# S+Wavelets Version 2

The new methodology implemented in S+Wavelets 2.0 stems almost entirely from Don Percivals book entitled, Wavelet Methods for Time Series Analysis, co-authored by Andrew Walden and published by Cambridge University Press in 2001. In addition, some functionality was provided by Derek Stanfords team (consisting of Alan Gibbs and Keith Davidson). A summary of the relevant new methodology is as follows:

### New wavelet transforms

Specifically, these transforms are DWT, MODWT, DWPT, MODWPT. These differ from those in S+Wavelets 1.x in that (1) they are based on convolution style filtering, (2) they only use periodic boundary conditions (making them energy conservative and usable for wavelet variance estimation), and (3) they are designed so that the boundary coefficients (those subject to circular filter operations) are easily identifiable, making them useful for certain wavelet-based statistics. We also have implemented the Dual Tree Wavelet Transform (DTWT) in both 1-D and 2-D. The DTWT has the advantage of being approximately shift-invariant without only a two-fold degeneracy. The corresponding multi-resolution decompositions and approximations are also coded in S+Wavelets 2.0.

### Wavelet variance-covariance estimators

If you integrate (over frequency) the spectral density function (SDF), you obtain the variance. The wavelet variance is a regularization of the SDF in that it represents the variance of the process on a scale-by-scale basis (each scale is associated with a particular octave band of frequencies). Both biased and unbiased versions of the MODWT and DWT variance estimators have been implemented.

### Fractionally Differenced (FD) Process parameter estimators

If the SDF follows a power law, then it is known as a colored noise process (e.g., white noise, pink or 1/f noise, random walk or red noise, etc.). The level of the SDF and the log-log slope (alpha) of the SDF are the two parameters that govern the model. Many real-world processes exhibit colored noise behavior, whose alpha fluctuates as a function of time. Of all the power law models, the so-called FD model is the best (no restriction on alpha, no continuity problems at pink

noise border, simple mathematical model). In S+Wavelets, we have implemented wavelet-based estimators of FD model parameters, ranging from instantaneous to block estimators. We also have two types: weighted least squares and maximum likelihood estimators. This technology has been used successfully on the analysis of aerothermal turbulence data. But it may also have a place in the finance community, where (possibly short term) SDF power law behavior exists.

### Hidden Markov Tree Modeling (2D)

This is used to model clutter in synthetic aperture radar (SAR) images. The idea is to model the clutter via the 2-D DWT so that it captures cross-scale relationships (Keith Davidson and Derek Stanford)

S+WAVELETS Version 2へのバージョンアップに関しては、NTTデータ数理システムまでお問合せください。