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From: Ray Koopman on 14 Jul 2010 13:57
On Jul 14, 7:36 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!

Subtract the column means, so that each column has a mean of zero,
then get *any* orthonormal basis for the column space (by SVD, QR,
Gram-Schmidt, ...).
From: Luna Moon on 14 Jul 2010 14:10
On Jul 14, 1:57 pm, Ray Koopman <koop...(a)sfu.ca> wrote:
> On Jul 14, 7:36 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!
>
> Subtract the column means, so that each column has a mean of zero,
> then get *any* orthonormal basis for the column space (by SVD, QR,
> Gram-Schmidt, ...).

As mentioned, complete orthonormality is not what are after. We want
the set to be as large as possible. There can be more than N vectors
if we allow non-orthognormal
From: Ray Koopman on 14 Jul 2010 14:30
On Jul 14, 11:10 am, Luna Moon <lunamoonm...(a)gmail.com> wrote:
> On Jul 14, 1:57 pm, Ray Koopman <koop...(a)sfu.ca> wrote:
>> On Jul 14, 7:36 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!
>>
>> Subtract the column means, so that each column has a mean of zero,
>> then get *any* orthonormal basis for the column space (by SVD, QR,
>> Gram-Schmidt, ...).
>
> As mentioned, complete orthonormality is not what are after. We want
> the set to be as large as possible. There can be more than N vectors
> if we allow non-orthognormal

Is this related to the sci.math thread "putting points on a unit
shpere?"
From: Paige Miller on 14 Jul 2010 14:41
On Jul 14, 1:57 pm, Luna Moon <lunamoonm...(a)gmail.com> wrote:
> On Jul 14, 1:43 pm, Paige Miller <paige.mil...(a)kodak.com> wrote:
> 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...

I know of no such method.

--
Paige Miller
paige\dot\miller \at\ kodak\dot\com
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