Approximate entropy

<h2 id="algorithm">Algorithm</h2>
A comprehensive step-by-step tutorial with an explanation of the theoretical foundations of Approximate Entropy is available.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> The algorithm is:

Step 1
Assume a time series of data 
 
 
 
 u
 (
 1
 )
 ,
 u
 (
 2
 )
 ,
 …
 ,
 u
 (
 N
 )
 
 
 {\displaystyle u(1),u(2),\ldots ,u(N)}
 
. These are 
 
 
 
 N
 
 
 {\displaystyle N}
 
 raw data values from measurements equally spaced in time.
Step 2
Let 
 
 
 
 m
 ∈
 
 
 Z
 
 
 +
 
 
 
 
 {\displaystyle m\in \mathbb {Z} ^{+}}
 
 be a positive <a href="/facts/Integer/PUwwrawx">integer</a>, with 
 
 
 
 m
 ≤
 N
 
 
 {\displaystyle m\leq N}
 
, which represents the length of a run of data (essentially a <a href="/facts/Window_function/5Xj7h2ah">window</a>).Let 
 
 
 
 r
 ∈
 
 
 R
 
 
 +
 
 
 
 
 {\displaystyle r\in \mathbb {R} ^{+}}
 
 be a positive <a href="/facts/Real_number/R02gw5Pb">real number</a>, which specifies a filtering level.Let 
 
 
 
 n
 =
 N
 −
 m
 +
 1
 
 
 {\displaystyle n=N-m+1}
 
.
Step 3
Define 
 
 
 
 
 x
 
 (
 i
 )
 =
 
 
 [
 
 
 u
 (
 i
 )
 ,
 u
 (
 i
 +
 1
 )
 ,
 …
 ,
 u
 (
 i
 +
 m
 −
 1
 )
 
 
 ]
 
 
 
 
 {\displaystyle \mathbf {x} (i)={\big [}u(i),u(i+1),\ldots ,u(i+m-1){\big ]}}
 
 for each 
 
 
 
 i
 
 
 {\displaystyle i}
 
 where 
 
 
 
 1
 ≤
 i
 ≤
 n
 
 
 {\displaystyle 1\leq i\leq n}
 
. In other words, 
 
 
 
 
 x
 
 (
 i
 )
 
 
 {\displaystyle \mathbf {x} (i)}
 
 is an 
 
 
 
 m
 
 
 {\displaystyle m}
 
-dimensional <a href="/facts/Vector_(mathematics_and_physics)/U8mM7A5F">vector</a> that contains the run of data starting with 
 
 
 
 u
 (
 i
 )
 
 
 {\displaystyle u(i)}
 
.Define the distance between two vectors 
 
 
 
 
 x
 
 (
 i
 )
 
 
 {\displaystyle \mathbf {x} (i)}
 
 and 
 
 
 
 
 x
 
 (
 j
 )
 
 
 {\displaystyle \mathbf {x} (j)}
 
 as the maximum of the distances between their respective components, given by

d
                [
                
                  x
                
                (
                i
                )
                ,
                
                  x
                
                (
                j
                )
                ]
              
              
                
                =
                
                  max
                  
                    k
                  
                
                
                  
                    (
                  
                
                
                  |
                
                
                  x
                
                (
                i
                
                  )
                  
                    k
                  
                
                −
                
                  x
                
                (
                j
                
                  )
                  
                    k
                  
                
                
                  |
                
                
                  
                    )
                  
                
              
            
            
              
              
                
                =
                
                  max
                  
                    k
                  
                
                
                  
                    (
                  
                
                
                  |
                
                u
                (
                i
                +
                k
                −
                1
                )
                −
                u
                (
                j
                +
                k
                −
                1
                )
                
                  |
                
                
                  
                    )
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}d[\mathbf {x} (i),\mathbf {x} (j)]&=\max _{k}{\big (}|\mathbf {x} (i)_{k}-\mathbf {x} (j)_{k}|{\big )}\\&=\max _{k}{\big (}|u(i+k-1)-u(j+k-1)|{\big )}\\\end{aligned}}}

for 
  
    
      
