I'm working with Gaussian process regression (GPR) and have a question about the behavior of the mean function when using the exponentiated quadratic kernel (or RBF) kernel.
RBF Formula:
$K(x, y) = \exp\left(-\frac{1}{2}\frac{{\|x - y\|}^2}{l^2}\right)$
I've noticed that the mean function in GPR, when using the RBF kernel, tends to continue the slope from the observed data points when making predictions in regions without data. In other words, if the slope of the function was increasing at the last observed data point, the mean function will continue to increase in the region beyond that point, and vice versa if the slope was decreasing.
Look at this graph for example, where mean line increased after last data point Example Graph
I've also noticed that this is not the case when we are using Exponential Kernel:
$K(x, y) = \exp\left(-\frac{{\|x - y\|}}{l}\right)$
could someone please explain this behavior ? why it follows the slope ?