Sample entropy (SampEn; more appropriately K_2 entropy or Takens–Grassberger–Procaccia correlation entropy ) is a modification of approximate entropy (ApEn; more appropriately "Procaccia–Cohen entropy"), used for assessing the complexity of physiological and other time-series signals, diagnosing e.g. diseased states. SampEn has two advantages over ApEn: data length independence and a relatively trouble-free implementation. Also, there is a small computational difference: In ApEn, the comparison between the template vector (see below) and the rest of the vectors also includes comparison with itself. This guarantees that probabilities C i ′ m ( r ) {\displaystyle C_{i}'^{m}(r)} are never zero. Consequently, it is always possible to take a logarithm of probabilities. Because template comparisons with itself lower ApEn values, the signals are interpreted to be more regular than they actually are. These self-matches are not included in SampEn. However, since SampEn makes direct use of the correlation integrals, it is not a real measure of information but an approximation. The foundations and differences with ApEn, as well as a step-by-step tutorial for its application is available at. SampEn is indeed identical to the "correlation entropy" K_2 of Grassberger & Procaccia, except that it is suggested in the latter that certain limits should be taken in order to achieve a result invariant under changes of variables. No such limits and no invariance properties are considered in SampEn. There is a multiscale version of SampEn as well, suggested by Costa and others. SampEn can be used in biomedical and biomechanical research, for example to evaluate postural control.
Definition Like approximate entropy (ApEn), Sample entropy (SampEn) is a measure of complexity. But it does not include self-similar patterns as ApEn does. For a given embedding dimension m {\displaystyle m} , tolerance r {\displaystyle r} and number of data points N {\displaystyle N} , SampEn is the negative natural logarithm of the probability that if two sets of simultaneous data points of length m {\displaystyle m} have distance < r {\displaystyle <r} then two sets of simultaneous data points of length m + 1 {\displaystyle m+1} also have distance < r {\displaystyle <r} . And we represent it by S a m p E n ( m , r , N ) {\displaystyle SampEn(m,r,N)} (or by S a m p E n ( m , r , τ , N ) {\displaystyle SampEn(m,r,\tau ,N)} including sampling time τ {\displaystyle \tau } ). Now assume we have a time-series data set of length N = { x 1 , x 2 , x 3 , . . . , x N } {\displaystyle N={\{x_{1},x_{2},x_{3},...,x_{N}\}}} with a constant time interval τ {\displaystyle \tau } . We define a template vector of length m {\displaystyle m} , such that X m ( i ) = { x i , x i + 1 , x i + 2 , . . . , x i + m − 1 } {\displaystyle X_{m}(i)={\{x_{i},x_{i+1},x_{i+2},...,x_{i+m-1}\}}} and the distance function d [ X m ( i ) , X m ( j ) ] {\displaystyle d[X_{m}(i),X_{m}(j)]} (i≠j) is to be the Chebyshev distance (but it could be any distance function, including Euclidean distance). We define the sample entropy to be
S a m p E n = − ln A B {\displaystyle SampEn=-\ln {A \over B}}
Where
A {\displaystyle A} = number of template vector pairs having d [ X m + 1 ( i ) , X m + 1 ( j ) ] < r {\displaystyle d[X_{m+1}(i),X_{m+1}(j)]<r}
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