Diverging Moments

Diverging Moments

Paulo GonçalvèsINRIA Rhône-Alpes

Rolf RiediRice 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


The typical question

• Are these signals multifractal?


Black box scaling analysis

• Easy…• Choose a wavelet: (t)• Compute wavelet decomposition:

– T(a,b) = < x , 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) = infq (qa- (q))


The wavelet transform

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Challenge: Choice of wavelet


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The Lines of Maxima

Challenge: Finding local maxima is difficult


The Partition Sum

Challenges: All coefficients/only maxima? Which q’s?

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The Partition Function

Challenges: Range of scaling. Quality of scaling.

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The Legendre transform

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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?


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?


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 athttp://www.inrialpes.fr/is2/

(http://www.inria-rocq.fr/fractales)


The Existence of Moments

• Random variable: X– Characteristic function: (f)= E[exp(ifX)]

• Intuitive (well-known): (n)(0)= in 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 )


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<


Wavelet Transform of

• Assume Fourier Transform is real.• Parseval:

T(a,b) = <,a,b> = <F,a,b> = E a,b(x)

• Corollary: |T(a,b)| <= |T(a,0)|, for all b

W(a) := T(a,0) = E a.x


Extension to orders > 2• Consider fractional Wavelets: (x) = c |x| exp(-x2)• Parseval:

T(a,0) a- = a- (f)(f/a) df

= a- (ax) dFX(x)

= c |x| exp(-(ax)2) dFX(x)

c |x| dFX(x)

• Lemma: If either side exists then

Supa T(a,0) a-= c E[ |X| ]

Proof: Monotonous convergence (Beppo-Levy Thm)


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


Numerical Implementation

The estimator of T(a,0) of is• Simple (Parseval):

T(a,0) = (f)(f/a) df = (ax) dFX(x) = E[(aX)]

estimator: (1/N) k (aXk)

• Unbiased– E[(1/N) k (aXk)] = E[(aX)] = T(a,0)

• Non-parametric!• Robust


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


Cutoff scales

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Upper Bound

slope = slope = -

slope = N

Compound process:

• x ~ (), x < • E |x|r < Inf, r > -

• x ~ -stable (,=1), x >=

• E |x|r < Inf, r <

Log W(a)

Log a


Application to fat tail estimation

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Gamma Laws

Alpha-stable laws


Application to Multifractal Analysis

We are now able to distinguish the mono- from the multi-fractal signal

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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


Traité Information - Commande - Communication Hermès 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)

Diverging Moments

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