How does VMD determine the number of decomposition layers (3): through certain criteria (formula standard)

For VMD-related knowledge, please refer to the article "On VMD---Variational Mode Decomposition" I wrote before.

This article talks about determining the number of decomposition layers of VMD through certain guidelines. Refer to the SCI paper Optimization scheme of wind energy prediction based on artificial intelligence https://link.springer.com/article/10.1007/s11356-021-13516-2

Go directly to the criterion (formula):

 Parameter meaning:

K is the number of decomposition layers; i is the specific sample of each IMF after decomposition; n is the number of samples;

Here is a translation in vernacular, this formula.

For the samples of each IMF after decomposition, first calculate the sum of squares, then divide by the sample size, and then take the square root. In this way, as many IMFs as there are, there are as many specific numbers obtained through the above calculations (that is, K numbers), and these numbers are added together, which is EK (defined by the author as energy). This formula means two EK The difference, when K is 8, is |E8-E7|/E7.

This is the meaning of the formula, how to judge it specifically?

"From the optimal theoretical results of VMD decomposition, the sum of the energy of each component is equal to the original signal. When the k value is too large, the generation of virtual components will cause the sum of the energy of each component to be too high. Based on this principle, According to the calculation of signal energy and energy difference (formula (12)), when the value of θk, k-1 is small, the signal will be under-decomposed. As the value of θk, k-1 increases, VMD is obviously over-decomposed. Therefore , as the parameter k increases, there will be a decomposition phenomenon, and the corresponding value θk- k1 will increase significantly. In this case, it can be considered that k-1 is the optimal mode number for VMD decomposition."

The original author's words may not be easy to understand. Everyone needs to read the original text and think about it carefully. To put it bluntly, it is θk, when the value of k-1 suddenly increases, just take K-1 as the decomposition layer.

Or I personally think that I can understand it from the perspective of EK. The value of θk, k-1 has increased, that is, EK is much larger than Ek-1. Let’s take a picture from the original text to illustrate.

The ordinate of this figure is EK. It can be seen that when it is decomposed into 6 layers, its EK suddenly increases compared to when the decomposition layer is 5, so choose the number of decomposition layers to be 5. This is roughly what it means. I hope you all read the original text carefully.

 Here the above formula is linked to the VMD decomposition. Go directly to the code.

 It should also be emphasized here that regarding the decomposition problem, the same decomposition method is not applicable to all data. Here is only a way of thinking, which you can refer to for learning.

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Origin blog.csdn.net/weixin_46062179/article/details/124786926