We establish a definition of ordinal patterns for multivariate time series data based on the concept of Tukey's halfspace depth. Given the definition of these \emph{depth patterns}, we are interested in the probabilities of observing specific patterns 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 under the assumption of weakly dependent time series data. Since our concept only depends on ordinal depth information, the resulting values are robust under small perturbations and measurement errors.

翻译：我们基于Tukey半空间深度的概念，建立了多变量时间序列数据的序模式定义。在定义这些“深度模式”之后，我们关注时间序列中观测到特定模式的概率。为此，我们将深度模式的相对频率作为其发生概率的自然估计量。根据参考分布的选择以及参考分布与数据分布之间的关系，我们区分了不同情形并分别予以考虑。在这些情形下，我们研究了序模式概率的统计性质，在弱相依时间序列数据假设下建立了相合性和渐近正态性。由于我们的概念仅依赖于序深度信息，所得结果对微小扰动和测量误差具有稳健性。