Sample entropy

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) 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.…

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) 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. Now assume we have a time-series data set of length N = { \{ x_1 , x_2 , x_3 , . . . , x_N \} } with a constant time interval \tau . We define a template vector of length m , such that 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)] (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 SampEn=-\ln {A \over B} Where A = number of template vector pairs having d[X_{m+1}(i),X_{m+1}(j)] B = number of template vector pairs having d[X_m(i),X_m(j)] It is clear from the definition that A will always have a value smaller or equal to B . Therefore, SampEn(m,r,\tau) will be always either be zero or positive value. A smaller value of SampEn also indicates more self-similarity in data set or less noise. Generally we take the value of m to be 2 and the value of r to be 0.2 \times std . Where std stands for standard deviation which should be taken over a very large dataset. For instance, the r value of 6 ms is appropriate for sample entropy calculations of heart rate intervals, since this corresponds to 0.2 \times std for a very large population. The definition mentioned above is a special case of multi scale sampEn with \delta=1 , where \delta is called skipping parameter. In multiscale SampEn template vectors are defined with a certain interval between its elements, specified by the value of \delta . And modified template vector is defined as X_{m,\delta}(i)={x_i,x_{i+\delta},x_{i+2\times\delta},...,x_{i+(m-1)\times\delta} } and sampEn can be written as SampEn \left ( m,r,\delta \right )=-\ln { A_\delta \over B_\delta } And we calculate A_\delta and B_\delta like before. from itertools import combinations from math import log def construct_templates(timeseries_data: list, m: int = 2): num_windows = len(timeseries_data) - m + 1 return [timeseries_data[x : x + m] for x in range(0, num_windows)] def get_matches(templates: list, r: float) -> int: return len( list(filter(lambda x: is_match(x[0], x[1], r), combinations(templates, 2))) ) def is_match(template_1: list, template_2: list, r: float) -> bool: return all([abs(x - y) An equivalent example in numerical Python. import numpy def construct_templates(timeseries_data, m): num_windows = len(timeseries_data) - m + 1 return numpy.array([timeseries_data[x : x + m] for x in range(0, num_windows)]) def get_matches(templates, r) -> int: return len( list(filter(lambda x: is_match(x[0], x[1], r), combinations(templates))) ) def combinations(x): idx = numpy.stack(numpy.triu_indices(len(x), k=1), axis=-1) return x[idx] def is_match(template_1, template_2, r) -> bool: return numpy.all([abs(x - y) An example written in other languages can be found: Matlab R. Rust

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