There are a lot of papers out there which make attempts to forecast or discuss the benefits of wavelets for frequency decomposition.

Oddly, very few discuss the huge boundary effects that are present when used with a non infinite series (most cases I can think of in finance).

Do any practical usages exist for wavelets without the benefit of hindsight?

I've gone through MODWT, SWT, DWT and CWT these all look great on an existing series but terrible when used in a causal sense.

Has anyone out there had any luck through extending their end point via Kalman or something else before applying a wavelet?

  • $\begingroup$ You can use a causal wavelet that isn't analytic to forecast. They exist in the literature, but unless you're going 100% hardcore into wavelets (e.g. trying to do band-pass phase difference analysis) you would just usually make your own in the real world. $\endgroup$ Oct 2 '13 at 10:47
  • $\begingroup$ The issue is pretty well known and a good discussion can be found in amazon.com/Neural-Novel-Hybrid-Algorithms-Prediction/dp/…. In the text, he discusses boundary effects and transformations to mitigate them. $\endgroup$
    – pat
    Oct 13 '13 at 0:40

The issue with any extension method you may use is that it has influence on the coefficents post boundary one way or another. Traditional methods (zero padding, symetric, polynomial extrapolation etc) carry because of the circularity of the wavelet function incluence into the coefficients. You can attempt to use any of the extensions commented if the signal you are considering is symetrical or if the signal ends and start with 0 or if it is periodic, however be very carefull in considering the boundary coefficients if your signal is not of any cathegories mentioned as in forecasting the signal forward you may have no edge effect but you will carry an error when you extrapolate (extrapolated signal vs you dont know actual). So your edge coefficients will be as good as your forecast, in addition depending which one you use orthogonatily may be lost. Therefore if your interest is in obtaining non affected coefficient on the edges for prediction to my knowledge is not possible with orthogonal wavelets. To forecast on the edges you can use the following methods:

A linear pass filter - This is using the redundant Haar wavelet replacing the symmetric low pass filter (B3 Spline filter) by a simple non symmetric filter. Then the scaling coefficients are calculated { ½ * (smoothed data t-1 -1(t)+smoothed data t-1 (t-2^j-1))} while the wavelet coefficients is obtained by calculating the difference between two successive smoothed versions (two neighboring resolution levels), effectively shifting the wavelet function on the time series data one point at a time, instead of 2j points at a time as with the normal DWT, where j denotes the current level of resolution.

You can also use non boundary coefficients for prediction. However you will have to exclude a large number of coefficients at each side of the signal per level of resolution (more on the right than on the left due to the window used by the WT) and boundary coefficients increase the higher the level. So you are realistically left with the lowest resolution levels which have the lower number of boundary coefficients (However this is where most of the energy of the signal is contained).

Alternatively, you can design an special filter to replace original filters at the signal borders (including the boundary elements). Such filters are adapted to interval boundaries and thus, do not require signal values outside the interval. However the function employed carries assumptions and can make the comparison of edge coeficients with rest of the transform function not straight forward or correct.

Other considerations:

  1. Is your signal continous or discrete (MRD is good for high frequency while discrete for less frequent evently spaced signals)

  2. Selection of length of the filter

  3. How good is the frequency representation of the different wavelets.

  4. Shift sensitivity, small shifts in the input signal can cause major variations in the distribution of energy between DWT coefficients at different scales.

I hope this helps

  • $\begingroup$ This is really great. $\endgroup$ Mar 27 '14 at 9:23

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