I have $n$ observations of $p$ random variables $\{Y_1,\dots,Y_p \}$. I know that each observation has been generated from a linear network model of the form $$ Y = WY+\epsilon $$ and that the graph described by $W$ is likely to be sparse and to have cycles. What is the current state of the art for estimating $W$? I have been reading about the graphical lasso but it seems limited to undirected graphs or acyclic directed graphs.

  • $\begingroup$ Are $Y$ on the left hand side and $Y$ on the right hand side the same? Could you clarify what $Y$, $W$ and $\epsilon$ are? $\endgroup$ – Juho Kokkala Dec 8 '17 at 18:44
  • $\begingroup$ Yes the $Y$ on the left and right hand side are the same. $W$ is a $p\times p$ matrix and $\epsilon$ is a vector of iid normal variable. $\endgroup$ – user_lambda Dec 8 '17 at 19:46
  • $\begingroup$ So does this mean $Y_k = (I-W)^{-1}\epsilon_k$ or am I misunderstanding the setting completely? $\endgroup$ – Juho Kokkala Dec 8 '17 at 19:52
  • $\begingroup$ Exactly (if $k$ is a vector of observations)! The issue is now to estimate $W$. $\endgroup$ – user_lambda Dec 8 '17 at 19:56
  • $\begingroup$ Um, $k$ is an index(ing integer), $Y_k$ is a $p$-dimensional vector. $\endgroup$ – Juho Kokkala Dec 8 '17 at 20:04

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