Re: SVD : Covariance

cov(a_j, a_k) = sum(V_ji * V_ki / w_i^2, {i, 0, M-1})
It should probably have been written like:

cov(x_j, x_k) = sum(V_ji * V_ki / w_i^2, {i, 0, M-1})

(changed a_j and a_k to x_j and x_k)

However, I use SVD on the matrix A and not (A^T A), is the formula above
still correct?

The usual regression model for your data would be b = A X + e,
where b is measured, A is known exactly, X is to be estimated,
and the elements of e are independent identically-distributed
zero-mean random variables.

Yes, that would be the normal model. However, in this case, I'm more or less solving the total least squares (TLS) problem, and the errors are associated with A and b (in your version, Ax = b; in my version, only errors on A, since b_yours = A_mine^T b_mine).


The usual estimate of X is x = (A^T A)^-1 A^T b = V W^-1 U^T b.
The covariance matrix of x is to proportional to (A^T A)^-1 =
V W^-2 V^T . The constant of proportionality is the variance of
the sampling distribution of e.

The usual estimate of the constant of proportionality is
f = d^T d /(n-k), where d = b - A x and A had n rows and
k columns. With real-world data, f will usually be bigger than
the variance of the measurement errors unless the model is
exactly correct, which is rarely the case.

Do you have a reference on all this! I could really use somewhere to read about it.



Best regards

Preben
.

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