processregression

Process regression is a framework in statistics and machine learning for modeling relationships where the data are stochastic processes, such as time-indexed or spatially indexed observations, rather than independent scalar points. The aim is to infer how a response process relates to one or more predictor processes and to predict future or unobserved trajectories while accounting for temporal, spatial, and stochastic structure.

The most widely used approach within process regression is Gaussian process regression, where the unknown regression function is assumed to be a sample from a Gaussian process. This provides a flexible, nonparametric way to model smooth functions with uncertainty. Other important methods include state-space or Kalman filter models for linear dynamical systems with Gaussian noise, which capture evolving latent states and observation processes. Functional regression extends regression to functional data, where either predictors or responses are curves or functions measured over a continuum, often using basis expansions and dimensionality reduction.

Estimation in process regression typically combines prior assumptions about smoothness or dynamics with observed trajectories. In Gaussian process settings, hyperparameters are learned by maximizing the marginal likelihood or via Bayesian inference, yielding posterior predictive distributions for new trajectories. Computations can be intensive, especially with large datasets or high-dimensional processes, and approximations or sparse representations are common remedies.

Applications of process regression appear in time-series forecasting, environmental and geophysical modelling, finance, engineering, and any domain where objects of interest are naturally described as functions over time or space. It emphasizes uncertainty quantification and the explicit incorporation of temporal or spatial correlation, offering a principled framework for learning from sequential data. See also Gaussian process regression, functional data analysis, and state-space models.