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

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

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

IST-ISR, January 2004 Black box scaling analysis Easy… Choose a wavelet:  (t) Compute wavelet decomposition: –T(a,b) = Compute partition sum: –S(a,q) =  b | T(a,b) | q Compute partition function  –log S(a,q) ~  (q) log a Compute Legendre transform: –f(a) = inf q (qa-  (q))

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

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

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

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

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

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

IST-ISR, January 2004 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 linear  (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

Testing for Diverging Moments All software freely available at (

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

IST-ISR, January 2004 Estimating the Regularity of  Motivation: exact regularity of  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  and is maximal at 0 then the following are equivalent:.|  (f)-P(f)| = O(|f| r ) for some polynomial P as (f  0 ) for all r<.|T(a,0)| = O(|a| r ) as (a  0 ) for all r<

IST-ISR, January 2004 Wavelet Transform of  Assume Fourier Transform  is real. Parseval: T(a,b) = = = E  a,b (x) Corollary: |T(a,b)| <= |T(a,0)|, for all b W(a) := T(a,0) = E  a.x 

IST-ISR, January 2004 Extension to orders > 2 Consider fractional Wavelets:  (x) = c |x| exp(-x 2 ) Parseval: T(a,0) a - = a -   (f)  (f/a) df = a -   (ax) dF X (x) = c  |x| exp(-(ax) 2 ) dF X (x)  c  |x| dF X (x) Lemma: If either side exists then Sup a T(a,0) a -  = c E[ |X| ] Proof: Monotonous convergence (Beppo-Levy Thm)

IST-ISR, January 2004 Bounding the range of finite moments Hölder regularity of  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  exist up to order  h  Implies non degenerated Taylor expansion of  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, then derivatives of  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 Numerical Implementation The estimator of T(a,0) of  is Simple (Parseval): T(a,0) =   (f)  (f/a) df =   (ax) dF X (x) = E[  (aX)] estimator: (1/N)  k  (aX k ) Unbiased –E[ (1/N)  k  (aX k )] = E[  (aX)] = T(a,0) Non-parametric! Robust

IST-ISR, January 2004 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 {x i } : lower bound –Body / Tail frontier : upper bound

IST-ISR, January 2004 Cutoff scales Compound process: x ~  (  ), x <  E |x| r -  x ~  -stable ( ,  =1), x >=  E |x| r < Inf, r <  Log W(a) Log a

IST-ISR, January 2004 Application to fat tail estimation  -      Gamma Laws Alpha-stable laws

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

IST-ISR, January 2004 Summary: Light in the Black Box Run several wavelets of increasing regularity –You should see  = min( +, N  ) 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  (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 test hypothesis of linear partition function

IST-ISR, January 2004 Traité Information - Commande - Communication Hermès Science Publications, Paris [ ] 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)