CN109582913B - Method for empirical mode decomposition screening iteration process termination criterion - Google Patents

Abstract

The invention discloses a method for determining the climatic state of data by using the termination criterion of an empirical mode decomposition screening iterative process, wherein the climatic state is represented by linear fitting of data x (t); iteratively solving, and calculating the relative variance of the margin after each iteration to obtain the rudiment of the IMF; and finding out the corresponding iteration times when the relative variance is minimum, wherein the IMF embryonic form at the moment is the end point of the iteration process. Entering the next iteration loop process to obtain the next IMF until the allowance x _i+1 And (t) is a trend term. The invention has the beneficial effects that the decomposition process of the EMD can be more standard and objective, the operability of the EMD decomposition and the consistency of the result are improved, and the 'experience' characteristic of the EMD is weakened.

Description

A Method of Empirical Mode Decomposition to Screen Iterative Process Termination Criteria

technical field

The invention belongs to the technical field of data analysis, and relates to the relative variance of the screening iteration margin as the termination criterion of the screening iteration process of the empirical mode decomposition method.

Background technique

One-dimensional Fourier decomposition, wavelet analysis, two-dimensional PCA/EOF and other methods all decompose from low frequency to obtain different mode/eigenfunctions, and the low frequency part usually obtains the largest variance contribution rate. Empirical Mode Decomposition (EMD) is an adaptive data analysis method without fixed basis functions. EMD decomposes a time series into a series of IMFs from high frequency to low frequency, and a nonlinear trend term. The decomposed intrinsic mode functions (IMFs) are usually not orthogonal. Therefore, the sum of the variance percentages of each IMF in the data may be greater than 1, which cannot be interpreted as the variance contribution rate of each IMF. However, it is still possible to determine whether the IMF decomposition result is reasonable according to the relative variance of the low frequency part. The solution of each IMF in the EMD decomposition process is obtained through an iterative process called "sifting". The current termination criteria for EMD screening iterations are as follows:

(1) Cauchy-type criterion (Huang et al., 1998): the relative root mean square error of the IMF prototype of two adjacent iterations is less than a predetermined sufficient small amount;

(2) Average value criterion (Flandrin, 2004): the deviation of the IMF prototype of two adjacent iterations at each time point is less than a predetermined sufficient small amount;

(3) S criterion (Huang et al., 2003): For the IMF prototype of S consecutive iterations, the number of extreme points and zero crossing points remains unchanged, or the difference is at most 1;

(4) Fixed number of iterations (Wu and Huang, 2010): Regardless of the characteristics of the data or the mode of the IMF to be solved, the number of iterations for all screening takes a fixed value. This fixed value is usually taken as 8-12.

Among the termination criteria of the above screening iterations, the first three criteria are usually relatively subjective. For example, how to choose the "sufficiently small amount" in the Cauchy criterion and the mean criterion? How to choose S in the S criterion? In general, a small "sufficiently small amount" and a large "S" require more iterations to reach the condition for iteration termination. Wang et al. (2010) proved that the increase in the number of iterations in the screening process will cause the amplitude of the decomposed IMF to tend to a constant, that is, the IMF is close to a linear function. This is contrary to EMD's pursuit of the "non-linear and unstable" feature of IMF. The fourth criterion (that is, the termination criterion of a fixed number of iterations) is also likely to cause confusion in terms of the selection of the number of iterations. Iterating 8 or 10 times in the screening, the IMF prototype obtained may have great differences, and it cannot be guaranteed to meet any one of the first three criteria.

In summary, Empirical Mode Decomposition (EMD) decomposes data into Intrinsic Mode Functions (IMFs) at multiple frequency bands and a nonlinear trend through a series of iterative screening processes. The generation of each IMF has to go through an iterative process of screening. The four iteration termination criteria currently used lack objective judgment criteria, and the selection of the number of iterations must be given by experience, and there may be insufficient iterations or excessive iterations.

Contents of the invention

The object of the present invention is to provide a kind of method of the termination criterion of empirical mode decomposition screening iteration process, the beneficial effect of the present invention is that the present invention can make the decomposition process of EMD more normative, objective, improve the operability and the result of EMD decomposition Consistency, downplaying its "experiential" character.

The technical scheme adopted in the present invention is carried out according to the following steps:

Step 1: Determine the climate state of the data. Use the linear fit of the data x(t) to represent its climate state;

Step 2: Iterative solution. After obtaining the prototype of IMF in each iteration, calculate the relative variance of the margin;

Step 3: Find the number of iterations corresponding to the minimum relative variance, and the IMF prototype at this time is the end of this iteration process;

Step 4: Enter the next iterative cycle process, and obtain the next IMF until the residual x _i+1 (t) is a trend item.

Further, in step 1 in matlab, c(t)=x(t)-detrend(x(t)) is used as the linear fitting of the time series x(t), namely the climate state.

Further, the relative variance of step 2 is defined as

Where x _i (t) is the data state before the start of screening iteration when solving the i-th IMF (i.e. m _i (t)), that is, x ₁ (t)=x(t), x ₂ (t)=x(t )-m ₁ (t), . . . , x _i (t)=x(t)-Σ _{k=1 . . . i} m _k (t). Let x _i (t) be called the margin, m _i,j (t) is the IMF prototype obtained after j iterations during the screening process of solving m _i (t), that is, m _i (t) is m _i,j (t) The state at the end of the screening iteration process.

