Modeling of and inference on multivariate data that have been measured in space, such as temperature and pressure, are challenging tasks in environmental sciences, physics and materials science. We give an overview over and some background on modeling with crosscovariance models. The R package RandomFields supports the simulation, the parameter estimation and the prediction in particular for the linear model of coregionalization, the multivariate Matérn models, the delay model, and a spectrum of physically motivated vector valued models. An example on weather data is considered, illustrating the use of RandomFields for parameter estimation and prediction.

Estimation of extreme value parameters from observations in the max-domain of attraction of a multivariate max-stable distribution commonly uses aggregated data such as block maxima. Multivariate peaks-over-threshold methods, in contrast, exploit additional information from the non-aggregated 'large' observations. We introduce an approach based on peaks over thresholds that provides several new estimators for processes η in the max-domain of attraction of the frequently used Hüsler-Reiss model and its spatial extension: Brown-Resnick processes. The method relies on increments η. / η.t 0 / conditional on η.t 0 / exceeding a high threshold, where t 0 is a fixed location. When the marginals are standardized to the Gumbel distribution, these increments asymptotically form a Gaussian process resulting in computationally simple estimates of the Hüsler-Reiss parameter matrix and particularly enables parametric inference for Brown-Resnick processes based on (high dimensional) multivariate densities. This is a major advantage over composite likelihood methods that are commonly used in spatial extreme value statistics since they rely only on bivariate densities. A simulation study compares the performance of the new estimators with other commonly used methods. As an application, we fit a non-isotropic Brown-Resnick process to the extremes of 12-year data of daily wind speed measurements.

For a non-stationary or non-ergodic marked point process (MPP) on R d , the definition of averages becomes ambiguous as the process might have a different stochastic behavior in different realizations (non-ergodicity) or in different areas of the observation window (non-stationarity). We investigate different definitions for the moments, including a new hierarchical definition for nonergodic MPPs, and embed them into a family of weighted mean marks. We point out examples of application in which different weighted mean marks all have a sensible meaning. Further, asymptotic properties of the corresponding estimators are investigated as well as optimal weighting procedures.

In this paper we provide the basis for new methods of inference for max-stable processes ξ on general spaces that admit a certain incremental representation, which, in important cases, has a much simpler structure than the max-stable process itself. A corresponding peaks-over-threshold approach will incorporate all single events that are extreme in some sense and will therefore rely on a substantially larger amount of data in comparison to estimation procedures based on block maxima. Conditioning a process η in the max-domain of attraction of ξ on being extremal, several convergence results for the increments of η are proved. In a similar way, the shape functions of mixed moving maxima (M3) processes can be extracted from suitably conditioned single events η. Connecting the two approaches, transformation formulae for processes that admit both an incremental and an M3 representation are identified.