Category: Advanced Indicator Set 2

Input parameters

Calculations

The implementation is based on the work of Dr. Murtagh (see reference below), who made a major contribution to the research of wavelet transformations of financial time series. The Redundant Haar Wavelet (RHW) calculations are as follows:

Resolution level 0:

RHW[0, i] = 0.5*(Time Series[i] + Time Series[i-1])

Resolution levels 1-16:

RHW[reslevel, i] = 0.5*(RHW[reslevel-1, i] + RHW[reslevel-1, i-2^reslevel])

Here reslevel is the resolution level, and i is the current bar.

Discussion

RHW is based on a classical continuous Haar wavelet transform invented in the first part of the 20th century. The RHW transform was specifically designed to work with discrete financial time series without introducing too much lag. Each resolution level of RHW is formed by way of convolution of the previous resolution level with a wavelet function. The RHW’s wavelet function is just an average between the current bar and a bar from past history, whose distance from the current bar is 2^reslevel.

The increasing “dilation” of the wavelet function makes RHW a good candidate for filtering out discrete time series without introducing too much lag into the smoothed curve. The RHW output represents a smoothed input time series. The lowest smoothing effect is achieved at resolution level = 0. The higher the resolution level, the greater the smoothing. The smoothing effect rises as 2^reslevel.

The redundancy of RHW means that it requires as many data points on each resolution level as the underlying time series has. Moreover, as far as each resolution level is computed from the previous one, the amount of time and memory resources to compute RHW grows linearly with the resolution level number. This may be a factor for very large time series on some older computers.

A consequence of the dilated wavelet function is that the RHW calculation requires a number of past bars, which increases with resolution level. This number is equal to 2^(reslevel+1). For example, for reslevel=3 it requires only 16 bars. However, for reslevel=10 it already requires 2048 bars, and for reslevel=16 this number is 131072 bars. In most cases, the resolution level of 3 is a very good default setting.

Note that this indicator, at low resolution levels, works more like an adaptive moving average – a moving average with low lag.

Another Haar wavelet indicator available from this group is the Redundant Haar Wavelet Coefficient.

Reference

G. Zheng, J.L. Starck, J.G. Campbell and F. Murtagh, "Multiscale transforms for filtering financial data streams", Journal of Computational Intelligence in Finance, 7, 18-35, 1999. Online version is available at http://strule.cs.qub.ac.uk/~gzheng/financial-engineering/finpapermay99.html