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 \(n \rightarrow \infty\) of a random walk of length \(n\) ( p213). An interesting variant is the brownian bridge obtained by conditioning \(X(t_0)=0\) ( 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.

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