scipy.stats.nct — SciPy v1.15.2 Manual (original) (raw)
scipy.stats.nct = <scipy.stats._continuous_distns.nct_gen object>[source]#
A non-central Student’s t continuous random variable.
As an instance of the rv_continuous class, nct 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
If \(Y\) is a standard normal random variable and \(V\) is an independent chi-square random variable (chi2) with \(k\) degrees of freedom, then
\[X = \frac{Y + c}{\sqrt{V/k}}\]
has a non-central Student’s t distribution on the real line. The degrees of freedom parameter \(k\) (denoted df in the implementation) satisfies \(k > 0\) and the noncentrality parameter\(c\) (denoted nc in the implementation) is a real number.
This distribution uses routines from the Boost Math C++ library for the computation of the pdf, cdf, ppf, sf and isfmethods. [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, nct.pdf(x, df, nc, loc, scale) is identically equivalent to nct.pdf(y, df, 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 nct import matplotlib.pyplot as plt fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
df, nc = 14, 0.24 mean, var, skew, kurt = nct.stats(df, nc, moments='mvsk')
Display the probability density function (pdf):
x = np.linspace(nct.ppf(0.01, df, nc), ... nct.ppf(0.99, df, nc), 100) ax.plot(x, nct.pdf(x, df, nc), ... 'r-', lw=5, alpha=0.6, label='nct 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 = nct(df, nc) ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
Check accuracy of cdf and ppf:
vals = nct.ppf([0.001, 0.5, 0.999], df, nc) np.allclose([0.001, 0.5, 0.999], nct.cdf(vals, df, nc)) True
Generate random numbers:
r = nct.rvs(df, 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(df, nc, loc=0, scale=1, size=1, random_state=None) | Random variates. |
|---|---|
| pdf(x, df, nc, loc=0, scale=1) | Probability density function. |
| logpdf(x, df, nc, loc=0, scale=1) | Log of the probability density function. |
| cdf(x, df, nc, loc=0, scale=1) | Cumulative distribution function. |
| logcdf(x, df, nc, loc=0, scale=1) | Log of the cumulative distribution function. |
| sf(x, df, nc, loc=0, scale=1) | Survival function (also defined as 1 - cdf, but sf is sometimes more accurate). |
| logsf(x, df, nc, loc=0, scale=1) | Log of the survival function. |
| ppf(q, df, nc, loc=0, scale=1) | Percent point function (inverse of cdf — percentiles). |
| isf(q, df, nc, loc=0, scale=1) | Inverse survival function (inverse of sf). |
| moment(order, df, nc, loc=0, scale=1) | Non-central moment of the specified order. |
| stats(df, nc, loc=0, scale=1, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). |
| entropy(df, 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=(df, 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(df, nc, loc=0, scale=1) | Median of the distribution. |
| mean(df, nc, loc=0, scale=1) | Mean of the distribution. |
| var(df, nc, loc=0, scale=1) | Variance of the distribution. |
| std(df, nc, loc=0, scale=1) | Standard deviation of the distribution. |
| interval(confidence, df, nc, loc=0, scale=1) | Confidence interval with equal areas around the median. |