ASCI 896 Statistical Genomics
Gota Morota
1/28/16
Whole-genome regression - ridge regression 2
Singular value decomposition
\( \mathbf{X} \) is full rank
\( \mathbf{X} = \mathbf{UDV}^{'} \)
\( \hat{\boldsymbol{\beta}}^{ols} = \mathbf{VD^{-1}U'}\mathbf{y} \)
\( \hat{\mathbf{y}}^{ols} = \mathbf{UU'}\mathbf{y} \)
\( \hat{\boldsymbol{\beta}}^{ridge} = \mathbf{V} (\mathbf{D^2} + \lambda \mathbf{I})^{-1}\mathbf{DU'}\mathbf{y} \)
\( \hat{\mathbf{y}}^{ridge} = \mathbf{UD} (\mathbf{D^2} + \lambda \mathbf{I})^{-1}\mathbf{DU'}\mathbf{y} \)