Multivariate functional data can be intrinsically multivariate like movement trajectories in 2D or complementary such as precipitation, temperature and wind speeds over time at a given weather station. We propose a multivariate functional additive mixed model (multiFAMM) and show its application to both data situations using examples from sports science (movement trajectories of snooker players) and phonetic science (acoustic signals and articulation of consonants). The approach includes linear and nonlinear covariate effects and models the dependency structure between the dimensions of the responses using multivariate functional principal component analysis. Multivariate functional random intercepts capture both the auto-correlation within a given function and cross-correlations between the multivariate functional dimensions. They also allow us to model between-function correlations as induced by, for example, repeated measurements or crossed study designs. Modelling the dependency structure between the dimensions can generate additional insight into the properties of the multivariate functional process, improves the estimation of random effects, and yields corrected confidence bands for covariate effects. Extensive simulation studies indicate that a multivariate modelling approach is more parsimonious than fitting independent univariate models to the data while maintaining or improving model fit.

Functional data, as observations of complexe processes, pose challenges when it comes to modeling spatial data in the form of curves, shapes, images, and more complex structures. This talk delves into Principal Component Analysis (PCA) for multivariate functional spatial data. We explore the synergy between spatial, and functional dimensions, unraveling hidden structures and patterns within complex functional spatial datasets.

The talk will give an overview of multivariate functional spatial data, highlighting its ubiquity in diverse fields such as environmental monitoring, geostatistics, and biomedical research. We will then delve into the theoretical underpinnings of Multivariate Functional Principal Component Analysis, emphasizing its adaptability to spatial datasets.

Practical applications of functional PCA in uncovering spatial dependencies, capturing variability, and dimensionality reduction will be illustrated. Case studies will showcase the effectiveness of the proposed methodology.

Additionally, we will touch upon challenges and considerations when applying PCA to multivariate functional spatial data, including handling large datasets, addressing computational complexities, and interpreting results in the context of real-world applications.

We discuss a system of spatial autoregressive models for stock returns, in which general dependencies, dependencies between industrial branches and country-specific dependencies are included. The model parameters are estimated with the method of moments, the estimators are consistent and asymptotically normal. It is shown that the model allows for convincing VaR-forecasting due to its sparsity. Furthermore, we present a fluctuation test for structural breaks in the model parameters and a specification test, which is based on the magnitude of the estimation residuals. (The talk is based on several publications with different coauthors.)

This work introduces the novel Space-Time Integer AutoRegressive and Moving Average (STINARMA) class of statistical models, aimed to cope with the temporal and spatial dependency of integer-valued processes. These models are inspired by key ideas of the continuous Space-Time ARMA (STARMA) and the Integer-valued ARMA (INARMA) models. To ensure the discrete nature of the process, the binomial thinning operator (BTO) replaces the multiplication of the continuous STARMA. Furthermore, the Gaussian distributed innovations are replaced by discrete random variables. The space-time information is introduced through a weight matrix embedded into the matrix-BTO. The focus of this work is on the full STINARMA(p{f1,...,fp}, q{m1,...,mq}) model, for which the univariate INARMA formulation appears as a special STINARMA case for fp=mq=0. First- and second-order moments as well as space-time autocorrelation functions are derived to characterise the process. The STINARMA(1{1},1{1}) with Poisson innovations is studied in detail and theoretical estimation results are derived via the method of moments (MM) and conditional maximum likelihood (CML). The finite-sample performance of MM is evaluated via simulation. Finally, the STINARMA(1{1},1{1}) model is applied to analyse real count data consisting of the daily number of hospital admissions, over time, in three different Portuguese locations.

We illustrate the main results of a non-stationary spatio-temporal precipitation model interpolation process of three different precipitation scenarios distributed throughout Austria for the years 1973-1092 and 2013-2022. We model mean and maximum precipitation
as well as the length of dry spells with a Gamma, blended generalized extreme value
and negative Binomial distribution. A generalized additive model accounts for influencing covariates as elevation and the coordinates of the monitoring stations which is then rewritten in a Bayesian hierarchical form. The spatial dependencies within the data are modelled through a non-stationary Matérn covariance function and the temporal ones through AR(1) dynamics. The stochastic partial differential equation (SPDE) approach is used to tackle the "big n" problem. Inference is performed through integrated nested Laplace approximation (INLA) which comes along with a user friendly R-INLA package. The model outputs are visualised and give insights into changes in precipitation patterns over two different time periods.