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Diverging Moments

This paper investigates the significance of multiscale analysis, focusing on diverging moments and multifractal structures in diverse domains such as economics, biology, and physics. The authors, Paulo Gonçalvès and Rolf Riedi, present challenges in wavelet decomposition, scaling analysis, and moment estimation. They discuss the importance of selecting appropriate wavelets for effective analysis and address issues related to the interpretation of scaling functions. The findings aim to enhance our understanding of random signals and their multifractal nature.

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Diverging Moments

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  1. Diverging Moments Paulo Gonçalvès INRIA Rhône-Alpes Rolf Riedi Rice University IST - ISR january 2004

  2. Importance of multiscale analysis • General: • LRD • Self-similarity • Multi-fractal, multiplicative structure • Economics, Networking, Biology, Physics • Turbulence • K41 • K62 • Intermittency IST-ISR, January 2004

  3. The typical question • Are these signals multifractal? IST-ISR, January 2004

  4. Black box scaling analysis • Easy… • Choose a wavelet: y(t) • Compute wavelet decomposition: • T(a,b) = < x , ya,b > • Compute partition sum: • S(a,q) = Sb | T(a,b) |q • Compute partition function t: • log S(a,q) ~ t(q) log a • Compute Legendre transform: • f(a) = infq (qa- t(q)) IST-ISR, January 2004

  5. The wavelet transform Challenge: Choice of wavelet IST-ISR, January 2004

  6. The Lines of Maxima Challenge: Finding local maxima is difficult IST-ISR, January 2004

  7. S(a,q), q<0 S(a,q), q<0 The Partition Sum S(a,q), q>0 S(a,q), q>0 Challenges: All coefficients/only maxima? Which q’s? IST-ISR, January 2004

  8. The Partition Function Challenges: Range of scaling. Quality of scaling. IST-ISR, January 2004

  9. The Legendre transform Challenge: Interpretation. IST-ISR, January 2004

  10. Black Box Scaling Analysis: Summary • It could be easy, but it is not… • Choose a “good” wavelet • How much regularity, localization • Compute wavelet decomposition • Continuous or discrete? • Compute partition sum • On all coefficients, or only along lines of maxima? • For which range of order q • Compute partition function • Over which range of scales? • Is the scaling sufficiently close to a powerlaw • Compute Legendre transform • Interpretation: is it a point or a curve? IST-ISR, January 2004

  11. Waking up to Reality • Most essential difficulty: • Interpretation of t(q) and its Legendre transform • To make the point: • One of the signals is “mono-fractal” with linear t(q) • The other signal is multifractal with strictly convex t(q) • We found no indication for linear t(q) • What went wrong? IST-ISR, January 2004

  12. A Look into the Black Box • All wavelets with sufficient regularity show the same • Scaling is satisfactory for the partition sum • with all coefficients, (q > 0) • along the lines of maxima (q < 0) • Indication for lineart(q) in one signal • but only over a finite range of q. • S(a,q) is an estimator for the q-th moment • Are we measuring the scaling of moments, or • rather the rate of convergence/divergence of the estimator IST-ISR, January 2004

  13. Testing for Diverging Moments All software freely available at http://www.inrialpes.fr/is2/ (http://www.inria-rocq.fr/fractales)

  14. The Existence of Moments • Random variable: X • Characteristic function: f(f)= E[exp(ifX)] • Intuitive (well-known): f(n)(0)= in E[ X n ] • Rigorous: For l>0 equivalent conditions are • E[ |X|r ] <  for all r<l • P[|X| > u] = O(|u|-r) for all r<las (u   ) • in the case l<2: |f(f)| = O(|f|r) for all r<l a s (f  0 ) IST-ISR, January 2004

  15. Estimating the Regularity of f • Motivation: exact regularity of f at zero provides the cutoff value for finite moments (as long as smaller than 2) • Measuring tool: Wavelets! • Simplified criterion: If the wavelet has regularity larger l>0 and is maximal at 0 then the following are equivalent: .|f(f)-P(f)| = O(|f|r) for some polynomial P as (f  0 ) for all r<l .|T(a,0)| = O(|a|r) as (a  0 ) for all r<l IST-ISR, January 2004

