# Interpolate covariance matrix

I have measurements $z_i$ and associated covariance matrices $R_i$ separated in time by some sampling interval, and I want to interpolate between measurements.

For example, I have a measurement at time $t_1=1$: $$z_1=\begin{pmatrix}8794 \\ 4013 \\ 2998 \end{pmatrix}, R_1=\begin{pmatrix}77 & -60 & -45 \\ -60 & 103 & -36 \\ -45 & -36 & 125 \end{pmatrix}$$ and another measurement at time $t_2=2$: $$z_2=\begin{pmatrix}8786 \\ 4043 \\ 2957 \end{pmatrix}, R_2=\begin{pmatrix}82 & -59 & -43 \\ -59 & 97 & -41 \\ -43 & -41 & 120 \end{pmatrix}$$ I want to find $\tilde{z}$ and $\tilde{R}$ at time $t=1.5$. It's simple enough to compute $\tilde{z}$ using linear interpolation. My question is how do I interpolate the covariance matrices to get $\tilde{R}$?

• Sorry, I should have clarified that although the sensor uncertainties are known, I don't have access to those values; I'm given only the measurements and the covariance matrix for each measurement. I know the standard formula $cov(x) = \frac{1}{n-1} \sum (x_i - \bar{x})(x_i - \bar{x})^T$ to find the covariance of the entire data set, but I don't know how to go about computing the covariance for each interpolated point. Is this not possible to do without knowing the underlying sensor uncertainties? Thanks for your help! – Sheeze Jun 13 '18 at 0:42