility is composed of an indeterminate number of repairpersons. When a non-repairable failure occurs, the number of repairpersons may be modified. Finally, the third system includes preventive maintenance in combination with random inspections. Various measures are incorporated, in an algorithmic and computational form, in transient and stationary regimes. Costs and rewards are included in the model to optimize the system from different standpoints. The results of this study are implemented computationally with Matlab, and a numerical example shows the versatility of the modeling.