        1
        ≤
        k
        ≤
        m
      
    
    {\displaystyle 1\leq k\leq m}
  
.
Step 4
Define a count 
  
    
      
        
          C
          
            i
          
          
            m
          
        
      
    
    {\displaystyle C_{i}^{m}}
  
 as

C
          
            i
          
          
            m
          
        
        (
        r
        )
        =
        
          
            
              (
              
                number of 
              
              j
              
                 such that 
              
              d
              [
              
                x
              
              (
              i
              )
              ,
              
                x
              
              (
              j
              )
              ]
              ≤
              r
              )
            
            n
          
        
      
    
    {\displaystyle C_{i}^{m}(r)={({\text{number of }}j{\text{ such that }}d[\mathbf {x} (i),\mathbf {x} (j)]\leq r) \over n}}

for each 
  
    
      
        i
      
    
    {\displaystyle i}
  
 where 
  
    
      
        1
        ≤
        i
        ,
        j
        ≤
        n
      
    
    {\displaystyle 1\leq i,j\leq n}
  
. Note that since 
  
    
      
        j
      
    
    {\displaystyle j}
  
 takes on all values between 1 and 
  
    
      
        n
      
    
    {\displaystyle n}
  
, the match will be counted when 
  
    
      
        j
        =
        i
      
    
    {\displaystyle j=i}
  
 (i.e. when the test subsequence, 
  
    
      
        
          x
        
        (
        j
        )
      
    
    {\displaystyle \mathbf {x} (j)}
  
, is matched against itself, 
  
    
      
        
          x
        
        (
        i
        )
      
    
    {\displaystyle \mathbf {x} (i)}
  
).
Step 5
Define

ϕ
          
            m
          
        
        (
        r
        )
        =
        
          
            1
            n
          
        
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        log
        ⁡
        (
        
          C
          
            i
          
          
            m
          
        
        (
        r
        )
        )
      
    
    {\displaystyle \phi ^{m}(r)={1 \over n}\sum _{i=1}^{n}\log(C_{i}^{m}(r))}

where 
 
 
 
 log
 
 
 {\displaystyle \log }
 
 is the <a href="/facts/Natural_logarithm/PnreKEzI">natural logarithm</a>, and for a fixed 
 
 
 
 m
 
 
 {\displaystyle m}
 
, 
 
 
 
 r
 
 
 {\displaystyle r}
 
, and 
 
 
 
 n
 
 
 {\displaystyle n}
 
 as set in Step 2.
Step 6
Define approximate entropy (
 
 
 
 
 A
 p
 E
 n
 
 
 
 {\displaystyle \mathrm {ApEn} }
 
) as

A
          p
          E
          n
        
        (
        m
        ,
        r
        ,
        N
        )
        (
        u
        )
        =
        
          ϕ
          
            m
          
        
        (
        r
        )
        −
        
          ϕ
          
            m
            +
            1
          
        
        (
        r
        )
      
    
    {\displaystyle \mathrm {ApEn} (m,r,N)(u)=\phi ^{m}(r)-\phi ^{m+1}(r)}

Parameter selection
Typically, choose 
 
 
 
 m
 =
 2
 
 
 {\displaystyle m=2}
 
 or 
 
 
 
 m
 =
 3
 
 
 {\displaystyle m=3}
 
, whereas 
 
 
 
 r
 
 
 {\displaystyle r}
 
 depends greatly on the application.
An implementation on Physionet,<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> which is based on Pincus,<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> use 
 
 
 
 d
 [
 
 x
 
 (
 i
 )
 ,
 
 x
 
 (
 j
 )
 ]
 <
 r
 
 
 {\displaystyle d[\mathbf {x} (i),\mathbf {x} (j)]<r}
 
 instead of 
 
 
 
 d
 [
 
 x
 
 (
 i
 )
 ,
 
 x
 
 (
 j
 )
 ]
 ≤
 r
 
 
 {\displaystyle d[\mathbf {x} (i),\mathbf {x} (j)]\leq r}
 
 in Step 4. While a concern for artificially constructed examples, it is usually not a concern in practice.

