CN109582913B

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

Info

Publication number: CN109582913B
Application number: CN201811379876.6A
Authority: CN
Inventors: 王刚; 乔方利
Original assignee: Qingdao National Laboratory for Marine Science and Technology Development Center; First Institute of Oceanography SOA
Current assignee: Qingdao Marine Science And Technology Center; First Institute of Oceanography SOA
Priority date: 2018-11-20
Filing date: 2018-11-20
Publication date: 2023-08-22
Anticipated expiration: 2038-11-20
Also published as: CN109582913A

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

Method for empirical mode decomposition screening iteration process termination criterion

Technical Field

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

Background

The methods of one-dimensional Fourier decomposition, wavelet analysis, two-dimensional PCA/EOF and the like are all to obtain different modes/characteristic functions by decomposing from low frequency, and the low frequency part usually obtains the maximum variance contribution rate. Empirical Mode Decomposition (EMD) is an adaptive data analysis method with no fixed basis functions. EMD breaks down a time series into a series of IMFs from high frequency to low frequency, and a nonlinear trend term. The decomposed eigenmode functions (IMFs) are typically not orthogonal. Thus, the sum of the variance percentages of each IMF in the data may be greater than 1 and 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 based on the relative variance of the low frequency portions. The solution of each IMF in the EMD decomposition process is obtained by an iterative process called "screening". The end criteria for the current EMD screening iteration are as follows:

(1) Cauchy type criterion (Huang et al, 1998): the relative root mean square error of the IMF embryonic form of two adjacent iterations is smaller than a certain preset sufficient small amount;

(2) Average criterion (Flandrin, 2004): the IMF rudiments of two adjacent iterations have a deviation at each time point of less than a predetermined sufficiently small amount;

(3) S criterion (Huang et al, 2003): the number of extreme points and zero crossing points of the IMF rudiment iterated for continuous S times is kept unchanged or the difference is at most 1;

(4) Fixed number of iterations (Wu and Huang, 2010): the characteristics of the data are not considered, the mode of the IMF to be solved is also not considered, and the iteration times of all the screening are fixed values. This fixed value is typically taken to be 8-12.

Of the above termination criteria for the screening iteration, the first 3 criteria are typically more subjective, such as how "sufficiently small" of the Cauchy-criteria and the average criteria is chosen? How does S in the S criteria choose? In general, a small "sufficiently small" and a large "S" require more iterations to reach the condition of iteration termination. Wang et al (2010) demonstrate that an increase in the number of iterations in the screening process will result in the decomposed IMF amplitude tending to be constant, i.e. the IMF approaches a linear function. This is a departure from EMD pursuing IMF "nonlinear non-stable" features. The 4 th criterion (i.e., the fixed number of iteration termination criterion) is also easily confused in terms of its choice of iteration number. The obtained IMF embryonic forms may have larger difference in 8 or 10 iterations in the screening, and any one of the first 3 criteria cannot be guaranteed to be met.

In summary, empirical Mode Decomposition (EMD) decomposes data into a plurality of eigenmode functions (IMFs) of different frequency bins and a nonlinear trend through a series of iterative screening processes. Each IMF is generated through a screening iterative process. The current adopted 4 iteration termination criteria lack objective judgment criteria, the iteration times are selected by experience, and the situations of insufficient iteration or excessive iteration possibly exist.

Disclosure of Invention

The invention aims to provide a method for stopping criteria in an empirical mode decomposition screening iteration process, which has the beneficial effects that the EMD decomposition process can be more standard and objective, the operability of EMD decomposition and the consistency of results are improved, and the 'experience' characteristics of the EMD decomposition process are reduced.

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

step 1: the climate state of the data is determined. Representing its climatic state by a linear fit of the data x (t);

step 2: and (5) carrying out iterative solution. After each iteration obtains the rudiment of the IMF, calculating the relative variance of the allowance;

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.

Further, step 1 uses c (t) =x (t) -detrend (x (t)) as a linear fit of the time series x (t), i.e. the climatic state, in matlab.

Further, step 2 relative variance is defined as

Wherein x is _i (t) to solve the ith IMF (i.e., 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). Will x _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.

Further, in step 3, for the ith IMF (i.e., m _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 (t)。

In practice, an appropriate maximum number of iterations (e.g., 50) may be predetermined, and then a minimum value may be found within this range.

Drawings

FIG. 1 is a schematic flow chart of the method of the present invention;

FIG. 2 is a schematic diagram of annual average global surface temperature anomalies;

fig. 3 is an IMF diagram obtained by EMD using the N-termination criterion (n=10);

fig. 4 is an IMF diagram obtained by EMD according to the present method.

Detailed Description

The present invention will be described in detail with reference to the following embodiments.

(1) Take the example data gsta. Dat (annual average global surface temperature anomaly, see fig. 2) of the RCADA website as an example. The number of screening iterations is fixed at 10 (i.e., N in the N-termination criterion is fixed at 10). For each screening process, the state of the data before screening is recorded in advance, the climate state (linear trend) is calculated, the maximum iteration number is set to be 100, and the relative variance after each iteration is calculated.

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

(3) And (3) re-screening and solving the IMF according to the optimal iteration times of the current IMF for the data state recorded in the step (2) before the screening starts.

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

(5) The steps obtain 6 IMFs, and the optimal iteration times corresponding to each IMF are respectively as follows: 56,100,60,6,1,5. Overall, when the average frequency of the data is low, the number of optimal iterations corresponding to the screening is also relatively small. For example, the optimal iteration number corresponding to IMFs 4 through 6 is less than the minimum iteration number 10 set in the original code.

Comparing the IMFs decomposed by the two methods of fig. 3 and fig. 4, the IMFs obtained by the method of the present invention (minimum relative variance criterion) significantly reduce the modal aliasing.

The invention has the advantages that:

(1) The screening iteration times are determined according to the relative variance of the margin, and objective standard standards are provided. Since the iteration number considers the characteristics of the data, such as different data lengths, average periods of the data, and the like, the required iteration number is different when solving the IMF.

(2) The operability of the termination criterion is strong. The standard is objective, the quantitative characterization can be realized, and the code automatically determines the iteration times without depending on experience selection.

(3) The method accords with the basic characteristics of EMD 'self-adaption', and is consistent with the basic ideas of Fourier, wavelet and other decomposition methods: the low frequency modality (/ component/feature function) should be able to interpret more variance.

The foregoing description is only of a preferred embodiment of the invention and is not intended to limit the invention in any way. Any simple modification, equivalent variation and modification of the above embodiments according to the technical substance of the present invention are all within the scope of the technical solution of the present invention.

Claims

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;

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.