Algorithm 2: Hybrid learning classifier algorithm for classification of sleep stages.

Inputs: Training feature matrix (${\mathbf{F}}_{tr}\in {\mathbf{R}}^{I_{tr}\times q}$), training class label (${\mathbf{L}}_{tr}\in {\mathbf{R}}^{I_{tr}}$), test feature matrix (${\mathbf{F}}_{te}\in {\mathbb{R}}^{I_{te}\times q}$), number of nearest neighbors (nn), desired sparsity level ($\rho$). Output: Predicted class label, ${\mathbf{L}}_P\in {\mathbf{R}}^{I_{te}}$ Step 1: The training feature matrix ${\mathbf{F}}_{tr}$ is taken as a dictionary for the sparse representation of the test feature vector. The rth test instance ${\mathbf{f}}_r$ can be written as ${\mathbf{f}}_r={\alpha }^1{\mathbf{F}}_{tr}^1+{\alpha }^2{\mathbf{F}}_{tr}^2+\cdots \cdots {\alpha }^e{\mathbf{F}}_{tr}^e$ [35]. where ${\mathbf{F}}_{tr}^e$ is the feature matrix for eth class. ${\alpha }^1,{\alpha }^2,\cdots \cdots {\alpha }^e$ are the class-specific sparse representation vectors. Step 2: In this step, the combined sparse representation vector $\alpha =\left[{\alpha }^1,{\alpha }^2\cdots \cdots {\alpha }^e\right]$ is evaluated using the orthogonal matching pursuit (OMP) method as the optimization problem based on the fact that the minimization of $L_0$-norm $\alpha =arg{min}_{\alpha }{∥\alpha ∥}_0$ subjected to ${\mathbf{f}}_r=\alpha {\mathbf{F}}_{tr}$ is NP-hard [45]. Step 3: The residual for the eth class is computed as ${\mathrm{Res}}^e={∥{\mathbf{f}}_r-{\alpha }^e{\mathbf{F}}_{tr}^e∥}_2$ [35]. Step 4: In this step, the distances between ${\mathbf{f}}_r$ and all training instances for the eth class are computed and these distances are given as ${\mathrm{dis}}^e(j)={∥{\mathbf{f}}_r-{\mathbf{f}}_{t{r}_j}^e∥}_2$. Then, the nearest distances for each class are selected. The median value of these distances for each class are evaluated, and they are given by ${\mathrm{D}}^e=\mathrm{median}\left({\mathrm{dis}}^e\left(1:\mathrm{nn}\right)\right)$, where nn is the number of nearest neighbors for each class. Step 5: The residual and distance for each class are summed up and the total distance (TD) is computed. The total distance for the eth class is given by ${\mathrm{TD}}^e={\mathrm{Res}}^e+{\mathrm{D}}^e$ [35]. Step 6: The distance vector is evaluated as $\mathrm{TD}=[{\mathrm{TD}}^1,{\mathrm{TD}}^2\cdots \cdots {\mathrm{TD}}^e]$. The predicted class label for each feature vector in the test feature matrix is computed as $l_p=\arg {\min }_e\mathrm{TD}$ [35]. For all test instances, the predicted class label vector is given as ${\mathbf{L}}_P$.