400 lines
14 KiB
Python
400 lines
14 KiB
Python
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# -*- coding: utf-8 -*-
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"""
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Created on Fri Apr 2 09:06:05 2021
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@author: matth
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"""
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from __future__ import annotations
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import math
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import numpy as np
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from scipy import special
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from typing import Optional, Union
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__all__ = ['entropy', 'differential_entropy']
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def entropy(pk: np.typing.ArrayLike,
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qk: Optional[np.typing.ArrayLike] = None,
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base: Optional[float] = None,
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axis: int = 0
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) -> Union[np.number, np.ndarray]:
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"""
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Calculate the Shannon entropy/relative entropy of given distribution(s).
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If only probabilities `pk` are given, the Shannon entropy is calculated as
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``H = -sum(pk * log(pk))``.
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If `qk` is not None, then compute the relative entropy
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``D = sum(pk * log(pk / qk))``. This quantity is also known
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as the Kullback-Leibler divergence.
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This routine will normalize `pk` and `qk` if they don't sum to 1.
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Parameters
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----------
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pk : array_like
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Defines the (discrete) distribution. Along each axis-slice of ``pk``,
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element ``i`` is the (possibly unnormalized) probability of event
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``i``.
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qk : array_like, optional
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Sequence against which the relative entropy is computed. Should be in
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the same format as `pk`.
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base : float, optional
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The logarithmic base to use, defaults to ``e`` (natural logarithm).
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axis : int, optional
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The axis along which the entropy is calculated. Default is 0.
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Returns
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-------
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S : {float, array_like}
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The calculated entropy.
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Notes
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-----
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Informally, the Shannon entropy quantifies the expected uncertainty
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inherent in the possible outcomes of a discrete random variable.
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For example,
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if messages consisting of sequences of symbols from a set are to be
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encoded and transmitted over a noiseless channel, then the Shannon entropy
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``H(pk)`` gives a tight lower bound for the average number of units of
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information needed per symbol if the symbols occur with frequencies
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governed by the discrete distribution `pk` [1]_. The choice of base
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determines the choice of units; e.g., ``e`` for nats, ``2`` for bits, etc.
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The relative entropy, ``D(pk|qk)``, quantifies the increase in the average
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number of units of information needed per symbol if the encoding is
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optimized for the probability distribution `qk` instead of the true
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distribution `pk`. Informally, the relative entropy quantifies the expected
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excess in surprise experienced if one believes the true distribution is
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`qk` when it is actually `pk`.
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A related quantity, the cross entropy ``CE(pk, qk)``, satisfies the
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equation ``CE(pk, qk) = H(pk) + D(pk|qk)`` and can also be calculated with
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the formula ``CE = -sum(pk * log(qk))``. It gives the average
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number of units of information needed per symbol if an encoding is
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optimized for the probability distribution `qk` when the true distribution
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is `pk`. It is not computed directly by `entropy`, but it can be computed
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using two calls to the function (see Examples).
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See [2]_ for more information.
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References
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----------
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.. [1] Shannon, C.E. (1948), A Mathematical Theory of Communication.
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Bell System Technical Journal, 27: 379-423.
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https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
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.. [2] Thomas M. Cover and Joy A. Thomas. 2006. Elements of Information
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Theory (Wiley Series in Telecommunications and Signal Processing).
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Wiley-Interscience, USA.
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Examples
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--------
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The outcome of a fair coin is the most uncertain:
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>>> import numpy as np
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>>> from scipy.stats import entropy
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>>> base = 2 # work in units of bits
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>>> pk = np.array([1/2, 1/2]) # fair coin
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>>> H = entropy(pk, base=base)
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>>> H
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1.0
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>>> H == -np.sum(pk * np.log(pk)) / np.log(base)
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True
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The outcome of a biased coin is less uncertain:
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>>> qk = np.array([9/10, 1/10]) # biased coin
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>>> entropy(qk, base=base)
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0.46899559358928117
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The relative entropy between the fair coin and biased coin is calculated
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as:
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>>> D = entropy(pk, qk, base=base)
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>>> D
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0.7369655941662062
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>>> D == np.sum(pk * np.log(pk/qk)) / np.log(base)
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True
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The cross entropy can be calculated as the sum of the entropy and
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relative entropy`:
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>>> CE = entropy(pk, base=base) + entropy(pk, qk, base=base)
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>>> CE
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1.736965594166206
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>>> CE == -np.sum(pk * np.log(qk)) / np.log(base)
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True
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"""
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if base is not None and base <= 0:
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raise ValueError("`base` must be a positive number or `None`.")
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pk = np.asarray(pk)
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pk = 1.0*pk / np.sum(pk, axis=axis, keepdims=True)
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if qk is None:
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vec = special.entr(pk)
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else:
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qk = np.asarray(qk)
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pk, qk = np.broadcast_arrays(pk, qk)
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qk = 1.0*qk / np.sum(qk, axis=axis, keepdims=True)
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vec = special.rel_entr(pk, qk)
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S = np.sum(vec, axis=axis)
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if base is not None:
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S /= np.log(base)
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return S
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def differential_entropy(
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values: np.typing.ArrayLike,
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*,
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window_length: Optional[int] = None,
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base: Optional[float] = None,
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axis: int = 0,
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method: str = "auto",
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) -> Union[np.number, np.ndarray]:
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r"""Given a sample of a distribution, estimate the differential entropy.
