Functional Principle Components for Derivatives and Higher Dimensions

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There is an increasing number of applications where the collected data are curves. Simple, onedimensional structures are well understood both from numerical and theoretical perspective. High dimensional data, however, with a complicated spatio-temporal correlation structure need more advanced statistical tools for exploration, inference and prediction. An example is neuroeconomics requiring quantitative analysis of large sets of dynamically recorded fMRI data. Also weather derivative markets and energy industry require spatio-temporal temperature modeling. In many applications, objects that are not directly observed are the real objects of interest. An example is the evolution of pricing kernels that gives information about investment and risk patterns, or the market price or risk that allows pricing of weather derivatives. While a lot of applied work has been done in this context, there is still a need to establish a sound theoretical basis. The challenge that we face here is to summarize curves by low dimensional components and dynamic parameters which can be interpreted within the time series framework, to explain variability. By combining tools of mathematical statistics with modern econometric techniques, we propose to gain a higher degree of insight into the challenging problem of high dimensional and indirectly observed curve object.