A Holistic Strategy for Classification of Sleep Stages with EEG

. 2022 May 7;22(9):3557. doi: 10.3390/s22093557

Algorithm 5: Ridge regression with limiting optimal weights.

Input: Data matrices $(n_{i} \times p)$ and outcomes $(n_{i} \times 1)$ , $(X_{i}, Y_{i})$ distributed across $q$ sites.
Output: Distributed ridge estimator ${\hat{β}}_{d i s t}$ of regression coefficients $β$ indicating the best features extracted and selected.
- 1. For $i \leftarrow 1$ to $q$ do.
  
  Calculate the MLE ${\hat{θ}}_{i} = ({\hat{σ}}_{i}^{2}, {\hat{α}}_{i}^{2})$ locally.
  
  Progress ${\hat{θ}}_{i}$ to the global data center
  
  End
- 2. Get a global estimator $\hat{θ} = ({\hat{σ}}^{2}, {\hat{α}}^{2}) = q^{- 1} \sum_{i = 1}^{q} {\hat{θ}}_{i}$ .
- 3. Tuning parameters $S$ is chosen around the initial guess $λ_{0} = q p / (n {\hat{α}}^{2})$ .
- 4. For $λ \in S$ do.
  
  For $i \leftarrow \begin{matrix} 1 & t o \begin{matrix} q \begin{matrix} d o \end{matrix} \end{matrix} \end{matrix}$ .
  
  Compute the local ridge estimator ${\hat{β}}_{i} (λ) = {(X_{i}^{T} X_{i} + n_{i} λ I_{p})}^{- 1} X_{i}^{T} Y$ .
  
  The weight $w_{i}$ is computed for the $i^{t h}$ local estimator as $w_{i} (λ) = \frac{{\hat{σ}}^{2} {\hat{α}}^{2} (1 - λ m)}{F + q G}$ .
  
  Progress ${\hat{β}}_{i} (λ)$ and $w_{i} (λ)$ to the global data center.
  
  End
  
  Terminate the performance of the distributed ridge estimator.
  
  End
- 5. Select the best tuning parameter $λ^{*}$ .
- 6. Output the respective distributed ridge estimator
  ${\hat{β}}_{d i s t} (λ^{*}) = \sum_{i = 1}^{q} w_{i} (λ^{*}) {\hat{β}}_{i} (λ^{*})$