Re: covariance

From: Stephen Miller (anon_at_hotmail.com)
Date: 06/22/04 

Date: Tue, 22 Jun 2004 16:49:28 +0000 (UTC)

On 22 Jun 04 05:30:56 -0400 (EDT), jax wrote:
>On Mon, 21 Jun 2004 11:13:16 -0400, Richard Ulrich wrote:
>>On Mon, 21 Jun 2004 11:59:09 +0000 (UTC), jax79sg@msn.com (jax)
>wrote:
>>
>>> Hi, i am new to stats and wondering if 2 variables with different
>>> amount of readings can be computed for covariance?
>>>
>>> Eg:
>>> X= 12,24,43,55,47
>>> Y= 24,35,49
>>> cov(X,Y)= ??
>>
>>umm, the notion of covariance is that the observations
>>are paired -- matched in someway.
>>
>>People do things sometimes with (say) two time series
>>with unequal Ns, by making assumptions (and estimations)
>>about intermediate points.
>>
>>Otherwise, no.
>>--
>>Rich Ulrich, wpilib@pitt.edu
>><a
>href="http://www.pitt.edu/~wpilib/index.html">http://www.pitt.edu/~wpilib/index.html>
>
>
>
>Well, i am considering making some huge assumptions to my dataset to
>compute this problem. But before that, let me place my problem here
>and see if an alternative/advice can be given. I am reading this
paper
>on Concurrent Mapping and localisation and one of the portion covers
>the its state covariance matrix as below.
>
>Lets say P be covariance, so Pab is Covariance matrix between
>variables a and b.
>
>a is a variable of 3 dimensions, b is of 2 dimensions. Eg:
>a(1,2,3),b(1,2)
>So Paa and Pbb are actually 3x3 and 2x2 matrices respectively. I can
>retrieve Paa and Pbb based on my dataset. Now i wish to find P, which
>is of the following 5x5 matrix.
>
>Paa Pab
>transpose(Pab) Pbb
>
>is there a way to derive Pab (3x2) from Paa and Pbb itself? Or is
>there supposed to be another way?
>
>rdgs,
>jax

This is a completely different question! This is how I read it:

For any one sample point (say t, as it might be a point in time) you
claim to have

a1(t),a2(t),a3(t)
b1(t),b2(t)

but maybe you have not actually recorded them - in any case you have
values for

Cov(ai,aj) i,j=1,2,3

and

Cov(bi,bj) i,j=1,2

Can you calculate

Cov(ai,bj) i=1,2,3 j=1,2

based on just these values?

NO! You need the original dataset. Any positive semi-definite
symmetric matrix could be the covariance matrix, so fixing the corners
will not give you the other entries.

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