#StackBounty: #machine-learning #lasso Clarification for replication in Adaptive Lasso

Bounty: 50

My question is regarding the Adaptive LASSO procedure, developed by Zou, the details of which can be found here.

In this paper, I wish to replicate Models 0 and 1. Let us focus on model 0 for the moment. But before that, let us clarify some notation. In this paper, we have a true model, consisting of $p_{0}$ variables,
lesser than the total n umber of predictors, $p.$ I could be wrong,
but the variance-covariance matrix of regressors,
$$
frac{1}{n}boldsymbol{X’Xrightarrow}boldsymbol{C}
$$

is partitioned as:
$$
boldsymbol{C=left[begin{array}{cc}
boldsymbol{C_{11}} & boldsymbol{C_{12}}\
boldsymbol{C_{21}} & boldsymbol{C_{22}}
end{array}right]}
$$

Here, the author states on page 2 that $boldsymbol{C_{11}}$is a
$p_{0}times p_{0}$ matrix. This, I would imagine, corresponds to
the variances, of the regressors in the true model. $boldsymbol{C_{22}}$
would correspond to the $p-p_{0}$ extraneous regressors, and the
off-diagonal elements correspond to the covariances. My question pertains
to page 6, wherein we replicate Model 0. In model 0, the author says
that they simulate data, $y=boldsymbol{x’beta+Nleft(0,sigma^{2}right)}$where
the true regression coefficients are $beta=left(5.6,5.6,5.6,0right)$
. The predictors $boldsymbol{x_{i}}(i=1…n)$ are i.i.d N(0,textbf{$boldsymbol{C)}$
}, where textbf{$boldsymbol{C}$ }is the corollary matrix with $rho_{1}$
and $rho_{2}$ given. The values of $rho_{1}=-0.39$ and $rho_{2}=0.23.$

My question is :- how do $rho_{1}$ and $rho_{2}$ correspond to
the matrix $boldsymbol{C?}$ Although it is clear that $boldsymbol{x_{i}}$
are distributed according to mean 0, how does $rho_{1}$ and $rho_{2}$
map to variances and covariances? Normally, $rho$ corresponds to
correlation. Without any further information, can we conclude that
these are standardized regressors and that the variances are 1? I
would greatly appreciate if someone could guide me towards what the
matrix $boldsymbol{C?}$ would look like in this case, so that I can replicate
this simulation in Model 0.


Get this bounty!!!

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.