scipy.stats.fisk — SciPy v1.15.2 Manual (original) (raw)

scipy.stats.fisk = <scipy.stats._continuous_distns.fisk_gen object>[source]#

A Fisk continuous random variable.

The Fisk distribution is also known as the log-logistic distribution.

As an instance of the rv_continuous class, fisk object 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 fisk is:

\[f(x, c) = \frac{c x^{c-1}} {(1 + x^c)^2}\]

for \(x >= 0\) and \(c > 0\).

Please note that the above expression can be transformed into the following one, which is also commonly used:

\[f(x, c) = \frac{c x^{-c-1}} {(1 + x^{-c})^2}\]

fisk takes c as a shape parameter for \(c\).

fisk is a special case of burr or burr12 with d=1.

Suppose X is a logistic random variable with location land scale s. Then Y = exp(X) is a Fisk (log-logistic) random variable with scale = exp(l) and shape c = 1/s.

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, fisk.pdf(x, c, loc, scale) is identically equivalent to fisk.pdf(y, c) / scale withy = (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 fisk import matplotlib.pyplot as plt fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

c = 3.09 mean, var, skew, kurt = fisk.stats(c, moments='mvsk')

Display the probability density function (pdf):

x = np.linspace(fisk.ppf(0.01, c), ... fisk.ppf(0.99, c), 100) ax.plot(x, fisk.pdf(x, c), ... 'r-', lw=5, alpha=0.6, label='fisk 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 = fisk(c) ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

vals = fisk.ppf([0.001, 0.5, 0.999], c) np.allclose([0.001, 0.5, 0.999], fisk.cdf(vals, c)) True

Generate random numbers:

r = fisk.rvs(c, 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