IDEAS home Printed from https://ideas.repec.org/p/zbw/sfb649/sfb649dp2016-040.html
   My bibliography  Save this paper

Principal component analysis in an asymmetric norm

Author

Listed:
  • Tran, Ngoc Mai
  • Burdejová, Petra
  • Osipenko, Maria
  • Härdle, Wolfgang Karl

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 quantification in finance 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

  • Tran, Ngoc Mai & Burdejová, Petra & Osipenko, Maria & Härdle, Wolfgang Karl, 2016. "Principal component analysis in an asymmetric norm," SFB 649 Discussion Papers 2016-040, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
  • Handle: RePEc:zbw:sfb649:sfb649dp2016-040
    as

    Download full text from publisher

    File URL: https://www.econstor.eu/bitstream/10419/148876/1/871011107.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    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. 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.
    3. 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.
    4. Abdelaati Daouia & Stéphane Girard & Gilles Stupfler, 2018. "Estimation of tail risk based on extreme expectiles," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(2), pages 263-292, March.
    5. López Cabrera, Brenda & Schulz, Franziska, 2014. "Forecasting generalized quantiles of electricity demand: A functional data approach," SFB 649 Discussion Papers 2014-030, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    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. 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.
    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. Majer, Piotr & Mohr, Peter N. C. & Heekeren, Hauke R. & Härdle, Wolfgang Karl, 2014. "Portfolio decisions and brain reactions via the CEAD method," SFB 649 Discussion Papers 2014-036, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    10. Zhang, Lingsong & Lu, Shu & Marron, J.S., 2015. "Nested nonnegative cone analysis," Computational Statistics & Data Analysis, Elsevier, vol. 88(C), pages 100-110.
    11. Philippe Jorion, 2000. "Risk management lessons from Long‐Term Capital Management," European Financial Management, European Financial Management Association, vol. 6(3), pages 277-300, September.
    12. Gao, Yafeng & Xu, Jiangmin & Yang, Shichao & Tang, Xiaomin & Zhou, Quan & Ge, Jing & Xu, Tengfang & Levinson, Ronnen, 2014. "Cool roofs in China: Policy review, building simulations, and proof-of-concept experiments," Energy Policy, Elsevier, vol. 74(C), pages 190-214.
    13. Burdejova, Petra & Härdle, Wolfgang Karl & Kokoszka, Piotr & Xiong, Q., 2015. "Change point and trend analyses of annual expectile curves of tropical storms," SFB 649 Discussion Papers 2015-029, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    14. 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.
    15. Fraiman, Ricardo & Pateiro-López, Beatriz, 2012. "Quantiles for finite and infinite dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 108(C), pages 1-14.
    16. Burdejova, P. & Härdle, W. & Kokoszka, P. & Xiong, Q., 2017. "Change point and trend analyses of annual expectile curves of tropical storms," Econometrics and Statistics, Elsevier, vol. 1(C), pages 101-117.
    17. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-847, July.
    18. James W. Taylor, 2008. "Estimating Value at Risk and Expected Shortfall Using Expectiles," Journal of Financial Econometrics, Oxford University Press, vol. 6(2), pages 231-252, Spring.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. repec:hum:wpaper:sfb649dp2017-027 is not listed on IDEAS
    2. López Cabrera, Brenda & Schulz, Franziska, 2014. "Forecasting generalized quantiles of electricity demand: A functional data approach," SFB 649 Discussion Papers 2014-030, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    3. repec:hum:wpaper:sfb649dp2016-058 is not listed on IDEAS
    4. repec:hum:wpaper:sfb649dp2014-030 is not listed on IDEAS
    5. Lin, Liang-Ching & Chen, Ray-Bing & Huang, Mong-Na Lo & Guo, Meihui, 2020. "Huber-type principal expectile component analysis," Computational Statistics & Data Analysis, Elsevier, vol. 151(C).
    6. Chao, Shih-Kang & Härdle, Wolfgang Karl & Huang, Chen, 2016. "Multivariate factorisable sparse asymmetric least squares regression," SFB 649 Discussion Papers 2016-058, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    7. 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.
    8. Wang, Bingling & Li, Yingxing & Härdle, Wolfgang Karl, 2022. "K-expectiles clustering," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    9. Chen, Cathy Yi-Hsuan & Härdle, Wolfgang Karl & Okhrin, Yarema, 2017. "Tail event driven networks of SIFIs," SFB 649 Discussion Papers 2017-004, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    10. Petra Burdejová & Wolfgang K. Härdle, 2019. "Dynamic semi-parametric factor model for functional expectiles," Computational Statistics, Springer, vol. 34(2), pages 489-502, June.
    11. Chen, Cathy Yi-Hsuan & Härdle, Wolfgang Karl & Okhrin, Yarema, 2019. "Tail event driven networks of SIFIs," Journal of Econometrics, Elsevier, vol. 208(1), pages 282-298.
    12. repec:hum:wpaper:sfb649dp2017-004 is not listed on IDEAS
