Table 2.

Definition of approximate Entropy (ApEn) as proposed in [35]. We used parameter values of $m=2$ and $r=0.2\times sd\left(\mathit{x}\right)$ as recommended in [36].

$ApEn(m,r,k)\mathrm{applied}\mathrm{to}\mathrm{a}\text{ time-series }\mathit{x}=\{x_1,x_2,\dots ,x_K\}$

$\mathrm{Define}\mathrm{a}\mathrm{vector}\mathrm{of}\mathrm{length}m\mathrm{that}\mathrm{begins}\mathrm{with}{x}_i,\mathrm{as}{\mathit{u}}_i=\left\{x_i,x_{i+1},\dots ,x_{i+m-1}\right\},\mathrm{where}1\le i\le K-m+1.\mathrm{Then},\mathrm{form}\mathrm{a}$ $\mathrm{sequence}\mathrm{of}\mathrm{these}\mathrm{vectors}\mathrm{that}\mathrm{covers}\mathrm{all}\mathrm{points}\mathrm{in}\mathrm{the}\mathrm{time}\mathrm{series}:\left\{{\mathit{u}}_1,{\mathit{u}}_2,\dots ,{\mathit{u}}_{K-m+1}\right\}$

$\mathrm{Define}\mathrm{the}\mathrm{distance}\mathrm{d}\mathrm{between}{\mathit{u}}_i\mathrm{and}{\mathit{u}}_j\mathrm{as}d\left[{\mathit{u}}_i,{\mathit{u}}_j\right]=\underset{1\le l\le m}{\max }\left|x_{i+l-1}-x_{j+l-1}\right|$

$\mathrm{Define}{C}_i^m\left(r\right)=\left(\mathrm{number}\mathrm{of}{u}_j\mathrm{such}\mathrm{that}d\left[u_i,u_j\right]\le r\right)/\left(K-m+1\right),\mathrm{where}1\le i\le K-m+1$

$\mathrm{The}\mathrm{scalar}\mathrm{defined}\mathrm{as}{\varnothing }^m\left(r\right)=\frac{1}{K-m+1}{\displaystyle \sum }_{i=1}^{K-m+1}\log C_i^m\left(r\right)\mathrm{then}\mathrm{gives}:ApEn\left(m,r,K\right)=\left[{\varnothing }^m\left(r\right)-{\varnothing }^{m+1}\left(r\right)\right]$