Abstract
Undirected graphs are often used to describe high dimensional distributions. Under sparsity conditions, the graph can be estimated using ℓ 1 penalization methods. However, current methods assume that the data are independent and identically distributed. If the distribution, and hence the graph, evolves over time then the data are not longer identically distributed. In this paper we develop a nonparametric method for estimating time varying graphical structure for multivariate Gaussian distributions using an ℓ 1 regularization method, and show that, as long as the covariances change smoothly over time, we can estimate the covariance matrix well (in predictive risk) even when p is large.
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Banerjee, O., Ghaoui, L. E., & d’Aspremont, A. (2008). Model selection through sparse maximum likelihood estimation. Journal of Machine Learning Research, 9, 485–516.
Bickel, P., & Levina, E. (2008). Covariance regularization by thresholding. The Annals of Statistics, 36(1), 199–227.
Drton, M., & Perlman, M. (2004). Model selection for Gaussian concentration graphs. Biometrika, 91(3), 591–602.
Friedman, J., Hastie, T., & Tibshirani, R. (2008). Sparse inverse covariance estimation with the graphical Lasso. Biostatistics, 9(3), 432–441. doi:10.1093/biostatistics/kxm045.
Greenshtein, E., & Ritov, Y. (2004). Persistency in high dimensional linear predictor-selection and the virtue of over-parametrization. Bernoulli, 10, 971–988.
Lam, C., & Fan, J. (2009). Sparsistency and rates of convergence in large covariance matrices estimation. Annals of Statistics, 37(6B), 4254–4278.
Meinshausen, N., & Bühlmann, P. (2006). High dimensional graphs and variable selection with the Lasso. Annals of Statistics, 34(3), 1436–1462.
Ravikumar, P., Wainwright, M., Raskutti, G., & Yu, B. (2008) High-dimensional covariance estimation by minimizing ℓ 1 -penalized log-determinant divergence (Tech. Rep. 767). UC Berkeley, Department of Statistics.
Rothman, A., Bickel, P., Levina, E., & Zhu, J. (2008). Sparse permutation invariant covariance estimation. Electronic Journal of Statistics, 2, 494–515.
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Editors: Sham Kakade and Ping Li.
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Zhou, S., Lafferty, J. & Wasserman, L. Time varying undirected graphs. Mach Learn 80, 295–319 (2010). https://doi.org/10.1007/s10994-010-5180-0
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DOI: https://doi.org/10.1007/s10994-010-5180-0