scipy.stats.mielke#
- scipy.stats.mielke = <scipy.stats._continuous_distns.mielke_gen object>[source]#
- A Mielke Beta-Kappa / Dagum continuous random variable. - As an instance of the - rv_continuousclass,- mielkeobject inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.- Notes - The probability density function for - mielkeis:\[f(x, k, s) = \frac{k x^{k-1}}{(1+x^s)^{1+k/s}}\]- for \(x > 0\) and \(k, s > 0\). The distribution is sometimes called Dagum distribution ([2]). It was already defined in [3], called a Burr Type III distribution ( - burrwith parameters- c=sand- d=k/s).- mielketakes- kand- sas shape parameters.- The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the - locand- scaleparameters. Specifically,- mielke.pdf(x, k, s, loc, scale)is identically equivalent to- mielke.pdf(y, k, s) / scalewith- y = (x - loc) / scale. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.- References [1]- Mielke, P.W., 1973 “Another Family of Distributions for Describing and Analyzing Precipitation Data.” J. Appl. Meteor., 12, 275-280 [2]- Dagum, C., 1977 “A new model for personal income distribution.” Economie Appliquee, 33, 327-367. [3]- Burr, I. W. “Cumulative frequency functions”, Annals of Mathematical Statistics, 13(2), pp 215-232 (1942). - Examples - >>> import numpy as np >>> from scipy.stats import mielke >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) - Calculate the first four moments: - >>> k, s = 10.4, 4.6 >>> mean, var, skew, kurt = mielke.stats(k, s, moments='mvsk') - Display the probability density function ( - pdf):- >>> x = np.linspace(mielke.ppf(0.01, k, s), ... mielke.ppf(0.99, k, s), 100) >>> ax.plot(x, mielke.pdf(x, k, s), ... 'r-', lw=5, alpha=0.6, label='mielke pdf') - Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed. - Freeze the distribution and display the frozen - pdf:- >>> rv = mielke(k, s) >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') - Check accuracy of - cdfand- ppf:- >>> vals = mielke.ppf([0.001, 0.5, 0.999], k, s) >>> np.allclose([0.001, 0.5, 0.999], mielke.cdf(vals, k, s)) True - Generate random numbers: - >>> r = mielke.rvs(k, s, size=1000) - And compare the histogram: - >>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2) >>> ax.set_xlim([x[0], x[-1]]) >>> ax.legend(loc='best', frameon=False) >>> plt.show()   - Methods - rvs(k, s, loc=0, scale=1, size=1, random_state=None) - Random variates. - pdf(x, k, s, loc=0, scale=1) - Probability density function. - logpdf(x, k, s, loc=0, scale=1) - Log of the probability density function. - cdf(x, k, s, loc=0, scale=1) - Cumulative distribution function. - logcdf(x, k, s, loc=0, scale=1) - Log of the cumulative distribution function. - sf(x, k, s, loc=0, scale=1) - Survival function (also defined as - 1 - cdf, but sf is sometimes more accurate).- logsf(x, k, s, loc=0, scale=1) - Log of the survival function. - ppf(q, k, s, loc=0, scale=1) - Percent point function (inverse of - cdf— percentiles).- isf(q, k, s, loc=0, scale=1) - Inverse survival function (inverse of - sf).- moment(order, k, s, loc=0, scale=1) - Non-central moment of the specified order. - stats(k, s, loc=0, scale=1, moments=’mv’) - Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). - entropy(k, s, loc=0, scale=1) - (Differential) entropy of the RV. - fit(data) - Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments. - expect(func, args=(k, s), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds) - Expected value of a function (of one argument) with respect to the distribution. - median(k, s, loc=0, scale=1) - Median of the distribution. - mean(k, s, loc=0, scale=1) - Mean of the distribution. - var(k, s, loc=0, scale=1) - Variance of the distribution. - std(k, s, loc=0, scale=1) - Standard deviation of the distribution. - interval(confidence, k, s, loc=0, scale=1) - Confidence interval with equal areas around the median.