Approximate and Sample Entropy Measures for Brain Death Diagnosis

is formed by the m-dimension vectors $X_{i} = [u_{i},u_{i+1},\ldots,u_{i+m-1}]$ and $X_{j} = [u_{j},u_{j+1},\ldots,u_{j+m-1}]$ , where $i,j \leq N - m + 1$ . The max distance between X _iand X _jcan be calculated by

$\displaystyle{ d[X_{i},X_{j}] = max_{k=1,2,\ldots,m}[\vert u_{i+k-1} - u_{j+k-1}\vert ]. }$

(1)

Given a threshold r and each $i \leq N - m + 1$ , let $B_{i}^{m}$ be the number of vectors X _jwithin r of X _i, and we define

$\displaystyle{ C_{i}^{m}(r) = \frac{B_{i}^{m}} {N - m + 1},\ where\ i \leq N - m + 1, }$

(2)

and ϕ ^m(r) as mean of $C_{i}^{m}(r)$

$\displaystyle{ \phi ^{m}(r) = \frac{1} {N - m + 1}\sum _{i=1}^{N-m+1}\ln C_{ i}^{m}(r). }$

(3)

Equation (2) is mainly defined to calculate the possibility that for each Xi and Xj, the two vectors are similar within the threshold r, while Eq. (3) is used to calculate the average.

By finding ϕ ^m+1(r), ApEn(r, m, N) takes the form as

$\displaystyle{ ApEn(m,r,N) =\phi ^{m}(r) -\phi ^{m+1}(r). }$

(4)

This is how ApEn is defined to measure the self-similarity of the time series [11].

2.2 Sample Entropy (SampEn)

SampEn deals with same m-dimension vectors X _iand X _jas defined in ApEn. The distance between two vectors is calculated by Eq. (1). In SampEn, let $A_{i}^{m}$ denotes the number of vectors X _jwithin r of X _itimes $(N - m)^{-1}$ , for j ranges from 1 to