  16. Wavelet Transform of f • Assume Fourier Transform Y is real. • Parseval: T(a,b) = <f,ya,b> = <F,Ya,b> = E Ya,b(x) • Corollary: |T(a,b)| <= |T(a,0)|, for all b W(a) := T(a,0) = E Y(a.x) IST-ISR, January 2004

  17. Extension to orders > 2 • Consider fractional Wavelets: Y(x) = c |x|n exp(-x2) • Parseval: T(a,0) a-n = a-nf(f)y(f/a) df = a-nY(ax) dFX(x) = c  |x|n exp(-(ax)2) dFX(x)  c  |x|n dFX(x) • Lemma: If either side exists then Supa T(a,0) a-n = c E[ |X|n ] Proof: Monotonous convergence (Beppo-Levy Thm) IST-ISR, January 2004

  18. Bounding the range of finite moments • Hölder regularity of f at zero: h • Theorem: • Moments are finite at least up to order h • Moment of order h +1 is infinite. • Proof 1: • Lemma implies moments up to h exist • Thus derivatives of f exist up to order h Implies non degenerated Taylor expansion of f at zero (does not follow in general from wavelet analysis) • Kawata criterion: moments up to order h exist. • Proof 2: • If the moment of order h +1 was finite, thenderivatives of f would exist up to order h +1, in contradiction to regularity h. h is the largest integer <= h Note that h+1 is strictly larger than h IST-ISR, January 2004

  19. Numerical Implementation The estimator of T(a,0) of f is • Simple (Parseval): T(a,0) = f(f)y(f/a) df = Y(ax) dFX(x) = E[Y(aX)] estimator: (1/N) SkY(aXk) • Unbiased • E[(1/N) SkY(aXk)] = E[Y(aX)] = T(a,0) • Non-parametric! • Robust IST-ISR, January 2004

  20. Practical Considerations • Choose a wavelet • With high enough regularity • With real positive Fourier transform (ex: even derivatives of gaussian window) • Cutoff scales • Shannon argument on max {xi} : lower bound • Body / Tail frontier : upper bound IST-ISR, January 2004

  21. Cutoff scales Log W(a) Compound process: • x ~ G(g), x < d • E |x|r < Inf, r > -g • x ~ a-stable (a,b=1), • x >= d • E |x|r < Inf, r < a Log a IST-ISR, January 2004

  22. Application to fat tail estimation Gamma Laws g -l- a a Alpha-stable laws a l a a IST-ISR, January 2004

  23. Application to Multifractal Analysis We are now able to distinguish the mono- from the multi-fractal signal IST-ISR, January 2004

  24. Summary: Light in the Black Box • Run several wavelets of increasing regularity • You should see b = min(l+, Ny) • Partition sum over all / only maximal coefficients • Scaling should improve for negative q over maxima • Report the scaling region (should be same for all q) • Compute error of t(q) using several traces • To provide statistical significance • Estimate the range of finite moments • Confine the Legendre transform to this range of q • Provides additional statistics on the process per se • If desired testhypothesis of linear partition function IST-ISR, January 2004

  25. Traité Information - Commande - CommunicationHermès Science Publications, Paris [ http://www.editions-hermes.fr/trait_ic2.htm ] Lois d’Echelle, Fractales et Ondelettes – (vol. 1,2) (P. Abry, P. Gonçalvès, J. Lévy Véhel) • Analyse multifractale et ondelettes (S. Jaffard) • Analyse Multifractale : développements mathématiques (R. Riedi) • Processus Auto-Similaires (J. Istas et A. Benassi) • Processus Localement Auto-Similaires (S. Cohen) • Calcul Fractionnaire (D. Matignon) • Analyse fractale et multifractale en traitement du signal (J. Lévy Véhel et C. Tricot) • Analyses en ondelettes et lois d'échelle (P. Flandrin, P. Abry et P. Gonçalvès) • Synthèse fractionnaire - Filtres fractals (L. Bel, G. Oppenheim, L. Robbiano, M-C. Viano) • IFS et applications en traitement d'images (J-.M. Chassery et F. Davoine) • IFS, IFS généralisés et applications en traitement du signal (K. Daoudi) • Lois d'échelles en télétrafic informatique (D. Veitch) • Analyse fractale d'images (A. Saucier) • Lois d'échelles en finance (C. Walter) • Relativité d'échelle, nondifférentiabilité et espace-temps fractal (L. Nottale) IST-ISR, January 2004

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