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Covariance decomposition in undirected Gaussian graphical models

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  • Beatrix Jones
  • Mike West

Abstract

The covariance between two variables in a multivariate Gaussian distribution is decomposed into a sum of path weights for all paths connecting the two variables in an undirected independence graph. These weights are useful in determining which variables are important in mediating correlation between the two path endpoints. The decomposition arises in undirected Gaussian graphical models and does not require or involve any assumptions of causality. This covariance decomposition is derived using basic linear algebra. The decomposition is feasible for very large numbers of variables if the corresponding precision matrix is sparse, a circumstance that arises in examples such as gene expression studies in functional genomics. Additional computational efficiences are possible when the undirected graph is derived from an acyclic directed graph. Copyright 2005, Oxford University Press.

Suggested Citation

  • Beatrix Jones & Mike West, 2005. "Covariance decomposition in undirected Gaussian graphical models," Biometrika, Biometrika Trust, vol. 92(4), pages 779-786, December.
    • Handle: RePEc:oup:biomet:v:92:y:2005:i:4:p:779-786
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    File URL: http://hdl.handle.net/10.1093/biomet/92.4.779
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    Cited by:

    1. Pratik Misra & Seth Sullivant, 2021. "Gaussian graphical models with toric vanishing ideals," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(4), pages 757-785, August.
    2. Alberto Roverato, 2021. "On the interpretation of inflated correlation path weights in concentration graphs," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(5), pages 1485-1505, December.
    3. Bianchi, Daniele & Billio, Monica & Casarin, Roberto & Guidolin, Massimo, 2019. "Modeling systemic risk with Markov Switching Graphical SUR models," Journal of Econometrics, Elsevier, vol. 210(1), pages 58-74.
    4. Liang Yulan & Kelemen Arpad, 2016. "Bayesian state space models for dynamic genetic network construction across multiple tissues," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 15(4), pages 273-290, August.
    5. Alberto Roverato & Robert Castelo, 2017. "The networked partial correlation and its application to the analysis of genetic interactions," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 66(3), pages 647-665, April.
    6. Tan, Kean Ming & Witten, Daniela & Shojaie, Ali, 2015. "The cluster graphical lasso for improved estimation of Gaussian graphical models," Computational Statistics & Data Analysis, Elsevier, vol. 85(C), pages 23-36.
    7. Mike West, 2020. "Bayesian forecasting of multivariate time series: scalability, structure uncertainty and decisions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(1), pages 1-31, February.
    8. Chiranjit Mukherjee & Prasad Kasibhatla & Mike West, 2014. "Spatially varying SAR models and Bayesian inference for high-resolution lattice data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(3), pages 473-494, June.
    9. Feher Kristen & Whelan James & Müller Samuel, 2011. "Assessing Modularity Using a Random Matrix Theory Approach," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 10(1), pages 1-34, September.
    10. Steele, Fiona & Clarke, Paul & Kuha, Jouni, 2019. "Modeling within-household associations in household panel studies," LSE Research Online Documents on Economics 88162, London School of Economics and Political Science, LSE Library.

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