The Short-Time Multifractal Formalism: Definition and Implement

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singularity distribu-tion indicates the spatial dynamics character of system. Therefore, the definition and implement of the short-time multifractal formal-.
The Short-Time Multifractal Formalism: Definition and Implement* Xiong Gang1,2, Yang Xiaoniu1, and Zhao Huichang2 1

NO.36 Research Institute of CETC, National Laboratory of Information Control Technology For Communication System, Jiaxing, Zhe-Jiang, 314001, China 2 Electronic engineering dept., NJUST, Nanjing 210094, China [email protected], [email protected]

Abstract. Although multifractal descirbles the singularity distribution of SE, there is no time information in the multifractal formalism, and the time-varying singularity distribu-tion indicates the spatial dynamics character of system. Therefore, the definition and implement of the short-time multifractal formalism is proposed, which is the prelude of time time-singularity spectra distribution.In this paper, the singularity analysis of windowed signal was given, further the short-time hausdorff spectum was deduced. The Partition Function and Short-time Legendre Spectrum was fractal statistical distribution of SE. WTMM method is popular in implement of MFA, and in section ,Short-time multifractal spectra based on WTMM is brough forward..



1 Introduction Biosignals such as electroencephalogram (EEG), electrocardiogram (ECG), as well as other signals such as turbulent flows, lightning strikes, DNA sequences, and geographical objects represent some of many natural phenomena which are very difficult to characterize using traditional signal processing. Such signals are mostly nonstationary in time and/or space, and have a nonlinear behaviour. Thus, spectral methods (e.g., Fourier transform) are insufficient for analyzing them. There is strong evidence indicating that such signals have similar behaviour at multiple scales. This property is often referred to as fractality (aka self-affinity, longrange dependence, or long-range autocorrelation).Characterization of such fractal signals can be achieved through a measure of singularity α (aka Holder, or Lipschitz exponent), Mandelbrot singularity spectrum (MS) ) f (a) , and the generalized fractal dimensions D(q). For a monofractal signal, the MS shows only one point in the spectrum. The MS of a multifractal signal represents a spectrum of singularities and their dimension. The characteristics of α, f (a) are deeply-rooted in thermodynamics, and have been discussed extensively from the mathematical point of view. * This paper is supported by Post-doctoral Research Foundation of Province Zhejiang (No. 2006-bsh-27) and National Science foundation Research: Time-Dimension Spectral Distribution and the Affine Class Time-Frequency Processing of Stochastic Multifratal (No. 60702016). D.-S. Huang et al. (Eds.): ICIC 2008, CCIS 15, pp. 541–548, 2008. © Springer-Verlag Berlin Heidelberg 2008

542

X. Gang, Y. Xiaoniu, and Z. Huichang

Although multifractal descirbles the singularity distribution of SE, there is no time information in the multifractal formalism, and the time-varying singularity distribution indicates the spatial dynamics character of system. Therefore, the definition and implement of the short-time multifractal formalism is proposed, which is the prelude of time time-singularity spectra distribution. In Section II, the singularity analysis of windowed signal was given, further the short-time hausdorff spectum was deduced. The Partition Function and Short-time Legendre Spectrum in section was fractal statistical distribution of SE. WTMM method is popular in implement of MFA, and in section Short-time multifractal spectra based on WTMM is brough forward.



Ⅳ,

2 Short-Time Singularity Exponent and Hausdorff Spectrum In macroscopical sense, The short-time multifractals provides the instantaneous singularity distribution, which arouses the difficulty of definiton. Given a cutty time interval, the singularity distribution of them is time-varying analysis. But the premise of previous analysis is that several singular signal poseese the linear adding character of time-varying analysis. 2.1 Singularity Analysis of Windowed Signal Assume the characteristic window of singular signal is h(τ − t ) , the windowed signal is

v(t ,τ ) = u (τ )h(τ − t ) Definition 2.1. A function or the path of a process

v(t ,τ ) is said to be in Cτh if there

is a polynomial Pu (τ ) such that

| v(t ,τ ) − Pu (τ ) |≤ C | u − τ | h for u suffciently close to τ . Then, the degree of local Holder regularity of

τ

v(t ,τ ) at

is H (t ,τ ) := sup{h : v (t ,τ ) ∈ Cτh } Of special interest for our purpose is the case when the approximating polynomial Pt is a constant, i.e., Pu (τ ) = v(t ,τ ) , in which case H (t ,τ ) can be computed easily.

To the end: Definition 2.2. Let us agree on the convention log(0) = −∞ and set

h(t,τ ) = liminf ε →0

1 log2 sup | v(t ,τ ) − v(t , u) | log2 (2ε ) |u −τ | h , when X ∈ C (t0 ) then there exists a constant C > 0 such that | d X ( j , k ) |≤ C 2 jh (1+ | 2 − j t0 − k |h ). Loosely speaking, it is commonly read as the fact that when X has Holder exponent h at

t0 = 2 j k ,the corresponding wavelet coefficients d X ( j , k ) are of the order of magnitude | d X ( j , k ) |~ 2 jh . This is precisely the case of the cusp like function mentioned above. Further results relating the decrease along scales of wavelet coefficients and Holder exponent can be found in e.g., [8]. 4.2 Wavelet Coefficient Based Multifractal Formalism Wavelet coefficient based structure functions and scaling exponents are defined as:

