I spent the afternoon in Amsterdam talking one-on-one with people about Gaussian processes. There are a lot of time-domain projects in astrophysics that involve both stochastic and quasi-periodic variability. Aside from eclipsing binaries, almost no variables are perfectly periodic. Indeed even eclipsing binaries are complicated if the members of the binary are themselves stochastically variable. Gaussian processes are among the simplest methods for modeling stochastically varying and quasi-periodically varying objects, but the simplicity is only visible once you have confronted some conceptual blocks! It is simultaneously very simple and not obvious that you can think of your entire data set as a single draw from an enormous, multi-variate Gaussian (or even better, a finite-dimensional sampling of an infinite-dimensional one). Once you make that leap—the leap of seing not each point as being a Gaussian draw but the whole data set as being a Gaussian draw—you get a huge amount of power to describe non-trivial phenomena in the time domain.
I approve of this post.
ReplyDeleteThe last bit about how to think about Gaussian processes is very important - a function is one point in parameter space. No curve is itself a GP, a GP describes a probability distribution over the space of possible curves. Just like how 1.112 is not a standard normal distribution, even though it could plausibly have been generated from one.
I like to completely avoid the language of things being "drawn from" distributions except in the context of monte carlo simulations, where you can see the actual code. Otherwise it's a confusing concept. My state of knowledge about x might be f(x) and yours might be g(x). We observe x. x wasn't "drawn" from f or g, it just is. If anything, it was drawn from a delta function distribution.