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.
$\begingroup$
$\endgroup$
8
-
$\begingroup$ 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 KokkalaCommented Dec 8, 2017 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_lambdaCommented Dec 8, 2017 at 19:46
-
$\begingroup$ So does this mean $Y_k = (I-W)^{-1}\epsilon_k$ or am I misunderstanding the setting completely? $\endgroup$– Juho KokkalaCommented Dec 8, 2017 at 19:52
-
$\begingroup$ Exactly (if $k$ is a vector of observations)! The issue is now to estimate $W$. $\endgroup$– user_lambdaCommented Dec 8, 2017 at 19:56
-
$\begingroup$ Um, $k$ is an index(ing integer), $Y_k$ is a $p$-dimensional vector. $\endgroup$– Juho KokkalaCommented Dec 8, 2017 at 20:04
|
Show 3 more comments