Wavelet local analysis

Wavelet local analysis Kemel-Gaussian wavelet... 

A small scale value permits us to perform a local analysis a large scale value is used for a global analysis. Combining local and global is a useful feature of the wavelet analysis. The wavelets having a compact support are used in local analysis. This is the case for Haar and Daubechies wavelets. As wavelet analysis constraint to Heisenberg uncertainty principle, it is impossible to reduce arbitrarily both time and frequency localisation. The resolution increases as the scale decreases. The greater the resolution, the smaller and finer are the details that can be accessed. 

Now, thanks to the development of the local wavelet analysis [3], it is possible to represent mathematically, according to the observations, finite waves with a well-defined energy. Therefore it is possible to represent the 0 wave, the extended yet localized part of the particle, with a defined energy by a wavelet. The wavelets, or finite waves, were developed in geophysics by Morlet in the early 1980s, to avoid some shortcomings of the nonlocal Fourier analysis. 

If, instead of the nonlocal Fourier analysis, one uses the local wavelet analysis to represent a quantum particle, the uncertainty relationships may change in form. On the other hand, this process has the advantage of containing the usual uncertainty relations when the size of the basic gaussian wavelet increases indefinitely. 

From Fig. 18 it is seen that the new the uncertainty relations derived with the local wavelet analysis exhibit the form... 

In this representation, integer j indexes the scale or resolution of analysis, i.e., smaller j corresponds to a higher resolution, and jo indicates the coarsest scale or the lowest resolution, k indicates the time location of the analysis. For a wavelet 4> t) centered at time zero and frequency u>o, the wavelet coefficient dj k measures the signal content around time 2 k and frequency 2 uio- The scaling coefficient Ck measures the local mean around time 2 °k. The DWT represents a function by a countable set of wavelet coefficients, which correspond to points on a 2-D grid of discrete points in the scale-time domain. 

In the case of Fourier analysis, the coherence critical value is independent of the processes to be compared, if they sufficiently well follow a linear description [1, 15]. This independency, however, holds exactly only in the limit of long time series. As wavelet analysis is a localized measure, this condition is not fullfilled. Hence, for different AR[1] processes (from white noise to almost nonstationary processes), we found a marginal dependency on the process parameters. 

I. Daubechies, The Wavelet Transform, Time Frequency Localization and Signal Analysis, IEEE Transactions on Information Theory. 36 (1990), 961-1005. 

Generally, WT is superior to FT in many respects. In Fourier analysis, only sine and cosine functions are available as filters [13], However, many wavelet filter families have been proposed. They include the Meyer wavelet, Coiflet wavelet, spline wavelet, the orthogonal wavelet, and Daubechies wavelet [14,15]. Both Daubechies and spline wavelets are widely employed in chemical studies. Furthermore, there is a well-known drawback in Fourier analysis (Fig. 1). Since the filters chosen for the Fourier analysis are localized in the frequency domain, the time-information is hidden after transformation. It is impossible to tell where a particular signal, for example as that shown in Fig. 1(b), takes place [13]. A small frequency change in FT produces changes everywhere in the Fourier domain. On the other hand, wavelet functions are localized both in frequency (or scale) and in time, via dilations and translations of the mother wavelet, respectively. Both time and frequency information are maintained after transformation (Figs. 1(c) and (d)). 

There are few possible strategies of library compression. Each of them has its own advantages and drawbacks. The most efficient method of data set compression, i.e. Principal Component Analysis (PCA), leads to use of global features. As demonstrated in [15] global features such as PCs (or Fourier coefficients) are not best suited for a calibration or classification purposes. Often, quite small, well-localized differences between objects determine the very possibility of their proper classification. For this reason wavelet transforms seem to be promising tools for compression of data sets which are meant to be further processed. However, even if we limit ourselves only to wavelet transforms, still the problem of an approach optimally selected for a particular purpose remains. There is no single method, which fulfills all requirements associated with a spectral library s compression at once. Here we present comparison of different methods in a systematic way. The approaches A1-A4 above were applied to library compression using 21 filters (9 filters from the Daubechies family, 5 Coiflets and 7 Symmlets, denoted, respectively as filters Nos. 2-10, 11-15 and 16-22). 

Wavelets possess two properties that make them especially valuable for data analysis they reveal local properties of the data and they allow multi-scale analysis. Their locality is useful e.g. for applications that require online response to changes. If the typical time scales of these changes are not known in advance a multi-scale approach is advantageous. 

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