Understanding how to do an Empirical Mode Decomposition

discrete mathematicsempirical-processessignal processing

I am trying to create my own code for Empirical Mode Decomposition and extraction of IMFs.

As far as I understand, the first part of the process must be done like this:

  1. find local maxima and minima. I understand that I have to find all peaks and valleys of the time series. It will be a peak if a value is bigger than the previous and bigger than the next point. It will be a valley if a value is smaller than the previous and smaller than the next.

  2. then I must create a cubic spline connecting these points.

Here is my first problem. I have created a time series for a sinusoidal wave and found 20 points as peaks and 19 as valleys. This time series has 1000 values and covers 15 seconds of data.

Now I have several questions:

  1. Suppose the first peak is at t=1.2s and the last peak is at t=13s. I create the first cubic spline using the peaks. So, now this spline can be used o find any value for any time between 1.2 and 13s. But the data goes from t=0 to t= 14. What about the times outside the spline interval?

  2. I have a different number of points for maxima and minima, 20 peaks and 19 valleys. So, the spline representing the minima will have 19 valleys and will represent a different interval. In my case, from t=0.5 to t=13.8. What about times outside this interval?

  3. It is time now to calculate the "average spline", but I have to do it for the full time covered by the data, that is 15 seconds. How do I do that if the splines cover smaller intervals?

Please explain it like I am five years old. My math sucks at this time.

thanks

Best Answer

I think it's common to just take the function value at each endpoint and consider it as a maximum and minimum. Then your piecewise function defined by the splines will be defined on the entire interval $[0,15]$. It doesn't matter than there are an unequal number of maxima and minima, at the end of this process we have an upper envelope (from the maxima) and a lower envelope (from the minima) which are defined on the entire interval $[0,15]$; the number of maxima or minima only defines how many "pieces" we have in our piecewise defined envelope.

Starting around slide 5 ish you can see the envelopes calculated.

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