Английская Википедия:Graphical lasso
In statistics, the graphical lasso[1] is a sparse penalized maximum likelihood estimator for the concentration or precision matrix (inverse of covariance matrix) of a multivariate elliptical distribution. The original variant was formulated to solve Dempster's covariance selection problem[2][3] for the multivariate Gaussian distribution when observations were limited. Subsequently, the optimization algorithms to solve this problem were improved[4] and extended[5] to other types of estimators and distributions.
Setting
Consider observations <math>X_1, X_2, \ldots, X_n</math> from multivariate Gaussian distribution <math>X \sim N(0, \Sigma)</math>. We are interested in estimating the precision matrix <math>\Theta = \Sigma^{-1}</math>.
The graphical lasso estimator is the <math>\hat{\Theta}</math> such that:
- <math>
\hat{\Theta} = \operatorname{argmin}_{\Theta \ge 0} \left(\operatorname{tr}(S \Theta) - \log \det(\Theta) + \lambda \sum_{j \ne k} |\Theta_{jk}| \right)</math>
where <math>S</math> is the sample covariance, and <math>\lambda</math> is the penalizing parameter.[4]
Application
To obtain the estimator in programs, users could use the R package glasso,[6] GraphicalLasso() class in the scikit-learn Python library,[7] or the skggm Python package[8] (similar to scikit-learn).
See also
References
