Wavelet series transform

Nofe For clarification of terminology, we refer to the wavelet transform" as being the procedure for producing the wavelet coefficients. When the function f(t) is represented as a linear combination of the wavelet coefficients and wavelet basis functions this is referred to as the "wavelet series representation" or "wavelet decomposition" of f(t). This is discussed in greater detail in Section 5. 

As stated previously, with most applications in analytical chemistry and chemometrics, the data we wish to transform are not continuous and infinite in size but discrete and finite. We cannot simply discretise the continuous wavelet transform equations to provide us with the lattice decomposition and reconstruction equations. Furthermore it is not possible to define a MRA for discrete data. One approach taken is similar to that of the continuous Fourier transform and its associated discrete Fourier series and discrete Fourier transform. That is, we can define a discrete wavelet series by using the fact that discrete data can be viewed as a sequence of weights of a set of continuous scaling functions. This can then be extended to defining a discrete wavelet transform (over a finite interval) by equating it to one period of the data length and generating a discrete wavelet series by its infinite periodic extension. This can be conveniently done in a matrix framework. 

When discrete wavelets are used to transform a continuous signal, the resulting set of coefficients is called the wavelet series decomposition. For this transformation to be useful it must be invertible, and for synthesis to take place, Eq. 9.11 must be satisfied... 

The discrete parameter wavelet transform (DPWT, or wavelet series). Here the... 

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. 

Wavelets and the wavelet transformation refer the representation of a spectral data set in terms of a finite spectral range or a rapidly decaying oscillating waveform. This waveform is scaled and translated to match the original spectmm. Wavelet transformation may be considered to calculate the time-frequency representation, related to the subject of harmonic analysis. The projection of a spectmm on a single wavelet or a series of wavelets reduces the dimensionality of the data set. Wavelet transforms are broadly divided into three classes, the continuous wavelet transform, the discrete wavelet transform and multiresolution-based wavelet transforms. Each class has advantages and disadvantages in terms of the wanted information. 

This vast spectral bandwidth illustrates the necessity of a reliable scale and time resolved decomposition of available observations to separate and describe single processes as individual parts of the whole system. Often, the comlex interplay between climate subsystems plays an essential role and the understanding of coupling mechanisms is of crucial importance for the study and prediction of at first sight independent phenomena. Continuous wavelet transformation (CWT) is the prototypic instrument to address these tasks As an important application, it transforms time series to the time/scale domain for estimating the linear non-stationary spectral properties of the underlying process. 

Given a time series s t), its wavelet transformation Wgs b, a) at time b and scale a (scale refers to 1 /frequency) with respect to the chosen wavelet g t) is given as... 

The first applications of wavelet transforms to analyse time series in the field of chemical dynamics were those of Permann and Hamilton [47,48]. Their interest lay in modelling diatomic molecules, close to dissociation, perturbed by a photon. They modelled the reaction using the equation of motion for a forced and damped Morse oscillator, given by ... 

Application of the Discrete Wavelet Transformation for Online Detection of Transitions in Time Series... 

Wavelets are very useful for applications that require local and multi-scale information. In this chapter an application of the discrete wavelet transform is discussed where these properties are used for online detection of transitions in time series. This method named Early Transition Detection is demonstrated on data from a chemical sensor array. 

In this chapter it will be demonstrated how wavelets can be used in data analysis by discussing a specific data set from a chemical sensor array. The data is a time series where each point in time belongs to a certain unknown class and needs to be classified. It will be discussed what problems arise when a common classifier like SIMCA [I] or Nearest Neighbour [2] is used. Further, it will be shown that an extended classifier named Early Transition Detection (ETD) [3] can be used to overcome these problems. For the construction of the ETD classifier the Discrete Wavelet Transform (DWT) is used. It will be shown that the DWT provides an expedient tool to solve this problem. 

Let us consider the discrete wavelet transform with the Haar wavelet. If the DWT with the Haar wavelet is applied to a time series the detail coefficients supply information about the temporal change of the time series. The detail coefficients of different levels correspond to changes on different time scales. Hence, these coefficients may serve as a measure of change of a time series. If a transition occurs in a time series the detail coefficients take values that are larger than the values they take during stable states. Below it is described how this property is used for detection of transitions. 

Akay, M. Time Frequency and wavelets. In Akay, M. (ed.) Biomedical Signal Processing IEEE Press Series in Biomedical Engineering. Wiley—IEEE Press, Piscataway (1997) Alsberg, B.K., Woodward, A.M., Kell, D.B. An introduction to wavelet transform for chemometricians a time-frequency approach. Chemometr. Intell. Lab. Syst. 37, 215-239... 

Chau, F.-T., Liang, Y.-Z., Gao, J., and Shao, X.-G. (2004) Chemo-metrics From Basics to Wavelet Transform (Chemiced Analysis A Series of Monographs on Analytical Chemistry and Its Applications), John Wiley ... 

Two other references can be recommended for their brevity. A monograph on wavelets transforms contains a chapter by van den Bogaert-" which provides a rather elegant and intuitive introduction to FT. In another volume from the same series, Eveleigh- also introduces the FT and related transformations in a concise and clear but more classically mathematical way. Finally, the paper by Angelidis is also quite pedagogical but also more in-depth and a little longer. 

The continuous wavelet transform of a time series can be defined as ... 

According to HALIM et al. (2008), some steps can be evaluated in order to calculate the TDAS if x(n) represents the vibration signals on time domain, and the number of wavelet scales r is S, then the wavelet transformation of x(n) would generate the wavelet coefficient W(s, n), which is a matrix of SxN dimension. The matrix W (s, n) may be a complex matrix depending on the wavelet used. By taking the absolute value of each of the elements of the matrix W(s, n), the matrix V(s,n) is produced, where all elements of V (s, n) are real. Each row of this matrix is a time series corresponding to one scale s with period P. Recall that the period of the time series is P and the time series has exactly M periods, so that N = P x M. Each of these time series can be synchronously averaged based on... 

Figure 9 This series illustrates multiresolution analysis, separating out the high-frequency information at each level of transformation in the pyramid algorithm (illustrated in Figure 8). Note that the approximation (a,) signal in higher level iterations contain much less detailed information, because this has been removed and encoded into wavelet detail coefficients at each DWT deconstruction step. To reconstruct the original signal, the inverse DWT needs the wavelet coefficients of a given approximation level / (ai) and all detail information leading to that level (di i).

Specifically, given a signal represented by a series of 2 values, the signal is divided into N discrete levels of detail. The transformation turns the original signal into a set of 2 wavelet coefficients, which reflect the individual contributions of their associated scaled wavelet. The original signal is really a combination of the dilated and translated wavelets as prescribed by the wavelet coefficients. 

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