ASCI 896 Statistical Genomics

Gota Morota
1/28/16

Whole-genome regression - ridge regression 2

Singular value decomposition

  • \( \mathbf{X} \) is full rank
  • \( \mathbf{X} = \mathbf{UDV}^{'} \)
  • \( \hat{\boldsymbol{\beta}}^{ols} = \mathbf{VD^{-1}U'}\mathbf{y} \)
  • \( \hat{\mathbf{y}}^{ols} = \mathbf{UU'}\mathbf{y} \)
  • \( \hat{\boldsymbol{\beta}}^{ridge} = \mathbf{V} (\mathbf{D^2} + \lambda \mathbf{I})^{-1}\mathbf{DU'}\mathbf{y} \)
  • \( \hat{\mathbf{y}}^{ridge} = \mathbf{UD} (\mathbf{D^2} + \lambda \mathbf{I})^{-1}\mathbf{DU'}\mathbf{y} \)