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Covariance matrix
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saneman  
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 More options May 16, 7:52 am
Newsgroups: sci.math
From: "saneman" <a...@ad.com>
Date: Fri, 16 May 2008 13:52:16 +0200
Local: Fri, May 16 2008 7:52 am
Subject: Covariance matrix
I have this matrix:

A =

           1        1000
           2        1010
           3        1020
           4        1030
           5        1040
           6        1050
           7        1060
           8        1070
           9        1080
          10       1090

If the first column is the variable X and the second column is the variable
Y I would like to see how correlated the data is. Computing the covariance
matrix gives:

C =

    9.1667   91.6667
   91.6667  916.6667

I have read that if an element in C is 0 there is no covariance between the
corresponding variables, if its positive there is a covariance between the
two variables and if its negative there is a covariance between X and -Y.

In C the covariance between X and Y is 91.6667. But how should that number
be interpretated and does the magnitude have anything to say?

If I plot the above data X and Y does not seem very correlated (close to
each other).


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Ray Vickson  
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 More options May 16, 11:19 am
Newsgroups: sci.math
From: Ray Vickson <RGVick...@shaw.ca>
Date: Fri, 16 May 2008 08:19:27 -0700 (PDT)
Local: Fri, May 16 2008 11:19 am
Subject: Re: Covariance matrix
On May 16, 4:52 am, "saneman" <a...@ad.com> wrote:

You are not thinking. Look at the data: the values of Y = y and X = x
are related as y = 1000 + 10*(x-1) for x = 1, 2, ..., 10. Y and X are
perfectly, 100% correlated. You could also get this from the
covariance matrix: the standard deviations of X and Y are sx =
sqrt(9.1667), sy = sqrt(916.6667), so the correlation coefficient of X
and Y is r(X,Y) = Cov(X,Y)/[sx * sy} = 0.999999, which would have be 1
exactly if you computed infinitely many decimal places in C.

You seem to misunderstand the interpretation of covariance. See, eg.,
http://en.wikipedia.org/wiki/Correlation

R.G. Vickson


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