Approximate entropy

A comprehensive step-by-step tutorial with an explanation of the theoretical foundations of Approximate Entropy is available at:[7]

Step 1

Form a time series of data ${\displaystyle \ u(1),u(2),\ldots ,u(N)}$. These are N raw data values from measurement equally spaced in time.

Step 2

Fix m, an integer, and r, a positive real number. The value of m represents the length of compared run of data, and r specifies a filtering level.

Step 3

Form a sequence of vectors ${\displaystyle \mathbf {x} (1)}$,${\displaystyle \mathbf {x} (2),\ldots ,\mathbf {x} (N-m+1)}$, in ${\displaystyle \mathbf {R} ^{m}}$, real ${\displaystyle \ m}$-dimensional space defined by ${\displaystyle \mathbf {x} (i)=[u(i),u(i+1),\ldots ,u(i+m-1)]}$.

Step 4

Use the sequence ${\displaystyle \mathbf {x} (1)}$,${\displaystyle \mathbf {x} (2),\ldots ,\mathbf {x} (N-m+1)}$ to construct, for each I, ${\displaystyle 1\leq i\leq N-m+1}$

${\displaystyle C_{i}^{m}(r)=({\text{number of }}x(j){\text{ such that }}d[x(i),x(j)]\leq r/(N-m+1))}$

in which ${\displaystyle \ d[x,x^{*}]}$ is defined as

${\displaystyle d[x,x^{*}]=\max _{a}|u(a)-u^{*}(a)|\,}$

The ${\displaystyle u(a)}$ are the m scalar components of ${\displaystyle \mathbf {x} }$. d represents the distance between the vectors ${\displaystyle \mathbf {x} (i)}$ and ${\displaystyle \mathbf {x} (j)}$, given by the maximum difference in their respective scalar components. Note that ${\displaystyle j}$ takes on all values, so the match provided when ${\displaystyle i=j}$ will be counted (the subsequence is matched against itself).

Step 5

Define

${\displaystyle \Phi ^{m}(r)=(N-m+1)^{-1}\sum _{i=1}^{N-m+1}\log(C_{i}^{m}(r))}$,

Step 6

Define approximate entropy ${\displaystyle \ (\mathrm {ApEn} )}$ as

${\displaystyle \mathrm {ApEn} =\Phi ^{m}(r)-\Phi ^{m+1}(r).}$

where ${\displaystyle \log }$ is the natural logarithm, for m and r fixed as in Step 2.

Parameter selection

typically choose ${\displaystyle m=2}$ or ${\displaystyle m=3}$, and r depends greatly on the application.

An implementation on Physionet,[8] which is based on Pincus [2] use ${\displaystyle d[x(i),x(j)]<r}$ whereas the original article uses ${\displaystyle d[x(i),x(j)]\leq r}$ in Step 4. While a concern for artificially constructed examples, it is usually not a concern in practice.

The presence of repetitive patterns of fluctuation in a time series renders it more predictable than a time series in which such patterns are absent. ApEn reflects the likelihood that similar patterns of observations will not be followed by additional similar observations.[9] A time series containing many repetitive patterns has a relatively small ApEn; a less predictable process has a higher ApEn.

Illustration of the Heart Rate Sequence

Suppose ${\displaystyle \ N=51}$, and the sequence consists of 51 samples of heart rate equally spaced in time:

${\displaystyle \ S_{N}=\{85,80,89,85,80,89,\ldots \}}$

(i.e., the sequence is periodic with a period of 3). Let's choose ${\displaystyle \ m=2}$ and ${\displaystyle \ r=3}$ (the values of ${\displaystyle \ m}$ and ${\displaystyle \ r}$ can be varied without affecting the result).

Form a sequence of vectors:

${\displaystyle \mathbf {x} (1)=[u(1)\,u(2)]=[85\,80]}$

${\displaystyle \mathbf {x} (2)=[u(2)\,u(3)]=[80\,89]}$

${\displaystyle \mathbf {x} (3)=[u(3)\,u(4)]=[89\,85]}$

${\displaystyle \mathbf {x} (4)=[u(4)\,u(5)]=[85\,80]}$…

Distance is calculated as follows:

${\displaystyle \ d[\mathbf {x} (1),\mathbf {x} (1)]=\max _{a}|u(a)-u^{*}(a)|=0<r=3}$

