(ph=SrSSKrSSKrSSKJr SSKJrJr SSK J r J r SS/r \"S S SS S S S9S$S\RRS\RRS-S\S-S\S\R"\R$-4 Sjj5rS%Sjr\"SSS\S9SSSSS.S\RRS\S-S\S-S\S\S\R"\R$-4 Sjj5rSrS rS!rS"rS#rg)&z5 Created on Fri Apr 2 09:06:05 2021 @author: matth N)special)_axis_nan_policy_factory_broadcast_arrays)array_namespacexp_moveaxis_to_endentropydifferential_entropycU$Nxs G/var/www/html/venv/lib/python3.13/site-packages/scipy/stats/_entropy.pyrac"SU;aUSbS$S$)Nqkrr )kwgss rrrs!dltDz5   rcU4$r r rs rrrsA4rT) n_samples n_outputsresult_to_tuplepaired too_smallpkrbaseaxisreturncDUbUS::a [S5eUc [U5O [X5nURU5n[R"SS9 SU-UR XSS9- nSSS5 Uc[ R"U5nOXURU5n[X4SUS 9up[USS9nSU-UR "U40UD6- n[ R"X5nUR XSS 9nUbU[R"U5-nU$!,(df  N=f) 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 >>> np.isclose(D, np.sum(pk * np.log(pk/qk)) / np.log(base), rtol=4e-16, atol=0) 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)invalid?Tr!keepdims)r!xpr!) ValueErrorrasarraynperrstatesumrentrrdictrel_entrmathlog)rrr r!r*vec sum_kwargsSs rr r s v DAIFGG "  0GB BB X & VbffRTf: : ' zll2 ZZ^"B8$2>td3 VbffR.:. .r& sA  TXXd^ H ' &s D DcUSnURUnURS[R"[R"U5S-55nSSU-s=::aU:dg gg)Nr window_length?rTF)shapegetr4floorsqrt)sampleskwargsr!valuesnr:s r"_differential_entropy_is_too_smallrDsb QZF TAJJ#zz$))A,*<=?M M! %A % & rcU$r r rs rrrrrcU4$r r rs rrrsr)rrrauto)r:r r!methodrBr:rHc:[U5nURU5nURURS5(a*UR XRS5R5n[ XUS9nUR SnUc-[R"[R"U5S-5nSSU-s=::aU:dO [SUSUS 35eUbUS ::a [S 5eURUSS 9n[[[[[S .nUR!5nXH;aS[#U53n [U 5eUS:XaUS::aSnO US::aSnOSnX"XqUS9n UbU [R$"U5-n UR XR5$)aGiven 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)') integralr'r*rr;rzWindow length (z7) must be positive and less than half the sample size (z).rr$r+)vasicekvan escorreaebrahimirGz`method` must be one of rG rMirOrL)rr-isdtypedtypeastyperr<r4r>r?r,sort_vasicek_entropy_van_es_entropy_correa_entropy_ebrahimi_entropylowersetr5) rBr:r r!rHr*rC sorted_datamethodsmessageress rr r sh  B ZZ F zz&,, ++6::b>#7#78  4F RA 499Q<##56 M! %A %m_-**+B 0   DAIFGG''&r'*K*((,' )G \\^F ,S\N;!!  7F $YFF /+ rsl  IE , -.2!%E  ##E  ##d*E $,E E RZZ' E E P1n0!% ~( II  ~(:~( $, ~(  ~(  ~(YY~( ~(B+"="".r