How to build a Gaussian Process regression model for observations that are constrained to be positive

Question

I'm currently trying to train a GP regression model in GPflow which will predict precipitation values given some meteorological inputs. I'm using a Linear+RBF+WhiteNoise kernel, which seems appropriate given the set of predictors I'm using.

My problem at the moment is that when I get the model to predict new values, it has a tendency to predict negative precipitation - see attached figure.

How can I "enforce" physical constraints when building the model? The training data doesn't contain any negative precipitation values, but it does contain a lot of values close to zero, which I assume means the GPR model isn't learning the "precipitation must be >=0" constraint very well.

If there's a way of explicitly enforcing a constraint like this it'd be perfect, but I'm not sure how that would work. Would this require a different optimization algorithm? Or is it possible to somehow build this constraint into the kernel structure?

Answer 1

This is more of a question for CrossValidated ... A Gaussian process is essentially a distribution over functions with Gaussian marginals: the predictive distribution of f(x) at any point is by construction a Gaussian, not constrained. E.g. if you have lots of observations close to zero, your model expects that something just below zero must also be very likely.

If your observations are strictly positive, you could use a different likelihood, e.g. Exponential (gpflow.likelihoods.Exponential) or Beta (gpflow.likelihoods.Beta). Note that model.predict_y() always returns mean and variance, and for non-Gaussian likelihoods the variance may not actually be what you want. In practice, you're more likely to care about quantiles (e.g. 10%-90% confidence interval); there is an open issue on the GPflow github that relates to this. Which likelihood you use is part of your modelling choice, and depends on your data.

The simplest practical answer to your problem is to consider modelling the log-precipitation: if your original dataset is X and Y (with Y > 0 for all entries), compute logY = np.log(Y) and create your GP model e.g. using gpflow.models.GPR((X, logY), kernel). You then predict logY at test points, and can then convert it back from log-precipitation into precipitation space. (This is equivalent to a LogNormal likelihood, which isn't currently implemented in GPflow, though this would be straightforward.)