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From: Luna Moon on 14 Jul 2010 10:36
Hi all,

Let's say we have M x N matrix, which represents N time series, each
having
M observations in order of time.

How do we find maximal number of linear combinations of these N time
series,
their mutual correlation has to be less than certain pre-specified
constraints.

That's to say, we would like to find as many combinations of the N
time
series as possible, such that their mutual correlation remains below a
bound.

Our understanding is that with the help from PCA, we will be able to
find
probably N such combinations, expressed in the form of eigenvectors,
such
that the N resultant newly constructed time series have 0 correlation
(orthogonal).

But we now want to relax the problem from 0 correlation to within a
certain
bound.

Your thoughts and pointers are highly appreciated. Thank you!
From: Paige Miller on 14 Jul 2010 11:25
On Jul 14, 10:36 am, Luna Moon <lunamoonm...(a)gmail.com> wrote:
> Hi all,
>
> Let's say we have M x N matrix, which represents N time series, each
> having
> M observations in order of time.
>
> How do we find maximal number of linear combinations of these N time
> series,
> their mutual correlation has to be less than certain pre-specified
> constraints.
>
> That's to say, we would like to find as many combinations of the N
> time
> series as possible, such that their mutual correlation remains below a
> bound.
>
> Our understanding is that with the help from PCA, we will be able to
> find
> probably N such combinations, expressed in the form of eigenvectors,
> such
> that the N resultant newly constructed time series have 0 correlation
> (orthogonal).
>
> But we now want to relax the problem from 0 correlation to within a
> certain
> bound.
>
> Your thoughts and pointers are highly appreciated. Thank you!

This really doesn't fit into a PCA framework, and I also think the
problem has an infinite number of solutions.

Since the PCA eigenvectors represent linear combinations of variables
that have zero correlation with other PCA eigenvectors, simply
changing slightly one coefficient in an eigenvector will most likely
give you a new variable that has very slight correlation with the
other eigenvectors. It seems to me you can do this an infinite number
of times, and I can't imagine a way to “search” for all such
combinations.

--
Paige Miller
paige\dot\miller \at\ kodak\dot\com
From: Luna Moon on 14 Jul 2010 12:41
On Jul 14, 11:25 am, Paige Miller <paige.mil...(a)kodak.com> wrote:
> On Jul 14, 10:36 am, Luna Moon <lunamoonm...(a)gmail.com> wrote:
>
>
>
> > Hi all,
>
> > Let's say we have M x N matrix, which represents N time series, each
> > having
> > M observations in order of time.
>
> > How do we find maximal number of linear combinations of these N time
> > series,
> > their mutual correlation has to be less than certain pre-specified
> > constraints.
>
> > That's to say, we would like to find as many combinations of the N
> > time
> > series as possible, such that their mutual correlation remains below a
> > bound.
>
> > Our understanding is that with the help from PCA, we will be able to
> > find
> > probably N such combinations, expressed in the form of eigenvectors,
> > such
> > that the N resultant newly constructed time series have 0 correlation
> > (orthogonal).
>
> > But we now want to relax the problem from 0 correlation to within a
> > certain
> > bound.
>
> > Your thoughts and pointers are highly appreciated. Thank you!
>
> This really doesn't fit into a PCA framework, and I also think the
> problem has an infinite number of solutions.
>
> Since the PCA eigenvectors represent linear combinations of variables
> that have zero correlation with other PCA eigenvectors, simply
> changing slightly one coefficient in an eigenvector will most likely
> give you a new variable that has very slight correlation with the
> other eigenvectors. It seems to me you can do this an infinite number
> of times, and I can't imagine a way to “search” for all such
> combinations.
>
> --
> Paige Miller
> paige\dot\miller \at\ kodak\dot\com

yeah, but the rotated new vector will have a high correlation with the
old one.

so in this case you are only obtaining N eigenvectors (N being the
dimension of PCA)

but we relax the correlation constraint because we want more
combinations...

Thank you!
From: Paige Miller on 14 Jul 2010 13:43
On Jul 14, 12:41 pm, Luna Moon <lunamoonm...(a)gmail.com> wrote:

> yeah, but the rotated new vector will have a high correlation with the
> old one.

You drop the old (unrotated) vector from the set, you keep the new
(rotated) vector, and you now have N vectors with low correlation with
each other.

Is this not what you want?

--
Paige Miller
paige\dot\miller \at\ kodak\dot\com
From: Luna Moon on 14 Jul 2010 13:57
On Jul 14, 1:43 pm, Paige Miller <paige.mil...(a)kodak.com> wrote:
> On Jul 14, 12:41 pm, Luna Moon <lunamoonm...(a)gmail.com> wrote:
>
> > yeah, but the rotated new vector will have a high correlation with the
> > old one.
>
> You drop the old (unrotated) vector from the set, you keep the new
> (rotated) vector, and you now have N vectors with low correlation with
> each other.
>
> Is this not what you want?
>
> --
> Paige Miller
> paige\dot\miller \at\ kodak\dot\com

not at all,
we want a set of linear combinations of the old variables,
and we want this set to be as large as possible, of course constrained
by the the maximal allowed mutual correlation...
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