# Determine noise distribution [closed]

I'm trying to solve the following least squares problem:

$\underset{x}{\text{min}} ||Ax - \tilde{b}||_2$

where $Ax = b$ and $\tilde{b} = b + w$

### Question:

How do I determine which probability distribution fits $w$ best?

Also, $A \in \mathbb{R}^{n\times n}$ is a large, and sparse Toeplitz matrix, $\tilde{b} \in \mathbb{R}^{n \times 1}$

$w$ is not your typical measurement noise and is probably not Gaussian. Also, I have access to $b$ (training data) and therefore, $w$.

Also, in a related question, I'm using LSQR to determine $x$ --- since I have my training data, is there any way to do cross-validation to determine how to best solve for $x$?

For example, I tried Tikhonov regularization to find the best $\lambda$ but I'm looking for a better method.

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## closed as off-topic by Ricardo Andrade, Chris Godsil, Stefan Kohl, Andrey Rekalo, Ryan BudneyApr 14 '14 at 16:22

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Ricardo Andrade, Chris Godsil, Stefan Kohl, Andrey Rekalo, Ryan Budney
If this question can be reworded to fit the rules in the help center, please edit the question.

I think you should reformulate your question. it seems like a poorly formulated homework question. – robin girard Jul 28 '10 at 17:16
@Robin: It's not homework. Which part is poorly formulated? – Jacob Jul 28 '10 at 17:27
To start with, you could improve the question : "Given a vector w, how do I determine which distribution fits it best?" distribution of what ? what is "it". – robin girard Jul 28 '10 at 18:16

## 1 Answer

There's no way to answer in general what distribution data has. It has whatever distribution it has, but what you really want to know is whether a distribution which someone has identified adequately fits your data. So you have to propose a specific distribution first, then test for goodness of fit. Searching on "goodness of fit" will give you lots of resources, but a good general-purposed goodness of fit test is Kolmogorov-Smirnov, often abbreviated K-S.

You could look at a histogram of your data to start your search. Is it symmetric or skewed? Long tails or short tails? Unimodal or bimodal? If you post an image, someone may be able to suggest some distributions to test.

Your question may be more appropriate for the Statistical Analysis Stack Exchange site.

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@John: Thank you! "goodness of fit" seems to be a good place to start. – Jacob Jul 28 '10 at 18:09
@jacob: another good place to start is trying to improve the way you formulate your question. – robin girard Jul 28 '10 at 18:18
@Robin: Thank you for the suggestion, how does it look now? – Jacob Aug 6 '10 at 15:55