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Principal Component Analysis in an Asymmetric Norm

Listed author(s):
  • Ngoc Mai Tran
  • Maria Osipenko
  • Wolfgang Karl Härdle

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.

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Paper provided by Sonderforschungsbereich 649, Humboldt University, Berlin, Germany in its series SFB 649 Discussion Papers with number SFB649DP2014-001.

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Length: 33 pages
Date of creation: Jan 2014
Handle: RePEc:hum:wpaper:sfb649dp2014-001
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  1. 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.
  2. 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.
  3. Brenda Lopez Cabrera & Franziska Schulz, 2014. "Forecasting Generalized Quantiles of Electricity Demand: A Functional Data Approach," SFB 649 Discussion Papers SFB649DP2014-030, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  4. Piotr Majer & Peter N. C. Mohr & Hauke R. Heekeren & Wolfgang K. Härdle, 2016. "Portfolio Decisions and Brain Reactions via the CEAD method," Psychometrika, Springer;The Psychometric Society, vol. 81(3), pages 881-903, September.
  5. 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.
  6. 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, January.
  7. Daouia, Abdelaati & Girard, Stéphane & Stupfler, Gilles, 2015. "Estimation of Tail Risk based on Extreme Expectiles," TSE Working Papers 15-566, Toulouse School of Economics (TSE), revised Jul 2017.
  8. 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.
  9. 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.
  10. 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.
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