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
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/TeX Live for SUSE Linux) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2015/01/01> patch level 2
Babel <3.9m> and hyphenation patterns for 79 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2014/09/29 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.173 \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
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/TeX Live for SUSE Linux) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2015/01/01> patch level 2
Babel <3.9m> and hyphenation patterns for 79 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2014/09/29 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.173 \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
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/TeX Live for SUSE Linux) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2015/01/01> patch level 2
Babel <3.9m> and hyphenation patterns for 79 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2014/09/29 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.173 \endinput
^^M
No pages of output.
Transcript written on math.log.
and
System Message: WARNING/2 (x_j)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/TeX Live for SUSE Linux) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2015/01/01> patch level 2
Babel <3.9m> and hyphenation patterns for 79 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2014/09/29 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.173 \endinput
^^M
No pages of output.
Transcript written on math.log.
. The element
System Message: WARNING/2 (C_{ii})
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/TeX Live for SUSE Linux) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2015/01/01> patch level 2
Babel <3.9m> and hyphenation patterns for 79 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2014/09/29 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.173 \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
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/TeX Live for SUSE Linux) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2015/01/01> patch level 2
Babel <3.9m> and hyphenation patterns for 79 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2014/09/29 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.173 \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
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,
System Message: WARNING/2 (x_0)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/TeX Live for SUSE Linux) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2015/01/01> patch level 2
Babel <3.9m> and hyphenation patterns for 79 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2014/09/29 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.173 \endinput
^^M
No pages of output.
Transcript written on math.log.
and
System Message: WARNING/2 (x_1)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/TeX Live for SUSE Linux) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2015/01/01> patch level 2
Babel <3.9m> and hyphenation patterns for 79 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2014/09/29 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.173 \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)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/TeX Live for SUSE Linux) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2015/01/01> patch level 2
Babel <3.9m> and hyphenation patterns for 79 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2014/09/29 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.173 \endinput
^^M
No pages of output.
Transcript written on math.log.
increases while
System Message: WARNING/2 (x_1)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/TeX Live for SUSE Linux) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2015/01/01> patch level 2
Babel <3.9m> and hyphenation patterns for 79 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2014/09/29 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.173 \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
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/TeX Live for SUSE Linux) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2015/01/01> patch level 2
Babel <3.9m> and hyphenation patterns for 79 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2014/09/29 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.173 \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
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/TeX Live for SUSE Linux) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2015/01/01> patch level 2
Babel <3.9m> and hyphenation patterns for 79 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2014/09/29 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.173 \endinput
^^M
No pages of output.
Transcript written on math.log.
and
System Message: WARNING/2 (x_1)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/TeX Live for SUSE Linux) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2015/01/01> patch level 2
Babel <3.9m> and hyphenation patterns for 79 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2014/09/29 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.173 \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