| Algorithmic advances in the form of Friedman's Coordinate Descent algorithm have allowed the Lasso REML penalty to see fruitful use in selection and regularization problems, such as Tibshirani's 2007 "Glasso" or "Graphical-Lasso" procedure. If we posit a sparse multiplicative inverse of the correlation matrix, or the "concentration matrix", we can interpret this as a Gaussian graphical network.;We begin with a treatment of the linear regression problem and our modified penalty, dubbed the "Two-Lasso", a hierarchical mixture of large and small variance LaPlacian priors. This mixture takes advantage of pre-existing Lasso algorithms, as well as the EM missing-data framework, allowing statistician to input familiar selection parameters, such as a prior probability of activation. Though the Two-Lasso threshold function is not necessarily continuous, the second chapter includes bounds for continuity. Six theorems in the first chapter's appendix establish the asymptotic properties of the Two-Lasso estimate and the "Limit-Lasso", which is a Two-Lasso estimate achieved for widely disparate active and de-active set prior variances.;Target data for a Two-Lasso penalized Glasso model, or "Two-Glasso", is financial markets data collected during a change in correlation structure. This includes the 2008 credit crisis which has recently wrecked havoc with endowments and institutional pension investment funds. We make three select modifications to the Glasso : a Two-Lasso selection penalty, constrained maximization rather than a strong prior on volatility to guarantee positive definite matrix estimates, and a non-zero baseline concentration level added to a sparse network. We fit Two-Glasso estimates for small-sample-size historical financial data and depict output visually with informative figures of our own design. From these graphs, we can posit a hedging strategy to the benefit of locked-in investors during a correlation-heavy crisis. |
