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An intercept-unpenalized Lasso estimator solves the constrained optimizationFor centered predictors, eliminate the intercept as in ridge regression and minimize $\Vert Y_c-X\beta {\Vert }_2^2$ under the same constraint. Equivalently, with a suitable tuning parameter $\lambda \ge 0$, use $\Vert Y-\alpha 1-X\beta {\Vert }_2^2+\lambda {\sum }_j|{\beta }_j|$. The constraint radius and penalty parameter are different parametrizations; larger radius permits less shrinkage.

The Lasso plot parametrizes the Lasso regularization path by the fraction of the maximum $l_1$ norm of the standardized slopes. Unlike the ridge quadratic penalty, the corners of the $l_1$ constraint can place some slopes exactly at zero, providing variable selection. The maximum norm refers to the path's unpenalized endpoint; centering and scaling conventions must match those used to construct that path.

At fraction $0.8$, the Lasso regularization path lies between the vertical lines marked steps 4 and 5. In that interval the first five predictors have nonzero slopes: $X_1,X_2,X_4,X_5$ are positive and $X_3$ is negative. The sixth predictor remains at zero until the later part of the path, beyond this fraction. Therefore the chosen Lasso model includesThe regression intercept is retained. In particular the negative $X_3$ trace has already left zero at fraction $0.8$; a small slope is not the same as a zero slope. The selected fraction came from ten-fold K-fold cross-validation, and is not a numerical value of the ridge penalty in the preceding parts.

Variable selection chooses which predictor variables enter a statistical model. The Lasso produces exact zero slopes along its Lasso regularization path; ridge regression generally shrinks without excluding variables. Selection for prediction should use predictive cross-validation or another stated criterion, rather than equating a coefficient p-value with practical usefulness.