Irregularities and Scaling in Signal and Image Processing: Multifractal Analysis

Transcription

1 Irregularities and Scaling in Signal and Image Processing: Multiractal Analysis P. Abry, S. Jaard, H. Wendt November 7, Abstract: B. Mandelbrot gave a new birth to the notions o scale invariance, selsimilarity and non-integer dimensions, gathering them as the ounding corner-stones used to build up ractal geometry. The irst purpose o the present contribution is to review and relate together these key notions, explore their interplay and show that they are dierent acets o a same intuition. Second, it will explain how these notions lead to the derivation o the mathematical tools underlying multiractal analysis. Third, it will reormulate these theoretical tools into a wavelet ramework, hence enabling their better theoretical understanding as well as their eicient practical implementation. B. Mandelbrot used his concept o ractal geometry to analyze real-world applications o very dierent natures. As a tribute to his work, applications o various origins, and where multiractal analysis proved ruitul, are revisited to illustrate the theoretical developments proposed here. Keywords: Scaling, Scale Invariance, Fractal, Multiractal, Fractional dimensions, Hölder regularity, Oscillations, Wavelet, Wavelet Leader, Multiractal Spectrum, Hydrodynamic Turbulence, Heart Rate Variability, Internet Traic, Finance Time Series, Paintings Signal, Systems and Physics, Physics Dept., CNRS UMR 567, Ecole Normale Supérieure de Lyon, Lyon, France patrice.abry@ens-lyon.r Address: Laboratoire d Analyse et de Mathématiques Appliquées, CNRS, UMR 85, UPEC, Créteil, France jaard@u-pec.r Purdue University, Dept. o Mathematics, USA hwendt@purdue.edu

2 Contents Introduction 3. Scale invariance and sel similarity Selsimilarity and Wavelets Beyond selsimilarity: Multiractal analysis Data analysis and Signal Processing Goals and Outline From ractals to multiractals. Scaling and Fractals From graph box-dimensions to p-variation: The oscillation scaling unction 4 3 Geometry vs. Analysis 5 3. Pointwise Hölder regularity Scaling unctions vs. unction spaces Wavelets: A natural tool to measure global scale invariance 4. Wavelet bases Wavelet Scaling Function Uniorm Hölder exponent Wavelet leaders Beyond unctional analysis: Multiractal analysis Multiractal ormalism or oscillation A ractal interpretation o the leader scaling unction: The leader multiractal ormalism Local spectrum and homogeneity Mono vs. Multiractality? Real-World Applications 4 6. Hydrodynamic Turbulence Finance Time Series: Euro-USD rate luctuations Fetal Heart Rate Variability Aggregated Internet Traic Texture classiication: Vincent Van Gogh meets Benoît Mandelbrot Conclusion 5

3 Introduction As many students, beore and ater us, we irst met Benoît Mandelbrot through his books [7, 8]; indeed, they were scientiic bestsellers, and could be ound in evidence in most scientiic bookshops. A quick look was suicient to realize that Benoît was an unorthodox scientist; these books radically diered rom what we were used to, and were trespassing several taboos o the time: First, they could not be straightorwardly associated with any o the standardly labelled ields o science, which had been taught to us as separate subjects. This could already be inerred rom their titles: The ractal geometry o nature [8] constitutes a statement that mathematics and natural sciences are intrinsically mixed, and that the purpose will not be to disentangle them artiicially, but rather to explore their interactions. The contents o these books conirmed this irst eeling: Though they contained a large proportion o very serious mathematics, they did not ollow the old deinition-theorem-proo articulation we were used to. The ocus was directly on the observation o igures and on their pertinence or modeling; the mathematics were there to help and comort the deep geometric intuition that Benoît had developed. This does not mean that he considered mathematics as an ancillary discipline. To the contrary, many testimonies conirm how enthusiastic he was when one o his mathematical intuitions was proven correct, and how he immediately incorporated the new mathematical methods in his own toolbox, which then helped him to sharpen his intuition and go urther. Another revolutionary point was that these books reversed the classical ocusses: While one did ind amiliar mathematical objects (the nowhere dierentiable Weierstrass unctions, or the devil s staircase had been encountered beore), the role he made them play was radically dierent: So ar, they had only been counterexamples, or pathological objects pedagogically used to shed light on possible pitalls; instead o this restrictive role, Benoît used them as the key examples and cornerstones to ound a new geometry based on roughness and selsimilarity. These books were opening a subject, and drawing tracks through a new territory; their purpose was not to give the usual polished, inal description o a well understood area, but to open windows on how science advances, and they had a strong inluence on our vocations to research. Later, personal meetings played a decisive role; let us only mention one such occasion: At the end o his PhD, one o us (S. J.) met Benoît who was visiting Ecole Polytechnique. Benoît had heard o wavelets, which, at the time, were a new tool still in inancy, and immediately envisaged possible interactions between wavelets and ractal analysis. A key eature o ractals is scaling invariance, and, because wavelets can be deduced one rom the others by translations and dilations o a given unction, their particular algorithmic structure should reveal the scaling invariance properties o the analyzed objects. Driven by his insatiable curiosity, Benoît wanted to learn everything about wavelets (which was actually not so much at the time!) and then, with his characteristic generosity, he shared with this student he barely knew his intuitions on the subject. Thus, the present contribution can be read as a tribute to some o the scientiic lines Benoît had dreamed about at the time: It will show how and why wavelet techniques yield powerul tools or analyzing ractal properties o signals and images, and thus supply new collections o tools or their classiication and modeling. 3

