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Kullback-Leibler distance as a measure of the information filtered from multivariate data

Author

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  • Michele Tumminello
  • Fabrizio Lillo
  • Rosario Nunzio Mantegna

Abstract

We show that the Kullback-Leibler distance is a good measure of the statistical uncertainty of correlation matrices estimated by using a finite set of data. For correlation matrices of multivariate Gaussian variables we analytically determine the expected values of the Kullback-Leibler distance of a sample correlation matrix from a reference model and we show that the expected values are known also when the specific model is unknown. We propose to make use of the Kullback-Leibler distance to estimate the information extracted from a correlation matrix by correlation filtering procedures. We also show how to use this distance to measure the stability of filtering procedures with respect to statistical uncertainty. We explain the effectiveness of our method by comparing four filtering procedures, two of them being based on spectral analysis and the other two on hierarchical clustering. We compare these techniques as applied both to simulations of factor models and empirical data. We investigate the ability of these filtering procedures in recovering the correlation matrix of models from simulations. We discuss such an ability in terms of both the heterogeneity of model parameters and the length of data series. We also show that the two spectral techniques are typically more informative about the sample correlation matrix than techniques based on hierarchical clustering, whereas the latter are more stable with respect to statistical uncertainty.

Suggested Citation

  • Michele Tumminello & Fabrizio Lillo & Rosario Nunzio Mantegna, 2007. "Kullback-Leibler distance as a measure of the information filtered from multivariate data," Papers 0706.0168, arXiv.org.
  • Handle: RePEc:arx:papers:0706.0168
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    Cited by:

    1. Gautier Marti & S'ebastien Andler & Frank Nielsen & Philippe Donnat, 2016. "Clustering Financial Time Series: How Long is Enough?," Papers 1603.04017, arXiv.org, revised Apr 2016.
    2. Sandoval, Leonidas & Franca, Italo De Paula, 2012. "Correlation of financial markets in times of crisis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(1), pages 187-208.
    3. Musmeci, Nicoló & Aste, Tomaso & Di Matteo, T., 2015. "Relation between financial market structure and the real economy: comparison between clustering methods," LSE Research Online Documents on Economics 61644, London School of Economics and Political Science, LSE Library.
    4. Gautier Marti & Frank Nielsen & Miko{l}aj Bi'nkowski & Philippe Donnat, 2017. "A review of two decades of correlations, hierarchies, networks and clustering in financial markets," Papers 1703.00485, arXiv.org, revised Nov 2020.
    5. Tumminello, Michele & Lillo, Fabrizio & Mantegna, Rosario N., 2010. "Correlation, hierarchies, and networks in financial markets," Journal of Economic Behavior & Organization, Elsevier, vol. 75(1), pages 40-58, July.
    6. Choi, Insu & Kim, Woo Chang, 2023. "Estimating Historical Downside Risks of Global Financial Market Indices via Inflation Rate-Adjusted Dependence Graphs," Research in International Business and Finance, Elsevier, vol. 66(C).
    7. Nicolo Musmeci & Tomaso Aste & Tiziana Di Matteo, 2014. "Relation between Financial Market Structure and the Real Economy: Comparison between Clustering Methods," Papers 1406.0496, arXiv.org, revised Jan 2015.
    8. Sensoy, Ahmet & Tabak, Benjamin M., 2014. "Dynamic spanning trees in stock market networks: The case of Asia-Pacific," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 414(C), pages 387-402.
    9. Christian Bongiorno & Marco Berritta, 2023. "Optimal Covariance Cleaning for Heavy-Tailed Distributions: Insights from Information Theory," Papers 2304.14098, arXiv.org, revised Apr 2023.
    10. Zhang, Xin & Podobnik, Boris & Kenett, Dror Y. & Eugene Stanley, H., 2014. "Systemic risk and causality dynamics of the world international shipping market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 415(C), pages 43-53.
    11. Gautier Marti & Sébastien Andler & Frank Nielsen & Philippe Donnat, 2016. "Clustering Financial Time Series: How Long is Enough?," Post-Print hal-01400395, HAL.
    12. Jovanovic, Franck & Mantegna, Rosario N. & Schinckus, Christophe, 2019. "When financial economics influences physics: The role of Econophysics," International Review of Financial Analysis, Elsevier, vol. 65(C).
    13. Huang, Wei-Qiang & Yao, Shuang & Zhuang, Xin-Tian & Yuan, Ying, 2017. "Dynamic asset trees in the US stock market: Structure variation and market phenomena," Chaos, Solitons & Fractals, Elsevier, vol. 94(C), pages 44-53.
    14. Giacomo Bormetti & Maria Elena De Giuli & Danilo Delpini & Claudia Tarantola, 2008. "Bayesian Analysis of Value-at-Risk with Product Partition Models," Papers 0809.0241, arXiv.org, revised May 2009.
    15. Joel Bun & Jean-Philippe Bouchaud & Marc Potters, 2016. "Cleaning large correlation matrices: tools from random matrix theory," Papers 1610.08104, arXiv.org.
    16. Wu Zhu & Jian-an Fang & Yang Tang & Wenbing Zhang & Wei Du, 2012. "Digital IIR Filters Design Using Differential Evolution Algorithm with a Controllable Probabilistic Population Size," PLOS ONE, Public Library of Science, vol. 7(7), pages 1-9, July.
    17. Contreras-Reyes, Javier E., 2014. "Asymptotic form of the Kullback–Leibler divergence for multivariate asymmetric heavy-tailed distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 395(C), pages 200-208.
    18. Giacomo Bormetti & Maria Elena De Giuli & Danilo Delpini & Claudia Tarantola, 2012. "Bayesian Value-at-Risk with product partition models," Quantitative Finance, Taylor & Francis Journals, vol. 12(5), pages 769-780, November.
    19. Nicoló Musmeci & Tomaso Aste & T Di Matteo, 2015. "Relation between Financial Market Structure and the Real Economy: Comparison between Clustering Methods," PLOS ONE, Public Library of Science, vol. 10(3), pages 1-24, March.

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