scipy.stats.fisk#

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.

Methods

rvs(c, loc=0, scale=1, size=1, random_state=None)	Random variates.
pdf(x, c, loc=0, scale=1)	Probability density function.
logpdf(x, c, loc=0, scale=1)	Log of the probability density function.
cdf(x, c, loc=0, scale=1)	Cumulative distribution function.
logcdf(x, c, loc=0, scale=1)	Log of the cumulative distribution function.
sf(x, c, loc=0, scale=1)	Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
logsf(x, c, loc=0, scale=1)	Log of the survival function.
ppf(q, c, loc=0, scale=1)	Percent point function (inverse of `cdf` — percentiles).
isf(q, c, loc=0, scale=1)	Inverse survival function (inverse of `sf`).
moment(order, c, loc=0, scale=1)	Non-central moment of the specified order.
stats(c, loc=0, scale=1, moments=’mv’)	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(c, 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=(c,), 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(c, loc=0, scale=1)	Median of the distribution.
mean(c, loc=0, scale=1)	Mean of the distribution.
var(c, loc=0, scale=1)	Variance of the distribution.
std(c, loc=0, scale=1)	Standard deviation of the distribution.
interval(confidence, c, loc=0, scale=1)	Confidence interval with equal areas around the median.

See also

burr

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 l and 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 with 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 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()