MUTUALSPECTRA - Measure the mutual closeness of singularity spectra.

Contents

Description

Given two (possibly reduced) spectra (e.g. singularity spectra representing fractal dimensions), quantify the degree of mutual closeness following the approach of [PTP09].

Algorithm

The mutual closeness of the spectra $D_1(h)$ and $D_2(h)$ with associated uncertainties $b_1(h)$ and $b_2(h)$ is expressed by the Eq.(7) of [PTP09] as:

$$ \Delta = \min \{ \Delta(1\rightarrow 2), \Delta(2\rightarrow 1) \}$$

where:

$$ \Delta(1\rightarrow 2) = \sum_h \frac{|D_1(h) -D_2(h)|}{b_1(h)\cdot b_2(h)} \, \Big/ \, \sum_h \frac{1}{b_1(h)\cdot b_2(h)} $$

and similarly for $\Delta(2\rightarrow 1)$.

Syntax

   delta = MUTUALSPECTRA(h1, D1, b1, h2, D2, b2);

Inputs

h1, D1 : couple of estimation, ie. the spectrum D1 of a (set of) signal(s) is estimated over a set of values h1.

b1 : associated uncertainty in the estimation.

h2, D2, b2 : ibid with an estimation performed over another (set of) signals.

Output

delta : mutual closeness.

References

[TPG06] A. Turiel, C. Pérez-Vicente and J. Grazzini: "Numerical methods for the estimation of multifractal singularity spectra on sampled data: a comparative study", Journal of Computational Physics, 216(1):362?390, 2006. http://www.sciencedirect.com/science/article/pii/S0021999105005565

[PTP09] O. Pont, A. Turiel, C.J. Perez-Vicente: "Empirical evidences of a common multifractal signature in economic, biological and physical systems", Physica A, 388:3015-2035, 2009. http://www.sciencedirect.com/science/article/pii/S0378437109000880

See also

Ressembles: FRACTALWAVE, FRACTALWAVESTAT. Requires: INTERP1, SUM, MIN.

Function implementation

function delta = mutualspectra(H1, DH1, ErrDH1, H2, DH2, ErrDH2)

first, define the directed weighted average difference: compute delta(1->2)

DH = interp1(H2, DH2, H1, 'linear');
ErrDH = interp1(H2, ErrDH2, H1, 'linear');
delta1 = sum(abs(DH1 - DH)./(ErrDH1.*ErrDH)) / sum(ErrDH1.*ErrDH);

compute delta(1->2)

DH = interp1(H1, DH1, H2, 'linear');
ErrDH = interp1(H1, ErrDH1, H2, 'linear');
delta = sum(abs(DH2 - DH)./(ErrDH2.*ErrDH)) / sum(ErrDH2.*ErrDH);

define the weighted average difference between the two reduced spectra as the minimum of the two possible directed weighted average differences

delta = min(delta, delta1);

end % end of mutualspectra