2
$\begingroup$

A Gaussian process gives a probability distribution over functions that pass through the data points. Is there a way to parameterize the Gaussian process to give a probability distribution over closed curves instead of functions? Can the same approach be generalized to give a distribution over surfaces without boundary?

$\endgroup$
3
$\begingroup$

If you take 2 Gaussian processes, $X(t)$ and $Y(t)$ on the unit interval, and with the property that $X(0)=X(1)$ almost surely (similarly for $Y$), then $(X(t),Y(t))$ will be a random curve in $R^2$.

Is that what you are looking for?

$\endgroup$
  • $\begingroup$ How would you construct a process like this? Would you just condition on the process going through those points? (Would be great to know if there is another way of doing this.) $\endgroup$ – user27182 Jan 6 '18 at 17:02
  • 1
    $\begingroup$ A simple example is the Brownian bridge with covariance kernel $k(x,y)=min(x,y)-xy$. You can also use Green's Functions to build models with boundary constraints. See page 134 of Rasmussen and Williams "Gaussian Processes for Machine Learning". It's available online. $\endgroup$ – Placidia Jan 9 '18 at 2:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.