scipy.stats.ncf#

scipy.stats.ncf = <scipy.stats._continuous_distns.ncf_gen object>[source]#

A non-central F distribution continuous random variable.

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

See also

scipy.stats.f: Fisher distribution

Notes

The probability density function for ncf is:

\[\begin{split}f(x, n_1, n_2, \lambda) = \exp\left(\frac{\lambda}{2} + \lambda n_1 \frac{x}{2(n_1 x + n_2)} \right) n_1^{n_1/2} n_2^{n_2/2} x^{n_1/2 - 1} \\ (n_2 + n_1 x)^{-(n_1 + n_2)/2} \gamma(n_1/2) \gamma(1 + n_2/2) \\ \frac{L^{\frac{n_1}{2}-1}_{n_2/2} \left(-\lambda n_1 \frac{x}{2(n_1 x + n_2)}\right)} {B(n_1/2, n_2/2) \gamma\left(\frac{n_1 + n_2}{2}\right)}\end{split}\]

for \(n_1, n_2 > 0\), \(\lambda \ge 0\). Here \(n_1\) is the degrees of freedom in the numerator, \(n_2\) the degrees of freedom in the denominator, \(\lambda\) the non-centrality parameter, \(\gamma\) is the logarithm of the Gamma function, \(L_n^k\) is a generalized Laguerre polynomial and \(B\) is the beta function.

ncf takes dfn, dfd and nc as shape parameters. If nc=0, the distribution becomes equivalent to the Fisher distribution.

This distribution uses routines from the Boost Math C++ library for the computation of the pdf, cdf, ppf, stats, sf and isf methods. [1]

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, ncf.pdf(x, dfn, dfd, nc, loc, scale) is identically equivalent to ncf.pdf(y, dfn, dfd, nc) / 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.

References

[1]

The Boost Developers. “Boost C++ Libraries”. https://www.boost.org/.

Examples

>>> import numpy as np
>>> from scipy.stats import ncf
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> dfn, dfd, nc = 27, 27, 0.416
>>> mean, var, skew, kurt = ncf.stats(dfn, dfd, nc, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(ncf.ppf(0.01, dfn, dfd, nc),
...                 ncf.ppf(0.99, dfn, dfd, nc), 100)
>>> ax.plot(x, ncf.pdf(x, dfn, dfd, nc),
...        'r-', lw=5, alpha=0.6, label='ncf 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 = ncf(dfn, dfd, nc)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = ncf.ppf([0.001, 0.5, 0.999], dfn, dfd, nc)
>>> np.allclose([0.001, 0.5, 0.999], ncf.cdf(vals, dfn, dfd, nc))
True

Generate random numbers:

>>> r = ncf.rvs(dfn, dfd, nc, 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()