<h2 id="example">Example</h2>

Consider a sequence of 
 
 
 
 N
 =
 51
 
 
 {\displaystyle N=51}
 
 samples of heart rate equally spaced in time:

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

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

x
                
                (
                1
                )
              
              
                
                =
                [
                u
                (
                1
                )
                 
                u
                (
                2
                )
                ]
                =
                [
                85
                 
                80
                ]
              
            
            
              
                
                  x
                
                (
                2
                )
              
              
                
                =
                [
                u
                (
                2
                )
                 
                u
                (
                3
                )
                ]
                =
                [
                80
                 
                89
                ]
              
            
            
              
                
                  x
                
                (
                3
                )
              
              
                
                =
                [
                u
                (
                3
                )
                 
                u
                (
                4
                )
                ]
                =
                [
                89
                 
                85
                ]
              
            
            
              
                
                  x
                
                (
                4
                )
              
              
                
                =
                [
                u
                (
                4
                )
                 
                u
                (
                5
                )
                ]
                =
                [
                85
                 
                80
                ]
              
            
            
              
              
                 
                 
                ⋮
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\mathbf {x} (1)&=[u(1)\ u(2)]=[85\ 80]\\\mathbf {x} (2)&=[u(2)\ u(3)]=[80\ 89]\\\mathbf {x} (3)&=[u(3)\ u(4)]=[89\ 85]\\\mathbf {x} (4)&=[u(4)\ u(5)]=[85\ 80]\\&\ \ \vdots \end{aligned}}}

Distance is calculated repeatedly as follows. In the first calculation,

d
 [
 
 x
 
 (
 1
 )
 ,
 
 x
 
 (
 1
 )
 ]
 =
 
 max
 
 k
 
 
 
 |
 
 
 x
 
 (
 1
 
 )
 
 k
 
 
 −
 
 x
 
 (
 1
 
 )
 
 k
 
 
 
 |
 
 =
 0
 
 
 {\displaystyle \ d[\mathbf {x} (1),\mathbf {x} (1)]=\max _{k}|\mathbf {x} (1)_{k}-\mathbf {x} (1)_{k}|=0}
 
 which is less than 
 
 
 
 r
 
 
 {\displaystyle r}
 
.
In the second calculation, note that 
 
 
 
 
 |
 
 u
 (
 2
 )
 −
 u
 (
 3
 )
 
 |
 
 >
 
 |
 
 u
 (
 1
 )
 −
 u
 (
 2
 )
 
 |
 
 
 
 {\displaystyle |u(2)-u(3)|>|u(1)-u(2)|}
 
, so

d
 [
 
 x
 
 (
 1
 )
 ,
 
 x
 
 (
 2
 )
 ]
 =
 
 max
 
 k
 
 
 
 |
 
 
 x
 
 (
 1
 
 )
 
 k
 
 
 −
 
 x
 
 (
 2
 
 )
 
 k
 
 
 
 |
 
 =
 
 |
 
 u
 (
 2
 )
 −
 u
 (
 3
 )
 
 |
 
 =
 9
 
 
 {\displaystyle \ d[\mathbf {x} (1),\mathbf {x} (2)]=\max _{k}|\mathbf {x} (1)_{k}-\mathbf {x} (2)_{k}|=|u(2)-u(3)|=9}
 
 which is greater than 
 
 
 
 r
 
 
 {\displaystyle r}
 
.
Similarly,

d
 [
 
 x
 
 (
 1
 )
 
 
 
 ,
 
 x
 
 (
 3
 )
 ]
 =
 
 |
 
 u
 (
 2
 )
 −
 u
 (
 4
 )
 
 |
 
 =
 5
 >
 r
 
 
 
 
 d
 [
 
 x
 
 (
 1
 )
 
 
 
 ,
 
 x
 
 (
 4
 )
 ]
 =
 
 |
 
 u
 (
 1
 )
 −
 u
 (
 4
 )
 
 |
 
 =
 
 |
 
 u
 (
 2
 )
 −
 u
 (
 5
 )
 
 |
 
 =
 0
 <
 r
 
 
 
 
 
 
 ⋮
 
 
 
 
 d
 [
 
 x
 
 (
 1
 )
 
 
 
 ,
 
 x
 
 (
 j
 )
 ]
 =
 ⋯
 
 
 
 
 
 
 ⋮
 
 
 