4 . Scale invariance and sel similarity Exact (or deterministic) sel similarity. Fractal analysis can roughly be thought o as a way to characterize and investigate the properties o irregular sets. Note that irregularity is a negatively deined word; the description o the scientiic domain it covers is thereore not clearly easible, and had not been tried beore B. Mandelbrot developed the concepts o ractal geometry. One o his leading ideas was a notion o regularity in irregularity or selsimilarity: The object is irregular, but there is some invariance in its behavior across scales. For some sets, the notion o selsimilarity replaces the notion o smoothness. For instance, the triadic Cantor set is selsimilar, in the sense that it is made o two pieces which are the same as the whole set is shrunk by a actor 3. The purpose o multiractal analysis is to transpose this idea rom the context o geometrical sets to that o unctions. Sometimes, this transposition can be directly achieved, when the geometrical set consisting o the graph o the unction also displays some selsimilarity. Consider the example o the Weierstrass-Mandelbrot unctions (a slight variant o the amous Weierstrass unctions, see [8] pp ): W a,h (x) = + n= sin(a n x) a Hn or a > and H (, ) () (note that the series converges when n + because a > and when n because sin(a n x) a n x and H < ). Those unctions clearly satisy the exact selsimilarity relationship x R, W a,h (ax) = a H W a,h (x). () Note that selsimilar unctions associated with a selsimilarity exponent H > can be obtained using slight variants: For instance, the unction + n= cos(a n x) a Hn or a > and H (, ) is selsimilar with a selsimilarity exponent H, which can go up to ; and the unction + n= sin(a n x) a n x a Hn or a > and H (, 3) is selsimilar with a selsimilarity exponent H which can go up to 3. Note that this renormalization technique can be pushed urther to deal with higher and higher values o H, yielding smoother and smoother unctions: One thus obtains unctions that are everywhere C [H] and nowhere C [H]+. Random (or statistical) sel similarity. Functions satisying () are very rare; thereore the notion o exact selsimilarity can be considered too restrictive or real world data modeling. Fruitul generalizations o this concept can be developed into several directions. A irst possibility lies in weakening the exact deterministic relationship () into a probabilistic one: The (random) unctions a H (ax) do not coincide sample path by sample path, 4

5 but share the same statistical laws. A stochastic process X t is said to be selsimilar, with selsimilarity exponent H i a >, {X at, t R} L = {a H X t, t R} (3) (see Deinition 5 where the precise deinition o equality in law o processes is recalled). The irst example o such selsimilar processes is that o ractional Brownian motion (hereater reerred to as Bm), introduced by Kolmogorov [5]. Its importance or the modeling o scale invariance and ractal properties in data was made explicit and clear by Mandelbrot and Van Ness in [38]. This article is characteristic o Mandelbrot s genius, which could perceive similarities between very distant disciplines: Motivations simultaneously rose rom hydrology (cumulated water low), economic time series and luctuations in solids. Notably, B. Mandelbrot explained that Bm (with.5 < H < ) was well-suited to model long term dependencies, or long memory eects, in data, a property he liked to reer to as the Joseph eect in a colorul and sel-speaking metaphoric comparison to the character o the Bible (c. e.g., [39]). Selsimilar processes have since been the subject o numerous exhaustive studies, see or instance the books [4, 55, 65, 68]. Note also that B. Mandelbrot put selsimilarity (which he actually more accurately called selainity) as a central notion in the study stochastic processes in views o applications, see [36] or a panorama o his works in this direction.. Selsimilarity and Wavelets Let us now briely come back to B. Mandelbrot s intuition and see how wavelet analysis can reveal relationships such as () or (3), as well as other important probabilistic properties o stochastic processes. In its simplest orm, an orthonormal wavelet basis o L (R) is deined as the collection o unctions j/ ψ( j x k), where j, k Z, (4) and the wavelet ψ is a smooth, well localized unction. Wavelet coeicients are urther deined by c j,k = j (x) ψ( j x k) dx. A L -normalization is used (rather than the classical L -normalization), as it is better suited to express scale invariance relationships, as shown in (5) below. Let denote a unction satisying () with a =, i.e. such that x R, (x) = H (x); then, clearly, its wavelet coeicients satisy j, k, c j,k = H c j+,k. Similarly, in the probabilistic setting, i (3) holds, then the (ininite) sequences o random vectors C j ( Hj c j,k ) k Z satisy the ollowing property: {C j } j Z share the same law. (5) 5

6 Other probabilistic properties have simple wavelet counterparts; let us mention a ew o them. Let X t be a Markov process, i.e. a stochastic process satisying the property s >, X t+s X t is independent o the (X u ) u t ; typical examples o Markov processes are supplied by Brownian motion, or, more generally, by Lévy processes, i.e. processes with stationary and independent increments. Because wavelets have vanishing integrals, the Markov property implies that, i the supports o the ψ( j x k) are disjoint, then the corresponding coeicients c j,k are independent random variables. Recall that a stochastic process X t has stationary increments i s >, X t+s X t has the same law as X s ; typical examples are ractional Brownian motions, or Lévy processes. I X t has stationary increments, then j, the random variables (c j,k ) k Z share the same law. (6) Note that this property holds or each given j, but implies no relationship between wavelet coeicients at dierent scales; however, i X t is a selsimilar process o exponent H with stationary increments, then it ollows rom (5) and (6) that the random variables Hj c j,k all share the same law (or all j and k). A random variable X has a stable distribution i a, b, c > and d R such that ax + bx L = cx + d where X and X are independent copies o X. A random process is stable i all linear combinations o the X(t i ) are stable. This property implies that the wavelet coeicients o a stable process are stable random variables. Further, it can actually be shown that any inite collection o wavelet coeicients orms a stable vector (c. e.g., [7, 6, 63, 75]). Finally, i X t is a Gaussian process, then any inite collection o its wavelet coeicients is a Gaussian vector. Note however a common pitall: Unless some additional stationarity property holds, the sole Gaussianity hypothesis or the marginal distribution o the process does not imply that the empirical distributions o the coeicients (c j,k ) k Z are close to Gaussian: It can only be said that they will consist o Gaussian mixtures (which can strongly depart rom a Gaussian distribution). The statistical properties o wavelet coeicients o Bm were studied in depth in the seminal works o P. Flandrin [7, 7] that signiicantly contributed to the popularity o wavelet transorms in the analysis o scale invariance. Among the results that we mentioned above, it is important to draw a dierence between those that are a consequence o the sole act that wavelet coeicients are deined as a linear mapping, and thereore would hold or any other basis (Gaussianity and stability) and those that are the consequence o the particular algorithmic nature o wavelets (selsimilarity and stationarity). These (rather straightorward) results are either implicit or explicit in the 6

