Re: Summing signals with different noise variance

On Jul 29, 11:13 am, spasmous <spasm...@xxxxxxxxx> wrote:
I have 'm' time-series data sets consisting of 'n' points (m=1:10, and
t=1:1000):

S(t;m) = a(m) * true(t) + b(m) * noise(t)

where there is true signal and Gaussian noise with different scalar
multipliers a,b.

I'd like to sum the 'm' individual data sets into a single "best"
estimate of true but unfortunately I don't know a and b. If b were the
same for all m then, as far as I understand, the best estimate the
true signal (in the sense of minimizing noise in the estimate) comes
from taking the the principal component of matrix S(t;m).

From some testing without any true signal present, usually b is of the
same order of magnitude and the method above works OK. However I am
wondering if there may be an approach that can take account of
differently scaled noise. Any suggestions appreciated, thanks.

If you're willing to do a little statistical slumming, try factor
analysis. But first I would try the first principal component of
S*D, where D = sqrt(diag(inverse(S'S/T))) is a diagonal matrix in
which d(m) is an estimate of a lower bound for 1/b(m). (This is a
very old inequality, well known to factor analysts. I don't know
who discovered it.)

.