- Prof. Wenceslao González Manteiga – University Santiago Compostela / Raquel Menezes Leite – University Minho / Prof. Wolfgang Polasek Institute Advanced Studies, Austria – University Porto / Prof. Feridun Turkman – University Lisboa
- Anfiteatro Fundação FCUL (FCUL) – Bloco C1 – Piso 3 – Campo Grande
- Terça-feira, 11 de Janeiro de 2011

**Half-day Workshop on Spatio-temporal Modelling **

** **

** ****January 11th 2011 — 14:00-17:30 — Lisbon, Portugal **

** Organized by CEAUL (****http://www.ceaul.fc.ul.pt/****)**

**PROGRAM:**

14:00-14:45 —

**WENCESLAO GONZÁLEZ MANTEIGA — University of Santiago de Compostela — “Testing the Dependence Structure in Spatial and Spatio-temporal Models”**

ABSTRACT: In this talk, different techniques for testing about the dependence in spatial and spatio-temporal models will be introduced. The proposed methods are based on the spectral representation of the dependence structure: the spectral density. This function is the Fourier transform of the covariance for second-order stationary processes and nonparametric estimators are easily computed from samples obtained on regular grids. The most well-know nonparametric estimator of the spectral density is the periodogram. Periodogram values at different frequencies are asymptotically independent, an advantage with respect to covariance estimators. In addition, the periodogram can be seen as the response variable in a multiplicative regression model, where the regression function is the spectral density. Equivalently, the log-periodogram is the response in an additive regression model where the regression function is the log-spectral density. From this interpretation, statistical procedures from the regression context can be adapted into this setting for approaching different testing problems about the dependence structure. Specifically, we will talk about: 1. Goodness-of-fit tests for the spectral density in a spatial model (see [3] for goodness-of-fit tests based on the periodogram and log-periodogram representations). 2. Comparison of dependence structures (see [1] and [2], for an L2 test for comparing spectral densities). 3. Testing for separability in spatio-temporal models (see [4], for a nonparametric test for separability, based on additive regression models). For all these problems, detailed explanations of the theory developments can be found in references below. We will present some asymptotic results, as well as simulation studies and algorithms for the practical implementation of the proposed techniques. Illustration with real data examples will be also provided.

References: [1] Crujeiras, R., Fernández-Casal, R. and González-Manteiga, W. (2007). Comparing spatial dependence structures using spectral density estimators. Environmetrics, 18, 793-808. [2] Crujeiras, R., Fernández-Casal, R. and González-Manteiga, W. (2008). An L2-test for comparing spatial spectral densities. Statistics & Probability Letters, 78, 2543-2551. [3] Crujeiras, R., Fernández-Casal, R. and González-Manteiga, W. (2010). Goodness-of-fit tests for the spatial spectral density. Stochastic Environmental Research and Risk Assessment, 24, 67-79. [4] Crujeiras, R., Fernández-Casal, R. and González-Manteiga, W. (2010). Nonparametric test for separability of spatio-temporal processes. Environmetrics, 21, 382-399.

**14:45-15:30 — RAQUEL MENEZES — University of Minho — “A Kernel Indicator Variogram and its Application to Environmental Data” **

ABSTRACT: This talk deals with the estimation of the distribution function of a spatial random process. Firstly, we propose the application of the indicator kriging approach, which demands an indicator variogram (or the indicator covariance function) to be calculated. For the latter aim, we suggest a kernel-type estimator, whose consistency will be proved, under several assumptions. In addition, we will check that the approximation of the sill of the kernel indicator variogram provides another mechanism for estimation of the distribution function. Numerical studies will be presented to illustrate the performance of both approaches for approximation of the distribution function. Furthermore, we will present an application of proposed techniques to a real environmental data set, where the presence of nitrate in groundwater in Beja district (Portugal) is measured.

15:30-16:00 — Coffee break

**16:00-16:45 — WOLFGANG POLASEK — Institute of Advanced Studies, Austria and University of Porto — “Spatial Chow-Lin methods for data completion in econometric flow models” **

ABSTRACT: Flow data across regions can be modeled by spatial econometric model, see LeSage and Pace (2009). Recently, regional studies became interested in the aggregation and disaggregation of flows model, because trade data cannot be obtained at a disaggregated level but data are published on an aggregate level. Furthermore, missing data in disaggregated flow models occur quite often since detailed measurements are often not possible at all observation points in time and space. In this paper we develop classical and Bayesian methods to complete flow data. The Chow and Lin (1971) method was developed for completing disaggregated incomplete time series data. We will extend this method in a general framework to spatially correlated flow data using the cross-sectional Chow-Lin method of Polasek et al. (2009). The missing disaggregated data can be obtained either by feasible GLS prediction or by a Bayesian (posterior) predictive density. Furthermore, the approach is based on the MCMC estimation method of LeSage and Pace (2009) for flow data and systems of panel data. The new disaggregation method is demonstrated with examples involving regional trade in Europe.

**16:45-17:30 — FERIDUN TURKMAN — University of Lisboa — “Discrete-Continuous Time Extremes of Stationary Processes” **

ABSTRACT: In many applications, the primary interest is the supremum of some continuous time process over a specified period. However, data are usually available over a discrete set of times and the inference can only be made for the maximum of the process over this discrete set of times. If we want to estimate the extremes of the continuous time process based on discrete time data, we need to understand the relationship between the continuous and discrete extremes. Thus, we look at asymptotic joint distributions of the maxima of stationary processes and their discrete versions.

**Place:** Anfiteatro da Fundação FCUL (C1, 3º Piso), Faculdade de Ciências de Universidade de Lisboa.

**Free registration required:** please confirm your presence to

ceaul@fc.ul.pt until January 4.

For any questions, contact gsilva@math.ist.utl.pt (Organizer: Giovani Silva)