Source code for pyinform.entropyrate

# 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:


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)
    >>> 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)

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)
    >>> 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)