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let's consider the lasso problem as: $$ \frac{1}{2}\|y-X\beta\|_2^2 + \lambda\|\beta\|_1 $$ and referring to The lasso by Patrick Breheny enter image description here

he says the correlation between a predictor and residuals, must exceed a certain minimum threshold λ before it is included in the model.

what if we chose λ equals 2 or any value larger than 1. how that statement could be true? as we know the correlation (Pearson correlation) lies between -1,1. or let's say by that statement if the correlation between a predictor and residuals =0.99 and λ = 2 (arbitrary example) then none of the predictors of my model won't be included in the model

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    $\begingroup$ That's not a correlation, that's a covariance. I wonder if he just experienced a brief mental lapse while typing in his slides? $\endgroup$
    – jbowman
    Dec 27, 2022 at 18:29
  • $\begingroup$ I think it's a correlation, not a covariance as all variables are standardized. $\endgroup$
    – Nidal
    Dec 28, 2022 at 4:30
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    $\begingroup$ @Nidal a residual is not standardized. $\endgroup$
    – AdamO
    Dec 28, 2022 at 6:01
  • $\begingroup$ @AdamO "the variables that have equal (and maximal) absolute correlation with the residual" $\endgroup$
    – Nidal
    Dec 28, 2022 at 16:16
  • $\begingroup$ stats.stackexchange.com/questions/351552/… read this also $\endgroup$
    – Nidal
    Dec 28, 2022 at 16:22

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