Genomic prediction using machine learning: a comparison of the performance of regularized regression, ensemble, instance-based and deep learning methods on synthetic and empirical data

Lourenço, Vanda M.; Ogutu, Joseph O.; Rodrigues, Rui A.P.; Posekany, Alexandra; Piepho, Hans-Peter

doi:10.1186/s12864-023-09933-x

Table 4 A summary of the estimators and penalty functions for the bridge-type and adaptive bridge-type regularized regression methods used in this study. The adaptive methods have an a prefix in their names

From: Genomic prediction using machine learning: a comparison of the performance of regularized regression, ensemble, instance-based and deep learning methods on synthetic and empirical data

Method	Penalty	Estimator
bridge	\(p_{\lambda ,\gamma }(\varvec{\beta })=\lambda \sum \limits _{j=1}^p\|\beta _j\|^{\gamma }\)	\(\widehat{\varvec{\beta }}_{bridge}=\underset{\varvec{\beta }}{\textit{argmin}} \ \Big \{\text {RSS} + \lambda \sum \limits _{j=1}^p\|\beta _j\|^{\gamma } \Big \}, \ \gamma >0, \ \lambda \ge 0\)	(2)
\(\bullet \ \gamma =1\):
LASSO	\(p_{\lambda }(\varvec{\beta })=\lambda \Vert \varvec{\beta }\Vert _1\)	\(\widehat{\varvec{\beta }}_{lasso}=\underset{\varvec{\beta }}{\textit{argmin}} \ \Big \{\text {RSS} + \lambda \Vert \varvec{\beta }\Vert _1\Big \}\)	(3)
\(\bullet \ \gamma =2\):
ridge	\(p_{\lambda }(\varvec{\beta })=\lambda \Vert \varvec{\beta }\Vert _2^2\)	\(\widehat{\varvec{\beta }}_{ridge}=\underset{\varvec{\beta }}{\textit{argmin}} \ \Big \{\text {RSS} + \lambda \Vert \varvec{\beta }\Vert _2^2\Big \}\)	(4)
\(\bullet\) Combination of LASSO and ridge penalties (\(\gamma =1,2\), respectively):
ENET	\(p_{\varvec{\lambda }}(\varvec{\beta })=\lambda _1\Vert \varvec{\beta }\Vert _1+\lambda _2\Vert \varvec{\beta }\Vert _2^2\)	\(\widehat{\varvec{\beta }}_{enet}=(1+\lambda _2)\times \underset{\varvec{\beta }}{\textit{argmin}}\Big \{ \text {RSS} + \lambda _1\Vert \varvec{\beta }\Vert _1+ \lambda _2 \Vert \varvec{\beta }\Vert _2^2\Big \}\)	(6)
abridge	\(p_{\lambda ,\gamma }(\varvec{\beta })=\lambda \sum \limits _{j=1}^p w_j\|\beta _j\|^{\gamma }\)	\(\widehat{\varvec{\beta }}_{\texttt {a}bridge}=\underset{\varvec{\beta }}{\textit{argmin}} \ \Big \{\text {RSS} + \lambda \sum \limits _{j=1}^p{w}_j\|\beta _j\|^{\gamma }\Big \}\)	(7)
\(\bullet \ \gamma =1\):
aLASSO	\(p_{\lambda }(\varvec{\beta })=\lambda \Vert {\textbf {w}}\varvec{\beta }\Vert _1\)	\(\widehat{\varvec{\beta }}_{\texttt {a}lasso}=\underset{\varvec{\beta }}{argmin} \ \Big \{\text {RSS} + \lambda \Vert {\textbf {w}}\varvec{\beta }\Vert _1\Big \}\)	(8)
\(\bullet\) Combination of aLASSO and ridge penalties (\(\gamma =1,2\), respectively):
aENET	\(p_{\varvec{\lambda }}(\varvec{\beta }) = \lambda _1\Vert {\textbf {w}}\varvec{\beta }\Vert _1+ \lambda _2 \Vert \varvec{\beta }\Vert _2^2\)	\(\widehat{\varvec{\beta }}_{\texttt {a}enet}= k\times \underset{\varvec{\beta }}{argmin}\ \ \Big \{\text {RSS} + \lambda _1\Vert {\textbf {w}}\varvec{\beta }\Vert _1+ \lambda _2 \Vert \varvec{\beta }\Vert _2^2\Big \}\)	(9)

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