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