Manifold Extended T-Process Regression

Once data are available,it is necessary to investigate the relationship between different variables.Regression is one such method of determining the relationship between certain variables.There are usually parametric and non-parametric regressions.Most of the regressions mentioned in this article are non-parametric regressions.Non-parametric regression has the following advantages over parametric regression:firstly,non-parametric regression usually has a freer form than parametric regression,and is not constrained by the form in parametric regression,and also does not require the distribution of the data.Thirdly,for non-linear data with unknown distribution,it usually shows more accurate prediction ability than parametric regression.Gaussian process regression is a widely used non-parametric regression method.It has the advantage of being flexible and convenient in that one can apply different kernel functions to different data.Since its introduction,Gaussian process regression has been extended by scholars.Firstly,Gaussian process regression is not robust and sometimes over-fits when facing outliers in the data;secondly,the traditional Gaussian process regression model cannot be applied to vector spaces such as manifold spaces,and the model may not perform well if the data comes from manifold spaces.Wang combines the Gaussian process with the inverse gamma distribution and extends the Gaussian process to the t-process,proposing an extended t-process regression,which is a robust non-parametric regression method that retains most of the advantages of the Gaussian process regression and has good robustness.In order to further extend the use of regression to the manifold space,this paper combines the robust regression method of extended t-process regression with the manifold method,and proposes manifold extended t-process regression.It improves the robustness to outliers and extends the space of use to the manifold space.A manifold extended t-process regression model(meTPR)is developed to fit function data with a complicated input space.A manifold method is used to transform covariate data from input space into a feature space,and then an extended t-process regression is used to map feature from feature space into observation space.An estimation procedures is constructed to compute parameters in the model.Numerical studies are investigated by both synthetic data and real data,and show that the proposed meTPR performs well.