Copula based dependence analysis of functional data for validation and calibration of dynamic aircraft models

飞机运动的物理模型被转换成一组描述几个感兴趣的变量之间的关系的微分方程。它们是分析飞行风险的重要工具，如硬着陆或跑道偏离。我们调查-在统计意义上-如果目录目前使用的物理模型适当地捕捉记录变量之间的依赖关系，在真实的数据。为了全面了解模型的充分性，分析将在不同的时间尺度上运行。对依赖关系进行统计分析的最强大工具是copula。它捕获随机变量的有限维向量中的依赖关系。然而，在某些时间尺度上，我们的数据中记录的可变轨迹必须被解释为随机函数（即，无限维）的时间。针对这种情况，我们开发了一个基于Copula的随机函数之间的依赖性建模框架。我们发现有限维表示的无限维随机函数使用功能的主成分得分。然后，这些分数可以配备一个灵活的vine copula模型来描述依赖关系。使用来自操作飞行的真实的数据，该相关性模型可用于评估飞机运动的几种最新物理模型。一个目标是使用物理模型校准的相关性表征。因此，时间序列之间的依赖结构估计，记录的数据和模型输出。随后，在几次迭代中更新物理模型的初始参数，使得估计的依赖结构更紧密地匹配。它的调查，这种参数估计技术是否提高了国家的最先进的方法。在参数估计不能充分完成的情况下，相关结构的分析将用于通过修改相应的微分方程来增强物理模型，以给出变量之间的关系的更适当的表示。

Physical models of aircraft motion are cast into a set of differential equations describing the relationship between several variables of interest. They are crucial tools for the analysis of flight risks, such as Hard Landings or Runway Excursions. We investigate - in a statistical sense - if the catalog of currently used physical models appropriately captures the dependencies between recorded variables in real data. To get a comprehensive view of the models' adequacy, the analysis will be run on different time scales.The most powerful tool for the statistical analysis of dependencies is the copula. It captures the dependencies in a finite-dimensional vector of random variables. On some time scales, however, the recorded variable trajectories in our data have to be interpreted as random functions (i.e., infinite-dimensional) of time. For such situations, we develop a copula-based modeling framework for the dependence between random functions. We find finite-dimensional representations of the infinite-dimensional random functions using functional principal component scores. These scores can then be equipped with a flexible vine copula model that describes the dependencies. Using real data from operational flights, this dependence model can be used to assess several state of the art physical models of aircraft motion. One goal is to use the dependence characterization for physical model calibration. Thereby, the dependence structure between time series is estimated for both, the recorded data and the model output. Subsequently, initial parameters of the physical model are updated in several iterations so that the estimated dependence structures match more closely. It is investigated, whether this parameter estimation technique improves over state of the art methods. In case the parameter estimation can not be done adequately, the analyses of the dependence structure will be used to enhance the physical model by modifying the corresponding differential equations to give a more appropriate representation of relationships between the variables.