Further, for the i-th IMF (ie m _i (t)) in step 3, a time series r _i,j related to the iteration number j will be obtained after multiple iterations. The number of iterations _{j corresponding to the minimum value of r i,} j is the optimal number of iterations of this screening process, which is denoted as N(i). At this time, the corresponding m _{i, N(i)} (t) is the first i eigenfunctions m _i (t).

In actual operation, an appropriate maximum number of iterations (such as 50) can be given in advance, and then the minimum value can be found within this range.

Description of drawings

Fig. 1 is a schematic flow sheet of the method of the present invention;

Figure 2 is a schematic diagram of the annual mean global surface temperature anomaly;

Fig. 3 adopts N-termination criterion (N=10), the IMF figure that EMD obtains;

Figure 4 is the IMF diagram obtained by EMD using this method.

Detailed ways

The present invention will be described in detail below in combination with specific embodiments.

(1) Take the sample data gsta.dat (annual mean global surface temperature anomaly, see Figure 2) from the RCADA website as an example. The number of screening iterations is fixed at 10 (that is, N in the N-termination criterion is fixed at 10). For each screening process, pre-record the state of the data before screening, calculate its climate state (linear trend), and set the maximum number of iterations to 100, and calculate the relative variance after each iteration.

(2) For each screening process, 100 iterations will give a relative variance sequence, and find the corresponding iteration number when the relative variance is minimum (for convenience, it is called the optimal iteration number). The optimal number of iterations of IMF1 obtained here is 56.

(3) For the data state recorded in (2) before the screening starts, re-screen and solve the IMF according to the optimal number of iterations of the current IMF.

(4) Repeat (3) and (4) for the next IMF and the corresponding screening process.

(5) Six IMFs are obtained in the above steps, and the optimal iteration times corresponding to each IMF are: 56, 100, 60, 6, 1, 5. Overall, when the average frequency of the data is low, the optimal number of iterations for screening is relatively small. For example, the optimal number of iterations corresponding to IMF4 to IMF6 is less than the minimum number of iterations 10 set in the original code.

Comparing the IMF decomposed by the two methods in Fig. 3 and Fig. 4, the IMF obtained by the method of the present invention (minimum relative variance criterion) significantly reduces the modal confusion phenomenon.

The present invention has the advantages of:

(1) The number of screening iterations is determined according to the size of the relative variance of the margin, and there are objective normative standards. Since the number of iterations takes into account the characteristics of the data itself, such as the length of the data, the average period of the data, etc., the number of iterations required to solve the IMF will also be different.

(2) This termination criterion is highly operable. The standard is objective and can be quantitatively described, and the code automatically determines the number of iterations without relying on empirical selection.

(3) This method conforms to the basic characteristics of EMD "adaptive", and is also consistent with the basic ideas of Fourier, wavelet and other decomposition methods: low-frequency modes (/components/eigenfunctions) should be able to explain more variance.

The above description is only a preferred embodiment of the present invention, and does not limit the present invention in any form. All simple modifications, equivalent changes and modifications made to the above implementation methods according to the technical essence of the present invention fall within the scope of the technical solutions of the present invention.

Claims (4)

1. The method for the termination criterion of the empirical mode decomposition screening iterative process is characterized by comprising the following steps of:

step 1: determining the climate state of the data, and representing the climate state by using the linear fitting of the data x (t);

step 2: iteratively solving, and calculating the relative variance of the margin after each iteration to obtain the rudiment of the IMF;

step 3: finding out the corresponding iteration times when the relative variance is minimum, wherein the IMF embryonic form at the moment is the end point of the iteration process;

step 4: entering the next iteration loop process to obtain the next IMF until the allowance x _i+1 And (t) is a trend term.

2. A method of empirical mode decomposition screening an end criterion for an iterative process as claimed in claim 1 wherein: in matlab, c (t) =x (t) -detrend (x (t)) is used as a linear fit of the time series x (t), the climate.

3. A method of empirical mode decomposition screening an end criterion for an iterative process as claimed in claim 1 wherein: the step 2 relative variance is defined as

Wherein x is _i (t) is to solve the ith IMF (m _i (t)) screening the data state before the start of an iteration, i.e. x ₁ (t)＝x(t),x ₂ (t)＝x(t)-m ₁ (t),…,x _i (t)＝x(t)-∑ _k＝1…i m _k (t) x is _i (t) is called margin, m _i,j (t) solving for m _i In the screening process of (t), IMF embryonic form obtained through j times of iteration, namely m _i (t) is m _i,j (t) screening the state at the end of the iterative process.

4. A method of empirical mode decomposition screening an end criterion for an iterative process as claimed in claim 1 wherein: for the ith IMF (i.e., m in step 3 _i (t)) a time series r is obtained after a number of iterations, which is related to the number j of iterations _i,j Find r _i,j The iteration number j corresponding to the minimum value of (2) is the optimal iteration number of the screening process, and is marked as N (i), and the corresponding m _i，N(i) (t) is the ith eigenfunction m _i In actual operation, a suitable maximum number of iterations may be predefined, and then a minimum value of the relative variance is found within this range.