Noise reduction analysis of deformation data based on CEEMD-PE-SVD modeling

In the field of high slope deformation monitoring, the deformation data obtained are often characterized by high volatility, strong nonlinearity, and noise content, due to the influence of factors such as the surrounding environment, human operation, and the complexity of the project. To overcome these problems, this paper proposes deformation data based on the coupling of complementary ensemble empirical modal decomposition (CEEMD), permutation entropy (PE), and singular value decomposition (SVD). Firstly, CEEMD decomposition is performed on the deformation data. Secondly, a randomness test is performed on the obtained modal components, and if the entropy value of the component arrangement entropy is greater than 0.5, SVD decomposition will be used for noise reduction. Finally, the components that meet the conditions are reconstructed to realize the noise reduction of the deformation data. The results show that the algorithm model has an excellent noise elimination effect, which can reflect the internal detail information of deformation data and provide a well-established method for future research on the processing of deformation monitoring data.


Introduction
The deformation data obtained from long-term observation of high slopes often contain noise (error) and the true value of the two parts of the external environment (temperature and humidity).The accuracy of the instrument and human operation and other factors resulting in the true value of the noise are often blended because of the presence of noise, so the deformation of the data fluctuation increased, increasing the difficulty of the later prediction of the deformation data.Therefore, it is very necessary to carry out noise reduction processing of deformation data [1] .Currently, the commonly used noise reduction methods mainly include wavelet analysis and complementary ensemble empirical mode decomposition (CEEMD) [2] .Wavelet analysis is usually suitable for processing low-frequency signals, and it is difficult to extract useful information from medium-and high-frequency signals and the noise cancellation effect varies greatly with different wavelet bases, thresholds, and decomposition scales [3] .CEEMD, on the other hand, is a new type of signal decomposition method without a priori knowledge, which can effectively separate low-frequency and high-frequency signals.At the same time, it can also improve the repeatability and robustness of signal decomposition, so the noise cancellation effect is better compared with wavelet analysis [4] .To further screen the hidden low-frequency useful information in the deformation data, this paper proposes a noise reduction method based on the coupling of the CEEMD algorithm, Permutation Entropy (PE), and Singular Value Decomposition (SVD), and the experiments show that the coupling algorithm has excellent effect on the deformation data of high slopes.The experiments show that the coupling algorithm has an excellent effect on noise reduction of high slope deformation data, which has great development prospects in practical application [5][6] .

Complementary ensemble empirical modal decomposition (CEEMD)
CEEMD decomposition is a new adaptive signal processing algorithm that greatly improves the efficiency of signal decomposition as well as reduces the reconstruction error of the signal during the decomposition of the signal [7] .The steps of CEEMD decomposition are as follows: (1) We add a set of white noise ai(t) and -ai(t) of opposite sign to the original data sequence S(t) to obtain new signals Bi(t), Ci(t).
(2) We perform EMD decomposition for Bi(t) and Ci(t), respectively.( ) ( ) where J is the number of components, ij IMF  is the jth component of the signal after adding positive white noise reconstruction, and ij IMF  is the jth component of the signal after adding negative white noise reconstruction.

Ranking entropy (PE)
Alignment entropy is a metric used to characterize the stochasticity of dynamic sequences with strong robustness, and the principle of the algorithm is described [8] .

Singular value decomposition (SVD)
The SVD decomposition can effectively filter out the noise components in smooth and periodic signals, and the decomposition steps are as follows [9] : (1) We construct a Hankel matrix for an arbitrary deformed signal { (1), (2),... ( )} Y y y y N  . ( ( ) (2) If the noise signal is T and the real signal is Z, the reconstructed matrix W is: (3) W decomposes into: where the subscript t represents the noise signal matrix and the subscript z represents the true signal matrix.

Establishment of CEEMD-PE-SVD noise cancellation-based modeling
Aiming at the characteristics of monitoring data on high slope deformation with noise content and nonlinearity, this paper proposes a CEEMD-PE-SVD noise cancellation model based on the following steps: (1) CEEMD decomposes the deformation monitoring data to obtain the sample vector i y , , , n y y y to realize the noise reduction of deformation monitoring data.

Experimental analysis
In this paper, the deformation data of high slopes on both sides of the highway are used as the research object, and the cumulative horizontal displacement of monitoring point BD01 is selected as the research sample.The observation time is from January 23, 2020, to May 1, 2020, and the sampling interval is 1 d, with a total of 100 periods of data.The deformation monitoring data of monitoring point BD01 is shown in Figure 1.As can be seen from Figure 1, the influence of noise results in strong fluctuation of BD01 monitoring point deformation data, and there is no obvious regularity to follow.If the noise is not filtered out, it is difficult to get reliable results.Therefore, the deformation data for noise elimination is an essential part of the process.

