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
takesdf1
,df2
andnc
as shape parameters. Ifnc=0
, the distribution becomes equivalent to the Fisher distribution.The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the
loc
andscale
parameters. Specifically,ncf.pdf(x, dfn, dfd, nc, loc, scale)
is identically equivalent toncf.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.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
andppf
:>>> 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()