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Left plot are draws from the prior function distribution. The other kernel parameters are set directly at initialization and are kept fixed. Its purpose is to allow a convenient formulation of the model, and is removed integrated out during prediction.
Stationary kernels can further be subdivided into isotropic and anisotropic kernels, where isotropic kernels are also invariant to rotations in the input space. GaussianProcessClassifier supports multi-class classification by performing either one-versus-rest or one-versus-one based training and prediction.
This kernel is infinitely differentiable, which implies that GPs with this kernel as covariance function have mean square derivatives of all orders, and are thus very smooth. The figure shows also that the model makes very confident predictions until around 1.
The main use-case of the WhiteKernel kernel is as part of a sum-kernel where it explains the noise-component of the signal.
Based on Bayes theorem, a Gaussian posterior distribution over target functions is defined, whose mean is used for prediction. While the hyperparameters chosen by optimizing LML have a considerable larger LML, they perform slightly worse according to the log-loss on test data.
They lose efficiency in high dimensional spaces — namely when the number of features exceeds a few dozens. The parameter gamma is considered to be a hyperparameter and may be optimized.
According to [RW]these irregularities can better be explained by a RationalQuadratic than an RBF kernel component, probably because it can accommodate several length-scales. Rather, a non-Gaussian likelihood corresponding to the logistic link function logit is used.
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The data consists of the monthly average atmospheric CO2 concentrations in parts per million by volume ppmv collected at the Mauna Loa Observatory in Hawaii, between and In one-versus-rest, one binary Gaussian process classifier is fitted for each class, which is trained to separate this class from the rest.
The anisotropic RBF kernel obtains slightly higher log-marginal-likelihood by assigning different length-scales to the two feature dimensions.
GaussianProcessClassifier approximates the non-Gaussian posterior with a Gaussian based on the Laplace approximation.
GPR correctly identifies the periodicity of the function to be roughly 6.
The specification of each hyperparameter is stored in the form of an instance of Hyperparameter in the respective kernel.