 
 
 
 {\displaystyle {\begin{aligned}d[\mathbf {x} (1)&,\mathbf {x} (3)]=|u(2)-u(4)|=5>r\\d[\mathbf {x} (1)&,\mathbf {x} (4)]=|u(1)-u(4)|=|u(2)-u(5)|=0<r\\&\vdots \\d[\mathbf {x} (1)&,\mathbf {x} (j)]=\cdots \\&\vdots \\\end{aligned}}}

The result is a total of 17 terms 
 
 
 
 
 x
 
 (
 j
 )
 
 
 {\displaystyle \mathbf {x} (j)}
 
 such that 
 
 
 
 d
 [
 
 x
 
 (
 1
 )
 ,
 
 x
 
 (
 j
 )
 ]
 ≤
 r
 
 
 {\displaystyle d[\mathbf {x} (1),\mathbf {x} (j)]\leq r}
 
. These include 
 
 
 
 
 x
 
 (
 1
 )
 ,
 
 x
 
 (
 4
 )
 ,
 
 x
 
 (
 7
 )
 ,
 …
 ,
 
 x
 
 (
 49
 )
 
 
 {\displaystyle \mathbf {x} (1),\mathbf {x} (4),\mathbf {x} (7),\ldots ,\mathbf {x} (49)}
 
. In these cases, 
 
 
 
 
 C
 
 i
 
 
 m
 
 
 (
 r
 )
 
 
 {\displaystyle C_{i}^{m}(r)}
 
 is

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

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

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

C
          
            4
          
          
            2
          
        
        (
        3
        )
        =
        
          
            17
            50
          
        
         
        ⋯
      
    
    {\displaystyle \ C_{4}^{2}(3)={\frac {17}{50}}\ \cdots }

Note in Step 4, 
 
 
 
 1
 ≤
 i
 ≤
 n
 
 
 {\displaystyle 1\leq i\leq n}
 
 for 
 
 
 
 
 x
 
 (
 i
 )
 
 
 {\displaystyle \mathbf {x} (i)}
 
. So the terms 
 
 
 
 
 x
 
 (
 j
 )
 
 
 {\displaystyle \mathbf {x} (j)}
 
 such that 
 
 
 
 d
 [
 
 x
 
 (
 3
 )
 ,
 
 x
 
 (
 j
 )
 ]
 ≤
 r
 
 
 {\displaystyle d[\mathbf {x} (3),\mathbf {x} (j)]\leq r}
 
 include 
 
 
 
 
 x
 
 (
 3
 )
 ,
 
 x
 
 (
 6
 )
 ,
 
 x
 
 (
 9
 )
 ,
 …
 ,
 
 x
 
 (
 48
 )
 
 
 {\displaystyle \mathbf {x} (3),\mathbf {x} (6),\mathbf {x} (9),\ldots ,\mathbf {x} (48)}
 
, and the total number is 16.
At the end of these calculations, we have

ϕ
          
            2
          
        
        (
        3
        )
        =
        
          
            1
            50
          
        
        
          ∑
          
            i
            =
            1
          
          
            50
          
        
        log
        ⁡
        (
        
          C
          
            i
          
          
            2
          
        
        (
        3
        )
        )
        ≈
        −
        1.0982
      
    
    {\displaystyle \phi ^{2}(3)={1 \over 50}\sum _{i=1}^{50}\log(C_{i}^{2}(3))\approx -1.0982}

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

x
                
                (
                1
                )
              
              
                
                =
                [
                u
                (
                1
                )
                 
                u
                (
                2
                )
                 
                u
                (
                3
                )
                ]
                =
                [
                85
                 
                80
                 
                89
                ]
              
            
            
              
                
                  x
                
                (
                2
                )
              
              
                
                =
                [
                u
                (
                2
                )
                 
                u
                (
                3
                )
                 
                u
                (
                4
                )
                ]
                =
                [
                80
                 
                89
                 
                85
                ]
              
            
            
              
                
                  x
                
                (
                3
                )
              
              
                
                =
                [
                u
                (
                3
                )
                 
                u
                (
                4
                )
                 
                u
                (
                5
                )
                ]
                =
                [
                89
                 
