I'm looking through a slide deck (slide 9) about Gaussian Processes, and I came to a slide that describes one example of a covariance function: Matérn $\frac{3}{2}$ Covariance.

$$C(x_1,x_2) = (1+\sqrt{6}\frac{|x_1-x_2|}{\ell})*\exp(\sqrt{6}\frac{|x_1-x_2|}{\ell})$$

$\ell>0$ is the "correlation length parameter" and
$\sigma^2>0$ the variance parameter (though this isn't in the formula which confuses me).

Then they show a graph like this:

enter image description here

You can see that changing $\ell$ changes the shape of the function. However, my understanding is that covariance requires two random variables/vectors as inputs. So, what is $x$ (on the x axis) of the graph referring to in this case?


The horizontal-axis label there is wrong, it should be $r$, the "radius".

As an example, have a look at Rasmussen & Williams Fig. 4.1

Covariance Figure

The "radius" is essentially the distance between two multidimensional points $\bf {x_1}$ and $\bf {x_2}$, and this is usually written as:

$$k(x_1, x_2) = f(r) = f(||x_1 - x_2||_p)$$


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