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entropy.py
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entropy.py
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import math
from collections import Counter
import numpy as np
def time_delay_embedding(time_series, embedding_dimension, delay):
"""Calculate time-delayed embedding.
Parameters
----------
time_series : np.ndarray
The input time series, shape (n_times)
embedding_dimension : int
The embedding dimension (order).
delay : int
The delay between embedded points.
Returns
-------
embedded : ndarray
The embedded time series with shape (n_times - (order - 1) * delay, order).
"""
series_length = len(time_series)
embedded_series = np.empty((embedding_dimension, series_length - (embedding_dimension - 1) * delay))
for i in range(embedding_dimension):
embedded_series[i] = time_series[i * delay : i * delay + embedded_series.shape[1]]
return embedded_series.T
def util_pattern_space(time_series, lag, dim):
"""Create a set of sequences with a given lag and dimension.
Parameters
----------
time_series : np.ndarray
Vector or string of the sample data
lag : int
Lag between the beginning of sequences
dim : int
Dimension (number of patterns)
Returns
-------
pattern_space: np.ndarray
2D array of vectors
Raises
------
ValueError: If the lag is less than 1 or the result matrix exceeds the size limit.
"""
n = len(time_series)
if lag < 1:
raise ValueError("Lag should be greater than or equal to 1.")
if lag * dim > n:
raise ValueError("Result matrix size limit exceeded. Adjust the lag or dim value.")
pattern_space = np.zeros((n - lag * (dim - 1), dim))
for i in range(dim):
pattern_space[:, i] = time_series[i * lag : i * lag + n - lag * (dim - 1)]
return pattern_space
def util_granulate_time_series(time_series, scale):
"""Extract coarse-grained time series.
Parameters
----------
time_series : np.ndarray
Time series
scale : int
Scale factor
Returns
-------
cts : np.ndarray
Array of coarse-grained time series with a given scale factor
"""
if not isinstance(time_series, np.ndarray):
time_series = np.array(time_series)
n = time_series.shape[0]
b = n // scale
cts = np.mean(time_series[: b * scale].reshape(b, scale), axis=1)
return cts
def shannon_entropy(time_series):
"""Calculate Shannon Entropy of the sample data.
Parameters
----------
time_series: np.ndarray | list[str]
Vector or string of the sample data
Returns
-------
ent: float
The Shannon Entropy as float value
"""
if isinstance(time_series, str):
# Calculate frequency counts
counter = Counter(time_series)
total_count = len(time_series)
# Calculate frequencies and Shannon entropy
ent = 0.0
for count in counter.values():
freq = count / total_count
ent += freq * np.log2(freq)
ent = -ent
return ent
# Calculate frequency counts
_, counts = np.unique(time_series, return_counts=True)
total_count = len(time_series)
# Calculate frequencies and Shannon entropy
frequencies = counts / total_count
ent = -np.sum(frequencies * np.log2(frequencies))
return ent
def sample_entropy(time_series, sample_length, tolerance=None):
"""Calculate the sample entropy of degree m of a time_series.
This method uses Chebyshev norm.
It is quite fast for random data but can be slower is there is
structure in the input time series.
Parameters
----------
time_series : np.ndarray
Time series, 1-d vector
sample_length : int
length of longest template vector
tolerance : float
tolerance (defaults to 0.1 * std(time_series)))
Returns
-------
sampen: np.ndarray
Array of Sample Entropies SE.
SE[k] is the ratio `#templates of length k+1` / `#templates of length k`
where `#templates of length 0` = n*(n - 1) / 2, by definition
Notes
-----
The parameter 'sample_length' is equal to m + 1 in Ref[1].
