ftjz>dZddlZddlZddlmZddlmZddlm Z m Z m Z m Z m Z mZddgZeed d dd d d  d$dejjdejjdzdedzdedejejzf dZd%dZeedddeddddddejjdedzdedzdededejejzf dZdZd Zd!Zd"Zd#ZdS)&z5 Created on Fri Apr 2 09:06:05 2021 @author: matth N)special)_axis_nan_policy_factory)array_namespace xp_promote xp_device is_marray _share_masksxp_capabilitiesentropydifferential_entropyc|SNxs R/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/scipy/stats/_entropy.pyrac"d|vr |ddndS)Nqkrr)kwgss rrrs dlltDz5 rc|fSrrr_s rrrsqdrT) n_samples n_outputsresult_to_tuplepaired too_smallpkrbaseaxisreturnc||dkrtdt||}t||d|\}}tjd5|0t |||\}}||||d z }||||d z }dddn #1swxYwY|tj|}n`t|r>> import numpy as np >>> from scipy.stats import entropy >>> base = 2 # work in units of bits >>> pk = np.array([1/2, 1/2]) # fair coin >>> H = entropy(pk, base=base) >>> H 1.0 >>> H == -np.sum(pk * np.log(pk)) / np.log(base) True The outcome of a biased coin is less uncertain: >>> qk = np.array([9/10, 1/10]) # biased coin >>> entropy(qk, base=base) 0.46899559358928117 The relative entropy between the fair coin and biased coin is calculated as: >>> D = entropy(pk, qk, base=base) >>> D 0.7369655941662062 >>> np.isclose(D, np.sum(pk * np.log(pk/qk)) / np.log(base), rtol=4e-16, atol=0) True The cross entropy can be calculated as the sum of the entropy and relative entropy`: >>> CE = entropy(pk, base=base) + entropy(pk, qk, base=base) >>> CE 1.736965594166206 >>> CE == -np.sum(pk * np.log(qk)) / np.log(base) True Nr+`base` must be a positive number or `None`.T) broadcastxpignore)invalidr+r&keepdims)maskr&) ValueErrorrrnperrstater sumrentrr rel_entrdataasarrayr1mathlog)r$rr%r&r+vecSs rr r sx DAIIFGGG R B B$2 6 6 6FB X & & &77 >!"bR000FBbffRdTf:::B "&&$&66 6 777777777777777  zl2 R== +"27BG44C**Srw*//CC"2r**C sA  TXd^^ HsAB--B14B1c|d}|j|}|d}|)tjtj|dz}dd|zcxkr|ksndSdS)Nr window_length?rTF)shapegetr;floorsqrt)sampleskwargsr&valuesnr@s r"_differential_entropy_is_too_smallrJsx QZF TAJJ//M 49Q<<##566 M! % % % %A % % % %t 5rc|Srrrs rrrrrc|fSrrrs rrrsA4r)r r!r#auto)r@r%r&methodrHr@rNct|}t|d|}|||d}|jd}|)t jt j|dz}dd|zcxkr|ksntd|d|d ||d krtd ||d }tttttd }| }||vr!dt|} t| |dkr|dkrd}n |dkrd}nd}|||||} || t j|z} || |jS)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)') T)force_floatingr+rNrArzWindow length (z7) must be positive and less than half the sample size (z).rr)r2)vasicekvan escorreaebrahimirMz`method` must be one of rM rRirTrQr.)rrmoveaxisrBr;rDrEr3sort_vasicek_entropy_van_es_entropy_correa_entropy_ebrahimi_entropylowersetr<astypedtype) rHr@r%r&rNr+rI sorted_datamethodsmessageress rr r sj  B t ; ; ;F [[r * *F RA 49Q<<##566 M! % % % %A % % % % 0m 0 0*+ 0 0 0    DAIIFGGG''&r'**K*((,' ))G \\^^F W;S\\;;!!!  77FF $YYFFF '&/+} < < rs  666666MMMMMMMMMMMMMMMM , -K!2!24.2!%J J  #J  #d*J $,J J RZ' J J J J ZK1.?.?0!% |(|(|( I |(:|( $, |(  |(  |(Y|(|(|(  |(~+++"""=== " " " .....r