scipy.stats.zipf#
- scipy.stats.zipf = <scipy.stats._discrete_distns.zipf_gen object>[source]#
- A Zipf (Zeta) discrete random variable. - As an instance of the - rv_discreteclass,- zipfobject inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.- Methods - rvs(a, loc=0, size=1, random_state=None) - Random variates. - pmf(k, a, loc=0) - Probability mass function. - logpmf(k, a, loc=0) - Log of the probability mass function. - cdf(k, a, loc=0) - Cumulative distribution function. - logcdf(k, a, loc=0) - Log of the cumulative distribution function. - sf(k, a, loc=0) - Survival function (also defined as - 1 - cdf, but sf is sometimes more accurate).- logsf(k, a, loc=0) - Log of the survival function. - ppf(q, a, loc=0) - Percent point function (inverse of - cdf— percentiles).- isf(q, a, loc=0) - Inverse survival function (inverse of - sf).- stats(a, loc=0, moments=’mv’) - Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). - entropy(a, loc=0) - (Differential) entropy of the RV. - expect(func, args=(a,), loc=0, lb=None, ub=None, conditional=False) - Expected value of a function (of one argument) with respect to the distribution. - median(a, loc=0) - Median of the distribution. - mean(a, loc=0) - Mean of the distribution. - var(a, loc=0) - Variance of the distribution. - std(a, loc=0) - Standard deviation of the distribution. - interval(confidence, a, loc=0) - Confidence interval with equal areas around the median. - See also - Notes - The probability mass function for - zipfis:\[f(k, a) = \frac{1}{\zeta(a) k^a}\]- for \(k \ge 1\), \(a > 1\). - zipftakes \(a > 1\) as shape parameter. \(\zeta\) is the Riemann zeta function (- scipy.special.zeta)- The Zipf distribution is also known as the zeta distribution, which is a special case of the Zipfian distribution ( - zipfian).- The probability mass function above is defined in the “standardized” form. To shift distribution use the - locparameter. Specifically,- zipf.pmf(k, a, loc)is identically equivalent to- zipf.pmf(k - loc, a).- References [1]- “Zeta Distribution”, Wikipedia, https://en.wikipedia.org/wiki/Zeta_distribution - Examples - >>> import numpy as np >>> from scipy.stats import zipf >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) - Get the support: - >>> a = 6.6 >>> lb, ub = zipf.support(a) - Calculate the first four moments: - >>> mean, var, skew, kurt = zipf.stats(a, moments='mvsk') - Display the probability mass function ( - pmf):- >>> x = np.arange(zipf.ppf(0.01, a), ... zipf.ppf(0.99, a)) >>> ax.plot(x, zipf.pmf(x, a), 'bo', ms=8, label='zipf pmf') >>> ax.vlines(x, 0, zipf.pmf(x, a), colors='b', lw=5, alpha=0.5) - Alternatively, the distribution object can be called (as a function) to fix the shape and location. This returns a “frozen” RV object holding the given parameters fixed. - Freeze the distribution and display the frozen - pmf:- >>> rv = zipf(a) >>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1, ... label='frozen pmf') >>> ax.legend(loc='best', frameon=False) >>> plt.show()   - Check accuracy of - cdfand- ppf:- >>> prob = zipf.cdf(x, a) >>> np.allclose(x, zipf.ppf(prob, a)) True - Generate random numbers: - >>> r = zipf.rvs(a, size=1000) - Confirm that - zipfis the large n limit of- zipfian.- >>> import numpy as np >>> from scipy.stats import zipf, zipfian >>> k = np.arange(11) >>> np.allclose(zipf.pmf(k, a), zipfian.pmf(k, a, n=10000000)) True