Joint best-basis

Walczak, B. and Massart, D. L. (2000). Joint basis and joint best-basis for data. In "Wavelets in Chemistry", (B. Walczak, Ed.), pp. 165-176. Elsevier, Amsterdam. 

Joint Basis and Joint Best-basis for Data Sets... 

Wavelet packet transform and joint best-basis... 

In the second case, the uniform representation of signals in the wavelet domain is not possible and instead, the joint best-basis, in which a small number of wavelets coefficients can describe the majority of data variance, must be selected. This can be done, by applying Coifman and Wickerhausers algorithm of best-basis selection to the so-called variance tree", the elements of which represent the variance of wavelet coefficients with the same addresses (indices) (see Fig. 8) [5]. 

Once the variance tree is constructed, it can be searched for the joint best-basis, using the Coifman-Wickerhauser best-basis selection algorithm with, e.g., the entropy criterion. The entropy cost function (see Chapter 6) for the variance tree coefficients, which occur at the jth level in the i band of the signal decomposition is defined as ... 

The orthogonal basis minimizing entropy is considered as the joint best-basis for the data set. Of course, while entropy is a good measure of efficiency of an expansion, various other cost functions are possible. 

The main steps of the joint best-basis selection procedure [4] can be summarized as (Fig. 9) ... 

Updating of the joint best basis requires the following steps ... 

Searching the variance tree for the joint best basis. 

All signals from the NIR data set were decomposed by WPT. The joint best-basis selected for the variance tree, is presented in Fig. 10(a), whereas the variance vector in that basis is visualized in Fig. 10(b). 

The percentage of the explained variance in function of the number of the retained coefficients is presented in Fig. 10(c). In the selected joint best-basis, only 39 wavelet coefficients only are necessary to describe 99% of data variance. 

Fig. 10 (a) Joint best-hasLs. selected for the variance tree of NIR data set (Egain denotes entropy drop between parents and child hands) (h) Elements of vector v in the joint best-basis (c) Cumulative percentage of the explained variance versus the number of the... 

While dealing with the data set we are interested in finding a joint basis (oi joint best-basis) for compression of the whole set. Only joint basis allows... 

For the A4 approach we need to store m times N wavelet coefficients values, vector containing their addresses, and vector containing addresses of the joint best-basis, i.e. A4 matrix R, and vectors A and B. 

Joint basis and joint best-basis approaches to data set compression... 

In the discussed approach the compressed data set has uniform representation (see Fig. 2). Compression ratio equals 7.0621 for DWT, and 7.3669 for WPT. Both, joint basis and joint best-basis minimize the storage require-... 

Fig. 12 (a) The top 200 coefficients (.squared) in the joint basis, and the top 200 elements of variance tree in the joint best basis sorted according to their amplitude and (h) cumulative percentage of the explained variance in the join basis, and in the best joint-... 

---