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Diverging Moments

Diverging Moments. Paulo Gonçalvès INRIA Rhône-Alpes Rolf Riedi Rice University. IST - ISR january 2004. Importance of multiscale analysis. General: LRD Self-similarity Multi-fractal, multiplicative structure Economics, Networking, Biology, Physics Turbulence K41 K62 Intermittency.

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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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