Processing method of EMD endpoint effect based on SVRM extension

2.1. Data series extension method of SVRM

After constructing the regression training model of SVRM, the data points can be extended forward and backward by using it. The following is an example to introduce the extension method and steps of SVRM:

(a) Construct the training set according to the existing data sequence.

For a given data sequence $s\left(1\right)$, $s\left(2\right)$,…, $s\left(N\right)$, where $N$ is the number of sampling points in the data column. Firstly, the number of training samples $l$ is determined, and a training set $L=\left\{\right(x_1,y_1),\dots \left(x_l,y_l\right)\}$, where $x_i=[\begin{array}{ll}s\left(i\right)& s(i+1)\cdots \end{array}\cdots$$s(N-l+i-1){]}^T$, $y_i=s(N-l+i)$, $1\le i\le l$.

(b) Construct regression model.

The regression model is constructed according to the above method.

(c) Predict the first sequence value.

The first predicted value outside the boundary can be obtained:

(d) Step by step iteration to obtain predictive sequence values.

Then let $s(N+1)$ as the new boundary point of the original data, the extension value of the second data sequence $s(N+2)$ can be obtained, and so on. According to the required extension length $M$, the whole extension sequence can be obtained $s\left(N\right)$, $s(N+1)$,…, $s(N+M)$, the value of any extension $s(N+m)$ is:

where, the value of any extension $s(N+m)$is $x_{l+m}=[\begin{array}{ll}s(l+m)& s(l+m+1)\cdots s(N+m-1)\end{array}{]}^T$.

2.2. Setting of extension parameters

The extension precision of SVRM increases with the increase of the number of samples, while the operation efficiency decreases with the increase of the number of samples. Therefore, in order to ensure the extension accuracy, the number of samples $N_e$ must be increased or decreased with the extension length $N_s$. Set $k=N_e/N_s$. In general, short time extension requires a higher extension accuracy, and $k$ is set as a smaller value. In the case of long time extension, the extension accuracy requirements can be appropriately reduced, and a larger $K$ can be selected to improve the operational efficiency. Specifically, the length between the signal extreme points is taken as the reference, so that:

where, $L$ is the distance between extremum points; $\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{m}\mathrm{i}\mathrm{n}\left(i\right)$ and $\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{m}\mathrm{a}\mathrm{x}\left(i\right)$ are respectively the moment of the $i$th minimum and maximum value of the signal, $k_s$ is the sample length coefficient, $k_s>$0.

3.1. EMD primary extension algorithm based on SVRM

The primary extension method refers to the IMF which only conducts primary extension of data before EMD decomposition and then truncates the extension part of the obtained decomposition results to obtain the original data. The characteristic of this method is that the whole decomposition process only needs primary extension, the algorithm is simple to implement, and there is no accumulation of error caused by repeated extension. The simulation signal $y_2\left(t\right)$ is taken as an example, and the waveform is shown in Fig. 1. The SVRM method is adopted to carry out a long extension, and then EMD decomposition. Sample length $N_s$ and extension length $N_e$ were controlled by adjusting the values of $k_s$ and $k_e$. The decomposition of can is shown in Fig. 2:

Fig. 1Waveform of simulated signal y2t

As shown in Fig. 2(a), when $k_s=$1 and $k_e=$5, the sample length $N_s=$16 and the extension length $N_e=$90, the decomposition results show large deformation at both ends, and the endpoint effect is not effectively contained. When $k_s=$5 and $k_e=$5, the sample length $N_s$ increases to 90. At this time, the decomposition results also show a large deformation at both ends, that is, the endpoint effect is effectively contained without the increase of the sample length. It can be inferred that when $k_e=$5, the extension length is too short, so the endpoint effect is not effectively contained outside the real signal. When $k_s=$1 and $k_e=$20, the extension length $N_e$ increases to 320 points, and the decomposition results are greatly improved. There is no obvious distortion at the end points, but residual term $r_5$ is generated. It can be inferred that when $k_s=$1, the regression model cannot accurately predict the signal trend due to the small sample data. When $k_s=$5 and $k_e=$20, the sample length increased to 80 points. At this time, a more accurate decomposition result was obtained, but the running time of the program increased to 71.8776 s.

3.2. EMD decomposition and extension simultaneously based on SVRM

In order to avoid the problem of operation efficiency caused by SVRM long time extension, an effective method is to make full use of the accuracy of SVRM short time extension, and adopt appropriate algorithm to avoid long time extension. The effective method is to decompose while extension. The idea is to conduct a short time extension with a length of more than two extreme points before each screening of EMD, and then cut off the extension part after generating the envelope. Since the algorithm needs to be extended before each screening, it only needs to extend a pair (a maximum value and a minimum value) above the extreme point at both ends of the signal each time. Repeated simulation experiments verify that, under the same conditions, when the two ends extend more than four extreme points respectively, the decomposition results will no longer change significantly. In order to ensure the decomposition accuracy, the method of extending four extreme points at both ends is adopted in this paper. Set $k_s=$4, then the EMD processing results of $y\_2\left(t\right)$ are shown in Fig. 3.

Fig. 2Decomposition of simulated signal after one extension

a)$k_s=$1, $k_e=$5, $T=$0.7648 s

b)$k_s=$5, $k_e=$5, $T=$22.2817 s

c)$k_s=$1, $k_e=$20, $T=$2.4462 s

d)$k_s=$5, $k_e=$20, $T=$71.8776 s

In the Fig. 3, when the sample is 10, residual terms are generated in the decomposition result; when the sample is increased to 20, the decomposition result reaches a very high precision, and at this time, the whole decomposition process takes only 0.9045 s.

It can be seen from Fig. 3 that when $k_s=$1, due to the small sample number and poor extension accuracy, the endpoint effect is restrained to some extent, but residual terms appear. When $k_s=$2, a relatively accurate decomposition result was obtained, and the operation time reached 388.2863 s.

Fig. 3Decomposition of simulated signal after decomposition and extension simultaneously

a)$k_s=$1, $k_e=$4, $T=$139.4893 s

b)$k_s=$2, $k_e=$4, $T=$388.2863 s

3.3. EMD continuation based on SVRM extremum prediction

The extremum forecast and continuation process, SVRM extremum predict continuation of sample sequence is extremum sequence of original signal, so the methods require original signal have enough number of extreme value (experiments show general may not be less than 6), when the signal extremum number is small, SVRM based on hard less extreme value point accurately predict epitaxial signal extremum position and value. It should be noted that, with the progress of EMD decomposition, the frequency components of the obtained components gradually decrease. If this method is used for side decomposition and side continuation, the phenomenon of inaccurate continuation may occur in the low-frequency stage due to the small number of extreme points. Therefore, this method is more suitable for the previous continuation of decomposition. The decomposition results are shown in Fig. 4.

Fig. 4EMD results of continuation prediction based on SVRM extreme value

a)$N_s=$10, $N_e=$30, $T=$0.7239 s

b)$N_s=$20, $N_e=$30, $T=$0.9045 s

In order to verify the validity of the effect of the endpoint effect and the practical practicability of the actual hydraulic system, the sampling frequency of the measured hydraulic system is analyzed, and the sampling frequency is 5 kHz, and the sampling point is 2048. The results of EMD continuation based on SVRM extremum prediction results are as shown in the Fig. 5. The weak at both ends, and the similar components of the frequency are not visible in the endpoints, and the model of the IMF component is obviously suppressed.

Fig. 5EMD results of hydraulic fault signal