Covariance Matrix of various $x,y,z$ (cartesian coordinates)

Question

I have around 1000 values of gps receiver positions as follows: I have to calculate the covariance matrix with all these values.

All of the following values represent a SINGLE POINT.

How can I get a covariance matrix?

EDIT1:

What I have tried so far?

I know the formulae

Then I decided to find $\operatorname{Var}(Y)$ and $\operatorname{Cov}(X,Y)$

I know $\operatorname{Cov}(X,Y)$ formulae is this:

and $\operatorname{Var}(Y)$ I am trying to find using the following formulae:

Do you think this is the correct way to go? Do I need additional values to compute this.

Answer 1

What you wrote is the covariance matrix of the random variables $X$, $Y$ and $Z$, where the random variables $X$, $Y$ and $Z$ represent the stochastic processes that can produce the data you have collected. In your case they are the 3 coordinate positions recorded by the gps. The finite amount of data you have stored is a sampling of the population of all possible outcomes of the random variables $X$, $Y$ and $Z$.

As you are dealing with a sampling (of about 1000 observations per coordinate/ random variable), then you should consider the sample covariance matrix, instead.

The number $N$ is the number of coordinate observations: in your case about 1000. Any statistical software is able to compute the sample covariance matrix.