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.

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()

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.