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I will try to illustrate the power of graphical lasso with an example which extracts the co-varying structure in historical data for international ETFs. This experiment shows some interesting patterns which can be exploited for constructing more complex models in pairs trading, building hedging portfolios, etc.
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The Julia code here is for illustration purpose only and tries to be a literal translation of the algorithm steps in previous sections for easy understanding. It tries to be a minimalistic walkthrough of the algorithm rather than a numerically stable implementation, which would become topics for following up articles.
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Defining the inverse covariance matrix as \(\Theta\), and the empirical covariance matrix from the data as S, we have the problem to maximize the following L1-penalized likelihood:Published:
Let us look at one hypothetical undirected graph:
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As a first step, now we must figure out how to formulate our problem as a mathematical one so that we know where its solution will likely fall.
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One interesting concept in Computer Science is that of the Undirected Graphical Model, which is basically an undirected graph in which the vertices represent entities/variables and the edges show the correlation between the entities. Sounding abstract? You are definitely not alone. Let us look at some real-world examples first:
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This series of articles will try provide some intuition behind the important problem of inverse covariance estimation and related solutions such as Graphical Lasso and its variants. Mathematical details such as rigor proofs are sometimes omitted for simplicity, but hopefully enough pointers and background information have been provided so readers may follow the theoretical vines themselves.