noncentral_chisquare

noncentral_chisquare#

class brainstate.random.noncentral_chisquare(df, nonc, size=None, key=None, dtype=None)#

Draw samples from a noncentral chi-square distribution.

The noncentral \(\chi^2\) distribution is a generalization of the \(\chi^2\) distribution.

Parameters:

  • df (float or array_like of floats) – Degrees of freedom, must be > 0.

  • nonc (float or array_like of floats) – Non-centrality, must be non-negative.

  • size (int | Sequence[int] | integer | Sequence[integer] | None) – Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if df and nonc are both scalars. Otherwise, np.broadcast(df, nonc).size samples are drawn.

  • key (int | Array | ndarray | None) – The key for the random number generator. If not given, the default random number generator is used.

Returns:

out – Drawn samples from the parameterized noncentral chi-square distribution.

Return type:

ndarray or scalar

Notes

The probability density function for the noncentral Chi-square distribution is

\[P(x;df,nonc) = \sum^{\infty}_{i=0} \frac{e^{-nonc/2}(nonc/2)^{i}}{i!} P_{Y_{df+2i}}(x),\]

where \(Y_{q}\) is the Chi-square with q degrees of freedom.

References

Examples

Draw values from the distribution and plot the histogram

>>> import brainstate
>>> import matplotlib.pyplot as plt  # noqa
>>> values = plt.hist(brainstate.random.noncentral_chisquare(3, 20, 100000),
...                   bins=200, density=True)
>>> plt.show()

Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare.

>>> plt.figure()
>>> values = plt.hist(brainstate.random.noncentral_chisquare(3, .0000001, 100000),
...                   bins=np.arange(0., 25, .1), density=True)
>>> values2 = plt.hist(brainstate.random.chisquare(3, 100000),
...                    bins=np.arange(0., 25, .1), density=True)
>>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob')
>>> plt.show()

Demonstrate how large values of non-centrality lead to a more symmetric distribution.

>>> plt.figure()
>>> values = plt.hist(brainstate.random.noncentral_chisquare(3, 20, 100000),
...                   bins=200, density=True)
>>> plt.show()