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Assume I am looking for a normal distribution $\mathcal{N}(\mu,\Sigma)$. For simplicity let's say we only have 2 random variables $x$ and $y$ and a known $\mu=0$.

Is it possible to estimate $\Sigma$ by observing the variance along multiple directions?

For example, I measure the variance $\sigma_1$ along the vector $\mathbb{v}_1 = (x_1,y_1)^T$. In another step I obtain a different variance $\sigma_2$ from a different direction $\mathbb{v}_2 = (x_2,y_2)^T$. Ideally one would continue to observe these single variations in different directions and combine them in one multivariate normal distribution.

Does this make sense?

EDIT: Some additional background information might be useful: I have a sensor device with known position and orientation in 2D space (in a future step both may have an uncertainty). The sensor is able to measure only the distance of a point along its orientation. I'm also given the sensor model. So for each distance measure $d_i$, I obtain the standard error $\sigma(d_i)$ which depends on the distance.

Since I'm not able to manipulate the sensor position to my advantage or perform a large number of measurements, I'd like to combine these variances into one covariance matrix in order to make a more reliable prediction of the position of the measured point.

This is just a thought that is still under development with no guaranty to work out correctly. Hence my question of "making sense"...

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  • $\begingroup$ Whether it "makes sense" or not may depend on the costs and constraints on your data collection process. Conceivably, it is easy to collect lots of data along individual lines but expensive to set up each line: such a circumstance would suggest an approach like this one. What is optimal, though, depends on details of the costs and the specific constraints. Could you perhaps share this kind of information with us or, more generally, explain why you are contemplating such an approach? $\endgroup$
    – whuber ♦
    Oct 10, 2013 at 19:34
  • $\begingroup$ Thanks. I updated my question. Hope this makes it more clear. $\endgroup$
    – Steffen
    Oct 11, 2013 at 12:40
  • $\begingroup$ It does not sound like computing a covariance matrix would address your problem. Instead, it seems that you may have one or more measurements of the distance of something (along with its orientation), made with a known amount of radial error, and you wish to estimate the location and quantify the uncertainty in that location. Would this be an accurate interpretation? $\endgroup$
    – whuber ♦
    Oct 11, 2013 at 13:58
  • $\begingroup$ yes, that's exactly what I'm trying to do... $\endgroup$
    – Steffen
    Oct 11, 2013 at 14:08

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You could, but it doesn't make too much sense to do it this way. Since there are ${n+1}\choose 2$ independent dimensions in a covariance matrix, you'd need to measure the variance in ${n+1}\choose 2$ different directions, and then do some linear algebra to reconstruct the covariance matrix. It's more straightforward just to measure the covariance matrix all at once.

For example, in $2$ dimensions, you could measure the variance along $(0,1)$, $(0,1)$ and $(1,1)$, then since $v_{(1,1)}=\Sigma_{11}+\Sigma_{22}+2\Sigma_{12}$, you would end up with $$\Sigma=\left(\array{v_{(1,0)} & \frac{v_{(1,1)}-v_{(1,0)}-v_{(0,1)}}{2}\\ \frac{v_{(1,1)}-v_{(1,0)}-v_{(0,1)}}{2} & v_{(0,1)}}\right)$$

I suppose if somehow you were constrained in such a way that you couldn't sample from the entire distribution, but you were only able to sample from projections onto a single dimension at a time, then this approach could be useful.

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