scipy.stats.ncf¶
-
scipy.stats.
ncf
(*args, **kwds) = <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.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 \(n_1, n_2 > 0\), \(\lambda\geq 0\). 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
.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
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, histtype='stepfilled', alpha=0.2) >>> ax.legend(loc='best', frameon=False) >>> plt.show()
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(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)
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(alpha, dfn, dfd, nc, loc=0, scale=1)
Endpoints of the range that contains alpha percent of the distribution