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:

**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:

More on this later.

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