Mode of the Signal Envelope

One thing that struck me as clever with the HHT was the use of projecting a spline across the minima and maxima for a given harmonic.   In effect this defines the envelope for the series for a given harmonic (level of decomposition).   A posteri, the mean or mode should be more or less equivalent to the average of the envelope splines.   Interesting!

This is a very appropriate way to model the mean within the context of mean-reversion (ie oscillations around the mode within an envelope).   Instead of trying to model the mean directly as a stochastic process, why not model the envelope — this is more appropriate as we can fit the envelope into our view of mean reversion.

Version 1
I used a regressor to estimate the mean and connected minima and maxima with a spline for the envelope.  The approach has issues (such as what sort of bias does the mean regressor have with respect to the data).   There are some issues below:

Picture 1

Version 2
I took a dfference approach, estimating the inflection points with a regressing “oscillator”  (in green) and determining the mid-points between minima and maxima to produce a spline representing the mode (blue).   So far looks good.   Edge cases, consolidation, and jumps need to be considered:

Picture 2

More on this later.


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Filed under mean, regression, signal-processing, statistics, technical-analysis

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