How best to interpolate covariance matrices?

Question

What is the proper way to interpolate covariance matrices? I have a spatial 2d grid and know covariance between sparse points on that grid, but want to extrapolate that that to produce a covariance matrix that covers the whole mesh. I am assuming that covariance is inversely proportional to inter-point distance. For amplitudes I can apply inverse-distance weighted interpolation, but what about for the covariance? Should I first interpolate the amplitudes then compute the covariance, or interpolate the covariance directly. If the second, how?

Thank you.

Answer 1

In the absence of any further information about the situation I would probably use inverse distance weighted average of correlations.

Answer 2

Consider a function, \begin{eqnarray}\sigma^{'}=\begin{bmatrix}\sigma_{1}^{'}\\\sigma_{2}^{'}\end{bmatrix}=\sigma^{'}(\sigma_{1},\sigma_{2})\end{eqnarray} Let us suppose that $\sigma^{'}_{1}$ and $\sigma^{'}_{2}$ are in between $\sigma_{1}$ and $\sigma_{2}$. Then by linear-linear interpolation \begin{eqnarray}\sigma_{1}^{'}=\sigma_{1}\left(\frac{E_{2}-E_{1}^{'}}{E_{2}-E_{1}}\right)+\sigma_{2}\left(\frac{E_{1}^{'}-E_{1}}{E_{2}-E_{1}}\right)\end{eqnarray} and \begin{eqnarray}\sigma_{2}^{'}=\sigma_{1}\left(\frac{E_{2}-E_{2}^{'}}{E_{2}-E_{1}}\right)+\sigma_{2}\left(\frac{E_{2}^{'}-E_{1}}{E_{2}-E_{1}}\right)\end{eqnarray} Above two equations represent the linear-linear interpolation formula. According to Taylor series expansion, covariance error matrix $V_{\sigma^{'}}$ for the function $\sigma^{'}$ is given by \begin{eqnarray}V_{\sigma^{'}}=\begin{bmatrix}\frac{\partial \sigma^{'}_{1}}{\partial \sigma_{1}}&\frac{\partial \sigma^{'}_{1}}{\partial \sigma_{2}}\\ \frac{\partial \sigma^{'}_{2}}{\partial \sigma_{1}}&\frac{\partial \sigma^{'}_{2}}{\partial \sigma_{2}} \end{bmatrix}\begin{bmatrix}var{\sigma_{1}} & cov(\sigma_{1}, \sigma _{2})\\cov(\sigma_{2}, \sigma _{1}) & var{\sigma_{2}} \end{bmatrix}\begin{bmatrix}\frac{\partial \sigma^{'}_{1}}{\partial \sigma_{1}} & \frac{\partial \sigma^{'}_{2}}{\partial \sigma_{1}}\\ \frac{\partial \sigma^{'}_{1}}{\partial \sigma_{2}} & \frac{\partial \sigma^{'}_{2}}{\partial \sigma_{2}} \end{bmatrix}\end{eqnarray} This is how the covariance matrix of a function $\sigma^{'}$ is linear-linearly interpolated. Note that, in order to interpolate the given covariance matrix $\left(\text{for eg.} \begin{bmatrix}var{\sigma_{1}} & cov(\sigma_{1}, \sigma _{2})\\cov(\sigma_{2}, \sigma _{1}) & var{\sigma_{2}} \end{bmatrix}\right)$, we have to know the relation between the interpolated values (for eg. $\sigma^{'}_{1},\sigma^{'}_{2}$) and the given values (for eg. $\sigma_{1},\sigma_{2}$) and the relation between the two is given by the linear-linear interpolation formula.

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