scipy.stats.kappa4#
- scipy.stats.kappa4 = <scipy.stats._continuous_distns.kappa4_gen object>[source]#
- Kappa 4 parameter distribution. - As an instance of the - rv_continuousclass,- kappa4object 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(h, k, loc=0, scale=1, size=1, random_state=None) - Random variates. - pdf(x, h, k, loc=0, scale=1) - Probability density function. - logpdf(x, h, k, loc=0, scale=1) - Log of the probability density function. - cdf(x, h, k, loc=0, scale=1) - Cumulative distribution function. - logcdf(x, h, k, loc=0, scale=1) - Log of the cumulative distribution function. - sf(x, h, k, loc=0, scale=1) - Survival function (also defined as - 1 - cdf, but sf is sometimes more accurate).- logsf(x, h, k, loc=0, scale=1) - Log of the survival function. - ppf(q, h, k, loc=0, scale=1) - Percent point function (inverse of - cdf— percentiles).- isf(q, h, k, loc=0, scale=1) - Inverse survival function (inverse of - sf).- moment(order, h, k, loc=0, scale=1) - Non-central moment of the specified order. - stats(h, k, loc=0, scale=1, moments=’mv’) - Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). - entropy(h, k, 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=(h, k), 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(h, k, loc=0, scale=1) - Median of the distribution. - mean(h, k, loc=0, scale=1) - Mean of the distribution. - var(h, k, loc=0, scale=1) - Variance of the distribution. - std(h, k, loc=0, scale=1) - Standard deviation of the distribution. - interval(confidence, h, k, loc=0, scale=1) - Confidence interval with equal areas around the median. - Notes - The probability density function for kappa4 is: \[f(x, h, k) = (1 - k x)^{1/k - 1} (1 - h (1 - k x)^{1/k})^{1/h-1}\]- if \(h\) and \(k\) are not equal to 0. - If \(h\) or \(k\) are zero then the pdf can be simplified: - h = 0 and k != 0: - kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)* exp(-(1.0 - k*x)**(1.0/k)) - h != 0 and k = 0: - kappa4.pdf(x, h, k) = exp(-x)*(1.0 - h*exp(-x))**(1.0/h - 1.0) - h = 0 and k = 0: - kappa4.pdf(x, h, k) = exp(-x)*exp(-exp(-x)) - kappa4 takes \(h\) and \(k\) as shape parameters. - The kappa4 distribution returns other distributions when certain \(h\) and \(k\) values are used. - h - k=0.0 - k=1.0 - -inf<=k<=inf - -1.0 - Logistic - logistic(x) - Generalized Logistic(1) - 0.0 - Gumbel - gumbel_r(x) - Reverse Exponential(2) - Generalized Extreme Value - genextreme(x, k) - 1.0 - Exponential - expon(x) - Uniform - uniform(x) - Generalized Pareto - genpareto(x, -k) - There are at least five generalized logistic distributions. Four are described here: https://en.wikipedia.org/wiki/Generalized_logistic_distribution The “fifth” one is the one kappa4 should match which currently isn’t implemented in scipy: https://en.wikipedia.org/wiki/Talk:Generalized_logistic_distribution https://www.mathwave.com/help/easyfit/html/analyses/distributions/gen_logistic.html 
- This distribution is currently not in scipy. 
 - References - J.C. Finney, “Optimization of a Skewed Logistic Distribution With Respect to the Kolmogorov-Smirnov Test”, A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College, (August, 2004), https://digitalcommons.lsu.edu/gradschool_dissertations/3672 - J.R.M. Hosking, “The four-parameter kappa distribution”. IBM J. Res. Develop. 38 (3), 25 1-258 (1994). - B. Kumphon, A. Kaew-Man, P. Seenoi, “A Rainfall Distribution for the Lampao Site in the Chi River Basin, Thailand”, Journal of Water Resource and Protection, vol. 4, 866-869, (2012). DOI:10.4236/jwarp.2012.410101 - C. Winchester, “On Estimation of the Four-Parameter Kappa Distribution”, A Thesis Submitted to Dalhousie University, Halifax, Nova Scotia, (March 2000). http://www.nlc-bnc.ca/obj/s4/f2/dsk2/ftp01/MQ57336.pdf - The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the - locand- scaleparameters. Specifically,- kappa4.pdf(x, h, k, loc, scale)is identically equivalent to- kappa4.pdf(y, h, k) / 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.- Examples - >>> import numpy as np >>> from scipy.stats import kappa4 >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) - Get the support: - >>> h, k = 0.1, 0 >>> lb, ub = kappa4.support(h, k) - Calculate the first four moments: - >>> mean, var, skew, kurt = kappa4.stats(h, k, moments='mvsk') - Display the probability density function ( - pdf):- >>> x = np.linspace(kappa4.ppf(0.01, h, k), ... kappa4.ppf(0.99, h, k), 100) >>> ax.plot(x, kappa4.pdf(x, h, k), ... 'r-', lw=5, alpha=0.6, label='kappa4 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 = kappa4(h, k) >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') - Check accuracy of - cdfand- ppf:- >>> vals = kappa4.ppf([0.001, 0.5, 0.999], h, k) >>> np.allclose([0.001, 0.5, 0.999], kappa4.cdf(vals, h, k)) True - Generate random numbers: - >>> r = kappa4.rvs(h, k, 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() 