Chau Chemometrics From Basics to Wavelet Transform

chemical reactors and chromatographic columns. They indicated that the algorithm is stable and convergent and is a simple, accurate, fast, and unified approach. It is not necessary to analyze the model on the basis of particular physical laws to derive a scheme for tracking the steep front. Furthermore, wavelets have the capability of representing solutions at different levels of resolution, which makes them particularly useful for developing hierarchical solutions for chemical processes. The most important advantage of this method is that it simulates the chemical processes with steep fronts more effectively than do conventional numerical methods.

Nagy and Pipek [96] studied multiresolution analysis of density operators, electron density, and energy functionals and proved that, for real physical systems, neither arbitrarily fine nor arbitrarily rough details of the wavefunction and density operators can exist. They also showed that the calculation of both kinetic energy and interaction energy expectation values can be reduced to the determination of some universal functions defined on integer-valued arguments.

Leherte [97] proposed a wavelet-based multiresolution analysis approach to generate low-resolution molecular electron density (ED) distribution functions and compared the performance of the proposed method with that of the other two methods (a crystallography-based formalism and an analytical approach) using the critical point graph representations of the molecular ED distributions for pairwise molecular superpositions. The results showed that, for generation of the ED distribution, the crystallography-based method is the fastest. Wavelet-based multiresolution analysis (WMRA) is the most time-consuming, but it is applicable to any grid representation of a molecular property. The analytical approach offers the advantage of generating functions that are continuously scalable, but requires an analytical description of the molecular property considered. For molecular superpositions, the WMRA approach led to the most consistent superposition results.

5.6.7. Conclusion

Applications of WT in various fields of chemistry have been extensively studied. Besides the contents and the works reviewed in this chapter, there are many other works related to applications of WT in chemistry, such as discriminant analysis, multiscale analysis, multifractal analysis, transition detection, and protein sequences analysis. These studies, have proved that WT based techniques are very efficient tools and have played an important role in chemical signal processing. There is no doubt that

an increasing number of chemists will be interested in exploring more application of WT in chemistry and will derive more benefit from using the techniques. WT-based techniques will be one of the most popular methods in the future.

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LIST OF WAVELET-RELATED INTERNET WEBSITES

A wavelet tutorial: http://engineering.rowan.edu/ polikar/wavelets/wttutorial.html.

A wavelet tutorial: http://nt.eit.uni-kl.de/wavelet/.

A practical guide to wavelet analysis: http://paos.colorado.edu/research/wavelets/. A wavelet tutorial from S. Mallat’s book: http://cas.ensmp. fr/ chaplais/Wavetour_

presentation/Wavetour_presentation_US. html.

256 application of wavelet transform in chemistry

Amara’s wavelet page---wavelet resources: http://www.amara.com/current/wavelet. html.

Wavelets---internet resources: http://www.cosy.sbg.ac.at/ uhl/wav.html. Information on wavelets: http://lettuce.ms.u-tokyo.ac.jp/mei/wavelet.html. WaveLab: http://www-stat.stanford.edu/ wavelab/.

The MathWorks---wavelet toolbox: http://www.mathworks.com/products/wavelet/. Wavelet Explorer new-Generation signal and image analysis: http://www.wolfram.

com/products/applications/wavelet/.

Digital signal processing at Rice University: http://www-dsp.rice.edu/software/. XWPL, the X Wavelet Packet Laboratory: http://math.yale.edu/pub/wavelets/software/

xwpl/html/xwpl.html.

WAVEKIT---a wavelet toolbox for MATLAB: http://www.math.rutgers.edu/ ojanen/ wavekit/.

Wavelet toolbox: http://www.comsol.se/products/wavelet/index.php.

Bibliographies on wavelets: http://liinwww.ira.uka.de/bibliography/Theory/Wavelets/. Wavelet transform in chemistry: http://fg702-6.abct.polyu.edu.hk/wavelet.html. Wavelet papers: http://www.ee.umanitoba.ca/ ferens/wavelets.html.

The Wavelet Digest: http://www.wavelet.org/.

a=[a1 . . . ak . . . an]

Figure A.1. Illustration of a digitized spectrum.

3.Most important feature of the hyphenated instruments is that they combine a chromatography (e.g. CE, GC, HPLC) and a multichannel spectroscopic detector (e.g. UV, IR, MS). At each regular time interval, a complete spectrum is measured. Consequently, random errors that occur in the chromatogram also influence the spectrum. Apart from these inevitably related fluctuations, the data will also be contaminated by detector noise which could be correlated among adjacent channels. Moreover, noise that is proportional to the magnitude of the signal is more common than purely additive noise. As a result, the overall noise present in the ‘‘spectrochromatographic’’ data collected is complicated by different factors, and more importantly, the noise is heteroscedastic in nature. Thus, the pretreatment of the two-dimensional data is more difficult.

.

.

..

...

..

..

.

Figure A.2. The physical meanings of the rows and columns of two-dimensional data matrix as generated by a hyphenated instrument.