Source code for pyinform.activeinfo

# 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.
Active information (AI) was introduced in [Lizier2012]_ to quantify information
storage in distributed computation. Active information is defined in terms of
a temporally local variant

.. math::

    a_{X,i}(k) = \\log_2 \\frac{p(x^{(k)}_i, x_{i+1})}{p(x^{(k)}_i)p(x_{i+1})}.

where the probabilities are constructed empirically from the *entire* time
series. From the local variant, the temporally global active information as

.. math::

    A_X(k) = \\langle a_{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)p(x_{i+1})}.

Strictly speaking, the local and average active information are defined as

.. math::

    a_{X,i} = \\lim_{k \\rightarrow \\infty} a_{X,i}(k)
    \\quad \\textrm{and} \\quad
    A_X = \\lim_{k \\rightarrow \\infty} A_X(k),

but we do not provide limiting functionality in this library (yet!).


A Single Initial Condition

The typical usage is to provide the time series as a sequence (or
``numpy.ndarray``) and the history length as an integer and let the
:py:func:`active_info` sort out the rest:

.. doctest:: active_info

    >>> active_info([0,0,1,1,1,1,0,0,0], k=2)
    >>> active_info([0,0,1,1,1,1,0,0,0], k=2, local=True)
    array([[-0.19264508,  0.80735492,  0.22239242,  0.22239242, -0.36257008,
             1.22239242,  0.22239242]])

Multiple Initial Conditions

What about multiple initial conditions? We've got that covered!

.. doctest:: active_info

    >>> active_info([[0,0,1,1,1,1,0,0,0], [1,0,0,1,0,0,1,0,0]], k=2)
    >>> active_info([[0,0,1,1,1,1,0,0,0], [1,0,0,1,0,0,1,0,0]], k=2, local=True)
    array([[ 0.80735492, -0.36257008,  0.63742992,  0.63742992, -0.77760758,
             0.80735492, -1.19264508],
           [ 0.80735492,  0.80735492,  0.22239242,  0.80735492,  0.80735492,
             0.22239242,  0.80735492]])

As mentioned in :ref:`subtle-details`, averaging the AI for over the initial
conditions does not give the same result as constructing the distributions using
all of the initial conditions together.

.. doctest:: active_info

    >>> import numpy as np
    >>> series = np.asarray([[0,0,1,1,1,1,0,0,0], [1,0,0,1,0,0,1,0,0]])
    >>> np.apply_along_axis(active_info, 1, series, 2).mean()

Or if you are feeling verbose:

.. doctest:: active_info

    >>> ai = np.empty(len(series))
    >>> for i, xs in enumerate(series):
    ...     ai[i] = active_info(xs, k=2)
    >>> ai
    array([0.30595849, 0.86312057])
    >>> ai.mean()

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 active_info(series, k, local=False): """ Compute the average or local active information of a timeseries 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 active information :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) ai = np.empty((n, q), dtype=np.float64) out = ai.ctypes.data_as(POINTER(c_double)) _local_active_info(data, c_ulong(n), c_ulong(m), c_int(b), c_ulong(k), out, byref(e)) else: ai = _active_info(data, c_ulong(n), c_ulong(m), c_int(b), c_ulong(k), byref(e)) error_guard(e) return ai
_active_info = _inform.inform_active_info _active_info.argtypes = [POINTER(c_int), c_ulong, c_ulong, c_int, c_ulong, POINTER(c_int)] _active_info.restype = c_double _local_active_info = _inform.inform_local_active_info _local_active_info.argtypes = [POINTER(c_int), c_ulong, c_ulong, c_int, c_ulong, POINTER(c_double), POINTER(c_int)] _local_active_info.restype = POINTER(c_double)