It is very difficult to predict accurately non-linear deformation time series of surrounding rock using general methods expecially under complex geological condition. A new method based on Gaussian Process (GP) machine learning, which is a newly developing machine learning method with excellent capability for solving highly nonlinear problems with small samples, is proposed. The method was verified by modeling deformations of surrounding rock of Yangzong tunnel in Yunnan Province, China. The monitoring nonlinear deformation time series of surrounding rock during excavation are taken as learning samples for GP machine learning. Non-linear relation of the deformation time series can be obtained by learning process of GP. Then, the future nonlinear deformations are predicted by using time series analysis based on GP. The results indicate that GP can appropriately describe the evolutionary law of non-linear deformation of tunnel and provide accuracy predictions. Furthermore, the results also show that the method is feasible, effective and simple to implement.


Modeling the non-linear deformation behavior of surrounding rock is an important aspect of the safety assessment for tunnel engineering in complex conditions. The deformation behavior of tunnel is affected by complex geological conditions, human activities and their coupled effects. Modeling such dynamic and non-stationary time series is a challenging task. As a result, it is very difficult to give accurate prediction of deformation using traditional model. But we still can reveal underlying evolution law of tunnel system by making use of the information on the measured deformation. In recent years, some scholars have made some progress on the forecasting non-linear deformation time series of tunnel with aid of machine learning technique such Artificial Neural Networks (ANN) (Ma, 2003) and Support Vector Machines (SVM) (Zhao, 2005). Machine learning technology paves a new way in solving the problems mentioned above.

Gaussian process (GP) is a newly developed machine learning technology (Rasmussen and Williams, 2006) based on strict theoretical fundamentals and Bayesian theory. In recent years, GP has attracted much attention in the machine learning community, there are a lot of successful applications in the field of solving for nonlinear, small samples and high dimensions problems (Seeger, 2004; Girolami, 2006; Gramacy, 2007). GP is moderately simple to implement and use without loss of performance compared with ANN and SVM (Xiong, 2005).

In this paper, a new model based on GP is proposed to model and predict non-linear deformation time series of surround rock of tunnel. The model is verified by its application to study non-linear deformation time series of Yangzong Tunnel in China.


The Bayesian analysis of forecasting models is difficult because a simple prior over parameters implies a complex prior distribution over functions. Rather than expressing our prior knowledge in terms of a prior for the parameters, we can instead integrate over the parameters to obtain a prior distribution for the model outputs in any set of cases. The prediction operation is most easily carried out if all the distributions are Gaussian.

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