# 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!).
Examples
--------
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)
0.3059584928680418
>>> 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)
0.35987902873686084
>>> 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()
0.5845395307173363
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()
0.5845395307173363
"""
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)