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 process- ing, mechanical ingeneering, 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 asymmet- ric norm rather than the classical L2-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 k-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 pricing.
|Date of creation:||Jan 2014|
|Date of revision:|
|Contact details of provider:|| Postal: Spandauer Str. 1,10178 Berlin|
Web page: http://sfb649.wiwi.hu-berlin.de
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- Sean D. Campbell & Francis X. Diebold, 2005.
"Weather Forecasting for Weather Derivatives,"
Journal of the American Statistical Association,
American Statistical Association, vol. 100, pages 6-16, March.
- Sean D. Campbell & Francis X. Diebold, 2002. "Weather Forecasting for Weather Derivatives," Center for Financial Institutions Working Papers 02-42, Wharton School Center for Financial Institutions, University of Pennsylvania.
- Campbell, Sean D. & Diebold, Francis X., 2004. "Weather forecasting for weather derivatives," CFS Working Paper Series 2004/10, Center for Financial Studies (CFS).
- Sean D. Campbell & Francis X. Diebold, 2003. "Weather Forecasting for Weather Derivatives," NBER Working Papers 10141, National Bureau of Economic Research, Inc.
- Kehui Chen & Hans‐Georg Müller, 2012. "Conditional quantile analysis when covariates are functions, with application to growth data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 74(1), pages 67-89, 01.
- Kneip A. & Utikal K. J, 2001. "Inference for Density Families Using Functional Principal Component Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 519-542, June.
- Kuan, Chung-Ming & Yeh, Jin-Huei & Hsu, Yu-Chin, 2009. "Assessing value at risk with CARE, the Conditional Autoregressive Expectile models," Journal of Econometrics, Elsevier, vol. 150(2), pages 261-270, June.
- James W. Taylor, 2008. "Estimating Value at Risk and Expected Shortfall Using Expectiles," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 6(2), pages 231-252, Spring.
- Peter Alaton & Boualem Djehiche & David Stillberger, 2002. "On modelling and pricing weather derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 9(1), pages 1-20.
- Šaltytė Benth, Jūratė & Benth, Fred Espen, 2012. "A critical view on temperature modelling for application in weather derivatives markets," Energy Economics, Elsevier, vol. 34(2), pages 592-602.
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