mars.tensor.random.noncentral_chisquare#

mars.tensor.random.noncentral_chisquare(df, nonc, size=None, chunk_size=None, gpu=None, dtype=None)[source]#

Draw samples from a noncentral chi-square distribution.

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

Parameters

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

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

  • size (int or tuple of ints, optional) – 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, mt.broadcast(df, nonc).size samples are drawn.

  • chunk_size (int or tuple of int or tuple of ints, optional) – Desired chunk size on each dimension

  • gpu (bool, optional) – Allocate the tensor on GPU if True, False as default

  • dtype (data-type, optional) – Data-type of the returned tensor.

Returns

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

Return type

Tensor 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.

In Delhi (2007), it is noted that the noncentral chi-square is useful in bombing and coverage problems, the probability of killing the point target given by the noncentral chi-squared distribution.

References

1

Delhi, M.S. Holla, “On a noncentral chi-square distribution in the analysis of weapon systems effectiveness”, Metrika, Volume 15, Number 1 / December, 1970.

2

Wikipedia, “Noncentral chi-square distribution” http://en.wikipedia.org/wiki/Noncentral_chi-square_distribution

Examples

Draw values from the distribution and plot the histogram

>>> import matplotlib.pyplot as plt
>>> import mars.tensor as mt
>>> values = plt.hist(mt.random.noncentral_chisquare(3, 20, 100000).execute(),
...                   bins=200, normed=True)
>>> plt.show()

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

>>> plt.figure()
>>> values = plt.hist(mt.random.noncentral_chisquare(3, .0000001, 100000).execute(),
...                   bins=mt.arange(0., 25, .1).execute(), normed=True)
>>> values2 = plt.hist(mt.random.chisquare(3, 100000).execute(),
...                    bins=mt.arange(0., 25, .1).execute(), normed=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(mt.random.noncentral_chisquare(3, 20, 100000).execute(),
...                   bins=200, normed=True)
>>> plt.show()