Wavelet spectrum

Cl and C2 being constants. As expected, the Fourier spectrum is given by the wavelet spectrum plus a correction term. The latter depends on the localization of the used wavelets and on the slope of the wavelet spectrum. For the asymptotic behavior defined in Eq. (12.8), it vanishes for high frequencies. For details and derivations refer to [17]. 

Bias of the wavelet sample spectrum The bias of the estimated wavelet spectrum reads... 

The structure of the pointwise test is similar to that developed for the wavelet spectrum. As the coherence is normalized to the single wavelet spectra, the critical value becomes independent of the scale as long as the smoothing is done properly according to Sec. 12.4.2, i.e. when the geometry of the reproducing kernel is accounted for. 

Also here, an areawise test can be performed to sort out false positive patches being artefacts from time/frequency resolved analysis. The procedure is exactly the same as for the wavelet spectrum, only the critical patchsize Pcrit b, a) has to be reestimated. Areawise significant patches denote significant common oscillations of two processes. Here, common means that two processes exhibit a rather stable phase relation on a certain scale for a certain time intervall. 

Ta-Hsin Li. Wavelet Spectrum and Its Characterization Property for Random Processes. IEEE Trans. Inf. Th., 48(ll) 2922-2937, 2002. 

Wavelet Spectrum of Yura Y88121401 Surface Waves... 

S is a wavelets spectrum, W is an analyzing wavelets matrix, and is the transpose matrix of W. This study uses Coifman function of order 30 as the analyzing wavelets as shown in Fig. 2. This basie wavelet is expanded in the multi-scale to analyze the image D [9]. Because the wavelets transform is an orthonormal transformation, the inverse wavelets transform and its multiresolution is expressed by... 

For x,y (IR), the wavelet coherence of two signals x and y, u - (-,-)with a fixed smoothing parameter 5 E M > 0 and wavelet is defined as the cross-wavelet spectrum of the two signals normalized by their corresponding autospectra ... 

Zhang et al.14 develop a neural network approach to bacterial classification using MALDI MS. The developed neural network is used to classify bacteria and to classify culturing time for each bacterium. To avoid the problem of overfitting a neural network to the large number of channels present in a raw MALDI spectrum, the authors first normalize and then reduce the dimensionality of the spectra by performing a wavelet transformation. 

A disadvantage of Fourier compression is that it might not be optimal in cases where the dominant frequency components vary across the spectrum, which is often the case in NIR spectroscopy [40,41], This leads to the wavelet compression [26,27] method, which retains both position and frequency information. In contrast to Fourier compression, where the full spectral profile is fit to sine and cosine functions, wavelet compression involves variable-localized fitting of basis functions to various intervals of the spectrum. The... 

Usual procedures for the selection of the common best basis are based on maximum variance criteria (Walczak and Massart, 2000). For instance, the variance spectrum procedure computes at first the variance of all the variables and arranges them into a vector, which has the significance of a spectrum of the variance. The wavelet decomposition is applied onto this vector and the best basis obtained is used to transform and to compress all the objects. Instead, the variance tree procedure applies the wavelet decomposition to all of the objects, obtaining a wavelet tree for each of them. Then, the variance of each coefficient, approximation or detail, is computed, and the variance values are structured into a tree of variances. The best basis derived from this tree is used to transform and to compress all the objects. 

B. A. Hess Dr. Gaspard has introduced the vibrogram as a tool to extract periodic orbits from the spectrum by means of a windowed Fourier transform. This raises the question whether other recent techniques of signal analysis like multiresolution analysis or wavelet transforms of the spectrum could be used to separate the time scales and thus to disentangle the information pertinent to the quasiclassical, semiclassical, and long-time regimes. Has this ever been tried ... 

Although the Fourier compression method can be effective for reducing data into frequency components, it cannot effectively handle situations where the dominant frequency components vary as a function of position in the spectrum. For example, in Fourier transform near-infrared (FTNIR) spectroscopy, where wavenumber (cm-1) is used as the x-axis, the bandwidths of the combination bands at the lower wavenumbers can be much smaller than the bandwidths of the overtone bands at the higher wavenumbers.31,32 In any such case where relevant spectral information can exist at different frequencies for different positions, it can be advantageous to use a compression technique that compresses based on frequency but still preserves some position information. The Wavelet transform is one such technique.33... 

From the wavelet coefficients Ttl,[x](a, b) one can calculate the energy density Ethree-dimensional surface E(l,[x](a, t). Sections of this surface at fixed time moments t = b define the local energy spectrum Ev[x](f, t) with / = a 1. Finally, in order to obtain the mean spectral distribution of the time series x t) we may consider a so-called scalogram, i.e., the time-averaged energy spectrum. This is analogous to the classic Fourier spectrum. 

The evaluation of the measurements, the correlation between the medium components and the various ranges of the 2D-fluorescence spectrum was performed by Principal Component Analysis (PCA), Self Organized Map (SOM) and Discrete Wavelet Transformation (DWT), respectively. Back Propagation Network (BPN) was used for the estimation of the process variables [62]. By means of the SOM the courses of several process variables and the CPC concentration were determined. 

Figure 5 Use of metabolite basis functions to fit clinical MRS data. (A) Final metabolite + baseline fit (black) overlaid on raw data (grey). (B) Non-parametric baseline signal estimation (based on wavelet filtering). (C) Metabolite basis functions modulated via scaling, B0 shift, lineshape and phase 0 and phase 1 to optimally fit raw data. (D) Residual spectrum of metabolite + baseline minus the raw data.

Wavelet transforms (Section 3.6.2) are a hot topic, and involve fitting a spectrum or chromatogram to a series of functions based upon a basic shape called a wavelet, of which diere are several in the literature. These transforms have the advantage that, instead of storing, for example, 1024 spectral datapoints, it may be possible to retain only a few most significant wavelets and still not lose much information. This can result in both data decompression and denoising of data. 

Wavelet transforms are normally applied to datasets whose size is a power of two, for example consisting of 512 or 1024 datapoints. If a spectrum or chromatogram is longer, it is conventional simply to clip the data to a conveniently sized window. 

A wavelet is a general function, usually, but by no means exclusively, of time, g(t), which can be modified by translation (b) or dilation (expansion/contraction) (a). The function should add up to 0, and can be symmetric around its mid-point. A very simple example the first half of which has the value +1 and the second half —1. Consider a small spectrum eight datapoints in width. A very simple basic wavelet function consists of four —1 s followed by four —Is. This covers the entire spectrum and is said to be a wavelet of level 0. It is completely expanded and there is no room to translate this function as it covers the entire spectrum. The function can be halved in size (a = 2), to give a wavelet of level 1. This can now be translated (changing b), so there are two possible wavelets of level 1. The wavelets may be denoted by [n, m] where n is the level and m the translation. 

The key to the usefulness of wavelet transforms is that it is possible to express the data in terms of a sum of wavelets. For a spectrum 512 datapoints long, there will be 511, plus associated scaling factors. This transform is sometimes expressed by... 

The first involves smoothing. If the original data consist of 512 datapoints, and are exactly fitted by 511 wavelets, choose die most significant wavelets (those widi die highest coefficients), e.g. die top 50. In fact, if die nature of the wavelet function is selected with care only a small number of such wavelets may be necessary to model a spectium which, in itself, consists of only a small number of peaks. Replace the spectrum simply with that obtained using the most significant wavelets. 

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