Functional Principle Components for Derivatives and Higher Dimensions

收集的数据为曲线的应用越来越多。简单的一维结构从数值和理论的角度都得到了很好的理解。然而，具有复杂时空相关性结构的高维数据需要更先进的统计工具来进行勘探、推理和预测。神经经济学就是一个例子，它需要对大量动态记录的fMRI数据进行定量分析。此外，天气衍生品市场和能源行业需要时空温度模型。在许多应用中，没有被直接观察到的对象是真正感兴趣的对象。一个例子是定价核心的演变，它提供了关于投资和风险模式的信息，或者是允许对天气衍生品定价的市场价格或风险。虽然在这方面已经做了大量的应用工作，但仍然需要建立健全的理论基础。我们在这里面临的挑战是用低维分量和动态参数来总结曲线，这些参数可以在时间序列框架内解释，以解释变异性。通过将数理统计工具与现代计量经济学技术相结合，我们提出了对高维间接观测曲线对象这一挑战性问题的更高程度的洞察。

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.