# #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.

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