7 literature concerning wavelet analysis o stochastic processes; however, in order to give a lavor o the ideas and methods involved, we give a proo o (5) at the end o Section 4.. Further, let us mention that all these wavelet properties do not constitute exact characterizations o the probabilistic properties o the corresponding processes, but only implications, because o a deiciency o wavelet expansions that will be explained in Section Beyond selsimilarity: Multiractal analysis Beyond selsimilarity. An obvious drawback o using exact probabilistic properties, such as selsimilarity, or independence o increments is that, though they can relevantly model physical properties o the underlying data, they usually do not hold exactly or real-lie signals. Two limitations o selsimilarity can classically be pointed out. First, the scaling properties may not hold or all scales (as required by the mathematical deinition o selsimilarity). This may be due to noise corruption or it may also result rom the act that the physical phenomena implying those properties intrinsically act over a inite range o scales only. Second, the appealing property o selsimilarity (the sole selsimilarity parameter H contains the essence o the model) may also constitute its limitation: Can one really expect that the complexity o real-world data can be accurately encapsulated in one single parameter only? Let us examine such issues. Multiplicative Cascades. As mentioned above, B. Mandelbrot coined Bm as one o the major candidates to model scale invariance. However, combining selsimilarity and stationary increments implies that all inite moments o the increments o Bm (and o any process whose deinition gathers these two properties) veriy, or all ps such that IE( X() p ) < [68, 65], τ R, IE( X(t + τ) X(t) p ) = IE( X() p ) τ ph. (7) Though very powerul, this result can also be regarded as a severe limitation or applications, where empirical estimates o moments, X(t + τ) X(t) p dt, computed on the data assuming ergodicity, may instead behave as: X(t+τ) X(t) p dt = τ ζ(p), or a range o τ, and where ζ(p) is, by construction, a concave unction, but is not necessarily linear. This is notably the case when analyzing velocity or dissipation ields in the study o hydrodynamic turbulence where strictly concave scaling unctions were obtained (c. e.g., [76] or a review o experimental results). This was justiied by a celebrated argument due to Kolmogorov and Obukhov in 96 [8, 49, 86]: Navier-Stokes equation, governing luid motions, implies that the (third power o the) gradient o the velocity u(x + r) u(x) is proportional to the mean o the local energy ɛ r dissipated into a bulk o size r: IE(u(x+r) u(x)) 3 IE(ɛ r ) r. I ɛ r is constant, then any power scales as IE(u(x + r) u(x)) p ɛ p rr p/3. But, in turbulence, ɛ r is not constant over space and should rather be regarded as a random variable, hence implying: IE(u(x + r) u(x)) p IE(ɛ p r) r p/3 which naturally implies a general scaling o the orm IE(u(x + r) u(x)) p r ζ(p), with ζ(p) which is likely to depart rom the linear behavior ph which is implied by the selsimilarity hypothesis. 7

8 To account or this orm o observed scale invariance, new stochastic models have been proposed: Multiplicative cascades. Numerous declinations o cascades were proposed in the literature o hydrodynamic turbulence to model data, see e.g., [76] (on the mathematical side, see the book [6] or a detailed review, and also the contribution o J. Barral and J. Peyrière [3] in the present volume, or a lighter introduction). For the sake o simplicity, the simplest (D) ormulation is presented here; however, it contains the key ingredients present in all other more general cascade models. Starting rom an interval (arbitrarily chosen as the unit interval), a cascade is constructed as the repetition o the ollowing procedure (c being an arbitrary integer): At iteration j, each o the c j sub-intervals {I j,k, k =,... c j }, obtained at iteration j, is split into c intervals o the same length c j ; a mean- positive random variable w j,k (drawn independently and with identical distribution) is associated with each {I j,k, k =,... c j }. Ater an ininite number o iterations, the cascade W is deined as the limit (in the sense o measures) o the inite products W J (x)dx = w j,k dx. j J, x I j,k Mathematically, this sequence o measures converges under mild conditions on the distribution o the multipliers w (c. [3]). Such constructions and variations have been massively used to model the Richardson energy cascade implied by the Navier-Stokes equation [64] (c. [76] or a review o such models). One o the major contributions o B. Mandelbrot to the ield o hydrodynamic turbulence was proposed in [4], where he presented the themes and variations on multiplicative cascades in a single ramework and studied their common properties, notably in terms o scale invariance. This celebrated contribution paved the way towards the creation o the word multiractal, to the notions o multiractal processes and multiractal analysis, where multi here reers to the act that instead o a single scaling exponent H, a whole amily o exponents ζ(p) are needed or a better description o the data. B. Mandelbrot also discussed the importance o the distribution o the multipliers and how multiplicative cascades may signiicantly depart rom log-normal statistics that could be obtained rom a simplistic argument: Πw j,k = Πe log(w j,k) = e log w j,k, where log w j,k tends to a normal distribution, hence e log w j,k should tend to a log-normal distribution (c. [3, 3], and also [86]). This opened many research tracks proposing alternatives to the log-normal model, the most celebrated one being the log-poisson model [7]. This will be urther discussed in Section 6.. Much later, multiplicative cascades were connected to Compound Poisson cascades and urther to Ininitely Divisible Distributions (c. e.g., [48]). Mandelbrot himsel contributed to these latter developments with the earliest vision o Compound Poisson Cascades [3] and also with his celebrated Bm in multiractal time [35], which will be considered in Section 6.. This gave birth to multiple constructions o processes with multiractal properties (c. e.g., [9, 3, 4, 33, 34, 48, 49]). Compared with the concept o selsimilarity (and in particular Bm), which is naturally related to random walks and hence to additive constructions, cascades are, by construction, tied to a multiplicative structure. This deep dierence in nature explains why practitioners, 8

