# 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 $\tilde{R}$?

Many thanks in advance!

• Could you explain how a single measurement might produce an entire covariance matrix? How are these matrices estimated and what would be the purpose or intended interpretation of the interpolated values? – whuber Jun 12 '18 at 19:05
• These measurements are from a sensor tracking the position of an object in 3D space. The sensor has known uncertainties. The measurement uncertainties increase as the object moves further away from the sensor. The purpose of interpolating the measurements is to synchronize them in time with measurements from other sources so that all the measurements can be fused together. – Sheeze Jun 12 '18 at 20:13
• Thank you. That raises many possibilities that are not apparent in the post itself. It shows you don't have a single two-point interpolation problem: you are interpolating an entire series of data. Second, if the measurement uncertainties (in the form of covariance matrices) are truly known, then why not just use the known formulas to compute them for the interpolated locations? – whuber Jun 12 '18 at 20:31
• 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