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

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 \(\mathscr {P}_{a}(\theta ;\lambda ) = \left\{ \begin{array}{*{35}{l}} \lambda |\theta |-\frac{\theta ^2}{2a}, &{} |\theta |\le \lambda a, \\ \frac{\lambda ^2 a}{2}, &{} |\theta |>\lambda a. \\ \end{array} \right.\)