Source code for pyinform.entropyrate

# Copyright 2016-2019 Douglas G. Moore. All rights reserved.
# Use of this source code is governed by a MIT
"""
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)
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)