SciPy

scipy.stats.ncf

scipy.stats.ncf = <scipy.stats._continuous_distns.ncf_gen object at 0x7fa40e94fcd0>[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:

ncf.pdf(x, df1, df2, nc) = exp(nc/2 + nc*df1*x/(2*(df1*x+df2))) *
            df1**(df1/2) * df2**(df2/2) * x**(df1/2-1) *
            (df2+df1*x)**(-(df1+df2)/2) *
            gamma(df1/2)*gamma(1+df2/2) *
            L^{v1/2-1}^{v2/2}(-nc*v1*x/(2*(v1*x+v2))) /
            (B(v1/2, v2/2) * gamma((v1+v2)/2))

for df1, df2, nc > 0.

ncf takes df1, df2 and nc as shape parameters.

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.

Examples

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

Calculate a few first 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, normed=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

(Source code)

../_images/scipy-stats-ncf-1.png

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 density function.
logcdf(x, dfn, dfd, nc, loc=0, scale=1) Log of the cumulative density function.
sf(x, dfn, dfd, nc, loc=0, scale=1) Survival function (1 - cdf — 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(n, dfn, dfd, nc, loc=0, scale=1) Non-central moment of order n
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, dfn, dfd, nc, loc=0, scale=1) Parameter estimates for generic data.
expect(func, 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(alpha, dfn, dfd, nc, loc=0, scale=1) Endpoints of the range that contains alpha percent of the distribution

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