References
----------
.. [1] http://en.wikipedia.org/wiki/Sample_Entropy
.. [2] http://physionet.incor.usp.br/physiotools/sampen/
.. [3] Madalena Costa, Ary Goldberger, CK Peng. Multiscale entropy analysis of biological signals
"""
if not isinstance(time_series, np.ndarray):
time_series = np.array(time_series)
if tolerance is None:
tolerance = 0.1 * np.std(time_series)
# The code below follows the sample length convention of Ref [1] so:
sample_length = sample_length - 1
n = len(time_series)
# N_temp is a vector that holds the number of matches
# N_temp[k] holds matches templates of length k
N_temp = np.zeros(sample_length + 2)
# Templates of length 0 match by definition:
N_temp[0] = n * (n - 1) / 2
for i in range(n - sample_length - 1):
template = time_series[i : (i + sample_length + 1)] # We have `sample_length+1` elements in the template
rem_time_series = time_series[i + 1 :]
search_list = np.arange(len(rem_time_series) - sample_length, dtype=np.int32)
for length in range(1, len(template) + 1):
hit_list = np.abs(rem_time_series[search_list] - template[length - 1]) < tolerance
N_temp[length] += np.sum(hit_list)
search_list = search_list[hit_list] + 1
sampen = -np.log(N_temp[1:] / N_temp[:-1])
return sampen
def multiscale_entropy(time_series, sample_length, tolerance=None, maxscale=None):
"""Calculate Multiscale Entropy considering different time-scales of the time series.
Parameters
----------
time_series : np.ndarray
Input time series for analysis.
sample_length : int
Bandwidth or group of points
tolerance : float
Tolerance value (default is 0.1 times the standard deviation of the `time_series`)
maxscale : int
Maximum timescale (default is the length of the `time_series`)
Returns
-------
mse : np.ndarray
Array of Multiscale Entropies
References
----------
.. [1] http://en.pudn.com/downloads149/sourcecode/math/detail646216_en.html
Can be viewed at https://web.archive.org/web/20170207221539/http://en.pudn.com/downloads149/sourcecode/math/detail646216_en.html
"""
if tolerance is None:
tolerance = 0.1 * np.std(time_series)
if maxscale is None:
maxscale = len(time_series)
mse = np.zeros(maxscale)
for i in range(maxscale):
temp = util_granulate_time_series(time_series, i + 1)
mse[i] = sample_entropy(temp, sample_length, tolerance)[-1]
return mse
def permutation_entropy(time_series, order=3, delay=1, normalize=False):
"""Calculate Permutation Entropy.
Parameters
----------
time_series : list | np.ndarray
Time series
order : int
Order of permutation entropy
delay : int
Time delay
normalize : bool
If True, divide by log2(factorial(m)) to normalize the entropy
between 0 and 1. Otherwise, return the permutation entropy in bit.
Returns
-------
pe : float
Permutation Entropy
References
----------
.. [1] Massimiliano Zanin et al. Permutation Entropy and Its Main
Biomedical and Econophysics Applications: A Review.
http://www.mdpi.com/1099-4300/14/8/1553/pdf
.. [2] Christoph Bandt and Bernd Pompe. Permutation entropy — a natural
complexity measure for time series.
http://stubber.math-inf.uni-greifswald.de/pub/full/prep/2001/11.pdf
Notes
-----
Last updated (Oct 2018) by Raphael Vallat ([email protected]):
- Major speed improvements
- Use of base 2 instead of base e
- Added normalization
Examples
--------
1. Permutation entropy with order 2
>>> x = [4, 7, 9, 10, 6, 11, 3]
>>> # Return a value between 0 and log2(factorial(order))
>>> print(permutation_entropy(x, order=2))
0.918
2. Normalized permutation entropy with order 3
>>> x = [4, 7, 9, 10, 6, 11, 3]
>>> # Return a value comprised between 0 and 1.
>>> print(permutation_entropy(x, order=3, normalize=True))
0.589
"""
x = np.array(time_series)
hashmult = np.power(order, np.arange(order))
# Embed x and sort the order of permutations
sorted_idx = time_delay_embedding(x, embedding_dimension=order, delay=delay).argsort(kind="quicksort")
# Associate unique integer to each permutations
hashval = (np.multiply(sorted_idx, hashmult)).sum(1)
# Return the counts
_, c = np.unique(hashval, return_counts=True)
# Use np.true_divide for Python 2 compatibility
p = np.true_divide(c, c.sum())
pe = -np.multiply(p, np.log2(p)).sum()
if normalize:
factorial = math.factorial(order)
pe /= np.log2(factorial)
return pe
def multiscale_permutation_entropy(time_series, m, delay, scale):
"""Calculate the Multiscale Permutation Entropy.