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Several estimation methods are available using the `method` parameter. By
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default, a method is selected based the size of the sample.
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Parameters
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----------
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values : sequence
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Sample from a continuous distribution.
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window_length : int, optional
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Window length for computing Vasicek estimate. Must be an integer
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between 1 and half of the sample size. If ``None`` (the default), it
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uses the heuristic value
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.. math::
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\left \lfloor \sqrt{n} + 0.5 \right \rfloor
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where :math:`n` is the sample size. This heuristic was originally
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proposed in [2]_ and has become common in the literature.
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base : float, optional
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The logarithmic base to use, defaults to ``e`` (natural logarithm).
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axis : int, optional
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The axis along which the differential entropy is calculated.
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Default is 0.
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method : {'vasicek', 'van es', 'ebrahimi', 'correa', 'auto'}, optional
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The method used to estimate the differential entropy from the sample.
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Default is ``'auto'``. See Notes for more information.
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Returns
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-------
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entropy : float
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The calculated differential entropy.
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Notes
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-----
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This function will converge to the true differential entropy in the limit
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.. math::
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n \to \infty, \quad m \to \infty, \quad \frac{m}{n} \to 0
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The optimal choice of ``window_length`` for a given sample size depends on
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the (unknown) distribution. Typically, the smoother the density of the
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distribution, the larger the optimal value of ``window_length`` [1]_.
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The following options are available for the `method` parameter.
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* ``'vasicek'`` uses the estimator presented in [1]_. This is
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one of the first and most influential estimators of differential entropy.
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* ``'van es'`` uses the bias-corrected estimator presented in [3]_, which
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is not only consistent but, under some conditions, asymptotically normal.
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* ``'ebrahimi'`` uses an estimator presented in [4]_, which was shown
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in simulation to have smaller bias and mean squared error than
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the Vasicek estimator.
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* ``'correa'`` uses the estimator presented in [5]_ based on local linear
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regression. In a simulation study, it had consistently smaller mean
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square error than the Vasiceck estimator, but it is more expensive to
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compute.
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* ``'auto'`` selects the method automatically (default). Currently,
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this selects ``'van es'`` for very small samples (<10), ``'ebrahimi'``
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for moderate sample sizes (11-1000), and ``'vasicek'`` for larger
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samples, but this behavior is subject to change in future versions.
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All estimators are implemented as described in [6]_.
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References
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----------
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.. [1] Vasicek, O. (1976). A test for normality based on sample entropy.
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Journal of the Royal Statistical Society:
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Series B (Methodological), 38(1), 54-59.
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.. [2] Crzcgorzewski, P., & Wirczorkowski, R. (1999). Entropy-based
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goodness-of-fit test for exponentiality. Communications in
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Statistics-Theory and Methods, 28(5), 1183-1202.
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.. [3] Van Es, B. (1992). Estimating functionals related to a density by a
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class of statistics based on spacings. Scandinavian Journal of
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Statistics, 61-72.
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.. [4] Ebrahimi, N., Pflughoeft, K., & Soofi, E. S. (1994). Two measures
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of sample entropy. Statistics & Probability Letters, 20(3), 225-234.
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.. [5] Correa, J. C. (1995). A new estimator of entropy. Communications
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in Statistics-Theory and Methods, 24(10), 2439-2449.
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.. [6] Noughabi, H. A. (2015). Entropy Estimation Using Numerical Methods.
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Annals of Data Science, 2(2), 231-241.
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https://link.springer.com/article/10.1007/s40745-015-0045-9
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Examples
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--------
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>>> import numpy as np
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>>> from scipy.stats import differential_entropy, norm
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Entropy of a standard normal distribution:
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>>> rng = np.random.default_rng()
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>>> values = rng.standard_normal(100)
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>>> differential_entropy(values)
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1.3407817436640392
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Compare with the true entropy:
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>>> float(norm.entropy())
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1.4189385332046727
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For several sample sizes between 5 and 1000, compare the accuracy of
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the ``'vasicek'``, ``'van es'``, and ``'ebrahimi'`` methods. Specifically,
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compare the root mean squared error (over 1000 trials) between the estimate
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and the true differential entropy of the distribution.
