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_continuousclass,- ncfobject 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 - ncfis:\[\begin{split}f(x, n_1, n_2, \lambda) = \exp\left(\frac{\lambda}{2} + \lambda n_1 \frac{x}{2(n_1 x + n_2)} \right) n_1^{n_1/2} n_2^{n_2/2} x^{n_1/2 - 1} \\ (n_2 + n_1 x)^{-(n_1 + n_2)/2} \gamma(n_1/2) \gamma(1 + n_2/2) \\ \frac{L^{\frac{n_1}{2}-1}_{n_2/2} \left(-\lambda n_1 \frac{x}{2(n_1 x + n_2)}\right)} {B(n_1/2, n_2/2) \gamma\left(\frac{n_1 + n_2}{2}\right)}\end{split}\]- 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. - ncftakes- dfn,- dfdand- ncas 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,- sfand- isfmethods. [1]- The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the - locand- scaleparameters. Specifically,- ncf.pdf(x, dfn, dfd, nc, loc, scale)is identically equivalent to- ncf.pdf(y, dfn, dfd, nc) / scalewith- 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 - cdfand- 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 - 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.