Modelling and prediction of eye gaze movement can be highly desirable in many real-world scenarios, e.g. human-machine interaction and human behavior analysis. This challenging area largely remains unexplored. In this study we tackle this challenge and propose a method to predict eye-gaze movement of human observers. Eye gaze trajectories are separated into three components, where two of them are considered as noise or bias, which can be removed from the trajectory data. So the remaining component, principle movement, can be modelled by a proposed new method, GAR HMM, which stands for Gaussian Autoregression Hidden Markov Model based on AR HMM. Instead of the Beta Processes in AR HMM, GAR HMM introduces a Gaussian Process. So the model can predict the probability of occurrence of eye gaze in each region over time. By joining the predicted points together as a sequence, we can generate the eye gaze movement prediction as a time series. To evaluate GAR HMM we collected eye gaze movement data from over 20 volunteers. Experiments show that good prediction can be achieved by our proposed GAR HMM method. As a groundbreaking work GAR HMM can lead to much further extension to benefit real applications.