S d (t , q, j ) =

1 nj

nj

∑| d

X

(t , j, k ) |q

k =1

⎛ log2 Sd (t , q, j ) ⎞ ⎟⎟ j ⎝ ⎠

τ d (t , q) = lim inf ⎜⎜ j →0

where n j is the number of available d X (t , j , k ) at octave j : n j ≈ no 2− j . By definition of the multifractal spectrum, there are about

(2 j ) − D ( h ) points with Holder

exponent h, hence with wavelet coefficients of the order d X (t , j , k ) ≈ (2 j ) h . They contribute

to

Sd (t , q, j )

~ 2 j (2 j ) qh (2 j ) − D ( h ) = (2 j )1+ qh − D ( h ) .

as

j τ d (t , q )

S d (t , q, j ) will behave as ~ cq (2 )

Therefore,

and a standard steepest descent argument

yields a Legendre transform relationship between the multifractal spectrum D(h) and the scaling exponents τ d (t , q ) : τ d (t , q) = inf h (1 + qh − D(h)) . The Wavelet

The Short-Time Multifractal Formalism: Definition and Implement

547

Coefficient based Multifractal Formalism (hereafter WCMF) is standardly said to hold when the following equality is valid:

D(t , h) = inf (1 + qh − τ d (t , q)) − q ≠0

5 Experimental Results and Discussion Fig. 2 is the original data of sea clutter of radar, and the Fig.3 shows the partition function and the multifractal spectrum of fractal sea clutter. From the simulation of multifractal spectrum of sea clutter, we can see that the sea clutter is multifractal.

Fig. 1. The sea clutter of radar

τ (q)

f (α )

q

α

Fig. 2. The partition function and the multifractal spectrum of the sea clutter of radar

Fig. 3. The short time multifractal spectral of sea clutter

Fig.3 gives the short time multifractal spectral distribution of sea clutter, from which it’s can be seen that the multifractal of sea clutter is time-varying, and the STMFS can extract more characteristic of multifractal than the multifractal, especially when the multifractal is changed along the time.

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X. Gang, Y. Xiaoniu, and Z. Huichang

References 1. Arneodo, A., Audit, B., Bacry, E., Manneville, S., Muzy, J.F., Roux, S.G.: Thermodynamics of Fractal Signals Based on Wavelet Analysis: Application to Fully Developed Turbulence Data and DNA Sequences. Physica A 254, 24–45 (1998) 2. Arneodo, A., Bacry, E., Muzy, J.F.: The Thermodynamics of Fractals Revisited with Wavelets. Physica A 213(1-2), 232–275 (1995) 3. Bacry, E.: Lastwave Pakage. Web Document, Febraury 28, 2005 (1997), http://www.cmap.polytechnique.fr/~bacry/LastWave/ 4. Donoho, D., Duncan, M.R., Huo, X.: WaveLab Documents, (Febraury 28, 2005) [Online] (1999), http://www.stat.stanford.edu/~wavelab/ 5. Faghfouri, A., Kinsner, W.: 1D Mandelbrot Singularity Spectrum, Ver. 1.0, (Febraury 28, 2005) [Online] (2005), http://www.ee.umanitoba.ca/~kinsner/projects 6. Grassberger, P., Procaccia, I.: Dimensions and Entropies of Strange Aattractors from a Fluctuating Dynamics Approach. Physica D 13(1-2), 34–54 (1984) 7. Hentschel, H., Procaccia, I.: The Infinite Number of Generalized Dimensions of Fractals and Strange Attractors. Physica D 8D, 435–444 (1983) 8. Kinsner, W.: Fractal: Chaos Engineering Course Notes. Winnipeg, MB: Dept. Electrical & Computer Eng., University of Manitoba (2003) 9. Mallat, S.G., Hwang, W.L.: Singularity Detection and Processing with Wavelets. IEEE Trans. Infor. Theory 38, 617–643 (1992) 10. Mallat, S.: A Wavelet Tour of Signal Processing, 2nd edn. Academic Press, Chestnut Hill (2001) 11. Mandelbrot, B.B., Fractals, Multifractals.: Noise, Turbulence and Galaxies. Springer, New York (1989) 12. Muzy, J.F., Bacry, E., Arneodo, A.: Wavelets and Multifractal Formalism for Singular Signals: Application to turbulence data. Phys. Rev. Lett. 67(25), 3515–3518 (1991) 13. Muzy, J.F., Bacry, E., Arneodo, A.: Multifractal Formalism for Fractal Signals: The Structure Function Approach Versus the Wavelet-transform Modulus-maxima Method. Phys. Rev. E 47(2), 875–884 (1993) 14. Muzy, J.F., Bacry, E., Arneodo, A.: The Multifractal Formalism Revisited with Wavelets. Int. Jrnl. Bif. Chaos 4(2), 245–302 (1994) 15. Oppenheim, A.V., Schafer, R.W., Buck, J.R.: Discrete-Time Signal Processing, 2nd edn. Prentice Hall, Englewood Cliffs (1999) 16. Proakis, J.G., Manolakis, D.G.: Digital Signal Processing: Principles, Algorithms and Applications, 2nd edn. Macmillan, New York (1996) 17. Van den Berg, J.: Wavelets in physics, 2nd edn. Cambridge University Press, Cambridge (2004)