gptp_multi_output

This toolkit is used to implement multivariate Gaussian process regression (MV-GPR) and multivariate Student-t process regression (MV-TPR).

Setup

This toolkit is based on GPML MATLAB Code http://www.gaussianprocess.org/gpml/code/matlab/doc/, version 3.6.

You have to run startup.m in GPML 3.6 first, and then run add_path.m in this toolkit.

Code Structure

The main function is gptp_general.m, which relies on four functions in solver folder.

solver

These four functions are used to solve GPR, TPR, MV-GPR and MV-TPR, respectively.

• gp_solve_gpml.m
• tp_solve_gpml.m
• mvgp_solve_gpml.m
• mvtp_solve_gpml.m

init

These functions are used to generate initial hyperparameters for the corresponding covariance function.

BE CAREFUL TO these initialisation function:

Omega_init.m, SE_init.m, and nv_init.m (in TPR and MV-TPR).

These functions play an important role in the final results, if you would like to obtain considerable results, PLEASE USE YOUR OWN FUNCTIONS according to YOUR OWN EXPERT OPINIONS using training data.

Do not forget to replace SE_init.m by the corresponding kernel initialisation function if you write a new for yourself.

util and cov

covSEiso.m, covSEard.m and sq_dist.m are collected from GPML MATLAB Code.

covSEiso.m and covSEard.m are used as default covariance function. More covariance functions can be selected in the GPML Code toolbox.

MultiGamma.m and vec2mat_diag.m are two small functions, which are used in the mvgp_solve_gpml.m and mvtp_solve_gpml.m.

sample

gptp_sample.m is to generate a sample from GP or TP with specified row and column covariance and zero mean function.

mv_gptp_sample.m is to generate a sample from MV-GP or MV-TP with specified row and column covariance and zero mean function.

example

SimulatedExample.m is a simple example for multi-output prediction (MV-GP and MV-TP) compared with independent prediction using GP and TP .

SimulatedParameter.m is a simple example for multi-output prediction (MV-GP and MV-TP) parameter estimation .

You need to modify Omega_init.m, SE_init.m, and nv_init.m to obtain the best results for parameter simulation and prediction.

Note

This code is proof-of-concept, not optimized for speed.

Reference

[1] Chen, Zexun, and Bo Wang. "How priors of initial hyperparameters affect Gaussian process regression models." Neurocomputing 275 (2018): 1702-1710.

[2] Chen, Zexun, Bo Wang, and Alexander N. Gorban. "Multivariate Gaussian and Student $-t $ Process Regression for Multi-output Prediction." Neural Computing and Applications 32.8 (2020): 3005-3028.

[3] Chen, Zexun, Jun Fan, and Kuo Wang. "Remarks on multivariate Gaussian Process." arXiv preprint arXiv:2010.09830 (2020).