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

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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.