Fig. 7.

Example structured sparse multivariate multiple regression models, where only regression weight matrices W are shown here. Let X be genotype data and Y be imaging QT data. (a) Illustration of the G-SMuRFS model [148] (${\min }_{\mathbf{W}}\Vert \mathbf{Y}-\mathbf{XW}{\Vert }_F^2+{\lambda }_1\Vert \mathbf{W}{\Vert }_{G_{2,1}}+{\lambda }_2\Vert \mathbf{W}{\Vert }_{2,1}$), where the group l_2,1-norm regularization (∥W∥_G2,1) does feature selection at the group level (e.g., LD-block), and the l_2,1-norm regulesization (∥W∥_2,1) does feature selection at the individual SNP level. [Image is reproduced here with permission from Oxford University Press [148]]. (b) Illustration of the TSAL model [149] (${\min }_{\mathbf{W}}{\sum }_{k=1}^t\Vert \mathbf{X}-{\mathbf{Y}}_k{\mathbf{W}}_k{\Vert }_F^2+{\lambda }_1{\mathcal{R}}_1(\mathbf{W})+{\lambda }_2{\mathcal{R}}_2(\mathbf{W})$), where ${\mathcal{R}}_1(\mathbf{W})$ is a Schatten p-norm regularization term to identify low rank structures (e.g., four green boxes sharing similar patterns), and ${\mathcal{R}}_2(\mathbf{W})$ is an l_2,1-norm to select SNPs correlated to most QTs over time (e.g., the red box). [Image is reproduced here with permission from Mary Ann Liebert, Inc. [149]].