The aim of this paper is to propose three approaches to analyse spatio- temporal environmental data supposing that the above mentioned data can be easily represented using dynamic space-time interaction models. From the current literature on this topic, appear that the study of a space-time dynamic phenomenon, can be performed in the data domain, using space-time auto-correlation or  state- space approach and in the frequencies domain, through bi-dimensional spectral analysis.
Departing from this consideration we consider:

1) Generalized Spatio-Temporal Autoregressive Moving Average Models (GSTARMA) when the data
    are stationary both in time and space.

2) State- Space Spatio Temporal Dynamic Model (SSSTD) when the data are not stationary in time or
    and in space and equally spaced.

3) Bi-dimensional Spectral Analysis (BSA) when are unknown the dynamic mechanisms of the system.

The three proposed methodologies have been compared with an application to environmental data on sea-water pollution observed in the Adriatic sea.

MAURO COLI
Dipartimento di Metodi Quantitativi e Teoria Economica
University G. d'Annunzio
Viale Pindaro, 42
Pescara, 65127, Italy
coli@dmqte.unich.it

Gaussian Autoregressive models play a prominent role in the modeling of spatial lattice series.  Important applications include the analysis of the impacts of environment quality on risks for human health, production of goods and economic activity, to mention a few. In these cases, regional frequencies of interest are often modeled by a Binomial or Poisson distribution depending on a number of parameters and, according to the empirical bayesian paradigm, a suitable transformation of the aforementioned parameters is modeled with a spatial Autoregressive (or Auto-Gaussian) model depending on interaction iperparameters. Although these hierarchical  models are highly parametric, they often show a good performance, accounting for unequal spatial variation and spatial dependence of data.
Spatial Autoregressive models can be viewed as a particular case of the general linear model specification, with their main feature being that the covariance error matrix is modeled by some neighborhood structure among observation sites.
Unfortunately, though, in most application positing the correct correlational structure among errors is more the exception than the rule, as a specification error in the error covariance matrix may occur. Two of the most common cases are: (1) the neighborhood structure may be misspecified (e.g., links that exist are not included and/or links that do not exist have been included: neighborhood structure specification error, NSSE), or, more generally, (2) the dependence structure may be misspecified (e.g., spatial stationariety is misspecified for a non-stationary situation: correlational structure specification error, CSSE).
This communication is devoted to study the distributional properties of maximum likelihood estimators under NSSE or CSSE (e.g. bias, consistency, efficiency), and the relating distortion of confidence intervals.  After reviewing some known asymptotic results, the main focus will be on the finite lattice properties of these estimators under various specification errors, with discussion of analytical results and simulations studies.

FRANCESCO LAGONA
Centre for Economics of Institutions
University "Rome 3"
via Corrado Segre 4
Rome 00146, Italy
lagona@uniroma3.it

In the analysis of multivariate spatial data several difficulties arise. Often, because of the nature itself of the phenomena under study, we cannot assume independence between observations, and instead seek to model the specific type of dependence we observe.  We usually model these datasets as random realizations of a multivariate spatial random field (MSRF), with specific assumptions on the dependencies among the variables -- which comprise the combined observation vector and the spatial coordinates.  However, before any serious attempt to specify a model can be made, we have to explore the available data as deeply as possible. This  is made very difficult by the high dimensionality of the data. Standard multivariate exploratory techniques such as principal components analysis (PCA) and  multidimensional scaling, usually fail to give us a good representation, as they do not explicitly take into account the spatial arrangement of observations.
Here  an exploratory technique based on the diagonalization of cross-variogram matrices is described. Through the definition of a model  for the analysis and simulation of multivariate  spatial data, a test procedure for the assumption of isotropy of multivariate spatial data is proposed. Applications to simulated and real data will be briefly reported.

GIOVANNA JONA LASINIO
Dipartimento di Statistica, Probabilita e Stat. Appl.
University of Rome "La Sapienza"
P.le Aldo More 5
Rome 00185, Italy
jona@pow2.sta.uniroma1.it