The Atomic Decompositions of Signals

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6.1 A Hybrid Transformation: Evolution of the "Matching Pursuit" Towards the Mallat and Zhang Version

It is one of the applications of waveforms theory, which corresponds to the transformation by the "Pursuit" algorithm with adaptive window. This technique will be applied in this chapter to a stock index, i.e. the French stock index: Cac40. We know that this transformation decomposes the signal in a time-frequency plane, the analyzing function is usually Gaussian of a variable width. The variable elements in the decomposition are the frequency, the position of the window and the size of the window, and we know that these three elements are independent. This transformation is particularly adapted to the strongly non-stationary signals which contain very different components. The algorithm "Pursuit" seeks the best "accord" (i.e. concordance) for each component of the signal rather than for the entire signal. The encoding or the decomposition of a non-stationary signal with "Pursuit" is concise and invariant by translation.

The latest version of the Matching Pursuit (i.e. Mallat and Zhang), that we present at the end of this chapter, has allowed J.B. Ramsey and Z. Zhang to establish several statements about the nature of the analyzed signals. This also made it possible to have a new perspective concerning the time-frequency analysis due to the use of the time-frequency atom dictionaries (i.e. waveform dictionaries). A wave form dictionary is a class of transforms that generalizes both windowed Fourier transforms and wavelets. Each wave form is parameterized by location, frequency and scale. Such methods can analyze signals that have highly localized structures in either time or frequency spaces as well as broad band structures. The matching pursuit algorithm is used to implement the application of wave form dictionaries to decompose the signal in the stock market index. Over long period, a stock market index shows very localized bursts of high intensity energy (see Ramses and Zhang) and in the neighborhood of which the signal is on the contrary very stable. Later on we will see how these explosions are decomposed and analyzed by the algorithm.

T. Vialar, Complex and Chaotic Nonlinear Dynamics, © Springer-Verlag Berlin Heidelberg 2009

The traditional statistics still recently concluded1 that the long-period stock market indexes have a random nature, whether they are stationary or not. Generally, it is said that they follow a random walk and they are also compared to the behaviors of Brownian motions. In econometrics, it is said that the non-stationary series have heteroscedastic variances.2

The Ramsey and Zhang analysis is to consider according to their own terms that "the energy of a system is largely internally generated, rather than the result of external forcing". In fact, they consider that oscillations of quasi-periodic nature, inside of which exist all the frequencies, can appear in an explosion of a signal, in a very localized manner. They also consider that the first proof that the stock exchange signals do not follow a random walk is that the number of wave forms necessary to represent them is smaller than for an usual random series. The wave form dictionaries, which are also called time-frequency atoms dictionaries, are noted

Before this recent version, the algorithms of Matching Pursuit exploited separately the Fourier dictionaries and the wavelet dictionaries. We provide some representations of sinusoid packets (also called cosine packets) and wavelet packets in the following section. These collections of sinusoids and wavelets are used for the decomposition of signals.

6.1.1 Construction of Sinusoid and Wavelet Packets

Starting from the Fourier series, we produce sinusoid collections of different types, they are named sinusoid packets (sine or cosine packets) and will be used for the signal decomposition. The sinusoid characteristics of a collection are indexed.

A similar methods is used for the wavelets, we generate different wavelet collections, these are wavelet packets which will be used for the signal decomposition. The wavelet characteristics of a collection are indexed. We give an example of these sinusoid and wavelet packets in Figs. 6.1 and 6.2, first, in time, then their respective spreads in frequency.

The figures which follow are important (Figs. 6.3 and 6.4), because they make it possible to observe how each sinusoid and wavelet is represented in the time-frequency plane. We observe the spread in the time-frequency plane of the Fourier

and a sub-dictionary is noted:

1 Until the work of Lo and MacKinlay.

2 Ref: Fama and LeBaron.

Some models of sine of cosine packets

Some wavelet packets in time domain



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