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>>> from scipy import stats
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>>> import matplotlib.pyplot as plt
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>>>
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>>>
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>>> def rmse(res, expected):
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... '''Root mean squared error'''
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... return np.sqrt(np.mean((res - expected)**2))
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>>>
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>>>
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>>> a, b = np.log10(5), np.log10(1000)
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>>> ns = np.round(np.logspace(a, b, 10)).astype(int)
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>>> reps = 1000 # number of repetitions for each sample size
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>>> expected = stats.expon.entropy()
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>>>
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>>> method_errors = {'vasicek': [], 'van es': [], 'ebrahimi': []}
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>>> for method in method_errors:
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... for n in ns:
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... rvs = stats.expon.rvs(size=(reps, n), random_state=rng)
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... res = stats.differential_entropy(rvs, method=method, axis=-1)
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... error = rmse(res, expected)
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... method_errors[method].append(error)
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>>>
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>>> for method, errors in method_errors.items():
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... plt.loglog(ns, errors, label=method)
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>>>
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>>> plt.legend()
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>>> plt.xlabel('sample size')
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>>> plt.ylabel('RMSE (1000 trials)')
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>>> plt.title('Entropy Estimator Error (Exponential Distribution)')
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"""
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values = np.asarray(values)
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values = np.moveaxis(values, axis, -1)
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n = values.shape[-1] # number of observations
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if window_length is None:
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window_length = math.floor(math.sqrt(n) + 0.5)
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if not 2 <= 2 * window_length < n:
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raise ValueError(
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f"Window length ({window_length}) must be positive and less "
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f"than half the sample size ({n}).",
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)
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if base is not None and base <= 0:
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raise ValueError("`base` must be a positive number or `None`.")
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sorted_data = np.sort(values, axis=-1)
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methods = {"vasicek": _vasicek_entropy,
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"van es": _van_es_entropy,
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"correa": _correa_entropy,
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"ebrahimi": _ebrahimi_entropy,
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"auto": _vasicek_entropy}
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method = method.lower()
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if method not in methods:
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message = f"`method` must be one of {set(methods)}"
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raise ValueError(message)
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if method == "auto":
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if n <= 10:
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method = 'van es'
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elif n <= 1000:
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method = 'ebrahimi'
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else:
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method = 'vasicek'
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res = methods[method](sorted_data, window_length)
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if base is not None:
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res /= np.log(base)
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return res
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def _pad_along_last_axis(X, m):
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"""Pad the data for computing the rolling window difference."""
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# scales a bit better than method in _vasicek_like_entropy
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shape = np.array(X.shape)
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shape[-1] = m
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Xl = np.broadcast_to(X[..., [0]], shape) # [0] vs 0 to maintain shape
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Xr = np.broadcast_to(X[..., [-1]], shape)
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return np.concatenate((Xl, X, Xr), axis=-1)
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def _vasicek_entropy(X, m):
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"""Compute the Vasicek estimator as described in [6] Eq. 1.3."""
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n = X.shape[-1]
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X = _pad_along_last_axis(X, m)
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differences = X[..., 2 * m:] - X[..., : -2 * m:]
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logs = np.log(n/(2*m) * differences)
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return np.mean(logs, axis=-1)
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def _van_es_entropy(X, m):
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"""Compute the van Es estimator as described in [6]."""
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# No equation number, but referred to as HVE_mn.
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# Typo: there should be a log within the summation.
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n = X.shape[-1]
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difference = X[..., m:] - X[..., :-m]
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term1 = 1/(n-m) * np.sum(np.log((n+1)/m * difference), axis=-1)
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k = np.arange(m, n+1)
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return term1 + np.sum(1/k) + np.log(m) - np.log(n+1)
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def _ebrahimi_entropy(X, m):
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"""Compute the Ebrahimi estimator as described in [6]."""
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# No equation number, but referred to as HE_mn
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n = X.shape[-1]
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X = _pad_along_last_axis(X, m)
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differences = X[..., 2 * m:] - X[..., : -2 * m:]
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|
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|
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i = np.arange(1, n+1).astype(float)
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ci = np.ones_like(i)*2
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ci[i <= m] = 1 + (i[i <= m] - 1)/m
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|
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ci[i >= n - m + 1] = 1 + (n - i[i >= n-m+1])/m
|
||
|
|
|
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|
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logs = np.log(n * differences / (ci * m))
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||
|
|
return np.mean(logs, axis=-1)
|
||
|
|
|
||
|
|
|
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|
|
def _correa_entropy(X, m):
|
||
|
|
"""Compute the Correa estimator as described in [6]."""
|
||
|
|
# No equation number, but referred to as HC_mn
|
||
|
|
n = X.shape[-1]
|
||
|
|
X = _pad_along_last_axis(X, m)
|
||
|
|
|
||
|
|
i = np.arange(1, n+1)
|
||
|
|
dj = np.arange(-m, m+1)[:, None]
|
||
|
|
j = i + dj
|
||
|
|
j0 = j + m - 1 # 0-indexed version of j
|
||
|
|
|
||
|
|
Xibar = np.mean(X[..., j0], axis=-2, keepdims=True)
|
||
|
|
difference = X[..., j0] - Xibar
|
||
|
|
num = np.sum(difference*dj, axis=-2) # dj is d-i
|
||
|
|
den = n*np.sum(difference**2, axis=-2)
|
||
|
|
return -np.mean(np.log(num/den), axis=-1)
|