
Daily Speculations 
08May2006
Wavelets Part II, from
James Sogi
As a surfer, my lifelong pursuit is to ride the waves. The Wavelet analysis mentioned in my last post uses filtering algorithms to capture one cycle, one wave, showing the cycle in an otherwise noisy looking time series. The output seems well suited to identify changing cycles. The results return a multiresolution analysis with the ability to change scales. The pyramid algorithms are filters that transform the time series into multiple vectors that split the time series into high pass and low pass vectors, and then split those vectors again, and into third and fourth moments and more and the output vectors create a matrix. Each step has a wavelet filter (high pass) and a scaling filter (low pass), corresponding in most cases to a high pass and low pass filters. The LA(8) ("least asymmetric") algorithm seems to be the best smoothing. The schematic of the pyramid algorithm is tree formed. The matrices are plotted as multi resolution analysis (MRA) along different scales to identify the change in cycles by smoothing the times series in a way (zero sum and orthogonal) that helps match the peaks and valleys of the original series without the lags that plague moving averages or other Ehler filters. The zero phase filter lines up with the original time, or in real time for that matter, without lag. The matrices are down sampled by inserting 0's which means taking evens or odds (similar to the down sampling tick data into 1/2 hr bars). Circular filtering allows adjustment to tune the wave to match the underlying. Wavelet analysis can be to be used for financial series to identify critical turning points. Wavelets have boundary and length issues like moving averages with the annoying drop offs, but the cycle wave utilizes a zero phase filter which cures some of the problem with use of overlaps (modwt). This qualitative summary is rendered in mathematical symbols for manipulation and proof. This troglodyte will analyze the jet engine that has fallen out of the sky using applied object oriented computational methods rather than Greek symbols. It sounds all rather complicated, but boils down to pretty simple stuff for the small price of a book. The attached wavelet plot is actual R output example using IBM.
The question is what are wavelet MRA plots used for and are they predictive? In practical use, the stretch away from the smoothing, a technique used with other smoothers, might be profitable, or the trend following methods. The Haar maximum overlap dynamic wave transform (modwt) smoothing based on a zero filter tends to line up with original series with reduced the boundary effect and shows the turning points. Its good to have a bias which acts as a rudder. The wavelets help tell when its time to throw the tiller over on the ship.
Is it predictive? The discreet wave transform can be used with ANOVA. probability density functions, Bayesian analysis, chisquare distributions, autocorrelation and other statistical functions can be applied to the wavelets as a representation of a stochastic process, but a description must await the next installment as the grass is quite tall in here and hard to get through with such a dull blade.