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Science Forum Index » Statistics - Math Forum » Restore distribution from noisy observations
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| Guest |
 Posted: Sun Apr 06, 2008 3:27 am |
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I have set of pairs (y[i], sigma[i]), where
y[i] = x[i] + sigma[i] * noise[i]
noise ~ N(0,1)
x ~ rho
How to restore rho? |
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| aruzinsky |
| Posted: Sun Apr 06, 2008 12:58 pm |
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Guest
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On Apr 6, 7:27 am, npisa...@gmail.com wrote:
Quote: I have set of pairs (y[i], sigma[i]), where
y[i] = x[i] + sigma[i] * noise[i]
noise ~ N(0,1)
x ~ rho
How to restore rho?
Is this a homework problem? A weighted average of y[i] is the minimum
variance estimate of x. Derive the optimum weights, Wi, by using the
usual method involving setting derivatives d(variance of weighted
average of y[i])/dWi equal to zero subject to sum(Wi) = 1. The answer
is simple, but you should derive it yourself. |
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| Guest |
| Posted: Sun Apr 06, 2008 5:57 pm |
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On 7 ÁÐÒ, 01:58, aruzinsky <aruzin...@general-cathexis.com> wrote:
Quote: On Apr 6, 7:27 am, npisa...@gmail.com wrote:
I have set of pairs (y[i], sigma[i]), where
y[i] = x[i] + sigma[i] * noise[i]
noise ~ N(0,1)
x ~ rho
How to restore rho?
Is this a homework problem?
No, it is not. It is something I really need to solve. And your
solution does not work.
Quote: A weighted average of y[i] is the minimum
variance estimate of x.
First it is not. Suppose that all x=y=5, sigma=0. Then any weighted
average of y[i] is 5. But a variance must be 0.
Second, rho is not normal distribution, for example it may be (and
oftenly is) bimodal. So the problem is not as simple as estimating
mean and variance.
Quote: Derive the optimum weights, Wi, by using the
usual method involving setting derivatives d(variance of weighted
average of y[i])/dWi equal to zero subject to sum(Wi) = 1. The answer
is simple, but you should derive it yourself. |
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| Guest |
| Posted: Sun Apr 06, 2008 6:41 pm |
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On 7 ÁÐÒ, 01:58, aruzinsky <aruzin...@general-cathexis.com> wrote:
Quote: On Apr 6, 7:27 am, npisa...@gmail.com wrote:
I have set of pairs (y[i], sigma[i]), where
y[i] = x[i] + sigma[i] * noise[i]
noise ~ N(0,1)
x ~ rho
How to restore rho?
Is this a homework problem? A weighted average of y[i] is the minimum
variance estimate of x.
There is no x. There are x[1], x[2], ... , x[n], which are
distributed
as Rho(x). Rho is a distribution which I need to restore. |
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| aruzinsky |
| Posted: Mon Apr 07, 2008 5:53 am |
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Guest
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On Apr 6, 10:41 pm, npisa...@gmail.com wrote:
Quote: On 7 ÁÐÒ, 01:58, aruzinsky <aruzin...@general-cathexis.com> wrote:
On Apr 6, 7:27 am, npisa...@gmail.com wrote:
I have set of pairs (y[i], sigma[i]), where
y[i] = x[i] + sigma[i] * noise[i]
noise ~ N(0,1)
x ~ rho
How to restore rho?
Is this a homework problem? A weighted average of y[i] is the minimum
variance estimate of x.
There is no x. There are x[1], x[2], ... , x[n], which are
distributed
as Rho(x). Rho is a distribution which I need to restore.
I thought x was a constant. Is "Rho" some standard designation of
which I am unaware?
I also assumed sigma[i] was known and variable with i. Is it? |
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| Herman Rubin |
| Posted: Mon Apr 07, 2008 11:07 am |
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Guest
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In article <d77a3113-2451-486c-84ae-456fb4c56919@f63g2000hsf.googlegroups.com>,
<npisarev@gmail.com> wrote:
Quote: On 7 =C1=D0=D2, 01:58, aruzinsky <aruzin...@general-cathexis.com> wrote:
On Apr 6, 7:27 am, npisa...@gmail.com wrote:
I have set of pairs (y[i], sigma[i]), where
y[i] =3D x[i] + sigma[i] * noise[i]
noise ~ N(0,1)
x ~ rho
How to restore rho?
Is this a homework problem? A weighted average of y[i] is the minimum
variance estimate of x.
There is no x. There are x[1], x[2], ... , x[n], which are
distributed
as Rho(x). Rho is a distribution which I need to restore.
Both density estimate and deconvolution are difficult.
If you had no noise, you would still have the problem
of estimating rho. This is an infinite-parametric
(usually miscalled "non-parametric"); about the best
I can suggest if you have SOME idea about rho is
to use some sort of prior Bayes procedure. A prior
Bayes procedure does not have to compute a posterior
distribution, which has not been done for reasonable
priors (from the standpoint of the generation of the
data, NOT from that of computation).
The noise adds another problem, and there are ways
around it. However, do not look for easy algorithms;
they do not exist.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 |
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| Guest |
| Posted: Mon Apr 07, 2008 12:05 pm |
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Quote: I thought x was a constant. Is "Rho" some standard designation of
which I am unaware?
No, Rho, as I wrote, is a probability density.
Quote: I also assumed sigma[i] was known and variable with i. Is it?
Yes, it is. |
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