A Holistic Strategy for Classification of Sleep Stages with EEG

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

Algorithm 3: SVD for ranking and synchronization.

Input: Measurement graph $G = ([n], E)$ and pairwise measurement $R_{i j}$ for ${i, j} \in E$ assigned for the clustered values.
Output: Rank estimates: $\hat{π}$ and score estimates $\hat{r} \in ℜ^{n}$ considered as the dimensionally reduced values.
- 1. Measurement matrix formation $H \in ℜ^{n \times n}$ by utilizing $R_{i j}$ .
- 2. Trace the top 2 left singular vectors of $H$ , namely, ${\hat{u}}_{1}$ , ${\hat{u}}_{2}$ .
- 3. As an orthogonal projection of $u_{1} = e / \sqrt{n}$ onto space ${{\hat{u}}_{1}, {\hat{u}}_{2}}$ ,obtain vector ${\bar{u}}_{1}$ .
- 4. Unit vector ${\tilde{u}}_{2} \in s p a n {{\hat{u}}_{1}, {\hat{u}}_{2}}$ is obtained.
- 5. Rank recovery: induced by ${\tilde{u}}_{2}$ , the ranking $\tilde{π}$ is obtained.
- 6. Minimize the number of upsets and reconcile its global sign.
- 7. The ranking estimate $\hat{π}$ is found out.
- 8. Score recovery: To recover the scale $τ \in ℜ$ , ${\tilde{u}}_{2}, H$ is utilized and the output is expressed, giving the dimensionally reduced values as:
  $\hat{r} = r {\tilde{u}}_{2} - \frac{e^{T} (T {\tilde{u}}_{2})}{n} e$