The authors’ chief considerations with regard to the choice of mother wavelet are aliasing, implying preference for a redundant wavelet transform algorithm, and a wavelet function that respects the asymmetric nature of a time-varying signal, leading to use of the Haar wavelet function. A planning task involves considering the approximation of a time series at coarser and coarser resolutions, summarized in a multiresolution decomposition. The à trous wavelet transform is described simply by using successive convolutions with discrete low-pass filter h:

where the finest scale is the original series: c₀(t) = x(t). Wavelet coefficients w_i are obtained from the sequence of smoothings of the signal, taking the difference between successive smoothed versions Expansion of the original data produces For a fixed number of scales, the computational complexity is O(n) for an n-length input.

Based on S&P 500 and futures data, the authors examine a new wavelet transform. Its disadvantages (asymmetry, lack of smoothness properties) have been taken into consideration, as well as its advantages (most importantly, respect for temporal data). An approach to modeling and prediction, based on pattern finding in the time series data, is presented. The authors also examine how wavelet coefficients could provide useful features. The results obtained out-perform other methods.