Theory#

Here are presented some of the theoretical aspects required to understand multifractal analysis

Scale Invariance#

A process \(X\) is said scale invariant if its two-point function \(\mathcal{G}_{\alpha}(r)\) scales as:

\[\forall b \in \mathbb{R}_+^*, \quad \mathcal{G}_{\alpha}(r/b) = b^{2x_{\alpha}}\mathcal{G}(r)\]

In other words, the rescaling factor of the two-point function only depends on the scaling factor \(b\), and does not depend on the characteristic scale \(r\). As a consequence, by choosing \(b=r\), we can rewrite this relation as a power law:

\[\mathcal{G}_{\alpha}(r) = \frac{\mathrm{constant}}{r^{2x_{\alpha}}}\]

Self-similarity#

For a 1D time series, we can choose the autocorrelation function as the two-point function:

\[A(\tau) =\]

Taking the Fourier transform, we have the following relation on the power spectrum:

\[\Gamma(f) =\]

Thus we have a power spectrun with a power law relation, appearing linear in a log-log plot. The exponent of that power law is related to the Hurst exponent for self-similarity, with the following relation:

The power spectrum investigate the scaling of the second-order moments of the time series increments:

Multifractality#

We can investigate the power law scaling of any arbitrary moment \(q\) of \(\Delta X\), described using the scaling function \(\zeta(q)\):

\[\mathbb{E}\left[ \Delta X ^q \right] \propto\]

Time series where \(\zeta(q)=H\), i.e. there is only one exponent \(H\) common to every moment, are called monofractal.

In the case where \(\zeta(q)\) is not constant, then the time series is described as multifractal.

Regularity#

For a function, the notion of regularity captures the notion of . There are multiple characterisations of regularity, such as Lipschitz, etc. Here we will focus on those relevant for multifractal analysis

Hölder regularity#

\(p\)-exponent#

Multifractal formalism#

With discrete measurements, in order to characterize \(\zeta(q)\) we cannot directly measure the pointwise regularity of our time series. Instead, we need to rely on multi-resolution quantities (MRQ) which describe the statistics of the underlying process. From those MRQ we can then derive estimates of the scaling function, and of the multifractal spectrum.

We note the MRQ as \(T(k, j)\) where \(k\) is a shift term, and \(j\) is the scale.

Structure functions#

In order to estimate \(\zeta(q)\), we can look at the structure functions \(S\), defined as:

\[S(j, q) = 2^{dj} \sum_k T(k, j)^q\]

Then we have:

\[\zeta(q) = \varliminf_{j \to -\infty} \frac{\log \left( S(j, q) \right)}{\log(2^j)}\]

Multifractal Spectrum#

From the structure functions we can get an upper-bound estimate of the multifractal spectrum via the Legendre transform of the scaling function.

\[\mathcal{L}(h) = \inf_{q\in \mathbb{R}} \left(d + qh - \zeta(q) \right)\]

Cumulants#

The wavelet formalism provides a convenient way to characterize the properties of the multifractal spectrum. Using the cumulant expansion of \(\zeta(q)\):

\[\zeta(q) = \sum_{m\geq 1} c_m \frac{q^m}{m!}\]

We can estimate the \(c_m\) from the scaling of the cumulants of the log MRQ:

\[\forall j, \quad \log \operatorname{Cumulant}_m \left[\left( T(k, j)\right)_k \right] \propto j c_m\]

Those \(c_m\) provide a description of the moments of the spectrum:

  • \(c_1\) corresponds to the mode of the spectrum, and to self-similarity

  • \(c_2\) corresponds to twice the spread of the multifratal spectrum

  • \(c_3\) and \(c_4\) describe asymmetry and tailedness respectively.

As it turns out, each definition of regularity has its corresponding multi-resolution quantity.

Wavelet coefficient#

While the discrete wavelet transform is a useful tool for computing the Hurst exponent, it fails at properly capturing the regularity of time series

Wavelet leader#

The multi-resolution quantity associated with the Hölder exponent is the wavelet leader, defined as:

\[\]

Wavelet \(p\)-leader#

Models for scale invariant signals#

fBM#

MRW#

Practical considerations#

Fractional integration#

Higher order cumulants#

Sensitivity to noise#

Comparison with alternative methods#

References#

  1. Wendt, Contributions of Wavelet Leaders and Bootstrap to Multifractal Analysis, ENS Lyon, 2008. Available at https://www.irit.fr/~Herwig.Wendt/data/ThesisWendt.pdf