                85
                 
                80
                ]
              
            
            
              
                
                  x
                
                (
                4
                )
              
              
                
                =
                [
                u
                (
                4
                )
                 
                u
                (
                5
                )
                 
                u
                (
                6
                )
                ]
                =
                [
                85
                 
                80
                 
                89
                ]
              
            
            
              
              
                 
                 
                ⋮
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\mathbf {x} (1)&=[u(1)\ u(2)\ u(3)]=[85\ 80\ 89]\\\mathbf {x} (2)&=[u(2)\ u(3)\ u(4)]=[80\ 89\ 85]\\\mathbf {x} (3)&=[u(3)\ u(4)\ u(5)]=[89\ 85\ 80]\\\mathbf {x} (4)&=[u(4)\ u(5)\ u(6)]=[85\ 80\ 89]\\&\ \ \vdots \end{aligned}}}

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

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

Therefore,

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

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

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

C
          
            4
          
          
            3
          
        
        (
        3
        )
        =
        
          
            17
            49
          
        
         
        ⋯
      
    
    {\displaystyle \ C_{4}^{3}(3)={\frac {17}{49}}\ \cdots }

At the end of these calculations, we have

ϕ
          
            3
          
        
        (
        3
        )
        =
        
          
            1
            49
          
        
        
          ∑
          
            i
            =
            1
          
          
            49
          
        
        log
        ⁡
        (
        
          C
          
            i
          
          
            3
          
        
        (
        3
        )
        )
        ≈
        −
        1.0982
      
    
    {\displaystyle \phi ^{3}(3)={1 \over 49}\sum _{i=1}^{49}\log(C_{i}^{3}(3))\approx -1.0982}

Finally,

A
          p
          E
          n
        
        =
        
          ϕ
          
            2
          
        
        (
        3
        )
        −
        
          ϕ
          
            3
          
        
        (
        3
        )
        ≈
        0.000010997
      
    
    {\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.

<h2 id="python-implementation">Python implementation</h2>
import math

def approx_entropy(time_series, run_length, filter_level) -> float:
    """
    Approximate entropy

>>> import random
 >>> regularly = [85, 80, 89] * 17
 >>> print(f"{approx_entropy(regularly, 2, 3):e}")
 1.099654e-05
 >>> randomly = [random.choice([85, 80, 89]) for _ in range(17*3)]
 >>> 0.8 < approx_entropy(randomly, 2, 3) < 1
 True
 """

def _maxdist(x_i, x_j):
        return max(abs(ua - va) for ua, va in zip(x_i, x_j))

def _phi(m):
 n = time_series_length - m + 1
 x = [
 [time_series[j] for j in range(i, i + m - 1 + 1)]
 for i in range(time_series_length - m + 1)
 ]
 counts = [
 sum(1 for x_j in x if _maxdist(x_i, x_j) <= filter_level) / n for x_i in x
 ]
 return sum(math.log(c) for c in counts) / n

time_series_length = len(time_series)

return abs(_phi(run_length + 1) - _phi(run_length))

if __name__ == "__main__":
    import doctest

doctest.testmod()

<h2 id="matlab-implementation">MATLAB implementation</h2>
<ul><li><a href="https://mathworks.com/matlabcentral/fileexchange/32427-fast-approximate-entropy?s_tid=srchtitle_fast%20approximate%20entropy_3">Fast Approximate Entropy</a> from MatLab Central</li>
<li><a href="https://mathworks.com/help/predmaint/ref/approximateentropy.html">approximateEntropy</a></li></ul>
<h2 id="interpretation">Interpretation</h2>
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.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> A time series containing many repetitive patterns has a relatively small ApEn; a less predictable process has a higher ApEn.

