- #1
Niles
- 1,866
- 0
Hi
Say I have a linear problem Mp = d, where d is our data, p our parameters and M our "observation matrix" (see http://en.wikipedia.org/wiki/Inverse_problem#Linear_inverse_problems). So what we are dealing with is an overdetermined problem.
Now, I have read an example where we have a vector of data d whose standard deviation we don't know. Then we try and estimate it, and the estimate is given by
[tex]
\sigma ^2 _{estimate} = \frac{1}{N}\sum\limits_i {\left( {d_i - \left( {Mp} \right)_i } \right)^2 }
[/tex]
My question is: How can they estimate the standard deviation like this? Usually we would use the mean, but they use (Mp)i, and I can't quite see why this yields an estimate.
Say I have a linear problem Mp = d, where d is our data, p our parameters and M our "observation matrix" (see http://en.wikipedia.org/wiki/Inverse_problem#Linear_inverse_problems). So what we are dealing with is an overdetermined problem.
Now, I have read an example where we have a vector of data d whose standard deviation we don't know. Then we try and estimate it, and the estimate is given by
[tex]
\sigma ^2 _{estimate} = \frac{1}{N}\sum\limits_i {\left( {d_i - \left( {Mp} \right)_i } \right)^2 }
[/tex]
My question is: How can they estimate the standard deviation like this? Usually we would use the mean, but they use (Mp)i, and I can't quite see why this yields an estimate.