In this paper, several regression-type models for multivariate ordinal time series are developed. The regression equations are inspired by existing GARCH-type models for univariate discrete-valued time series and include feedback terms in addition to the usual lagged observations. The corresponding terms from other individuals are represented by weighted averages which are calculated based on a proximity matrix. The marginal conditional distributions are either binomial (employing the simplifying rank-count formulation) or multinomial. The approach can generalized to obtain VARMA-type models to allow for more specific dependence between individuals. Additionally, different copulas are considered to model possible cross-correlation explicitly. The main data example concerns the daily air quality (ordinal) in three cities in North China. Here, a spatial dimension is present, which can be exploited in the definition of the proximity matrix and the copulas.

We analyze data occurring in a regular two-dimensional grid for spatial dependence based on spatial ordinal patterns (SOPs). After having derived the asymptotic distribution of the SOP frequencies under the null hypothesis of spatial independence, we use the concept of the type of SOPs to define the statistics to test for spatial dependence. The proposed tests are not only implemented for real-valued random variables, but a solution for discrete-valued spatial processes in the plane is provided as well. The performances of the spatial-dependence tests are comprehensively analyzed by simulations, considering various data-generating processes. The results show that SOP-based dependence tests have good size properties and constitute an important and valuable complement to the spatial autocorrelation function. To be more specific, SOP-based tests can detect spatial dependence in non-linear processes, and they are robust with respect to outliers and zero inflation. To illustrate their application in practice, two real-world data examples from agricultural sciences are analyzed.

The classification of movement in space is one of the key tasks in environmental science. Various geospatial data such as rainfall or other weather data, data on animal movement or landslide data require a quantitative analysis of the probable movement in space to obtain information on potential risks, ecological developments or changes in future. Usually, machine-learning tools are applied for this task. Yet, machine-learning approaches also have some drawbacks, e.g. the often required large training sets and the fact that the algorithms are often hard to interpret. We propose a classification approach for spatial data based on ordinal patterns. Ordinal patterns have the advantage that they are easily applicable, even to small data sets, are robust in the presence of certain changes in the time series and deliver interpretative results. They, therefore, do not only offer an alternative to machine-learning in the case of small data sets but might also be used in pre-processing for a meaningful feature selection. In this talk, we introduce the basic concept of multivariate ordinal patterns and the corresponding limit theorem. We focus on the discrete case, that is, on movements on a two dimensional grid. The approach is applied to rainfall radar data.

In time series analysis, ordinal patterns describe the spatial ordering of consecutive observations in temporally-ordered data.These have been well-studied for univariate time series. Since a definition of ordinal patterns presupposes a total ordering of observations, there is, however, no straightforward extension of this notion to multivariate data. Nevertheless, applications often require an analysis of data in $mathbb{R}^d$, $d>1$. A lack of canonical ordering of $mathbb{R}^d$ can be overcome by the concept of statistical depth, i.e. by measuring how deep a data point lies in a given reference distribution. The corresponding center-outward ordering of observations in multivariate time series data naturally leads to the definition of ordinal patterns for multivariate data (depth patterns). Given the definition of depth patterns, we are interested in the probability of observing a specific pattern in a time series. For this, we consider the relative frequency of depth patterns as natural estimators for their occurrence probabilities. Depending on the choice of reference distribution and the relation between reference and data distribution, we distinguish different settings that are considered separately. Within these settings we study statistical properties of ordinal pattern probabilities, establishing consistency and asymptotic normality in specific cases under the assumption of weakly dependent time series data.