scipy.stats.argus#

scipy.stats.argus = <scipy.stats._continuous_distns.argus_gen object>[source]#

Argus distribution

As an instance of the rv_continuous class, argus 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(chi, loc=0, scale=1, size=1, random_state=None)	Random variates.
pdf(x, chi, loc=0, scale=1)	Probability density function.
logpdf(x, chi, loc=0, scale=1)	Log of the probability density function.
cdf(x, chi, loc=0, scale=1)	Cumulative distribution function.
logcdf(x, chi, loc=0, scale=1)	Log of the cumulative distribution function.
sf(x, chi, loc=0, scale=1)	Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
logsf(x, chi, loc=0, scale=1)	Log of the survival function.
ppf(q, chi, loc=0, scale=1)	Percent point function (inverse of `cdf` — percentiles).
isf(q, chi, loc=0, scale=1)	Inverse survival function (inverse of `sf`).
moment(order, chi, loc=0, scale=1)	Non-central moment of the specified order.
stats(chi, loc=0, scale=1, moments=’mv’)	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(chi, 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=(chi,), 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(chi, loc=0, scale=1)	Median of the distribution.
mean(chi, loc=0, scale=1)	Mean of the distribution.
var(chi, loc=0, scale=1)	Variance of the distribution.
std(chi, loc=0, scale=1)	Standard deviation of the distribution.
interval(confidence, chi, loc=0, scale=1)	Confidence interval with equal areas around the median.

Notes

The probability density function for argus is:

\[f(x, \chi) = \frac{\chi^3}{\sqrt{2\pi} \Psi(\chi)} x \sqrt{1-x^2} \exp(-\chi^2 (1 - x^2)/2)\]

for \(0 < x < 1\) and \(\chi > 0\), where

\[\Psi(\chi) = \Phi(\chi) - \chi \phi(\chi) - 1/2\]

with \(\Phi\) and \(\phi\) being the CDF and PDF of a standard normal distribution, respectively.

argus takes \(\chi\) as shape a parameter. Details about sampling from the ARGUS distribution can be found in [2].

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, argus.pdf(x, chi, loc, scale) is identically equivalent to argus.pdf(y, chi) / scale with 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]

“ARGUS distribution”, https://en.wikipedia.org/wiki/ARGUS_distribution

[2]

Christoph Baumgarten “Random variate generation by fast numerical inversion in the varying parameter case.” Research in Statistics, vol. 1, 2023, doi:10.1080/27684520.2023.2279060.

Added in version 0.19.0.

Examples

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

Calculate the first four moments:

>>> chi = 1
>>> mean, var, skew, kurt = argus.stats(chi, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(argus.ppf(0.01, chi),
...                 argus.ppf(0.99, chi), 100)
>>> ax.plot(x, argus.pdf(x, chi),
...        'r-', lw=5, alpha=0.6, label='argus 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 = argus(chi)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = argus.ppf([0.001, 0.5, 0.999], chi)
>>> np.allclose([0.001, 0.5, 0.999], argus.cdf(vals, chi))
True

Generate random numbers:

>>> r = argus.rvs(chi, 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()