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Principal Component Analysis in an Asymmetric Norm

Author

Listed:
  • Ngoc M. Tran
  • Petra Burdejová
  • Maria Osipenko
  • Wolfgang K. Härdle

Abstract

Principal component analysis (PCA) is a widely used dimension reduction tool in the analysis of high-dimensional data. However, in many applications such as risk quanti cation in nance or climatology, one is interested in capturing the tail variations rather than variation around the mean. In this paper, we develop Principal Expectile Analysis (PEC), which generalizes PCA for expectiles. It can be seen as a dimension reduction tool for extreme value theory, where one approximates uctuations in the -expectile level of the data by a low dimensional subspace. We provide algorithms based on iterative least squares, prove upper bounds on their convergence times, and compare their performances in a simulation study. We apply the algorithms to a Chinese weather dataset and fMRI data from an investment decision study.

Suggested Citation

  • Ngoc M. Tran & Petra Burdejová & Maria Osipenko & Wolfgang K. Härdle, 2016. "Principal Component Analysis in an Asymmetric Norm," SFB 649 Discussion Papers SFB649DP2016-040, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  • Handle: RePEc:hum:wpaper:sfb649dp2016-040
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    References listed on IDEAS

    as
    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. 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.
    3. 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.
    4. Sean D. Campbell & Francis X. Diebold, 2005. "Weather Forecasting for Weather Derivatives," Journal of the American Statistical Association, American Statistical Association, pages 6-16.
    5. 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.
    6. 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.
    7. repec:taf:jnlasa:v:112:y:2017:i:517:p:127-136 is not listed on IDEAS
    8. Brenda López Cabrera & Franziska Schulz, 2017. "Forecasting Generalized Quantiles of Electricity Demand: A Functional Data Approach," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(517), pages 127-136, January.
    9. 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.
    10. 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.
    11. 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.
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    Cited by:

    1. Brenda López Cabrera & Franziska Schulz, 2017. "Forecasting Generalized Quantiles of Electricity Demand: A Functional Data Approach," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(517), pages 127-136, January.
    2. Shih-Kang Chao & Wolfgang K. Härdle & Chen Huang, 2016. "Multivariate Factorisable Sparse Asymmetric Least Squares Regression," SFB 649 Discussion Papers SFB649DP2016-058, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    3. Cathy Yi-Hsuan Chen & Wolfgang Karl Härdle & Yarema Okhrin, 2017. "Tail event driven networks of SIFIs," SFB 649 Discussion Papers SFB649DP2017-004, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    4. Petra Burdejová & Wolfgang K. Härdle, 2017. "Dynamic semi-parametric factor model for functional expectiles," SFB 649 Discussion Papers SFB649DP2017-027, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.

    More about this item

    JEL classification:

    • C38 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Classification Methdos; Cluster Analysis; Principal Components; Factor Analysis
    • C55 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Large Data Sets: Modeling and Analysis
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

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