Heavy-tailed Process Regression Model For Dependent Response Variables

Gaussian process regression(GPR)is an important method in the field of func-tional data analysis.GPR has received much attention due to many virtues such as sim-plicity of implementation,strong interpretability,and statistical asymptotic properties.Extended t-process regression(eTPR)is another model for functional data regression,and inherit almost all advantages from GPR.Extended t-process regression is based on heavy-tailed extended multivariate t-distribction,thus proven to be more robust against data with outliers than GPR.In real life,multivariate functional responses are usually rather dependent with each other,whereas GPR does not consider the dependence within those response variables,which wastes much valuable information and fails to perform well,especially on sparse data.Dependent Gaussian process regression(DGPR)con-structs Gaussian processes with dependence by convolving white noise sources with kernels.DGPR uses covariance matrix to depict the dependent structure in the data,and manage to give more accurate predictions.Dependent t-process regression(DTPR)is introduced to model functional data with multivariate functional responses,where a dependent t-process(DTP)via bringing in a random effect and a convolution integration is constructed to build the correlated structure of multivariate responses.It shows that DTPR has more robustness against to outliers compared to Gaussian process regression,while it possesses the interpretability as GPR.Robustness and information consistency of the proposed prediction are stud-ied.Numerical studies including simulation and real examples show that the proposed DTPR performs well.