numpy.cov

numpy.cov(m, y=None, rowvar=1, bias=0, ddof=None)

Estimate a covariance matrix, given data.

Covariance indicates the level to which two variables vary together. If we examine N-dimensional samples,

System Message: WARNING/2 (X = [x_1, x_2, ... x_N]^T)

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.

, then the covariance matrix element

System Message: WARNING/2 (C_{ij})

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 covariance of

System Message: WARNING/2 (x_i)

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.

and

System Message: WARNING/2 (x_j)

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. The element

System Message: WARNING/2 (C_{ii})

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 variance of

System Message: WARNING/2 (x_i)

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.

.

Parameters :

m : array_like

A 1-D or 2-D array containing multiple variables and observations. Each row of m represents a variable, and each column a single observation of all those variables. Also see rowvar below.

y : array_like, optional

An additional set of variables and observations. y has the same form as that of m.

rowvar : int, optional

If rowvar is non-zero (default), then each row represents a variable, with observations in the columns. Otherwise, the relationship is transposed: each column represents a variable, while the rows contain observations.

bias : int, optional

Default normalization is by (N - 1), where N is the number of observations given (unbiased estimate). If bias is 1, then normalization is by N. These values can be overridden by using the keyword ddof in numpy versions >= 1.5.

ddof : int, optional

New in version 1.5.

If not None normalization is by (N - ddof), where N is the number of observations; this overrides the value implied by bias. The default value is None.

Returns :

out : ndarray

The covariance matrix of the variables.

See also

corrcoef

Normalized covariance matrix

Examples

Consider two variables,

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and

System Message: WARNING/2 (x_1)

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.

, which correlate perfectly, but in opposite directions:

>>> x = np.array([[0, 2], [1, 1], [2, 0]]).T
>>> x
array([[0, 1, 2],
       [2, 1, 0]])

Note how

System Message: WARNING/2 (x_0)

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increases while

System Message: WARNING/2 (x_1)

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.

decreases. The covariance matrix shows this clearly:

>>> np.cov(x)
array([[ 1., -1.],
       [-1.,  1.]])

Note that element

System Message: WARNING/2 (C_{0,1})

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.

, which shows the correlation between

System Message: WARNING/2 (x_0)

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.

and

System Message: WARNING/2 (x_1)

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 negative.

Further, note how x and y are combined:

>>> x = [-2.1, -1,  4.3]
>>> y = [3,  1.1,  0.12]
>>> X = np.vstack((x,y))
>>> print np.cov(X)
[[ 11.71        -4.286     ]
 [ -4.286        2.14413333]]
>>> print np.cov(x, y)
[[ 11.71        -4.286     ]
 [ -4.286        2.14413333]]
>>> print np.cov(x)
11.71