9 or real-world applications, are oten eager to distinguish between these two classes o models. This distinction is however oten misleadingly conused with the opposition o monoversus multi-ractal processes. This will be urther discussed in Section 5.4. Scaling unctions and scaling exponents. The empirical observations o strictly concave scaling exponents, made on hydrodynamic turbulence data, led to the design o a new tool or analysis: Scaling unctions. Let : R d R. The Kolmogorov scaling unction o (see [7]) is the unction η (p) implicitly deined by the relationship p, (x + h) (x) p dx h η (p), (8) the symbol meaning that ( log η (p) = lim h ) (x + h) (x) p dx log h. (9) Note that this limit does not necessarily always exist; when it does, it relects the act that shows clear scaling properties (and it is needed to hold in order to numerically compute scaling unctions). However, we are not aware o any simple and general mathematical assumption implying that it does; thereore, it is important to ormulate a mathematical and slightly more general deinition o the scaling unction, which is well deined or any unction L p : ( ) log (x + h) (x) p dx p, η (p) = lim in. () h log h It is important to note that, i data are smooth (i.e., i one obtains that η (p) p), then one has to use dierences o order or more in (9) and () in order to deine correctly the scaling unction. For application to real-world data, which are only available with a inite time or space resolution, (9) can be interpreted as a linear regression o the log o the let hand side o (8) vs. log( h ). The same remark holds or the other scaling unctions that we will meet. Multiractal analysis. In essence, multiractal analysis consists in the determination o scaling unctions (variants to the original proposition o Kolmogorov η will be considered later). Such scaling unctions can then be involved into classiication or model selection procedures. Scaling unctions constitute a much more lexible analysis tool than the strict relation (). For many dierent real-world data stemming rom applications o various natures, relationships such as (8) are ound to hold, while this is usually not the case or (), see Section 6 or illustrations on real world data. An obvious key advantage o the use o the scaling unction η (p) is that its dependence in p can take a large variety o orms, hence providing versatility in adjustment to data. Thereore, multiractal analysis, being based on a whole unction (the scaling unction η (p), 9

10 or some o its variants introduced below), rather than on the single selsimilarity exponent, yields much richer tools or classiication or model selection. The scaling unction however satisies a ew constraints, or example, it has to be a concave non-decreasing unction (c. e.g., [99, 83]). Another undamental dierence between the selsimilarity exponent and the scaling unction lies in the act that, or stochastic processes, the selsimilarity exponent is, by deinition, a deterministic quantity, whereas the scaling unction is constructed or each sample path, and thereore is a priori deined as a random quantity. And, indeed, even i the limit in () turns out to be deterministic or many examples o stochastic processes (such as Bm, Lévy processes, or multiplicative cascades, or instance), it is not always the case, as shown, or instance, by the example o airly general classes o Markov processes, such as the ones studied in [7]. Note that, i a stationarity assumption can be reasonably assumed, then one can deine a deterministic quantity by taking an expectation instead o a space average in (), which leads to a deinition o the scaling unction through moments o increments. Under appropriate ergodicity assumptions, time (or space) averages can be shown to coincide with ensemble averages, and, in this case, both approaches lead to the same scaling unctions, see [65] or a discussion (checking such ergodicity assumptions or experimental data does, however, not seem easible and can become an intricate issue, c. [])..4 Data analysis and Signal Processing In its early ages, the concept o scale invariance has been deeply tied to that o /processes, that is, to second order stationary random processes whose power spectral density behave as a power law with respect to requency over a broad range o requencies: Γ X () C β, where the requency satisies m M, with M / m. From that starting point, spectrum analysis was regarded as the natural tool to analyze scale invariance and estimate the corresponding scaling parameter β. In that context, the contribution o B. Mandelbrot and J. Van Ness [38] can be regarded as seminal since, elaborating on [85], it put orward Bm, and its increment process, ractional Gaussian noise (Gn), as a model that extends and encompasses /-processes: Gn is a /-process, with β = H, Bm is a non-stationary process that shares the same scale invariance intuition. This change o paradigm immediately raised the need or new and generic estimation tools or measuring the parameter H rom real world data, which were no longer based on classical spectrum estimation. Relying strongly on the intuitions beyond selsimilarity and long memory (a concept deeply tied with selsimilarity, c. e.g. [36]), the R/S estimator (or Range-Scale Statistics) has been the irst such tool proposed in the literature, to the study o which B. Mandelbrot also contributed [34, 39, 4]. The R/S estimator has notably been used in inance and economy, in particular by D. Strauss-Kahn and his collaborators, (see e.g., [8, 57]). Later, in the 9s, with the advent o wavelet transorms, seminal contributions studied the properties o the wavelet coeicients o Bm [7, 7, 77] and opened avenues towards accurate and robust wavelet-based estimations o the selsimilarity parameter H [3, 8, 79, 8].

