Approximate Entropy with Python/Numpy: https://en.wikipedia.org/wiki/Approximate_entropy

gistfile1.txt

import numpy as np

def ApEn(U, m, r):

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

    def _phi(m):
        x = [[U[j] for j in range(i, i + m - 1 + 1)] for i in range(N - m + 1)]
        C = [len([1 for x_j in x if _maxdist(x_i, x_j) <= r]) / (N - m + 1.0) for x_i in x]
        return (N - m + 1.0)**(-1) * sum(np.log(C))

    N = len(U)

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

# Usage example
U = np.array([85, 80, 89] * 17)
print ApEn(U, 2, 3)
1.0996541105257052e-05

A more efficient algorithm, using NumPy builtin functions:

def ApEn_new(U, m, r):
    U = np.array(U)
    N = U.shape[0]
            
    def _phi(m):
        z = N - m + 1.0
        x = np.array([U[i:i+m] for i in range(int(z))])
        X = np.repeat(x[:, np.newaxis], 1, axis=2)
        C = np.sum(np.absolute(x - X).max(axis=2) <= r, axis=0) / z
        return np.log(C).sum() / z
    
    return abs(_phi(m + 1) - _phi(m))

Comparing with the results of the original version:

a = [85, 80, 89] * 17
print('ApEn(a):    ', ApEn(a))
print('ApEn_new(a):', ApEn_new(a))
# ApEn(a):     1.0996541105257052e-05
# ApEn_new(a): 1.099654110658932e-05

a = np.random.choice([85, 80, 89], size=17 * 3)
print('ApEn(a):    ', ApEn(a))
print('ApEn_new(a):', ApEn_new(a))
# ApEn(a):     0.969167624973212
# ApEn_new(a): 0.969167624973212

Comparing with the efficiency of the original version:

import timeit
t = timeit.Timer(stmt="ApEn([85, 80, 89] * 17)", setup=setup)
print(t.timeit(20))    # 1.7361566790000325

t = timeit.Timer(stmt="ApEn_new([85, 80, 89] * 17)", setup=setup)
print(t.timeit(20))    # 0.12356296300004033

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