Wavelet coefficient method

The combination of PCA and LDA is often applied, in particular for ill-posed data (data where the number of variables exceeds the number of objects), e.g. Ref. [46], One first extracts a certain number of principal components, deleting the higher-order ones and thereby reducing to some degree the noise and then carries out the LDA. One should however be careful not to eliminate too many PCs, since in this way information important for the discrimination might be lost. A method in which both are merged in one step and which sometimes yields better results than the two-step procedure is reflected discriminant analysis. The Fourier transform is also sometimes used [14], and this is also the case for the wavelet transform (see Chapter 40) [13,16]. In that case, the information is included in the first few Fourier coefficients or in a restricted number of wavelet coefficients. 

The multiscale basis functions capture the fast changes in coefficients corresponding to the fine-scale basis functions, while the slower changes are captured by the coarse-scale basis functions. Thus, the wavelet thresholding method adapts its resolution to the nature of the signal features and reduces the contribution of errors with minimum distortion of the features retained in the rectified signal. 

The training problem determines the set of model parameters given above for an observed set of wavelet coefficients. In other words, one first obtains the wavelet coefficients for the time series data that we are interested in and then, the model parameters that best explain the observed data are found by using the maximum likelihood principle. The expectation maximization (EM) approach that jointly estimates the model parameters and the hidden state probabilities is used. This is essentially an upward and downward EM method, which is extended from the Baum-Welch method developed for the chain structure HMM [43, 286]. 

Chau and his co-workers have proposed some wavelet-based methods to compress UV-VIS spectra [24,37]. In their work, a UV-VIS spectrum was processed with the Daubechies wavelet function, Djfi. Then, all the Cj elements and selected Dj coefficients at different) resolution levels were stored as the compressed spectral data. A hard-thresholding method was adopted for the selection of coefficients from Dj. A compression ratio up to 83% was achieved. As mentioned in the previous section, the choice of mother wavelets is vast in WT, so one can select the best wavelet function for different applications. Flowever, most workers restrict their choices to the orthogonal wavelet bases such as Daubechies wavelet. Chau et al. chose the biorthogonal wavelet for UV VIS spectral data compression in another study [37]. Unlike the orthogonal case, which needs only one mother wavelet (p(t), the biorthogonal one requires two mother wavelets. (p(t) and (p(t), which satisfy the following biorthogonal property [38] ... 

All the wavelet coefficients above resolution level k are set to zero and do not contribute. For each combination i the chosen multivariate method is applied to the masked data. Fig. 5 illustrates the basic idea behind the method. 

The improved performance of the multiscale approach is due to the ability of orthonormal wavelets to approximately decorrelate most stochastic processes, and compress deterministic features in a small number of large wavelet coefficients. These properties permit representation of the prior probability distribution of the variables at each scale as a Gaussian or exponential function for stochastic and deterministic signals, respectively. Consequently, computationally expensive non-parametric methods need not be used for estimating the probability distribution of the coefficients at each scale. If the probability distribution of the contaminating errors and the prior can be represented as a Gaussian, the multiscale Bayesian approach provides... 

Thresholding techniques are successfully used in numerous data processing domains, since in most cases a small number of wavelet coefficients with large amplitudes preserves most of the information about the original data set. Different thresholding methods like... 

ABSTRACT This paper provides a short review of recent developments in crash pulse analysis methods and a short review of wavelet based data processing methods. A discrete wavelet transform can he performed in 0 n) operations, and it captures not only a frequency of the data, but also spatial informations. Moreover wavelet enables sparse representations of diverse types of data including those with discontinuities. And finally wavelet based compression, smoothing, denoising, and data reduction are performed by simple thresholding of wavelet coefficients. Combined, these properties make wavelets a very attractive tool in mary applications. Here, a noisy crash signals are analyzed, smoothed and denoised by means of the discrete wavelet transform. The optimal choice of wavelet is discussed and examples of crash pulse analysis are also given. 

All compression, denoising, and data reduction methods retain the largest Nx number of coefficients depending on the choice of appropriate threshold k. These methods follow the principle assuming that large wavelet coefficients characterize signal better in the sense of its energy and thus retain more information. 

In the signal processing, we can use either the linear approximation method or nonlinear approximation method. The linear wavelet-based approximation picks wavelet coefficients from the coarsest level to the finest level while the nonlinear wavelet-based approximation selects wavelet coefficients adaptively e.g. it takes N largest coefficients in absolute value. In both approaches N is taken to be fixed or for instance in the way to satisfy the predetermined error bound. The wavelet coefficients selected from the above approximation methods are usually treated as compressed data. 

Figure 2.1 gives a comparison between the steady state solution for the liquid and the vapour phase mole fractions calculated by the Wavelet-Galerkin method and by a full discrete model. As expected the composition profiles are almost identical, the deviations are resulting from the interpolation of the physical property routines due to the formulation (2.20) of the phase equilibrium coefficient in the wavelet flash. Note that this is no error but merely a... 

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