CN109582913B - Method for empirical mode decomposition screening iteration process termination criterion - Google Patents
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Abstract
Description
技术领域technical field
本发明属于数据分析技术领域,涉及筛选迭代余量的相对方差作为经验模态分解方法筛选迭代过程的终止准则。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.
背景技术Background technique
一维的Fourier分解、小波分析,二维的PCA/EOF等方法,都是从低频开始分解,获取不同的模态/特征函数,且低频部分通常取得最大的方差贡献率。经验模态分解(EMD)是一种自适应的数据分析方法,没有固定的基底函数。EMD将一个时间序列分解为一系列从高频到低频的IMF,以及一个非线性趋势项。分解得到的各本征模态函数(IMF)通常并不正交。因而,各IMF在数据中所占的方差百分比之和,可能大于1,不能解释为各IMF的方差贡献率。但是,仍然可以根据低频部分的相对方差来确定IMF分解结果是否合理。EMD分解过程中每个IMF的求解都是通过一个称为“筛选(sifting)”的迭代过程得到。目前EMD筛选迭代的终止准则有如下几种: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型准则(Huang et al.,1998):相邻两次迭代的IMF雏形的相对均方根误差小于某个预先给定的充分小量;(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)平均值准则(Flandrin,2004):相邻两次迭代的IMF雏形,在每一个时间点上的偏差都小于预先给定的充分小量;(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准则(Huang et al.,2003):连续S次迭代的IMF雏形,其极值点及跨零点的个数保持不变,或者最多相差1;(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)固定迭代次数(Wu and Huang,2010):不考虑数据的特征,也不考虑将要求解的IMF为第几模态,所有筛选的迭代次数取固定值。这个固定值通常取为8~12。(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.
以上筛选迭代的终止准则中,前3种准则通常有较大的主观性,比如Cauchy性准则和平均值准则中的“充分小量”如何选取?S准则中的S如何选取?一般来说,小的“充分小量”和大的“S”,需要更多的迭代次数才能达到迭代终止的条件。Wang et al.(2010)证明,筛选过程中迭代次数的增加将导致分解得到的IMF振幅趋于常数,即IMF接近线性函数。这与EMD追求IMF“非线性非稳定”特征是背离的。第4种准则(即固定次数的迭代终止准则),从其迭代次数的选择上,也容易引起困惑。筛选中迭代8次或10次,得到的IMF雏形可能有较大的差异,更不能保证满足前3种准则中的任意一个。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.
综上,经验模态分解(EMD)通过一系列迭代筛选过程将数据分解为多个不同频率段的本征模态函数(IMF)和一个非线性趋势。每个IMF的产生,都要经过一个筛选迭代过程。目前采用的4种迭代终止准则,缺乏客观的判定标准,迭代次数的选择要靠经验给出,可能存在迭代不足或者迭代过度的情况。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.
发明内容Contents of the invention
本发明的目的在于提供一种经验模态分解筛选迭代过程的终止准则的方法,本发明的有益效果是本发明能够使EMD的分解过程更加规范、客观,提高EMD分解的可操作性和结果的一致性,淡化其“经验”特征。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.
本发明所采用的技术方案按照以下步骤进行:The technical scheme adopted in the present invention is carried out according to the following steps:
步骤1:确定数据的气候态。用数据x(t)的线性拟合来代表其气候态;Step 1: Determine the climate state of the data. Use the linear fit of the data x(t) to represent its climate state;
步骤2:迭代求解。每次迭代得到IMF的雏形后,计算余量的相对方差;Step 2: Iterative solution. After obtaining the prototype of IMF in each iteration, calculate the relative variance of the margin;
步骤3:找出相对方差最小时对应的迭代次数,此时的IMF雏形就是本次迭代过程的终点;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;
步骤4:进入下一个迭代循环过程,获取下一个IMF,直至余量xi+1(t)为趋势项为止。Step 4: Enter the next iterative cycle process, and obtain the next IMF until the residual x i+1 (t) is a trend item.
