numpy.random.mtrand.RandomState.noncentral_chisquare

RandomState.noncentral_chisquare(df, nonc, size=None)

Draw samples from a noncentral chi-square distribution.

The noncentral

System Message: WARNING/2 (\chi^2)

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distribution is a generalisation of the

System Message: WARNING/2 (\chi^2)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, ge rman-x-2011-07-01, ngerman-x-2011-07-01, afrikaans, ancientgreek, ibycus, arabi c, armenian, basque, bulgarian, catalan, pinyin, coptic, croatian, czech, danis h, dutch, ukenglish, usenglishmax, esperanto, estonian, ethiopic, farsi, finnis h, french, friulan, galician, german, ngerman, swissgerman, monogreek, greek, h ungarian, icelandic, assamese, bengali, gujarati, hindi, kannada, malayalam, ma rathi, oriya, panjabi, tamil, telugu, indonesian, interlingua, irish, italian, kurmanji, lao, latin, latvian, lithuanian, mongolian, mongolianlmc, bokmal, nyn orsk, polish, portuguese, romanian, romansh, russian, sanskrit, serbian, serbia nc, slovak, slovenian, spanish, swedish, turkish, turkmen, ukrainian, uppersorb ian, welsh, loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

distribution.

Parameters :

df : int

Degrees of freedom, should be >= 1.

nonc : float

Non-centrality, should be > 0.

size : int or tuple of ints

Shape of the output.

Notes

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

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

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, ge rman-x-2011-07-01, ngerman-x-2011-07-01, afrikaans, ancientgreek, ibycus, arabi c, armenian, basque, bulgarian, catalan, pinyin, coptic, croatian, czech, danis h, dutch, ukenglish, usenglishmax, esperanto, estonian, ethiopic, farsi, finnis h, french, friulan, galician, german, ngerman, swissgerman, monogreek, greek, h ungarian, icelandic, assamese, bengali, gujarati, hindi, kannada, malayalam, ma rathi, oriya, panjabi, tamil, telugu, indonesian, interlingua, irish, italian, kurmanji, lao, latin, latvian, lithuanian, mongolian, mongolianlmc, bokmal, nyn orsk, polish, portuguese, romanian, romansh, russian, sanskrit, serbian, serbia nc, slovak, slovenian, spanish, swedish, turkish, turkmen, ukrainian, uppersorb ian, welsh, loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

where

System Message: WARNING/2 (Y_{q})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, ge rman-x-2011-07-01, ngerman-x-2011-07-01, afrikaans, ancientgreek, ibycus, arabi c, armenian, basque, bulgarian, catalan, pinyin, coptic, croatian, czech, danis h, dutch, ukenglish, usenglishmax, esperanto, estonian, ethiopic, farsi, finnis h, french, friulan, galician, german, ngerman, swissgerman, monogreek, greek, h ungarian, icelandic, assamese, bengali, gujarati, hindi, kannada, malayalam, ma rathi, oriya, panjabi, tamil, telugu, indonesian, interlingua, irish, italian, kurmanji, lao, latin, latvian, lithuanian, mongolian, mongolianlmc, bokmal, nyn orsk, polish, portuguese, romanian, romansh, russian, sanskrit, serbian, serbia nc, slovak, slovenian, spanish, swedish, turkish, turkmen, ukrainian, uppersorb ian, welsh, loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

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

[R178]

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

[R179]

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
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
...                   bins=200, normed=True)
>>> plt.show()

(Source code, png, pdf)

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

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

(png, pdf)

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

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

(png, pdf)