Table 1.

The detailed procedures of APIT-MEMD.

(1)Given an n-variate signal s(t), and its covariance matrix C is performed eigendecomposition, C = ΣΛΣT, Σ denotes the eigenvector matrix, and Λ denotes the eigenvalues matrix. The largest eigenvalue λ1 corresponds to eigenvector Σ1, which is the first principal component. (2)Construct another vector Σo1 along the opposite direction of Σ1 diametrically. (3)Using Hammerseley sequence, uniformly sample an (n−1) sphere to obtain K direction uniform projection vectors ${\left\{{\mathit{x}}^{{\theta }_k}\right\}}_{k=1}^K$. Then compute Euclidean distances of each direction vector to Σ1. (4)Relocate half of the uniform projection vectors ${\mathit{x}}_{{\mathsf{\Sigma }}_1}^{{\theta }_k}$, which are near to Σ1, using ${\widehat{\mathit{x}}}_{{\mathsf{\Sigma }}_1}^{{\theta }_k}=\frac{{\mathit{x}}_{{\mathsf{\Sigma }}_1}^{{\theta }_k}+\alpha {\mathsf{\Sigma }}_1}{\left|{\mathit{x}}_{{\mathsf{\Sigma }}_1}^{{\theta }_k}+\alpha {\mathsf{\Sigma }}_1\right|}$. Using ${\widehat{\mathit{x}}}_{{\mathsf{\Sigma }}_{o1}}^{{\theta }_k}=\frac{{\mathit{x}}_{{\mathsf{\Sigma }}_{o1}}^{{\theta }_k}+\alpha {\mathsf{\Sigma }}_{o1}}{\left|{\mathit{x}}_{{\mathsf{\Sigma }}_{o1}}^{{\theta }_k}+\alpha {\mathsf{\Sigma }}_{o1}\right|}$ to relocate another half of ${\mathit{x}}_{{\mathsf{\Sigma }}_{o1}}^{{\theta }_k}$, near to Σo1. The density of relocated vectors is controlled by α (The illustration of α is given hereinafter). (5)Conduct local mean estimation based on conventional MEMD algorithm [22], while employing adaptive direction vectors ${\mathit{x}}_{{\mathsf{\Sigma }}_1}^{{\theta }_k}$ and ${\mathit{x}}_{{\mathsf{\Sigma }}_{o1}}^{{\theta }_k}$.