import numpy as np
import pandas as pd
from .utils_complexity_embedding import complexity_embedding
from .utils_complexity_symbolize import complexity_symbolize
[docs]
def entropy_symbolicdynamic(signal, dimension=3, symbolize="MEP", c=6, **kwargs):
"""**Symbolic Dynamic Entropy (SyDyEn) and its Multiscale variants (MSSyDyEn)**
Symbolic Dynamic Entropy (SyDyEn) combines the merits of symbolic dynamic and information
theory.
Parameters
----------
signal : Union[list, np.array, pd.Series]
The signal (i.e., a time series) in the form of a vector of values.
dimension : int
Embedding Dimension (*m*, sometimes referred to as *d* or *order*). See
:func:`complexity_dimension` to estimate the optimal value for this parameter.
symbolize : str
Method to convert a continuous signal input into a symbolic (discrete) signal. Can be one
of ``"MEP"`` (default), ``"NCDF"``, ``"linear"``, ``"uniform"``, ``"kmeans"``, ``"equal"``,
or others. See :func:`complexity_symbolize` for details.
c : int
Number of symbols *c*.
**kwargs : optional
Other keyword arguments (currently not used).
Returns
-------
SyDyEn : float
Symbolic Dynamic Entropy (SyDyEn) of the signal.
info : dict
A dictionary containing additional information regarding the parameters used.
See Also
--------
entropy_shannon, entropy_multiscale, entropy_dispersion
Examples
----------
.. ipython:: python
import neurokit2 as nk
signal = [2, -7, -12, 5, -1, 9, 14]
# Simulate a Signal
signal = nk.signal_simulate(duration=2, sampling_rate=200, frequency=[5, 6], noise=0.5)
# Compute Symbolic Dynamic Entropy
sydyen, info = nk.entropy_symbolicdynamic(signal, c=3, symbolize="MEP")
sydyen
sydyen, info = nk.entropy_symbolicdynamic(signal, c=3, symbolize="kmeans")
sydyen
# Compute Multiscale Symbolic Dynamic Entropy (MSSyDyEn)
@savefig p_entropy_symbolicdynamic1.png scale=100%
mssydyen, info = nk.entropy_multiscale(signal, method="MSSyDyEn", show=True)
@suppress
plt.close()
# Compute Modified Multiscale Symbolic Dynamic Entropy (MMSyDyEn)
@savefig p_entropy_symbolicdynamic2.png scale=100%
mmsydyen, info = nk.entropy_multiscale(signal, method="MMSyDyEn", show=True)
@suppress
plt.close()
References
----------
* Matilla-García, M., Morales, I., Rodríguez, J. M., & Marín, M. R. (2021). Selection of
embedding dimension and delay time in phase space reconstruction via symbolic dynamics.
Entropy, 23(2), 221.
* Li, Y., Yang, Y., Li, G., Xu, M., & Huang, W. (2017). A fault diagnosis scheme for planetary
gearboxes using modified multi-scale symbolic dynamic entropy and mRMR feature selection.
Mechanical Systems and Signal Processing, 91, 295-312.
* Rajagopalan, V., & Ray, A. (2006). Symbolic time series analysis via wavelet-based
partitioning. Signal processing, 86(11), 3309-3320.
"""
# Sanity checks
if isinstance(signal, (np.ndarray, pd.DataFrame)) and signal.ndim > 1:
raise ValueError(
"Multidimensional inputs (e.g., matrices or multichannel data) are not supported yet."
)
# Store parameters
info = {"Dimension": dimension, "c": c, "Symbolization": symbolize}
# We could technically expose the Delay, but the paper is about consecutive differences so...
if "delay" in kwargs.keys():
delay = kwargs["delay"]
kwargs.pop("delay")
else:
delay = 1
n = len(signal)
# There are four main steps of SDE algorithm
# 1. Convert the time series into the symbol time series (called symbolization).
symbolic = complexity_symbolize(signal, method=symbolize, c=c)
# 2. Construct the embedding vectors based on the symbol time series and compute the potential
# state patterns probability
embedded = complexity_embedding(symbolic, dimension=dimension, delay=delay)
# 3. Construct the state transitions and compute the probability of state transitions.
unique = np.unique(embedded, axis=0)
counter1 = np.zeros(len(unique))
counter2 = np.zeros((len(unique), c))
Bins = np.arange(0.5, c + 1.5, 1)
for i in range(len(unique)):
Ordx = np.any(embedded - unique[i, :], axis=1) == 0
counter1[i] = sum(Ordx) / (n - ((dimension - 1) * delay))
Temp = embedded[
np.hstack((np.zeros(dimension * delay, dtype=bool), Ordx[: -(dimension * delay)])), 0
]
counter2[i, :], _ = np.histogram(Temp, Bins)
Temp = np.sum(counter2, axis=1)
counter2[Temp > 0, :] = counter2[Temp > 0, :] / np.tile(Temp[Temp > 0], (c, 1)).transpose()
counter2[np.isnan(counter2)] = 0
# 4. Based on the Shannon entropy [39], we define the SDE as the sum of the state entropy and
# the state transition entropy
with np.errstate(divide="ignore"):
P1 = -sum(counter1 * np.log(counter1))
P2 = np.log(np.tile(counter1, (c, 1)).transpose() * counter2)
P2[~np.isfinite(P2)] = 0
sydyen = P1 - sum(counter1 * np.sum(P2, axis=1))
# Normalize
sydyen = sydyen / np.log(c ** (dimension + 1))
return sydyen, info
