Source code for pyinform.relativeentropy

# 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.
`Relative entropy`_, also known as the Kullback-Leibler divergence, measures the
amount of information gained in switching from a prior :math:`q_X` to a
posterior distribution :math:`p_X` *over the same support*. That is :math:`q_X`
and :math:`P` represent hypotheses of the distribution of some random variable
:math:`X.` Time series data sampled from the posterior and prior can be used to
estiamte those distributions, and the relative entropy can the be computed via
a call to :py:func:`~.shannon.relative_entropy`. The result is

.. math::

    D_{KL}(p||q) = \\sum_{x_i} p(x_i) \\log_2 \\frac{p(x_i)}{q(x_i)}

which has as its local counterpart

.. math::

    d_{KL, i}(p||q) = \\log_2 \\frac{p(x_i)}{q(x_i)}.

Note that the average in moving from the local to the non-local relative entropy
is taken over the posterior distribution.

See [Kullback1951]_ and [Cover1991]_ for more information.

.. _Relative entropy:


.. doctest:: relative_entropy

        >>> xs = [0,1,0,0,0,0,0,0,0,1]
        >>> ys = [0,1,1,1,1,0,0,1,0,0]
        >>> relative_entropy(xs, ys)
        >>> relative_entropy(ys, xs)

.. doctest:: relative_entropy

        >>> xs = [0,0,0,0]
        >>> ys = [0,1,1,0]
        >>> relative_entropy(xs, ys)
        >>> relative_entropy(ys, xs)

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 relative_entropy(xs, ys, local=False): """ Compute the local or global relative entropy between two time series treating each as observations from a distribution. :param xs: the time series sampled from the posterior distribution :type xs: a sequence or ``numpy.ndarray`` :param ys: the time series sampled from the prior distribution :type ys: a sequence or ``numpy.ndarray`` :param bool local: compute the local relative entropy :return: the local or global relative entropy :rtype: float or ``numpy.ndarray`` :raises ValueError: if the time series have different shapes :raises InformError: if an error occurs within the ``inform`` C call """ us = np.ascontiguousarray(xs, dtype=np.int32) vs = np.ascontiguousarray(ys, dtype=np.int32) if us.shape != vs.shape: raise ValueError("timeseries lengths do not match") b = max(2, np.amax(us) + 1, np.amax(vs) + 1) xdata = us.ctypes.data_as(POINTER(c_int)) ydata = vs.ctypes.data_as(POINTER(c_int)) n = us.size e = ErrorCode(0) if local is True: re = np.empty(b, dtype=np.float64) out = re.ctypes.data_as(POINTER(c_double)) _local_relative_entropy(xdata, ydata, c_ulong(n), c_int(b), out, byref(e)) else: re = _relative_entropy(xdata, ydata, c_ulong(n), c_int(b), byref(e)) error_guard(e) return re
_relative_entropy = _inform.inform_relative_entropy _relative_entropy.argtypes = [POINTER(c_int), POINTER(c_int), c_ulong, c_int, POINTER(c_int)] _relative_entropy.restype = c_double _local_relative_entropy = _inform.inform_local_relative_entropy _local_relative_entropy.argtypes = [POINTER(c_int), POINTER(c_int), c_ulong, c_int, POINTER(c_double), POINTER(c_int)] _local_relative_entropy.restype = POINTER(c_double)