Parameters
----------
time_series : np.ndarray
Time series for analysis
m : int
Order of permutation entropy
delay : int
Time delay
scale : int
Scale factor
Returns
-------
mspe : np.ndarray
Array of Multiscale Permutation Entropies
References
----------
.. [1] Francesco Carlo Morabito et al. Multivariate Multi-Scale Permutation Entropy for
Complexity Analysis of Alzheimer`s Disease EEG. www.mdpi.com/1099-4300/14/7/1186
.. [2] http://www.mathworks.com/matlabcentral/fileexchange/37288-multiscale-permutation-entropy-mpe/content/MPerm.m
"""
mspe = np.empty(scale)
for i in range(scale):
coarse_time_series = util_granulate_time_series(time_series, i + 1)
mspe[i] = permutation_entropy(coarse_time_series, order=m, delay=delay)
return mspe
def weighted_permutation_entropy(time_series, order=2, delay=1, normalize=False):
"""Calculate the weighted permutation entropy.
Weighted permutation entropy captures the information in the amplitude of a signal where
standard permutation entropy only measures the information in the ordinal pattern, "motif".
Parameters
----------
time_series : list | np.ndarray
Time series
order : int
Order of permutation entropy
delay : int
Time delay
normalize : bool
If True, divide by log2(factorial(m)) to normalize the entropy
between 0 and 1. Otherwise, return the permutation entropy in bit.
Returns
-------
wpe : float
Weighted Permutation Entropy
References
----------
.. [1] Bilal Fadlallah, Badong Chen, Andreas Keil, and José Príncipe
Phys. Rev. E 87, 022911 - Published 20 February 2013
Notes
-----
- Updated in Jun 2023 by Nikolay Donets
- Updated in March 2021 by Samuel Dotson ([email protected])
Examples
--------
1. Weighted permutation entropy with order 2
>>> x = [4, 7, 9, 10, 6, 11, 3]
>>> # Return a value between 0 and log2(factorial(order))
>>> print(permutation_entropy(x, order=2))
0.912
2. Normalized weighted permutation entropy with order 3
>>> x = [4, 7, 9, 10, 6, 11, 3]
>>> # Return a value comprised between 0 and 1.
>>> print(permutation_entropy(x, order=3, normalize=True))
0.547
"""
x = time_delay_embedding(time_series, embedding_dimension=order, delay=delay)
weights = np.var(x, axis=1)
sorted_idx = x.argsort(kind="quicksort", axis=1)
motif_weights = {}
for weight, indices in zip(weights, sorted_idx): # noqa: B905
motif = tuple(indices)
if motif in motif_weights:
motif_weights[motif] += weight
else:
motif_weights[motif] = weight
pw = np.array(list(motif_weights.values()))
pw /= weights.sum()
b = np.log2(pw)
wpe = -np.dot(pw, b)
if normalize:
wpe /= np.log2(math.factorial(order))
return wpe
def composite_multiscale_entropy(time_series, sample_length, scale, tolerance=None):
"""Calculate Composite Multiscale Entropy.
Parameters
----------
time_series : np.ndarray
Time series for analysis
sample_length : int
Number of sequential points of the time series
scale : int
Scale factor
tolerance : float
Tolerance (default = 0.1 * std(time_series))
Returns
-------
cmse : np.ndarray
Array of Composite Multiscale Entropies
References
----------
.. [1] Wu, Shuen-De, et al. "Time series analysis using
composite multiscale entropy." Entropy 15.3 (2013): 1069-1084.
"""
if tolerance is None:
tolerance = 0.1 * np.std(time_series)
cmse = np.zeros(scale)
for i in range(scale):
for j in range(i + 1):
tmp = util_granulate_time_series(time_series[j:], i + 1)
se = sample_entropy(tmp, sample_length, tolerance)[-1]
cmse[i] += se / (i + 1)
return cmse