Wavelet transform modulus maxima method

The Wavelet Transform Modulus Maxima (WTMM) is a method for detecting the fractal dimension [1] of a signal.

More than this, the WTMM is capable of partitioning the time and scale domain of a signal into fractal dimension regions, and the method is sometimes referred to as a "mathematical microscope" due to its ability to inspect the multi-scale dimensional characteristics of a signal and possibly inform about the sources of these characteristics.

The WTMM method uses continuous wavelet transform [2] rather than fourier transforms[3] to detect singularities[4] - that is discontinuities, areas in the signal that are not continuous at a particular derivative.

In particular, this method is useful when analyzing multifractal [5] signals, that is, signals having multiple fractal dimensions.

Description

Consider a signal that can be represented by the following equation:

$f(t)=a_{0}+a_{1}(t-t_{i})+a_{2}(t-t_{i})^{2}+...+a_{h}(t-t_{i})^{h_{i}}$

where $t$ is close to $t_{i}$ and $h_{i}$ is a non-integer quantifying the local singularity. (Compare this to a Taylor series [6], where in practice only a limited number of low-order terms are used to approximate a continuous function.)

Generally, a continuous wavelet transform [7] decomposes a signal as a function of time, rather than assuming the signal is stationary (For example, the Fourier transform). Any continuous wavelet can be used, though the first derivative of the Gaussian and the mexican hat wavelet (2nd derivative of Gaussian)[8] are common. Choice of wavelet may depend on characteristics of the signal being investigated.

Below we see one possible wavelet basis given by the first derivative of the Gaussian:

$G'(t,a,b)={\frac {a}{(2\pi )^{-1/2}}}(t-b)e^{[{\frac {-(t-b)^{2}}{2a^{2}}}]}$

Once a "mother wavelet" is chosen, the continuous wavelet transform is carried out as a continuous, square-integrable [9] function that can be scaled and translated. Let $a>0$ be the scaling constant and $b\in \mathbb {R}$ be the translation of the wavelet along the signal:

$X_{w}(a,b)={\frac {1}{\sqrt {a}}}\int _{-\infty }^{\infty }x(t)\psi ^{\ast }\left({\frac {t-b}{a}}\right)\,dt$

where $\psi (t)$ is a continuous function in both the time domain and the frequency domain called the mother wavelet and $^{\ast }$ represents the operation of complex conjugate [10].

By calculating $X_{w}(a,b)$ for subsequent wavelets that are derivatives of the mother wavelet, singularities can be identified. Successive derivative wavelets remove the contribution of lower order terms in the signal, allowing the maximum $h_{i}$ to be detected. (Recall that when taking derivatives, lower order terms become 0.) This is the "modulus maxima".

Thus, this method idenitifies the singularity spectrum by convolving the signal with a wavelet at different scales and time offsets.

It is then capable of producing a "skeleton" that partitions the scale and time space by fractal dimension.

File:Fig to squel.gif

In the above figure, the x-axis is time. Two signals are analyzed. (top) Their fractal dimension at different scales is extracted using the WTMM method (middle), and finally a skeleton partitioning the signals by fractal dimension at different times and scales is drawn (bottom). (source Scholarpedia [11])

History

The WTMM was developed out of the larger field of continuous wavelet transforms, which arose in the 1980s, and it's contemporary fractal dimension "box counting" methods

At its essence, it is a combination of fractal dimension "box counting" methods and wavelet transforms, where wavelets at various scales are used instead of boxes.

WTMM was originally developed by Mallat and Hwang in 1992 and used for image processing. [12]

Bacry, Muzy, and Arneodo were early users of this methodology. [13][14] It has subsequently been greatly used in many fields that use signal processing.

Application to Analysis of Networks

As of this writing, with a few exceptions, this method has not been widely applied to network analysis. [15] [16] However, due to scaling factors and phase transition phenomena found widely in network analysis, it is not unreasonable that multi-fractal methods and the mathematics behind them can be brought to bear. Recent developments in fractal dimension of networks are compelling. [17]

References:

P. Abry, P. Flandrin, M.S. Taqqu, and D. Veitch, K. Park and W. Willinger, "Wavelets for the analysis, estimation and synthesis of scaling data", Self-Similar Network Traffic and Performance Evaluation, pp. 39 - 88, 2000. :Wiley [18]

Abry, P.; Baraniuk, R.; Flandrin, P.; Riedi, R.; Veitch, D.; , "Multiscale nature of network traffic," Signal Processing Magazine, IEEE , vol.19, no.3, pp.28-46, May 2002 doi: 10.1109/79.998080 [19]

Alain Arneodo et al. (2008), Scholarpedia, 3(3):4103. [20]

A Wavelet Tour of Signal Processing, by Stéphane Mallat; ISBN : 0-12-466606-X; Academic Press, 1999[21]

Mallat, S.; Hwang, W.L.; , "Singularity detection and processing with wavelets," Information Theory, IEEE Transactions on , vol.38, no.2, pp.617-643, Mar 1992 doi: 10.1109/18.119727 [22]

Arneodo on Wavelets [23]

Wavelets and multifractal formalism for singular signals : application to turbulence data J.F. Muzy, E. Bacry and A. Arneodo, Phys. Rev. Lett. 67, 3515 (1991). [24]

Multifractal formalism for fractal signals: the structure fonction approach versus the wavelet transform modulus maxima method. J.F. Muzy, E. Bacry and A. Arneodo, Phys. Rev. E 47, 875 [25]