    13. Ana C S Costa & Diego E Pereira & Caio M Veríssimo & Marcos A D Bomfim & Rita C R E Queiroga & Marta S Madruga & Susana Alves & Rui J B Bessa & Maria E G Oliveira & Juliana K B Soares, 2019. "Developing cookies formulated with goat cream enriched with conjugated linoleic acid," PLOS ONE, Public Library of Science, vol. 14(9), pages 1-15, September.
    14. Osipenko, Maria, 2021. "Directional assessment of traffic flow extremes," Transportation Research Part B: Methodological, Elsevier, vol. 150(C), pages 353-369.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. repec:hum:wpaper:sfb649dp2016-040 is not listed on IDEAS
    2. repec:hum:wpaper:sfb649dp2014-001 is not listed on IDEAS
    3. repec:hum:wpaper:sfb649dp2016-058 is not listed on IDEAS
    4. Chao, Shih-Kang & Härdle, Wolfgang Karl & Huang, Chen, 2016. "Multivariate factorisable sparse asymmetric least squares regression," SFB 649 Discussion Papers 2016-058, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    5. repec:hum:wpaper:sfb649dp2017-027 is not listed on IDEAS
    6. Petra Burdejová & Wolfgang K. Härdle, 2019. "Dynamic semi-parametric factor model for functional expectiles," Computational Statistics, Springer, vol. 34(2), pages 489-502, June.
    7. Taoufik Bouezmarni & Mohamed Doukali & Abderrahim Taamouti, 2024. "Testing Granger non-causality in expectiles," Econometric Reviews, Taylor & Francis Journals, vol. 43(1), pages 30-51, January.
    8. repec:hal:journl:hal-04672516 is not listed on IDEAS
    9. Daouia, Abdelaati & Girard, Stéphane & Stupfler, Gilles, 2018. "Tail expectile process and risk assessment," TSE Working Papers 18-944, Toulouse School of Economics (TSE).
    10. James Ming Chen, 2018. "On Exactitude in Financial Regulation: Value-at-Risk, Expected Shortfall, and Expectiles," Risks, MDPI, vol. 6(2), pages 1-28, June.
    11. Daouia, Abdelaati & Padoan, Simone A. & Stupfler, Gilles, 2023. "Extreme expectile estimation for short-tailed data, with an application to market risk assessment," TSE Working Papers 23-1414, Toulouse School of Economics (TSE), revised May 2024.
    12. Daouia, Abdelaati & Girard, Stéphane & Stupfler, Gilles, 2021. "ExpectHill estimation, extreme risk and heavy tails," Journal of Econometrics, Elsevier, vol. 221(1), pages 97-117.
    13. Man, Rebeka & Tan, Kean Ming & Wang, Zian & Zhou, Wen-Xin, 2024. "Retire: Robust expectile regression in high dimensions," Journal of Econometrics, Elsevier, vol. 239(2).
    14. Yundong Tu & Siwei Wang, 2023. "Variable Screening and Model Averaging for Expectile Regressions," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 85(3), pages 574-598, June.
    15. Daouia, Abdelaati & Stupfler, Gilles & Usseglio-Carleve, Antoine, 2023. "An expectile computation cookbook," TSE Working Papers 23-1458, Toulouse School of Economics (TSE).
    16. Girard, Stéphane & Stupfler, Gilles & Usseglio-Carleve, Antoine, 2022. "Functional estimation of extreme conditional expectiles," Econometrics and Statistics, Elsevier, vol. 21(C), pages 131-158.
    17. Härdle, Wolfgang Karl & Ling, Chengxiu, 2018. "How Sensitive are Tail-related Risk Measures in a Contamination Neighbourhood?," IRTG 1792 Discussion Papers 2018-010, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    18. Bingzhen Geng & Yang Liu & Yimiao Zhao, 2024. "Value-at-Risk- and Expectile-based Systemic Risk Measures and Second-order Asymptotics: With Applications to Diversification," Papers 2404.18029, arXiv.org.
    19. Mohammedi, Mustapha & Bouzebda, Salim & Laksaci, Ali, 2021. "The consistency and asymptotic normality of the kernel type expectile regression estimator for functional data," Journal of Multivariate Analysis, Elsevier, vol. 181(C).
    20. Stéphane Girard & Gilles Stupfler & Antoine Usseglio‐Carleve, 2022. "Nonparametric extreme conditional expectile estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 78-115, March.
    21. Stéphane Girard & Gilles Claude Stupfler & Antoine Usseglio-Carleve, 2021. "Extreme Conditional Expectile Estimation in Heavy-Tailed Heteroscedastic Regression Models," Post-Print hal-03306230, HAL.
    22. Litimein, Ouahiba & Laksaci, Ali & Mechab, Boubaker & Bouzebda, Salim, 2023. "Local linear estimate of the functional expectile regression," Statistics & Probability Letters, Elsevier, vol. 192(C).
    23. Qinyu Wu & Fan Yang & Ping Zhang, 2023. "Conditional generalized quantiles based on expected utility model and equivalent characterization of properties," Papers 2301.12420, arXiv.org.
    24. Zhang, Feipeng & Xu, Yixiong & Fan, Caiyun, 2023. "Nonparametric inference of expectile-based value-at-risk for financial time series with application to risk assessment," International Review of Financial Analysis, Elsevier, vol. 90(C).
    25. Daouia, Abdelaati & Paindaveine, Davy, 2019. "Multivariate Expectiles, Expectile Depth and Multiple-Output Expectile Regression," TSE Working Papers 19-1022, Toulouse School of Economics (TSE), revised Feb 2023.

    More about this item

    Keywords

    principal components; asymmetric norm; dimension reduction; quantile; expectile; fMRI; risk attitude; brain imaging; temperature; functional data;
    All these keywords.

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:zbw:sfb649:sfb649dp2016-040. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: ZBW - Leibniz Information Centre for Economics (email available below). General contact details of provider: https://edirc.repec.org/data/sohubde.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.