11 To some extent, the present contribution can be read as a tribute to ideas and tracks originally addressed by B. Mandelbrot (and others) aiming at estimating the parameters characterizing scale invariance in real world data: The wavelet based ormulations o multiractal analysis (i.e., o the estimation rom data o scaling unctions and multiractal spectra) devised in Sections 4 and 5 can be read as continuations and extensions o these pioneering works towards richer models and reined variations on the characterization o scale invariance..5 Goals and Outline In Section, it is explained how various paradigms pertinent in ractal geometry, such as selsimilarity or ractal dimension, can be shited to the setting o unctions, thus leading to the introductions o several scaling unctions. In Section 3, it is shown how such scaling unctions can be reinterpreted in terms o unction space regularity: On one hand, it allows to derive some o their properties; on other hand, it paves the way towards equivalent ormulations that will be derived later. Section 4 introduces wavelet techniques, which allow to reormulate these ormer notions in a more eicient setting, rom both a theoretical and pratical point o view. Section 5 introduces the multiractal ormalism, which oers a new interpretation or scaling unctions in terms o ractal dimensions o the sets o points where the unctions considered has a given pointwise smoothness. Applications o the tools and techniques developed here will be given in Section 6, aiming at illustrating the various aspects o multiractal analysis addressed in the course o this contribution. Such applications are chosen either because B. Mandelbrot signiicantly contributed to their analysis and/or because they illustrate particularly well some concepts which will be developed here. The selected applications stem rom widely dierent backgrounds, ranging rom Turbulence to Internet, Heart Beat Variability and Finance (in D), and rom natural textures to paintings by amous masters (in D), thus showing the extremely rich possibilities o these new techniques. From ractals to multiractals. Scaling and Fractals Dynamics versus Geometry. One o the major contributions o B. Mandelbrot has been to understand the beneits o using mathematical tools and objects already deined in earlier works but regarded as incidental, such as ractal dimensions, in order to explore the world o irregular sets, to classiy them and to study their properties. A key example is supplied by the Von Koch curve (c. e.g., [55, 7] and Fig. ). The important role that B. Mandelbrot gave to this curve is typical o his way o diverting a mathematical object rom its initial purpose and o making it it to his own views; indeed, it was initially introduced as an example o the graph o a continuous, nowhere dierentiable unction (see the book by G. Edgar [64], which made available many classic articles o historical importance in ractal analysis). B. Mandelbrot shited the status o this curve rom a

12 peripherical example o strange set to a central element in the construction o his theory, and he baptized it the Von Koch snowlake, a typical example o the poetic accuracy that B. Mandelbrot displayed or coining new names (other examples, being the devil s staircase, Lévy lights, Noah or Joseph eects,...). This example will be revisited below in Section 4. to validate a numerical method or multiractal analysis. Two eatures dynamics and geometry are intimately tied into the Von Koch snowlake: It is constructed rom (and hence invariant under) a dynamical system consisting o the iterative repetition o a split and reproduce procedure (highly reminiscent o the split and multiply procedure entering the deinition o multiplicative cascades, c. Section.3 earlier); the resulting geometrical set has well deined geometrical properties that can be put into light through the determination o its box dimension, whose deinition we now recall. Dimensions. Deinition Let A be a bounded subset o R d ; i ε >, let N ε (A) be the smallest number such that there exists a covering o A by N ε (A) balls o radius ε. The upper and lower box dimension o A are respectively given by dim B (A) = lim sup ε log N ε (A) log ε, and dim log N ε (A) B(A) = lim in ε log ε. () When both limits coincide (as it is the case or the Von Koch snowlake), they are reerred to as the box dimension o the set A: dim B (A) = lim ε log N ε (A) log ε. () This implies that N ε (A) displays an approximate power-law behavior with respect to scales: N ε (A) ε dim B(A), (3) a major eature that makes the box dimension useul or practical applications, as it can be computed through a linear regression in a log-log plot. The strong similarity (in a geometric setting) between the box dimension and the Kolmogorov scaling unction () can now clearly be seen. The latter is also deined as a linear regression in log-log plots (log o the p-th moments vs. the log o the scales): Both objects hence all under the general heading o scale invariance. This relationship between geometric and analytic quantities is not only ormal: On one hand, the determination o scaling unctions has important consequences in the numerical determination o the ractal dimensions o graphs, and related quantities, such as the p- variation, see Section. ; on other hand, the multiractal ormalism, exposed in Section 5, allows to draw relationships between scaling unctions and the dimensions o other types o sets, deined in a dierent way: They are the sets o points where the unction considered has a given pointwise smoothness (see Deinition 3). Note however that the notion o dimension used in the context o the multiractal ormalism diers rom the box dimension: One has to use the Hausdor dimension which is now deined (see [66]).