<h2 id="advantages">Advantages</h2>
The advantages of ApEn include:<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

<ul><li>Lower computational demand. ApEn can be designed to work for small data samples (
 
 
 
 N
 <
 50
 
 
 {\displaystyle N<50}
 
 points) and can be applied in real time.</li>
<li>Less effect from noise. If data is 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.</li></ul>
<h2 id="limitations">Limitations</h2>
The ApEn algorithm counts each sequence as matching itself to avoid the occurrence of 
 
 
 
 log
 ⁡
 (
 0
 )
 
 
 {\displaystyle \log(0)}
 
 in the calculations. This step might introduce bias in ApEn, which causes ApEn to have two poor properties in practice:<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> 

<ol><li>ApEn is heavily dependent on the record length and is uniformly lower than expected for short records.</li>
<li>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.</li></ol>
<h2 id="applications">Applications</h2>
ApEn has been applied to classify <a href="/facts/Electroencephalography/XGvpcBl5">electroencephalography</a> (EEG) in psychiatric diseases, such as schizophrenia,<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> epilepsy,<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> and addiction.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Recurrence_quantification_analysis/aKdsM7zT">Recurrence quantification analysis</a></li>
<li><a href="/facts/Sample_entropy/W2w9gIA0">Sample entropy</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Pincus, S. M.; Gladstone, I. M.; Ehrenkranz, R. A. (1991). "A regularity statistic for medical data analysis". Journal of Clinical Monitoring and Computing. 7 (4): 335–345. doi:10.1007/BF01619355. PMID 1744678. S2CID 23455856. <a href="/w/index.php?title=Journal_of_Clinical_Monitoring_and_Computing&action=edit&redlink=1" target="_blank">/w/index.php?title=Journal_of_Clinical_Monitoring_and_Computing&action=edit&redlink=1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Pincus, S. M.; Gladstone, I. M.; Ehrenkranz, R. A. (1991). "A regularity statistic for medical data analysis". Journal of Clinical Monitoring and Computing. 7 (4): 335–345. doi:10.1007/BF01619355. PMID 1744678. S2CID 23455856. <a href="/w/index.php?title=Journal_of_Clinical_Monitoring_and_Computing&action=edit&redlink=1" target="_blank">/w/index.php?title=Journal_of_Clinical_Monitoring_and_Computing&action=edit&redlink=1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Pincus, S. M. (1991). "Approximate entropy as a measure of system complexity". Proceedings of the National Academy of Sciences. 88 (6): 2297–2301. Bibcode:1991PNAS...88.2297P. doi:10.1073/pnas.88.6.2297. PMC 51218. PMID 11607165. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC51218" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC51218</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Cohen, A.; Procaccia, I. (1985). "Computing the Kolmogorov entropy from time signals of dissipative and conservative dynamical systems". Physical Review A. 28 (3): 2591(R). Bibcode:1985PhRvA..31.1872C. doi:10.1103/PhysRevA.31.1872. PMID 9895695. <a href="/wiki/Physical_Review_A" target="_blank">/wiki/Physical_Review_A</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Pincus, S. M.; Gladstone, I. M.; Ehrenkranz, R. A. (1991). "A regularity statistic for medical data analysis". Journal of Clinical Monitoring and Computing. 7 (4): 335–345. doi:10.1007/BF01619355. PMID 1744678. S2CID 23455856. <a href="/w/index.php?title=Journal_of_Clinical_Monitoring_and_Computing&action=edit&redlink=1" target="_blank">/w/index.php?title=Journal_of_Clinical_Monitoring_and_Computing&action=edit&redlink=1</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Pincus, S.M.; Kalman, E.K. (2004). "Irregularity, volatility, risk, and financial market time series". Proceedings of the National Academy of Sciences. 101 (38): 13709–13714. Bibcode:2004PNAS..10113709P. doi:10.1073/pnas.0405168101. PMC 518821. PMID 15358860. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC518821" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC518821</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Pincus, S.M.; Goldberger, A.L. (1994). "Physiological time-series analysis: what does regularity quantify?". The American Journal of Physiology. 266 (4): 1643–1656. doi:10.1152/ajpheart.1994.266.4.H1643. PMID 8184944. S2CID 362684. <a href="/wiki/The_American_Journal_of_Physiology" target="_blank">/wiki/The_American_Journal_of_Physiology</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">McKinley, R.A.; McIntire, L.K.; Schmidt, R; Repperger, D.W.; Caldwell, J.A. (2011). "Evaluation of Eye Metrics as a Detector of Fatigue". Human Factors. 