a j'= @sdZddlmZddlZddlZddlmZddlm Z m Z ddgZ e d d d d dd d d ddd,ddddddddZ d-ddZ e dd ddd e ddddddddddddd d!dZd"d#Zd$d%Zd&d'Zd(d)Zd*d+ZdS).z5 Created on Fri Apr 2 09:06:05 2021 @author: matth ) annotationsN)special)_axis_nan_policy_factory_broadcast_arraysentropydifferential_entropycCs|SNxr r :E:\soccer\ml_env\Lib\site-packages\scipy/stats/_entropy.pyrcCsd|vr|ddurdSdS)Nqkrr )Zkwgsr r r rscCs|fSr r r r r r rrT)Z n_samples n_outputsresult_to_tupleZpaired too_smallznp.typing.ArrayLikeznp.typing.ArrayLike | Nonez float | Noneintznp.number | np.ndarray)pkrbaseaxisreturncCs|dur|dkrtdt|}tjdd(d|tj||dd}Wdn1s\0Y|durzt|}nPt|}t||fdd \}}t|dd}d|tj|fi|}t ||}tj||d }|dur|t |}|S) a 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 Nr+`base` must be a positive number or `None`.ignore)invalidg?TrZkeepdimsr) ValueErrornpasarrayZerrstatesumrZentrrdictZrel_entrlog)rrrrZvecZ sum_kwargsSr r r rs { 6    cCsP|d}|j|}|dtt|d}dd|krF|ksLndSdS)Nr window_length?rTF)shapegetmathfloorsqrt)Zsampleskwargsrvaluesnr'r r r "_differential_entropy_is_too_smalls r1cCs|Sr r r r r r rrcCs|fSr r r r r r rr)rrrauto)r'rrmethodz int | Nonestr)r/r'rrr3rc Cst|}t||d}|jd}|dur>tt|d}dd|krV|ksnntd|d|d|dur|dkrtd tj|dd }t t t t t d }| }||vrd t|}t||d kr|dkrd}n|dkrd}nd}||||} |dur| t|} | S)aVGiven 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)') rNr(rzWindow length (z7) must be positive and less than half the sample size (z).rrr)vasicekvan esZcorreaebrahimir2z`method` must be one of r2 r6ir7r5)r!r"Zmoveaxisr)r+r,r-r sort_vasicek_entropy_van_es_entropy_correa_entropy_ebrahimi_entropylowersetr%) r/r'rrr3r0Z sorted_datamethodsmessageresr r r rsF   cCsTt|j}||d<t|ddgf|}t|ddgf|}tj|||fddS)z9Pad the data for computing the rolling window difference.r.rr)r!arrayr)Z broadcast_toZ concatenate)Xmr)ZXlZXrr r r _pad_along_last_axisis  rFcCs`|jd}t||}|dd|df|ddd|f}t|d||}tj|ddS)z:Compute the Vasicek estimator as described in [6] Eq. 1.3.r.rNr)r)rFr!r%mean)rDrEr0 differenceslogsr r r r:ss   (r:cCs|jd}|d|df|dd| f}d||tjt|d||dd}t||d}|td|t|t|dS)z1Compute the van Es estimator as described in [6].r.Nrr)r)r!r#r%arange)rDrEr0 differenceZterm1kr r r r;|s  ",r;cCs|jd}t||}|dd|df|ddd|f}td|dt}t|d}d|||kd||||k<d|||||dk|||||dk<t||||}tj|ddS)z3Compute the Ebrahimi estimator as described in [6].r.rNrGrr) r)rFr!rKZastypefloatZ ones_liker%rH)rDrEr0rIicirJr r r r=s  ( 0r=c Cs|jd}t||}td|d}t| |ddddf}||}||d}tj|d|fddd}|d|f|}tj||dd} |tj|d dd} tjt| | dd S) z1Compute the Correa estimator as described in [6].rrN.rGTrrr)r)rFr!rKrHr#r%) rDrEr0rOZdjjZj0ZXibarrLnumZdenr r r r<s   r<)NNr)r)__doc__ __future__rr+numpyr!ZscipyrZ_axis_nan_policyrr__all__rr1rrFr:r;r=r<r r r r sB     =