# Copyright 2016-2019 Douglas G. Moore. All rights reserved.
# Use of this source code is governed by a MIT
# license that can be found in the LICENSE file.
"""
`Entropy rate`_ (ER) quantifies the amount of information needed to describe the
:math:`X` given observations of :math:`X^{(k)}`. In other words, it is the
entropy of the time series conditioned on the :math:`k`-histories. The local
entropy rate
.. math::
h_{X,i}(k) = \\log_2 \\frac{p(x^{(k)}_i, x_{i+1})}{p(x^{(k)}_i)}
can be averaged to obtain the global entropy rate
.. math::
H_X(k) = \\langle h_{X,i}(k) \\rangle_{i}
= \\sum_{x^{(k)}_i,\\, x_{i+1}} p(x^{(k)}_i, x_{i+1}) \\log_2 \\frac{p(x^{(k)}_i, x_{i+1})}{p(x^{(k)}_i)}.
Much as with :ref:`active-information`, the local and average entropy rates are
formally obtained in the limit
.. math::
h_{X,i} = \\lim_{k \\rightarrow \\infty} h_{X,i}(k)
\\quad \\textrm{and} \\quad
H_X = \\lim_{k \\rightarrow \\infty} H_X(k),
but we do not provide limiting functionality in this library (yet!).
See [Cover1991]_ for more details.
.. _Entropy rate: https://en.wikipedia.org/wiki/Entropy_rate
Examples
--------
A Single Initial Condition
^^^^^^^^^^^^^^^^^^^^^^^^^^
Let's apply the entropy rate to a single initial condition. Typically, you will
just provide the time series and the history length, and let
:py:func:`.entropy_rate` take care of the rest:
.. doctest:: entropy_rate
>>> entropy_rate([0,0,1,1,1,1,0,0,0], k=2)
0.6792696431662095
>>> entropy_rate([0,0,1,1,1,1,0,0,0], k=2, local=True)
array([[1. , 0. , 0.5849625, 0.5849625, 1.5849625, 0. ,
1. ]])
>>> entropy_rate([0,0,1,1,1,1,2,2,2], k=2)
0.39355535745192416
Multiple Initial Conditions
^^^^^^^^^^^^^^^^^^^^^^^^^^^
Of course multiple initial conditions are handled.
.. doctest:: entropy_rate
>>> series = [[0,0,1,1,1,1,0,0,0], [1,0,0,1,0,0,1,0,0]]
>>> entropy_rate(series, k=2)
0.6253491072973907
>>> entropy_rate(series, k=2, local=True)
array([[0.4150375, 1.5849625, 0.5849625, 0.5849625, 1.5849625, 0. ,
2. ],
[0. , 0.4150375, 0.5849625, 0. , 0.4150375, 0.5849625,
0. ]])
"""
import numpy as np
from ctypes import byref, c_int, c_ulong, c_double, POINTER
from pyinform import _inform
from pyinform.error import ErrorCode, error_guard
[docs]def entropy_rate(series, k, local=False):
"""
Compute the average or local entropy rate of a time series with history
length *k*.
:param series: the time series
:type series: sequence or ``numpy.ndarray``
:param int k: the history length
:param bool local: compute the local active information
:returns: the average or local entropy rate
:rtype: float or ``numpy.ndarray``
:raises ValueError: if the time series has no initial conditions
:raises ValueError: if the time series is greater than 2-D
:raises InformError: if an error occurs within the ``inform`` C call
"""
xs = np.ascontiguousarray(series, np.int32)
if xs.ndim == 0:
raise ValueError("empty timeseries")
elif xs.ndim > 2:
raise ValueError("dimension greater than 2")
b = max(2, np.amax(xs) + 1)
data = xs.ctypes.data_as(POINTER(c_int))
if xs.ndim == 1:
n, m = 1, xs.shape[0]
else:
n, m = xs.shape
e = ErrorCode(0)
if local is True:
q = max(0, m - k)
er = np.empty((n, q), dtype=np.float64)
out = er.ctypes.data_as(POINTER(c_double))
_local_entropy_rate(data, c_ulong(n), c_ulong(m), c_int(b), c_ulong(k), out, byref(e))
else:
er = _entropy_rate(data, c_ulong(n), c_ulong(m), c_int(b), c_ulong(k), byref(e))
error_guard(e)
return er
_entropy_rate = _inform.inform_entropy_rate
_entropy_rate.argtypes = [POINTER(c_int), c_ulong, c_ulong, c_int, c_ulong, POINTER(c_int)]
_entropy_rate.restype = c_double
_local_entropy_rate = _inform.inform_local_entropy_rate
_local_entropy_rate.argtypes = [POINTER(c_int), c_ulong, c_ulong, c_int, c_ulong, POINTER(c_double), POINTER(c_int)]
_local_entropy_rate.restype = POINTER(c_double)