53 (4): 403–414. doi:10.1177/0018720811411297. PMID 21901937. S2CID 109251681. <a href="/wiki/Human_Factors_(journal)" target="_blank">/wiki/Human_Factors_(journal)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Delgado-Bonal, Alfonso; Marshak, Alexander; Yang, Yuekui; Holdaway, Daniel (2020-01-22). "Analyzing changes in the complexity of climate in the last four decades using MERRA-2 radiation data". Scientific Reports. 10 (1): 922. Bibcode:2020NatSR..10..922D. doi:10.1038/s41598-020-57917-8. ISSN 2045-2322. PMC 6976651. PMID 31969616. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6976651" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6976651</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Delgado-Bonal, Alfonso; Marshak, Alexander (June 2019). "Approximate Entropy and Sample Entropy: A Comprehensive Tutorial". Entropy. 21 (6): 541. Bibcode:2019Entrp..21..541D. doi:10.3390/e21060541. PMC 7515030. PMID 33267255. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7515030" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7515030</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">"PhysioNet". Archived from the original on 2012-06-18. Retrieved 2012-07-04. <a href="https://web.archive.org/web/20120618173711/http://physionet.org/physiotools/ApEn/" target="_blank">https://web.archive.org/web/20120618173711/http://physionet.org/physiotools/ApEn/</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Pincus, S. M. (1991). "Approximate entropy as a measure of system complexity". Proceedings of the National Academy of Sciences. 88 (6): 2297–2301. Bibcode:1991PNAS...88.2297P. doi:10.1073/pnas.88.6.2297. PMC 51218. PMID 11607165. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC51218" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC51218</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Ho, K. K.; Moody, G. B.; Peng, C.K.; Mietus, J. E.; Larson, M. G.; levy, D; Goldberger, A. L. (1997). "Predicting survival in heart failure case and control subjects by use of fully automated methods for deriving nonlinear and conventional indices of heart rate dynamics". Circulation. 96 (3): 842–848. doi:10.1161/01.cir.96.3.842. PMID 9264491. <a href="/wiki/Circulation_(journal)" target="_blank">/wiki/Circulation_(journal)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Pincus, S. M. (1991). "Approximate entropy as a measure of system complexity". Proceedings of the National Academy of Sciences. 88 (6): 2297–2301. Bibcode:1991PNAS...88.2297P. doi:10.1073/pnas.88.6.2297. PMC 51218. PMID 11607165. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC51218" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC51218</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Richman, J.S.; Moorman, J.R. (2000). "Physiological time-series analysis using approximate entropy and sample entropy". American Journal of Physiology. Heart and Circulatory Physiology. 278 (6): 2039–2049. doi:10.1152/ajpheart.2000.278.6.H2039. PMID 10843903. S2CID 2389971. <a href="/wiki/American_Journal_of_Physiology._Heart_and_Circulatory_Physiology" target="_blank">/wiki/American_Journal_of_Physiology._Heart_and_Circulatory_Physiology</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Sabeti, Malihe (2009). "Entropy and complexity measures for EEG signal classification of schizophrenic and control participants". Artificial Intelligence in Medicine. 47 (3): 263–274. doi:10.1016/j.artmed.2009.03.003. PMID 19403281. <a href="/wiki/Artificial_Intelligence_in_Medicine" target="_blank">/wiki/Artificial_Intelligence_in_Medicine</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Yuan, Qi (2011). "Epileptic EEG classification based on extreme learning machine and nonlinear features". Epilepsy Research. 96 (1–2): 29–38. doi:10.1016/j.eplepsyres.2011.04.013. PMID 21616643. S2CID 41730913. <a href="/w/index.php?title=Epilepsy_Research&action=edit&redlink=1" target="_blank">/w/index.php?title=Epilepsy_Research&action=edit&redlink=1</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Yun, Kyongsik (2012). "Decreased cortical complexity in methamphetamine abusers". Psychiatry Research: Neuroimaging. 201 (3): 226–32. doi:10.1016/j.pscychresns.2011.07.009. PMID 22445216. S2CID 30670300. <a href="/wiki/Psychiatry_Research:_Neuroimaging" target="_blank">/wiki/Psychiatry_Research:_Neuroimaging</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
</ol>

Approximate entropy open-in-new

Approximate entropy