Ordinal-Pattern-Dependence: Limit Theorems and Structural Breaks in the Long Range Dependent Case with Applications to Hydrology, Medicine and Finance

序型的起源是动力系统理论。我们只考虑n个连续的数据点的相对位置。这个顺序信息被编码在原型结构中。2014年，我介绍了一种方法，使用顺序模式来衡量时间序列之间的依赖程度：如果相同的模式经常出现在两个时间序列的同一时刻，这被称为正顺序模式依赖。在负依赖分析中，计算了反射模式。在我的项目开始之前，我们只能分析短期相关的时间序列。特别是在水文学中，众所周知，数据集是长期依赖的。因此，在这个框架下证明极限定理是很重要的。我们得到了这类结果，并证明了极限分布取决于赫斯特指数与所考虑的估计量的性质之间的相互作用。在水文学中，我们的方法用于规划测量网络和定义同质群。然而，我们也面临着新的问题：在水文学中，分类数据经常出现（比如洪水警报级别在0到5之间）。在这些数据集中，关系经常出现。到目前为止，这些被认为是“单调递增”。因此，联合运动被系统地高估了。唯一的解决方案是明确地考虑带联系的模式，并建立一个处理这个概念的新理论。这也使得处理高频财务数据成为可能。在这里，也出现了一些联系。这些广义模式的结构断裂的极限定理和测试是未来几年的主要目标。此外，还将介绍序数模式的多元推广。使用二维空间作为索引集，而不是一维时间，使我们能够分析地球表面的气候数据。由于存在其他依赖度量，我们分析了这些度量与顺序模式依赖之间的相互作用。与Kendall's Tau, Spearman's Rho和经典相关性的关系已经很好理解，然而，这些概念有多元对应，应该以同样的方式进行分析。此外，还将考虑联结和距离相关。除了我们的理论结果之外，我们还制作了一个用GNU R编写的软件包，并在CRAN上发布。其他研究人员可以免费使用它。每一个新的理论结果都将在包中实现。

The origin of ordinal patterns is the theory of dynamical systems. One considers n consecutive data points only using their relative positions. This ordinal information is encoded in an archetype structure. 2014 I have introduced a method which uses ordinal patterns to measure the degree of dependence between time series: if the same patterns do appear at the same instants of time in both time series very often, this is called positive ordinal pattern dependence. In the analysis of negative dependence, reflected patterns are counted.Before my project was started, we were only able to analyze short-range dependent time series. In particular in hydrology, data sets are well-known to be long-range dependent. Hence, it was important to prove limit theorems in this framework. We obtained results of this kind and showed that the limit distribution depends on the interplay between the Hurst exponent and properties of the estimator under consideration. In hydrology our methods are used in order to plan gauge networks and to define homogeneous groups. However, we are facing also new problems: in hydrology often categorial data shows up (like flood alert levels in the range 0 to 5). In these data sets ties appear quite often. By now, these are considered as ‚monotone increasing‘. Hence, co-movement is systematically overestimated. The only solution is to consider explicitly pattern with ties and built a new theory dealing with this concept. This makes it also possible to deal with high frequency financial data. Here, also several ties do appear. Limit theorems and tests for structural breaks for these generalized patterns are main objectives for the next years. In addition, multivariate generalizations for ordinal patterns will be introduced. Using two-dimensional space as index set instead of one-dimensional time, allows us to analyze climate data on the earth’s surface. Since there are other dependence measures as well, we analyze the interplay between these and ordinal pattern dependence. The relation to Kendall's Tau, Spearman's Rho and classical correlation is quite well understood, however, these concepts have multivariate counterparts, which should be analyzed in the same way. Furthermore, copulas and distance correlation will be considered.In addition to our theoretical results, we have produced a package written in GNU R which is published on CRAN. Other researchers can use it free of charge. Every new theoretical result will be implemented in the package.