Error Propagation: Calc Errors w/ Variance Covariance Matrix

[0xDEADBEEF]

I am confused about calculating errors. I have learned if you take the variance covariance matrix $\Sigma_{ij}$ of a fit of function f(x,p) to data for parameters $p_i$ (for example by using Levenberg-Marquart) that the one sigma error interval for $p_i$ is $$\sigma_{p_i}=\sqrt{\Sigma_{ii}}$$ I only understand this, if there are no covariance terms. Why do we do this? I would have thought a better way to find the error would be to do diagonalize $\Sigma$, say the diagonal form is $\Xi$ with normalized eigenvectors $(\vec{v})_k$. Then we would have independent variables that have a Gaussian distribution and one can calculate the error on $p_i$ using error propagation, i.e. $$\sigma_{p_i} = \sqrt{\sum \Xi_{kk}\left\langle(\vec{v})_k\mid l_i \right\rangle}$$ where $\left\langle(\vec{v})_k\mid l_i \right\rangle$ is the $i^\text{th}$ component of $(\vec{v})_k$. If this is permissible, is there a name for it?

 

[Stephen Tashi]

I am confused about calculating errors. I have learned if you take the variance covariance matrix $\Sigma_{ij}$ of a fit of function f(x,p) to data for parameters $p_i$ (for example by using Levenberg-Marquart) that the one sigma error interval for $p_i$ is $$\sigma_{p_i}=\sqrt{\Sigma_{ii}}$$

It is rather confusing how any process can purport to calculate a standard deviation for the paramters of a fit y = f(x,p) in the case when the data is of the form $(x_i,y_i)$. There is no random sample of the parameters. How can any variation be assigned to them? My best guess is in post #7 of the thread: https://www.physicsforums.com/showthread.php?t=645291&highlight=parameters

I'm not sure what you mean by "the variance covariance matrix $\Sigma_{i,j}$ of a fit of the function f(x,p) to the data for parameters $p_i$". What is the definition of that matrix?

 

[0xDEADBEEF]

Well I guess that you know the theory better than I do, but the idea is somehow a correspondence between least squares and maximum likelihood.
So you have the sum of the squares of a fit function $f(x,p_1,p_2,\dots)$ to data $x_i,y_i$

$$sq(p_1,p_2,\dots) = \sum_i (f(x_i,p_1,p_2,\dots)-y_i)^2$$

And the residuals
$$r_i=f(x_i,p_1,p_2,\dots)-y_i$$

for some optimal set of parameters $p_k$ that minimizes sq. If the residuals are gaussian then the variance of the residuals times the reciprocal of the Hessian of $sq(p_1,p_2,\dots)$ is somehow a measure of how confident one can be in the fitted parameters and it is also a variance-covariance matrix. This is how I understand it, but if I would really understand the theory I wouldn't be asking questions. Anyhow my question was why one only uses the diagonal elements of that matrix.

 

[Stephen Tashi]

Can your original question can be considered outside of the context of curve-fitting.

I would have thought a better way to find the error would be to do diagonalize $\Sigma$, say the diagonal form is $\Xi$ with normalized eigenvectors $(\vec{v})_k$. Then we would have independent variables that have a Gaussian distribution and one can calculate the error on $p_i$ using error propagation, i.e. $$\sigma_{p_i} = \sqrt{\sum \Xi_{kk}\left\langle(\vec{v})_k\mid l_i \right\rangle}$$ where $\left\langle(\vec{v})_k\mid l_i \right\rangle$ is the $i^\text{th}$ component of $(\vec{v})_k$. If this is permissible, is there a name for it?

Suppose the $p_i$ are simply a set of random variables, not necessarily having the meaning of parameters in a curve fit. If the covariance matrix is $\Sigma$, are you proposing a method to get a different estimate for each $\sigma^2_{p_i}$ than using the diagonal element $\Sigma_{i,i}$ ?

 

[0xDEADBEEF]

Exactly. Maybe the thing I am looking for already has a name. If we have a covariance matrix like this

$$\Sigma = \left( \begin{matrix} .1&100\\ 100&1000 \end{matrix} \right)$$

The first parameter is varying very little while the second one is varying a lot. But the second parameter also has a large influence on the first parameter, and it seems to me that this does not get captured if we use .1 as the variance for the first parameter. So I was suggesting to diagonalize the matrix to get independent parameters and then something like error propagation to determine the "real" uncertainty of the first parameter. I tried to make an example but I don't know how to make random numbers with a given covariance matrix.

 

[Stephen Tashi]

You could use a bivariate normal distribution and try to get the desired covariance matrix.

if you don't want to use the variance of a random variable to define its uncertainty, you'll have to state what definition for uncertainty that you want to use.

The variance of one random variable in a joint distribution, doesn't define a joint confidence interval for several variables. Perhaps you are trying to find a joint confidence interval.