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Abstract:
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Although the Gaussian process has been broadly used for flexible nonparametric modelling, it has a structural limitation that only one kernel function defines the whole process, i.e., the process variability is homogeneous. This drawback prevents such models from reflecting the change of smoothness or dependency of the underlying function to the outputs. The heteroscedastic Gaussian process regression is an attempt to address that drawback by assuming that residuals of outputs from the regression vary over inputs. We here proposed a generalized heteroscedastic Gaussian process, which relies on the Nadaraya-Watson kernel estimation for constructing a continuous covariance function to reflect errors' heteroscedasticity. The model allows us to carry out an explicit marginalization of the hidden function, resulting in the closed-form continuous covariance function. By doing so, our model is free from additional hidden variables for noise modelling and works consistently for the multivariate outputs, even if they consist of different types (either continuous or categorical), thus is more general. We demonstrate its advantages by both simulations and applications to real-data examples of regression, classification and state-space models.
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