13 Deinition Let A be a subset o R d. I ε > and δ [, d], let ( ) Mε δ = in A i δ, R where R is an ε-covering o A, i.e. a covering o A by bounded sets {A i } i N o diameters A i ε. (The inimum is thereore taken on all ε-coverings.) For any δ [, d], the δ-dimensional Hausdor measure o A is mes δ (A) = lim ε M δ ε. One can show that there exists δ [, d] such that δ < δ, mes δ (A) = + δ > δ, mes δ (A) =. This critical δ is called the Hausdor dimension o A, and is denoted by dim(a). An important convention, in view o the use o these dimensions in the context supplied by the multiractal ormalism, is that, i A is empty, then dim ( ) =. In contradistinction with the box dimension, the Hausdor dimension is not obtained through a regression on log-log plots o quantities that are computable on real-lie data; thereore it is itted to theoretical questions and can not be used directly in signal or image processing; however, we will see that the multiractal ormalism allows to bypass this problem by supplying an indirect way to compute such dimensions. Scaling, dimensions and dynamics. The discussions in this section aimed at illustrating the interplay between three dierent notions: The signal processing notion o scaling, or scale invariance, the geometrical notion o dimension and the dynamic notion o constructive split/multiply or split/reproduce procedures. Note that the relationships between ractals and dynamics have been investigated by B. Mandelbrot, see [37] or a panorama. The dynamic constructive procedures o geometrical sets (e.g., Von Koch snowlake) or o unctions (e.g., multiplicative cascades) result in geometrical properties o the constructed object. These geometrical properties can be theoretically characterized using dimensions. These dimensions can be measured practically i they additionally imply the existence o power law behaviours (or scaling) (such as those reported in (3) or (8)). When the constructive procedure underlying the geometrical sets or unctions analyzed are known and parametrized, the measure o geometrical dimension may enable to recover the corresponding dynamic parameters. However this is certainly no more true outside o a parametric model: Many dierent dynamic constructions may yield the same dimensions. Thereore, the inverse problem o identiying the nature o the dynamic or even its existence rom the sole measure o dimensions is ill-posed and can not be achieved in general. Speciically, a concave scaling unction, that hence departs rom linearity in p, does not prove the existence o an underlying cascade mechanism in data. i 3

14 . From graph box-dimensions to p-variation: The oscillation scaling unction From graphs to unctions. For data analysis purposes, an obvious dierence between the determination o the ractal dimension o a set A and the scaling unction η (p) o a unction has already been pointed out: While the ormer reduces to a single number, the latter consists o a whole unction o the variable p, hence yields a much richer tool or classiication and identiication. However, the gap can easily be bridged in the case where the geometrical set A consists o the graph, denoted Gr(), o the unction. Let us examine how a slight extension o the notion o box dimension o a graph allows to introduce a dependency on a variable p, and hence a amily o exponents whose deinition bears some similarity with the Kolmogorov scaling unction. Oscillations. Let : [, ] d R be a continuous unction (the set [, ] d plays no speciic role and is taken here just to ix ideas). I A [, ] d, let Os (A) denote the oscillation o on the set A, i.e. Os (A) = sup in. A A Note that this notion deines only irst order oscillation; i A is a convex set, second order oscillation (which has to be used or smooth unctions) is deined as ( ) Os (A) = sup x + y (x) + (y) x,y A (and so on or higher order oscillation). Let λ j (x ) denote the dyadic cube o width j that contains x ; Λ j will denote the set o such dyadic cubes λ R d o width j. Let now p be given. The p-oscillation at scale j o is deined as R (p, j) = dj λ Λ j (Os (λ)) p. (4) This quantity allows to deine the oscillation scaling unction o as O (p) = lim in j + log (R (p, j)) log( j. (5) ) Box dimension o graphs versus the oscillation scaling unction. Since is continuous, the graph o is a bounded subset o R d+. First, note that, in the deinition o the upper box dimension (), one can rather consider coverings by dyadic cubes o side j ; indeed each optimal ball o width ε used in the covering is included in at most 4 d dyadic cubes o width j = [log (ε)]; and, in the other way, a dyadic cube is contained in the ball o same center and width d j. Thereore using dyadic cubes (instead o optimally positioned balls) only changes preactors in N ε (Gr()) and not scaling exponents. Let us now consider the dyadic cubes used in the covering o the graph o and which stand above a dyadic cube λ o dimension d and width j included in [, ] d. Their number 4

15 N(λ) clearly satisies: (sup j in λ λ ) N(λ) j ( ) sup in +. λ λ This, together with the deinition o the oscillation scaling unction, implies that or p =, dim B (Gr()) = max(d, d + O ()). However, the oscillation scaling unction can be used or other (positive) values o p, and thereore yields a classiication tool with similar potentialities as those oered by the Kolmogorov scaling unction. We will see an example o its use in inance, in order to estimate quadratic variations, see Section 6.. An extension o the oscillation scaling unction will be deined later, deined through wavelet quantities: The leader scaling unction, c. Section 4.4. Box dimension and selsimilarity exponent. For some selsimilar unctions and processes, a direct relationship between the box dimension o their graphs and their selsimilarity exponent can be drawn. For instance, one-dimensional Bm o selsimilarity index H has a graph o box dimension H, and the same result holds in a deterministic setting or the graphs o the Weierstrass-Mandelbrot unctions, which have box dimension H (see [67, 68] where similar results can also be ound or Hausdor dimensions o graphs). Note that B. Mandelbrot himsel had already drawn connections between selsimilarity and ractal dimension (c. e.g., [9, 3, 3]). 3 Geometry vs. Analysis The shit rom ractal geometry to multiractal analysis essentially consists in replacing the study o selsimilarity, and o geometric properties o sets, by the study o analytic properties o unctions. This shit has several acets: First, in Section 3., we will explore relationships between selsimilarity and pointwise regularity, two concepts which are oten conused, and actually have subtle interplays which are ar rom being ully understood today. Second, in Section 3., we will give the interpretation o scaling unctions in terms o unction space regularity, which allows to use the ull apparatus o unctional analysis. Finally, in Section 5, the multiractal ormalism is introduced: It allows to interpret the (Legendre transorm o the) scaling unction in terms o dimensions o sets o points where has a given pointwise Hölder regularity. 3. Pointwise Hölder regularity Hölder exponent. Selsimilarity o either deterministic unctions or stochastic processes is intimately related with their local regularity. Beore explaining these relationships, let us start by recalling the notion o pointwise Hölder regularity (alternative notions o pointwise regularity, have also been considered, allowing to deal with unctions that are not locally bounded, see [95, 96, 97, ]). 5

