235 research outputs found
Principal Component Analysis in an Asymmetric Norm
Principal component analysis (PCA) is a widely used dimension reduction tool
in the analysis of many kind of high-dimensional data. It is used in signal
processing, mechanical engineering, psychometrics, and other fields under
different names. It still bears the same mathematical idea: the decomposition
of variation of a high dimensional object into uncorrelated factors or
components. However, in many of the above applications, one is interested in
capturing the tail variables of the data rather than variation around the mean.
Such applications include weather related event curves, expected shortfalls,
and speeding analysis among others. These are all high dimensional tail objects
which one would like to study in a PCA fashion. The tail character though
requires to do the dimension reduction in an asymmetric norm rather than the
classical -type orthogonal projection. We develop an analogue of PCA in an
asymmetric norm. These norms cover both quantiles and expectiles, another tail
event measure. The difficulty is that there is no natural basis, no `principal
components', to the -dimensional subspace found. We propose two definitions
of principal components and provide algorithms based on iterative least
squares. We prove upper bounds on their convergence times, and compare their
performances in a simulation study. We apply the algorithms to a Chinese
weather dataset with a view to weather derivative pricingComment: 31 pages, 5 figure
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