Note ${\displaystyle \ |u(2)-u(3)|>|u(1)-u(2)|}$, so

${\displaystyle \ d[\mathbf {x} (1),\mathbf {x} (2)]=\max _{a}|u(a)-u^{*}(a)|=|u(2)-u(3)|=9>r=3}$

Similarly,

${\displaystyle \ d[\mathbf {x} (1),\mathbf {x} (3)]=|u(2)-u(4)|=5>r}$

${\displaystyle \ d[\mathbf {x} (1),\mathbf {x} (4)]=|u(1)-u(4)|=|u(2)-u(5)|=0<r}$

Therefore, ${\displaystyle \mathbf {x} (j){\text{s}}}$ such that ${\displaystyle \ d[\mathbf {x} (1),\mathbf {x} (j)]\leq r}$ include ${\displaystyle \mathbf {x} (1),\mathbf {x} (4),\mathbf {x} (7),\ldots ,\mathbf {x} (49)}$, and the total number is 17.

${\displaystyle \ C_{1}^{2}(3)={\frac {17}{50}}}$

${\displaystyle \ C_{2}^{2}(3)={\frac {17}{50}}}$

${\displaystyle \ C_{3}^{2}(3)={\frac {16}{50}}}$

${\displaystyle \ C_{4}^{2}(3)={\frac {17}{50}}\ \ldots }$

Please note in Step 4, for ${\displaystyle \mathbf {x} (i)}$, ${\displaystyle \ 1\leq i\leq N-m+1}$. So the ${\displaystyle \mathbf {x} (j){\text{s}}}$ such that ${\displaystyle \ d[\mathbf {x} (3),\mathbf {x} (j)]<r}$ include ${\displaystyle \mathbf {x} (3),\mathbf {x} (6),\mathbf {x} (9),\ldots ,\mathbf {x} (48)}$, and the total number is 16.

${\displaystyle \Phi ^{2}(3)=(50)^{-1}\sum _{i=1}^{50}\log(C_{i}^{2}(3))\approx -1.0982}$

Then we repeat the above steps for m=3. First form a sequence of vectors:

${\displaystyle \mathbf {x} (1)=[u(1)\,u(2)\,u(3)]=[85\,80\,89]}$

${\displaystyle \mathbf {x} (2)=[u(2)\,u(3)\,u(4)]=[80\,89\,85]}$

${\displaystyle \mathbf {x} (3)=[u(3)\,u(4)\,u(5)]=[89\,85\,80]}$

${\displaystyle \mathbf {x} (4)=[u(4)\,u(5)\,u(6)]=[85\,80\,89]}$…

By calculating distances between vector ${\displaystyle \mathbf {x} (i),\mathbf {x} (j),1\leq i\leq 49}$, we find the vectors satisfying the filtering level have the following characteristic:

${\displaystyle \ d[\mathbf {x} (i),\mathbf {x} (i+3)]=0<r}$

Therefore,

${\displaystyle \ C_{1}^{3}(3)={\frac {17}{49}}}$

${\displaystyle \ C_{2}^{3}(3)={\frac {16}{49}}}$

${\displaystyle \ C_{3}^{3}(3)={\frac {16}{49}}}$

${\displaystyle \ C_{4}^{3}(3)={\frac {17}{49}}\ \ldots }$

${\displaystyle \Phi ^{3}(3)=(49)^{-1}\sum _{i=1}^{49}\log(C_{i}^{3}(3))\approx -1.0982}$

Finally,

${\displaystyle \mathrm {ApEn} =\Phi ^{2}(3)-\Phi ^{3}(3)\approx 0.000010997}$

The value is very small, so it implies the sequence is regular and predictable, which is consistent with the observation.

The advantages of ApEn include:[2]

• Lower computational demand. ApEn can be designed to work for small data samples (n < 50 points) and can be applied in real time.
• Less effect from noise. If data are noisy, the ApEn measure can be compared to the noise level in the data to determine what quality of true information may be present in the data.

ApEn has been applied to classify EEG in psychiatric diseases, such as schizophrenia,[10] epilepsy,[11] and addiction.[12]

The ApEn algorithm counts each sequence as matching itself to avoid the occurrence of ln(0) in the calculations. This step might cause bias of ApEn and this bias causes ApEn to have two poor properties in practice:[13]

1. ApEn is heavily dependent on the record length and is uniformly lower than expected for short records.
2. It lacks relative consistency. That is, if ApEn of one data set is higher than that of another, it should, but does not, remain higher for all conditions tested.

• Recurrence quantification analysis
• Sample entropy