Compositional kernel learning using tree-based genetic programming for Gaussian process regression
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RESEARCH PAPER
Compositional kernel learning using tree-based genetic programming for Gaussian process regression Seung-Seop Jin 1 Received: 21 August 2019 / Revised: 23 January 2020 / Accepted: 27 February 2020 # Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Although Gaussian process regression (GPR) is a powerful Bayesian nonparametric regression model for engineering problems, its predictive performance is highly dependent on a kernel for covariance function of GPR. However, choosing a proper kernel is still challenging even for experts. To choose proper kernel automatically, this study proposes a compositional kernel (CPK) learning using tree-based genetic programming (GEP). The optimal structure of the kernel is defined as a compositional representation based on sums and products of eight base-kernels. The CPK can be encoded as a tree-structure, so that treebased GEP is employed to discover an optimal tree-structure of the CPK. To avoid overly complex solution in GEP, the proposed method introduced a dynamic maximum tree-depth technique. The novelty of the proposed method is to utilize more flexible and efficient learning capability to learn the relationship between input and output than existing methods. To evaluate the learning capability of the proposed method, seven test functions were firstly investigated with various noise levels, and its predictive accuracy was compared with existing methods. Reliability problems in both parallel and series systems were introduced to evaluate the performance of the proposed method for efficient reliability assessment. The results show that the proposed method generally outperforms or performs similarly to the best one among existing methods. In addition, it is also shown that proper kernel function can significantly improve the performance of GPR as the training data increases. Stated differently, the proposed method can learn the function of being fitted efficiently with less training samples than existing methods. In this context, the proposed method can make powerful and automatic predictive modeling based on GPR in engineering problems. Keywords Gaussian processes regression . Compositional kernel learning . Tree-based genetic programming . Surrogate modeling . Reliability analysis
1 Introduction A Gaussian process regression (GPR) is a Bayesian nonparametric model to provide analytically tractable way of representing a complex function mapping from input to output (Rasmussen and Williams 2006). In engineering problems, GPR has been widely employed for predictive modeling (interpolation) (Costabal et al. 2019; DiazDelaO and Adhikari 2011; Forrester et al. 2008; Guo and Hesthaven 2018; Jin and Jung 2016; Kleijnen and van Beers 2004; Perdikaris et al. 2017) and regression (Babaee et al. 2016;
Responsible Editor: Mehmet Polat Saka * Seung-Seop Jin [email protected] 1
Sustainable Infrastructure Research Center, KICT, Goyang-si 10223, Republic of Korea
Raissi et al. 2017) by the following useful properties: (1) based on a covariance function
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