Graphical lasso
{{Orphan|date=July 2016}}
In statistics, the graphical lasso{{Cite journal|last1=Friedman|first1=Jerome|last2=Hastie|first2=Trevor|last3=Tibshirani|first3=Robert|date=2008-07-01|title=Sparse inverse covariance estimation with the graphical lasso|url= |journal=Biostatistics|language=en|volume=9|issue=3|pages=432–441|doi=10.1093/biostatistics/kxm045|pmid=18079126|pmc=3019769|issn=1465-4644}} 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{{Cite journal|last=Dempster|first=A. P.|date=1972|title=Covariance Selection|journal=Biometrics|volume=28|issue=1|pages=157–175|doi=10.2307/2528966|issn=0006-341X|jstor=2528966}}{{cite arXiv|last1=Banerjee|first1=Onureena|last2=d'Aspremont|first2=Alexandre|last3=Ghaoui|first3=Laurent El|date=2005-06-08|title=Sparse Covariance Selection via Robust Maximum Likelihood Estimation|eprint=cs/0506023}} for the multivariate Gaussian distribution when observations were limited. Subsequently, the optimization algorithms to solve this problem were improved{{cite journal
| author = Friedman, Jerome and Hastie, Trevor and Tibshirani, Robert
| title = Sparse inverse covariance estimation with the graphical lasso
| publisher = Biometrika Trust
| journal = Biostatistics
| volume = 9
| issue = 3
| pages = 432–41
| year = 2008
| url = http://statweb.stanford.edu/~tibs/ftp/graph.pdf
| doi = 10.1093/biostatistics/kxm045
| pmid = 18079126
| pmc = 3019769
}} and extended{{Cite journal|last1=Cai|first1=T. Tony|last2=Liu|first2=Weidong|last3=Zhou|first3=Harrison H.|date=April 2016|title=Estimating sparse precision matrix: Optimal rates of convergence and adaptive estimation|journal=The Annals of Statistics|volume=44|issue=2|pages=455–488|doi=10.1214/13-AOS1171|s2cid=14699773|issn=0090-5364|doi-access=free|arxiv=1212.2882}} to other types of estimators and distributions.
Setting
Consider observations from multivariate Gaussian distribution . We are interested in estimating the precision matrix .
The graphical lasso estimator is the such that:
:
\hat{\Theta} = \operatorname{argmin}_{\Theta \ge 0} \left(\operatorname{tr}(S \Theta) - \log \det(\Theta) + \lambda \sum_{j \ne k} |\Theta_{jk}| \right)
where is the sample covariance, and is the penalizing parameter.
Application
To obtain the estimator in programs, users could use the R package [https://cran.r-project.org/web/packages/glasso/glasso.pdf glasso],{{cite book
| title = glasso: Graphical lasso- estimation of Gaussian graphical models
| url = https://cran.r-project.org/package=glasso
| year = 2014
|author1=Jerome Friedman |author2=Trevor Hastie |author3=Rob Tibshirani }} [https://scikit-learn.org/stable/modules/generated/sklearn.covariance.GraphicalLasso.html GraphicalLasso() class] in the scikit-learn Python library,{{cite journal
| author = Pedregosa, F. and Varoquaux, G. and Gramfort, A. and Michel, V. and Thirion, B. and Grisel, O. and Blondel, M. and Prettenhofer, P. and Weiss, R. and Dubourg, V. and Vanderplas, J. and Passos, A. and Cournapeau, D. and Brucher, M. and Perrot, M. and Duchesnay, E.
| title = Scikit-learn: Machine Learning in Python
| journal = Journal of Machine Learning Research
| year = 2011
| volume = 12
| page = 2825
| url = http://scikit-learn.org/stable/about.html
| bibcode = 2011JMLR...12.2825P
| arxiv = 1201.0490
}} or the [https://github.com/skggm/skggm skggm] Python package{{cite journal
| author1 = Jason Laska
| author2 = Manjari Narayan
| title = skggm 0.2.7: A scikit-learn compatible package for Gaussian and related Graphical Models
| journal = Zenodo
| year = 2017
| doi = 10.5281/zenodo.830033
| bibcode = 2017zndo....830033L
}} (similar to scikit-learn).