Wiener kernel

The Wiener kernel defines a non-stationary gaussian process, the Wiener process:

\begin{equation*}
\displaystyle
K(x, x') = \min(x, x')
\end{equation*}

It is convenient to model time-series. It is the limit as $n \to \infty$ of a random walk of length $n$ ((rasmussen2003) p213). An interesting variant is the brownian bridge obtained by conditioning $X (t_{0}) = 0$ ((grimmett2020) p534).

The parameter $l$ is the characteristic lengthscale of the process. As one can see on the figure below, the larger the value of $l$ the “further” the kernel takes non-negligible values.

file:

Wiener kernel

Backlinks