Wavelet approach

To analyze the spectral composition of the pressure variations in more detail we have made use of a wavelet approach [9]. This approach, which allows us to determine instantaneous values of the frequencies and amplitudes of the various oscillatory components, is particularly useful for biological time series that often are neither homogeneous nor stationary. 

O.V. Sosnovtseva, A.N. Pavlov, E. Mosek-ilde, N.-H. Holstein-Rathlou, and D. J. Marsh, Double-Wavelet Approach to Study Frequency and Amplitude Modulation in Renal Autoregulation, Phys. Rev. E 70, 031915-8 (2004). 

Let us consider that the niunber of echoes M and the incident wavelet (/) (e.g., a normalized comer echo) are known. Least Squares approach for estimating parameter vectors x and requires the solution to the nonlinear least squares problem ... 

In most solids, the sound speed is an increasing function of pressure, and it is that property that causes a compression wave to steepen into a shock. The situation is similar to a shallow water wave, whose velocity increases with depth. As the wave approaches shore, a small wavelet on the trailing, deeper part of the wave moves faster, and eventually overtakes similar disturbances on the front part of the wave. Eventually, the water wave becomes gravitationally unstable and overturns. 

The space-frequency localization of wavelets has lead other researchers as well (Pati, 1992 Zhang and Benveniste, 1992) in considering their use in a NN scheme. In their schemes, however, the determination of the network involves the solution of complicated optimization problem where not only the coefficients but also the wavelet scales and positions in the input space are unknown. Such an approach evidently defies the on-line character of the learning problem and renders any structural adaptation procedure impractical. In that case, those networks suffer from all the deficiencies of NNs for which the network structure is a static decision. 

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. 

Other approaches Hamming networks, pattern recognition, wavelets, and neural network learning systems are sometimes discussed but have not been commercially implemented. 

While we have discussed pre-processing steps in a sequential manner, newer approaches are capable of integrating all the steps into single transformation approaches. For example, applications of wavelet transforms [77] can... 

One of the more challenging unsolved problems is the representation of transient events, such as attacks in musical percussive sounds and plosives in speech, which are neither quasi-periodic nor random. The residual which results from the deterministic/stochastic model generally contains everything which is not deterministic, i.e., everything that is not sine-wave-like. Treating this residual as stochastic when it contains transient events, however, can alter the timbre of the sound, as for example in time-scale expansion. A possible approach to improve the quality of such transformed sounds is to introduce a second layer of decomposition where transient events are separated and transformed with appropriate phase coherence as developed in section 4.4. One recent method performs a wavelet analysis on the residual to estimate and remove transients in the signal [Hamdy et al., 1996] the remainder is a broadband noise-like component. 

We elaborate on these in the following sections. The pipelines explored here have been used by Konig et al. (23), but these investigators used another feature extraction technique, the Haar wavelet transform. We explored a novel feature extraction technique based on a combinatorial approach. We confirm further the results obtained via the pipelines above using the Haar wavelet transform. This transform was done using the clusters due to the consecutive-ones clustering technique (23). 

The wave group remains compact and of constant extension, but the breadth and number of the wavelets vary with time, becoming narrower and more numerous as the centre (q = 0) is approached, and smoothed out at the turning points hence a function of velocity. 

Transformation — Several approaches are available for transformation of time domain data into the - frequency domain, including - Fourier transformation, the maximum entropy method (MEM) [i], and wavelet analysis [ii]. The latter two methods are particularly useful for nonstationary signals whose spectral composition vary over long periods of time or that exhibit transient or intermittent behavior or for time records with unevenly sampled data. In contrast to Fourier transformation which looks for perfect sine... 

Wavelet analysis is a rather new mathematical tool for the frequency analysis of nonstationary time series signals, such as ECN data. This approach simulates a complex time series by breaking up the ECN data into different frequency components or wave packets, yielding information on the amplitude of any periodic signals within the time series data and how this amplitude varies with time. This approach has been applied to the analysis of ECN data [v, vi]. Since electrochemical noise is 1/f (or flicker) noise, the new technique of -> flicker noise spectroscopy may also find increasing application. 

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