scipy.stats.ncf — SciPy v1.15.3 Manual (original) (raw)

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.

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 andisf 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 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.

References

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()

Methods