Applications of fractal-type kernels in Gaussian process regression and support vector machine regression

A symmetric positive semi-definite kernel defined by fractal interpolation functions is introduced and then applied to Gaussian process(GP) regression and support vector machine(SVM) regression. An example is given using the Crude Oil WTI Futures daily highest prices as the raw data to show the performance of each regression model. We choose a sample subset D & lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {D}<^>*$$\end{document} to build the models and show how well the prediction curve of each model fits the raw data. Optuna is applied in GP regression to estimate the parameters by maximizing the log marginal likelihood. In our results, the maximal log marginal likelihood of GP regression with the fractal-type kernel is larger than that of GP regressions with the Gaussian kernel and the Mat & eacute;rn kernel. On the other hand, we use Optuna to estimate the parameters in SVM regression by minimizing the mean squared error (MSE). Compared with the SVM regression models with the "rbf" kernel and the "poly" kernel in sklearn.svm.SVR, we see that SVM regression with the fractal-type kernel has the smallest MSE, and the prediction curve best fits the raw data.