16 Deinition 3 Let : R d R be a locally bounded unction, x R d and let γ ; belongs to C γ (x ) i there exist C >, R > and a polynomial P o degree less than γ such that i x x R, then (x) P (x x ) C x x γ. The Hölder exponent o at x is h (x ) = sup {γ : is C γ (x )}. When Hölder exponents lie between and, the Taylor polynomial P (x x ) boils down to (x ) and the deinition o the Hölder exponent heuristically means that, (x) (x ) x x h (x ). For these cases, a relationship can be drawn with the local oscillation considered in Section.. Indeed, let λ j (x ) denote the dyadic cube o width j that contains x, and 3λ j (x ) the cube o same center and 3 times wider (it thereore contains 3 d dyadic cubes o width j ). One easily checks that, i < h (x ) <, then h (x ) = lim in j + log (Os (3λ j (x ))) log( j. (6) ) This ormula puts into light the act that the Hölder exponent can theoretically be obtained through linear regressions on log-log plots o the oscillations Os (3λ) versus the log o the scale. However, in practice, this ormula has ound little direct applications or two reasons: One is that alternative ormulas based on wavelet leaders (c. Section 4.4) are numerically more stable, and the other is that many types o signals display an extremely erratic Hölder exponent, so that the instability o its determination is intrinsic, no matter which method is used. We will see in Section 4.4 how to turn this deeper obstruction. Implications o selsimilarity on regularity and dimensions. The Hölder exponent sometimes coincides with the selsimilarity exponent, hence resulting in a conusing identiication o the two quantities. This is the case or two requently used examples: Weierstrass-Mandelbrot unctions and Bm sample paths have everywhere the same and constant Hölder exponent, x : h (x) = H. The theoretical deinition o such examples essentially relies on a single parameter, and thereore it comes as no surprise that, or such simple unctions with a constant Hölder exponent, both their selsimilarity and their local regularity are governed by this sole parameter (and coincide). Similarly, their scaling unctions are linear in p, and thereore are governed by one parameter only, the slope o this linear unction, which, or these two examples, is H. However, one should not iner hasty conclusions rom these examples: Selsimilar processes with stationary increments do not always possess a single Hölder exponent that corresponds to the selsimilarity exponent, as shown by the example o stable Lévy processes: These processes are selsimilar o exponent H = /α, where α is the stability parameter. In addition, their scaling unction is linear in p only or p small enough: η (p) = Hp i p H = i p > H. 6

17 This is intimately related with the act that their Hölder exponent is not constant: Sample paths o a stable Lévy process have, or each h [, H], a dense set o points where their Hölder exponent takes the value h (c. [9] or Lévy processes, and [63] or Lévy sheets indexed by R d ). Thus, their Hölder exponents jump in a very erratic manner rom point to point, as a consequence o the high variability o the increments: Indeed, their distributions only have power-law decay, and thereore (in strong contradistinction with Gaussian variables) can take extremely large values with a relatively large probability; the importance o this phenomenon in applications was largely undervalued in the past, until B. Mandelbrot pointed out its possibly dramatic consequences, reerring to them in a colorul way as Noah s eect. Let us now be more explicit concerning the relationships between selsimilarity and Hölder regularity. A irst result relates selsimilarity with uniorm regularity o sample paths. Proposition Assume that X t is a selsimilar process o exponent H, with stationary increments, and such that E( X α ) < or some α >. Then, h < H /α, a.s. C, t, s, X t X s C t s h (7) The proo ollows directly rom the Kolmogorov-Chentsov theorem: Recall that this theorem asserts that, i X t is a random process satisying then, t, s, E( X t X s α ) C t s +β h < β/α, a.s. C, t, s, X t X s C t s h (see [55]). But, i X is selsimilar, then and the proposition ollows. E( X t X s α ) = t s Hα E( X α ), (8) We will reinterpret this result in Section 4.3 when the uniorm Hölder exponent will be introduced. The ollowing result goes in the opposite direction: It relates selsimilarity with irregularity o sample paths. Proposition Let X t be a selsimilar process o exponent H, with stationary increments, and which satisies P({X = }) =, i.e. the law o X at time t = (or at any other time t > ) does not contain a Dirac mass at the origin. Then a.s. a.e. h X (t) H. (9) Proo: We irst prove the result or t =. The assumption P({X = }) = implies the existence o a decreasing sequence (δ n ) n N which converges to and satisies P( X δ n ) n. 7

