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In traditional system identification methods, it is often assumed that the output data are corrupted by Gaussian white noise which is independent and identically distributed (i.i.d.). However, this assumption may lead to poor robustness since the noise characteristic often varies throughout the sampling process. In this work, output measurements affected by switching Gaussian noise are considered. In addition, a Markov chain model is utilized to describe the multi-mode behavior of the noises. Meanwhile, the collected data are usually incomplete in practice. Taking these circumstances into account, a new algorithm for Gaussian process regression (GPR) with switching noise mode and missing data is introduced. The parameters of the model are estimated by expectation maximization (EM) algorithm via conjugate gradient (CG) method. Two numerical examples along with a continuous stirred tank reactor simulation are employed to verify the effectiveness of the proposed algorithm. The superior performance is demonstrated by comparing the proposed algorithm with other existing relevant methods. [ABSTRACT FROM AUTHOR]