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Exploring Multicollinearity Using a Random Matrix Theory Approach

Author

Listed:
  • Feher Kristen

    (University of Western Australia)

  • Whelan James

    (University of Western Australia)

  • Müller Samuel

    (University of Sydney)

Abstract

Clustering of gene expression data is often done with the latent aim of dimension reduction, by finding groups of genes that have a common response to potentially unknown stimuli. However, what is poorly understood to date is the behaviour of a low dimensional signal embedded in high dimensions. This paper introduces a multicollinear model which is based on random matrix theory results, and shows potential for the characterisation of a gene cluster's correlation matrix. This model projects a one dimensional signal into many dimensions and is based on the spiked covariance model, but rather characterises the behaviour of the corresponding correlation matrix. The eigenspectrum of the correlation matrix is empirically examined by simulation, under the addition of noise to the original signal. The simulation results are then used to propose a dimension estimation procedure of clusters from data. Moreover, the simulation results warn against considering pairwise correlations in isolation, as the model provides a mechanism whereby a pair of genes with `low' correlation may simply be due to the interaction of high dimension and noise. Instead, collective information about all the variables is given by the eigenspectrum.

Suggested Citation

  • Feher Kristen & Whelan James & Müller Samuel, 2012. "Exploring Multicollinearity Using a Random Matrix Theory Approach," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 11(3), pages 1-35, May.
    • Handle: RePEc:bpj:sagmbi:v:11:y:2012:i:3:n:15
    DOI: 10.1515/1544-6115.1668
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    References listed on IDEAS

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    1. Nick Patterson & Alkes L Price & David Reich, 2006. "Population Structure and Eigenanalysis," PLOS Genetics, Public Library of Science, vol. 2(12), pages 1-20, December.
    2. Baik, Jinho & Silverstein, Jack W., 2006. "Eigenvalues of large sample covariance matrices of spiked population models," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1382-1408, July.
    3. 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.
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