Hyper-parameters Sample Clauses

Hyper-parameters. ‌ The parameters that are present in the kernel of the covariance are called hyperparame- ters, which is used to tune the predicted function form or regularise the function inference process. Two hyperparameters {σcov, σerror} are involved in the covariance (Eq.4.14) 3 y 2 1 0 5 4 3 2 y 1 0 2 3 x Figure 4.2: One-dimensional Gaussian Process with the black dots mark the data points. The solid curves marks the regressed mean function with the error bars indi- cating the prediction variance from the GP process. σerr =0.05 eV/˚A and two different σcov= 0.5 (upper) and1.5 (lower) respectively in the plot. used in this thesis. A demonstration of the inference for a one-dimensional function is given in Figs.4.2 and 4.3, with several different prior hyperparameters {σerror, σcov}. The σcov is significant in controlling the correlation length of any two data points along the distance scale. Smaller σcov makes the prediction more accurate locally near the data points, but increases the uncertainty in the longer extrapolation regime, in other words, decreases the weight of distant data points in the prediction. The larger σcov tends to put less weight on the local data points and usually yields a more general predictive form in the long range of data space. 3 y 2 1 0 4 3 2 y 1 0 1 x Figure 4.3: Two different noise assumed for the Gaussian processes, σerror of 0.05 (upper) and 1.0 (lower) eV/˚A. The other hyperparameter σcov were kept constant for the two cases as 1.0 The noise assumed on the data is indicated by σerror. Small values correspond to the rigid parameterisation with over fitting to the data points, which however, lack of the extrapolation capability beyond the data points. Larger σerror enables better extrapola- tions at a risk of losing the accuracy at the fitting data (with ‘blurring’ around the data of the magnitude of σerror) and predicted function takes simple form rather than cross the accurate data points. In the case of high-accuracy required calculations, suitable hyperparameters are key and thus optimisation procedures are often needed, such as by the maximising of the marginal likelihood to be discussed below.