Calculate uncertainty from (how to interpolate) a given covariance matrix

Question

Assume that there is a given covariance matrix of an evaluated quantity ( in my case it's a rection cross section $\sigma = f(E),ドル where $\sigma$ is the cross section and $E$ is each energy point where it was calculated) $\mathbf{V_\sigma}$

This means that I have a file with the following format

$E_1$ $\sigma_1$ $\delta\sigma_1$

$E_2$ $\sigma_2$ $\delta\sigma_2$

...

$E_m$ $\sigma_m$ $\delta\sigma_m$

and as I said the covariance matrix of $\sigma = f(E)$.

What I want to do is calculate the cross section $\sigma = f(E)$ at some energy points that are not in the file so I decided to do a linear interpolation between two evaluated points $(E_i, \sigma_i)$ and $(E_{i+1}, \sigma_{i+1})$. This means that the $\sigma$ at the desired energy $E$ will be

$\sigma = \dfrac{\sigma_i - \sigma_{i+1}}{E_i - E_{i+1}}(E-E_i)+\sigma_i$

This operation will be done total $n$-times, i.e. I would like to get the cross section at $n$ points by performing the interpolation at $n$ sets (each set consists of two data points).

The question is how to consruct the covariance matrix of the final $n$ points given the covariance matrix of the evaluated cross section?

Any idea would be more that welcome!

Answer 1

Here's a simple approach, which may be about as good, or not (but see caveat in last paragraph), as your linear interpolation.

Let $C$ be the m by m covariance matrix of $\sigma$ values at the m $E_i$ 's.

Let $A$ be the n by m matrix which produces the n interpolated points from the m $E_i$ 's. So for instance if the first interpolated point is 25% of the way from $E_1$ to $E_2,ドル the first row of $A$ would be [0.75 0.25 0 .... 0]. Given your interpolation scheme, $A$ will be a bi-diagonal matrix.

Then the covariance matrix of $\sigma$ at the interpolated points is taken as $ACA^T$.

One consequence, howwever, is that there will be non-zero off-diagonal covariances in $ACA^T,ドル even if the original covariance matrix $C$ has no non-zero off-diagonal covariances. If this is unacceptable, you may need an approach more specific to the "physics" of your problem.

Answer 2

Consider a function, enter image description here

Let us suppose that sig1p and sig2p are in between sig1 and sig2. Then by linear interpolation

enter image description here and

enter image description here

Then by Taylor series expansion, covariance error matrix for the function sigp is given by enter image description here

Vsigp can also be written as

enter image description here

This is how the covariance matrix of a function sigp is linearly interpolated.

For more details refer, https://www.researchgate.net/publication/361115402_A_note_on_the_examination_of_the_possibility_to_interpolate_the_covariance_uncertainty_matrix_in_nuclear_data_science