CEEMD-PE-SVD noise canceling experiment
To observe the internal characteristics of the deformation data more clearly and reduce the volatility of the observed data, the deformation data are first decomposed by CEEMD.As can be seen from Table 1, the entropy value of IMF1 and IMF2 components is greater than 0.5, which is considered to contain more noise.It is necessary to carry out the SVD decomposition of these two components.The entropy value of the rest of the modal components is less than 0.5, which is considered to be more stable and is defaulted to be a pure signal.It does not need to be further processed for noise reduction.As can be seen from Figure 3, after SVD decomposition, the volatility of IMF1 and IMF2 components are reduced, and the entropy value of the alignment entropy of these two components is less than 0.5 (entropy value after SVD decomposition of IMF1 component=0.4331and entropy value after SVD decomposition of IMF2 component=0.4803).It reflects that the effect of the noise reduction processing by SVD decomposition is obvious and the noise reduction process using SVD decomposition is very effective, which can effectively filter out the noise in the high-frequency components (IMF1 and IMF2).From the comparison of the image effect before and after noise reduction, it can be found that the IMF1 component contains white noise and cycle error in the deformation data, and the IMF2 component contains white noise and the residual amount of cycle error in the deformation data, which makes the two components show large ups and downs and it difficult to capture the useful information hidden in the high-frequency signal.After SVD decomposition, the waveform undulations of both components are slowed down, and the curves of SVD decomposition at the inflection points and extreme points are smoother.
In general, the CEEMD-PE-SVD noise cancellation experiment mainly includes three steps.Firstly, CEEMD decomposition of the high slope deformation data is performed to obtain seven IMF components and one RES component from high to low frequency.Secondly, a randomness test is carried out for the multiple components obtained by using the entropy of the permutation.For the components larger than a set threshold (entropy value larger than 0.5), SVD decomposition is carried out, and the components after SVD decomposition are recalculated.If it meets the requirements, the next step will be carried out directly; If it does not meet the requirements, we continue to carry out the SVD decomposition, until all the components have an entropy value that is less than 0.5 of the arrangement entropy.Finally, the reconstruction of each IMF component that meets the conditions can be realized to the deformation of the high slope data of noise elimination processing.

Comparison experiment of noise canceling effect
To verify the effectiveness of the method proposed in this paper (based on the CEEMD-PE-SVD noise cancellation model), it is compared and analyzed with the more popular wavelet threshold noise cancellation method.After many experiments, when the deformation data of the BD01 monitoring point selects the db4 wavelet base and the number of decomposition layers is 8, the noise cancellation effect is optimal.Figure 4 shows the comparison between this paper's noise cancellation method and wavelet threshold noise cancellation method.Compared with Figure 1 (deformation data of BD01 monitoring point), it is not difficult to see that these four noise cancellation methods have reduced the volatility of the deformation data, and the general trend of the curve is similar to that of the deformation data before noise cancellation.However, there are some shortcomings, for example, the curve after noise reduction in this paper is not smooth enough, and a large amount of deformation data details are lost after the soft threshold minimaxi, heursure, and sqtwolog noise reduction processes.To compare the advantages and disadvantages of several noise reduction methods more intuitively, the signal-to-noise ratio (SIGNAL-NOISE RATIO, SNR) and the root mean square error (RMSE) are introduced to judge which noise reduction method is superior [10][11] .
From Table 2, it can be found that compared with the wavelet thresholding noise cancellation method, the noise cancellation method in this paper is superior in SNR and RMSE.The curve after noise cancellation in this paper is not as smooth as the curve after wavelet thresholding noise cancellation, because the method in this paper not only filters out the high-frequency white noise in the deformation data but also preserves the detail information in the process of noise cancellation.On the other hand, the wavelet thresholding noise cancellation method has the problem of "over-filtering", in which the useful information is also filtered out in the process of noise cancellation, making it difficult to show the detailed information in the deformation data.The noise cancellation effect is not as good as the method proposed in this paper.

Conclusion
This paper proposes a CEEMD-PE-SVD noise cancellation model based on the real high slope deformation data through the noise cancellation process.The results show that the method proposed in this paper, to a certain extent, overcomes the wavelet threshold noise cancellation method for the high and medium frequency signal noise cancellation effect.The reconstructed signal better reflects the deformation monitoring data, more clearly observes the deformation data of the cyclic changes, and is closer to the real deformation of high slope deformation law.The later study of the deformation analysis and deformation prediction provides a strong reference basis.

3 )
Phase space reconstruction i y and normalization process are conducted to get the arrangement entropy value of n [0, 1];(4) We perform SVD decomposition for the components in (3) with entropy values greater than 0.5;(5) We repeat Steps (3) and (4) until the entropy values of i y are all less than 0.5;(6) We reconstruct   1 2

Figure 2 .
Figure 2. CEEMD decomposition of deformation data.As shown in Figure2, seven IMF components and one residual (RES) with frequencies from high to low are obtained by CEEMD decomposition.To extract the hidden detail information in the highfrequency IMF components, the entropy values of the aligned entropy of the seven IMF components and the RES are calculated respectively.Table1.Alignment entropy of each component.

Figure 3 .
Figure 3.Comparison of the effect of IMF1 and IMF2 before and after noise cancellation.

Figure 4 .
Figure 4. Comparison between the noise cancellation method in this paper and the wavelet thresholding noise cancellation method.

Table 1 .
Alignment entropy of each component.

Table 2 .
Comparison of several noise cancellation methods.