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Table 3 \({L_q}\) penalty functions of regularization term used in RCPH models

From: Detecting prognostic biomarkers of breast cancer by regularized Cox proportional hazards models

Methods Formulas References
Ridge \(\mathscr {P}(\varvec{\theta };\lambda ) = \lambda \sum \limits _{j=1}^{p} \theta _{j} ^{2}\) [73]
Lasso \(\mathscr {P}(\varvec{\theta };\lambda ) = \lambda \sum \limits _{j=1}^{p}|\theta _{j}|\) [74]
Enet \(\mathscr {P}(\varvec{\theta }; \lambda ) = \lambda \Big [\alpha \sum \limits _{j=1}^{p} | \theta _{j}|+ (1-\alpha )\sum \limits _{j=1}^{p}\theta _{j}^{2}\Big ]\) [75]
\(L_{0}\) \(\mathscr {P}(\varvec{\theta };\lambda ) = \lambda \sum \limits _{j=1}^{p}{1}\left[ {{\theta }_{j}}\ne 0 \right]\) [76]
\(L_{1/2}\) \(\mathscr {P}(\varvec{\theta };\lambda ) = \lambda \sum \limits _{j=1}^{p}|\theta _{j}|^{\frac{1}{2}}\) [77]
SCAD \(\mathscr {P}(\varvec{\theta };\lambda )=\sum \limits _{j=1}^{p}\mathscr {P}_{a}\left( |\theta _{j}|;\ \lambda \right)\), [78]
where \(\mathscr {P}_{a}(|\theta |;\lambda )=\left\{ \begin{array}{*{35}{l}} \lambda |\theta |, &{} |\theta |\le \lambda , \\ \frac{-\left( \theta ^{2} -2a\lambda |\theta |+ \lambda ^{2} \right) }{2(a-1)}, &{} \lambda < |\theta | \le a\lambda , \\ \frac{(a+1) \lambda ^2}{2}, &{} |\theta |>a\lambda . \\ \end{array} \right.\)
MCP \(\mathscr {P}(\varvec{\theta };\lambda )=\sum \limits _{j=1}^{p}\mathscr {P}_{a}\left( \theta _{j};\ \lambda \right)\), [79]
where \(\mathscr {P}_{a}(\theta ;\lambda ) = \left\{ \begin{array}{*{35}{l}} \lambda |\theta |-\frac{\theta ^2}{2a}, &{} |\theta |\le \lambda a, \\ \frac{\lambda ^2 a}{2}, &{} |\theta |>\lambda a. \\ \end{array} \right.\)