进一步,步骤1在matlab里,用c(t)=x(t)-detrend(x(t))作为时间序列x(t)的线性拟合,即气候态。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.
进一步,步骤2相对方差定义为Further, the relative variance of step 2 is defined as
其中xi(t)为求解第i个IMF(即mi(t))时筛选迭代开始前的数据状态,即x1(t)=x(t),x2(t)=x(t)-m1(t),…,xi(t)=x(t)-∑k=1…imk(t)。将xi(t)称为余量,mi,j(t)为求解mi(t)的筛选过程中,经过j次迭代得到的IMF雏形,即mi(t)为mi,j(t)筛选迭代过程结束时的状态。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.
进一步,步骤3中对于第i个IMF(即mi(t)),多次迭代后会得到一个与迭代次数j有关的时间序列ri,j。找出ri,j的最小值所对应的迭代次数j就是本次筛选过程的最佳迭代次数,记为N(i),此时对应的mi,N(i)(t)即为第i个本征函数mi(t)。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).
实际操作中,可预先给定一个适当的最大迭代次数(如50),然后在此范围内寻找最小值。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.
附图说明Description of drawings
图1是本发明方法流程示意图;Fig. 1 is a schematic flow sheet of the method of the present invention;
图2是年平均全球表面温度异常示意图;Figure 2 is a schematic diagram of the annual mean global surface temperature anomaly;
图3是采用N-终止准则(N=10),EMD得到的IMF图;Fig. 3 adopts N-termination criterion (N=10), the IMF figure that EMD obtains;
图4是采用本方法EMD得到的IMF图。Figure 4 is the IMF diagram obtained by EMD using this method.
具体实施方式Detailed ways
下面结合具体实施方式对本发明进行详细说明。The present invention will be described in detail below in combination with specific embodiments.
(1)以RCADA网站的示例数据gsta.dat(年平均全球表面温度异常,见图2)为例。筛选迭代的次数固定为10次(即N-终止准则中的N固定为10)。对于每个筛选过程,预先记录筛选之前数据的状态,计算其气候态(线性趋势),并设定最大迭代次数为100,计算每一次迭代之后的相对方差。(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)对于每个筛选过程,100次迭代将给出一个相对方差序列,求出相对方差最小时对应的迭代次数(为方便起见,称为最优迭代次数)。这里得到的IMF1的最优迭代次数为56。(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)对于(2)中记录的筛选开始前的数据状态,根据当前IMF的最优迭代次数,重新筛选求解IMF。(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)对下一个IMF及对应的筛选过程,重复(3)、(4)过程。(4) Repeat (3) and (4) for the next IMF and the corresponding screening process.
(5)以上各步骤得到6个IMF,每个IMF对应的最优迭代次数分别为:56,100,60,6,1,5。整体而言,当数据的平均频率较低时,筛选对应的最优迭代次数也相对较小。例如,IMF4至IMF6对应的最优迭代次数均小于原代码中设定的最小迭代次数10。(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.
比较图3和图4中两种方法分解得到的IMF,本发明方法(最小相对方差准则)得到的IMF显著减小了模态混淆现象。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.
本发明的优点还在于:The present invention has the advantages of:
(1)根据余量的相对方差的大小来确定筛选迭代次数,有客观的规范标准。由于迭代次数考虑了数据自身的特征,如数据长度、数据的平均周期等的不同,会导致求解IMF时,所需的迭代次数也不同。(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)本终止准则的可操作性强。标准客观,可以定量刻画,代码自动确定迭代次数,而不需要依赖经验选择。(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)该方法符合EMD“自适应”的基本特征,同时也与Fourier、小波等分解方法的基本思想一致:低频模态(/分量/特征函数)应当能够解释更多的方差。(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.
以上所述仅是对本发明的较佳实施方式而已,并非对本发明作任何形式上的限制。凡是依据本发明的技术实质对以上实施方式所做的任何简单修改,等同变化与修饰,均属于本发明技术方案的范围内。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.
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