How would you use Gaussian Processes in time series forecasting?

Official answer

A Gaussian process places a joint Gaussian prior on function values at any finite set of inputs. For a time series, specify a mean function, a covariance kernel over time (and justified covariates), and an observation-noise model. Conditioning on the historical observations gives a predictive distribution under those assumptions.

Choose kernels to express plausible structure: a smooth or Matérn component for local variation, a periodic component for repeating behavior, or combinations for trends and evolving cycles. Estimate hyperparameters from the training period, often through log marginal likelihood, and check sensitivity to initialization and assumptions. An RBF kernel alone does not promise useful long-range extrapolation.

Distinguish uncertainty about the latent function from uncertainty in a future noisy observation. If latent predictive variance is 4 and independent observation-noise variance is 9, future-observation variance is 13. A library's returned uncertainty must be interpreted according to how noise was specified; do not label every returned interval a calibrated prediction interval.

Use rolling forecast origins and the intended horizon to compare with simpler baselines. Check coverage and width as well as point error. Standard exact dense GP regression has O(n³) factorization cost and O(n²) storage, so long histories may need an approximation or exploitable kernel structure, with its error assessed.

Counts may require a suitable likelihood or transformation rather than uncritically accepting negative Gaussian predictions. Future covariates must be known or forecast with their uncertainty considered. A flexible kernel and an interval are model outputs, not guarantees of accuracy or calibration after a regime change.