## 1 Introduction
Wavelet decomposition (WD) and empirical modal decomposition (EMD) are two common signal decomposition techniques [1; 28; 29; 30; 31], and they are widely used in the processing of strongly fluctuating data. However, they have certain drawbacks, such as the difficulty in choosing wavelet bases and decomposition scales in wavelet transforms, and the basic disadvantage of EMD is the lack of mathematical foundation. To address these issues, the variational mode decomposition (VMD) [32] algorithm is introduced in this study. Compared with EMD, WD, and other methods, VMD is an accurate mathematical model, which can decompose the original data into a set of variational mode functions (VMFs) that fluctuate around the center frequency [33; 34; 35], with better decomposition effect and higher robustness [36; 37; 38]. Currently, the contribution of VMD in the field of prediction has now been confirmed by some scholars, such as energy power generation prediction [39], carbon price prediction [40], runoff prediction [41], container throughput prediction [42] and air quality index prediction [43]. However, to the best of the authors' knowledge, the application for DD prediction has not been extensively explored, so this study will provide a priori knowledge for the utilization of VMD in dam safety monitoring. In practical applications, the decomposition effect of VMD is affected by the decomposition modulus K. If the K value is too small, the obtained VMFs will lose information; on the contrary, it will lead to excessive decomposition and even make the decomposition results worse [44; 45]. In paper [41], the decomposition modulus was determined by central frequency. Here, we propose a more efficient method for determining the decomposition modulus--sample entropy (SE) method. The SE can characterize the complexity of the time series, and when the time series is perturbed and the uncertainty of its state value increases, the value of the SE will also increase accordingly [46]. If two or more subsequences have similar SE values, modal confusion occurs in the decomposition. Therefore, in this paper, we choose the maximum K value that allows a large difference between the SE values of any two subsequences as the final decomposition modulus of the VMD. In addition, there is an error sequence (ER) between the sum of the subsequences obtained by the VMD algorithm and the original sequence. In other words, the sum of the VMFs is not equal to the original sequence [47]. Since the ER contains the true fluctuation characteristics of the original sequence, considering only VMFs will not fully reflect the randomness, which will lead to distortion of the prediction results to some extent. In this paper, the ER of the DD after decomposition is extracted, and because the sequence is approximately noisy, it cannot be modeled directly using the machine learning model to obtain good prediction results. To solve this problem, the authors of paper [47] proposed the random number method for point prediction of ER, but the uncertainty of the results obtained by this method is significant. Therefore, we innovatively propose a VMD-based secondary decomposition for the ER in order to dig deeper into the temporal features embedded in it. The results show that the prediction obtained by considering the ER is closer to the observed values and has more practical engineering significance. In summary, a hybrid model, namely VMD-SE-ER-PACF-ELM, for the DD prediction is proposed, which makes full use of the advantages of the VMD, SE, PACF, and ELM neural network. First, the VMD-SE model is used to decompose a deformation sequence into K VMFs with good characteristics and an ER. For the ER, the same method is used to obtain a series of subsequences. Secondly, the partial autocorrelation function (PACF) method is used to determine the input variables of the subsequences. Then the ELM models corresponding to each subsequence is trained in the machine-learning process.