Fig. 3

Feature selection using LASSO to shrink some regression coefficients to exactly zero (loss function \({\text{minimize}}\left\{ {\frac{1}{2N}\sum\nolimits_{i = 1}^{N} {(y_{i} - \beta_{0} - \sum\nolimits_{j = 1}^{p} {x_{ij} } \beta_{j} )^{2} + \lambda \left\| \beta \right\|_{{l_{1} }} } } \right\}\). \(\lambda\) is used to limit \(\sum\nolimits_{j = 1}^{p} {\left| {\beta_{j} } \right| \le t}\)). Ten-time cross-validations were used to determine the optimal values of tuning parameter (λ). We selected λ via 1-SE (standard error). The optimal λ is the largest value for which the partial likelihood deviance is within one SE of the smallest value of partial likelihood deviance. a, c Tuning parameter (λ) selection in the LASSO model shown versus log (λ). Dotted vertical lines were drawn at the optimal values using the minimum binomial deviation value, log (λ) = − 3.38 in unenhanced CT and log (λ) = − 3.67 in enhanced CT; b, d LASSO coefficient profiles of the 180 texture features. A coefficient profile plot was produced against the log (λ) sequence. Option l resulted in four nonzero coefficients on unenhanced phase CT imaging and three nonzero coefficients on enhanced phase CT imaging