18 L Let l n = exp( /δ n ). Then X ln = (ln ) H X. Thereore P ( X ln (l n) H ) = P( X δ n ) log(/l n ) n. The Borel-Cantelli lemma implies that a.s. a inite number o these events happens, and thereore h X () H. By the stationarity o increments, this argument holds the same at each t, i.e. t a.s. h X (t) H; and Fubini s theorem allows to conclude that (9) holds. In particular, assume that X t satisies the assumptions o Proposition, and also o Proposition or any α >. Then, both results put together show that, in this case, (7) is optimal. Relationships between selsimilarity and Hölder exponents or selsimilar processes will be urther investigated in Section 5.4. Note that, without additional assumptions, one can not expect results more precise than those given by Propositions and, as shown by the subcase o stable, selsimilar processes with stationary increments, see [9, 68]: Let H denote the selsimilarity index o such a process, and α its stability index. First, note that there is no direct relationship between α, H and Hölder regularity: They always satisy < α < and H max(, /α); the case o Bm is misleadingly simple since it satisies α = and x, h BH (x) = H (, ); however Lévy processes satisy H = /α and h (x) takes all values in [, H]. Finally, Lévy ractional stable processes have nowhere locally bounded stable sample paths i H < /α, and continuous sample paths i H > /α (see Section 4.3 or a more precise result). Sample paths o selsimilar processes (with stationary increments) together with their scaling unctions (and multiractal spectra) are illustrated in Fig. in Section 5.. Let us now mention a ew results concerning the relationship between selsimilarity and ractional dimensions. We start with Bm which again is the simplest and best understood case. Such results are o two types: Dimensions o the graph o sample paths: Box and Hausdor dimensions coincide and take the value H, see [67, 68]. Relationship between the Hausdor dimension o a set A and o its image X(A) (see see [73] and reerences therein) a.s., or any Borel set, A R, dim X(A) = min(, dim(a)/h). () 8

19 We will see that this second result has important consequences or the multiractal analysis o some inance models proposed by L. Calvet, A. Fisher, and B. Mandelbrot, see (5). Partial results o these two types also hold or other selsimilar processes, see [73]. Regularity versus dimension: The mass distribution principle. The example o stable Lévy processes suggests that the most ruitul connection to be drawn does not involve the notion o selsimilarity, but rather should connect scaling unctions with pointwise regularity. This is precisely what the multiractal ormalism perorms, as it establishes a relationship between scaling unctions and the Hausdor dimensions o the isohölder sets o deined by E (h) = {x : h (x ) = h}. () The multiractal ormalism will be detailed in Section 5. A clue which shows that such relationships are natural can however be immediately given, via the mass distribution principle which goes back to Besicovitch (see Proposition 3 below). This principle establishes a connection between the dimension o a set A and the regularity o the measures that this set can carry. The motivation or it stems rom the ollowing remark. Mathematically, it is easy to obtain upper bounds or the Hausdor dimension o a set A: Indeed, any well chosen sequence o particular ε-coverings (or a sequence ε n ) yields an upper bound. To the opposite, obtaining a lower bound by a direct application o the deinition is more diicult, since it requires to consider all possible ε-coverings o A. The advantage o the mass distribution principle is that it allows to obtain lower bounds o Hausdor dimensions by just constructing one measure carried by the set A. Recall that a measure µ is carried by A i µ(a c ) =. Note, however, that a set A satisying such a deinition is not unique; In particular, this notion should not be mistaken with the more usual one o support o the measure, which is uniquely deined as the intersection o all closed sets carrying µ. For example, consider the measure µ = q 3 δ p/q p/q [,],p q= (where δ a denotes the Dirac mass at the point a). The support o this measure is the interval [, ], but it is (also) carried by the (much smaller) set A = Q [, ]. Proposition 3 Let µ be a probability measure carried by A. Assume that there exists δ [, d], C > and ε > such that, or any ball B o diameter less than ε, µ satisies the ollowing uniorm regularity assumption µ(b) C B δ. Then the δ-dimensional measure o A satisies: mes δ (A) /C, and thereore dim(a) δ. Proo: Let {B i } i N be an arbitrary ε-covering o A. Then ( ) = µ(a) = µ Bi µ(b i ) C B i δ. The result ollows by passing to the limit when ε. 9

20 3. Scaling unctions vs. unction spaces From scaling unctions to unction spaces. Kolmogorov scaling unction η (p) (as deined by ()) receives a unction space interpretation, which is important or several reasons. On one hand, it allows to derive a number o its mathematical properties, and on other hand, it points towards variants and extensions o this scaling unction, yielding sharper inormation on the singularities existing in data, and thereore oering a deeper understanding o the multiractal ormalism. Function space interpretation. The unction space interpretation o the Kolmogorov scaling unction can be obtained through the use o the spaces Lip(s, L p ) deined as ollows. Deinition 4 Let s (, ), and p [, ]; Lip(s, L p (R d )) i (x) p dx < and C >, h >, (x + h) (x) p dx C h sp. () Note that larger smoothness indices s are reached by replacing in the deinition the irstorder dierence (x+h) (x) by higher and higher order dierences: (x+h) (x+ h) + (x),... (which is coherent with the remark just ater the introduction o Kolmogorov scaling unction (), where, there too, higher order dierences have to be introduced in order to deal with smooth unctions ). It ollows rom (8) and () that, p, η (p) = sup{s : Lip(s/p, L p (R d ))}. (3) In other words, the scaling unction allows to determine within which spaces Lip(s, L p ) data are contained, or p [, ]. The reormulation supplied by (3) has several advantages: It will lead to a better numerical implementation, based on wavelet coeicients, and it will have the additional advantage o yielding a natural extension o these unction spaces to p (, ); this, in turn, will lead to a scaling unction also deined or p (, ), thereore supplying a more powerul tool or classiication, see Section 4.. Note that the reason why Kolmogorov scaling unction cannot be directly extended to p < is that L p spaces and the spaces which are derived rom them (such as Lip(s, L p )) do not make sense or p < ; indeed Deinition 4 or p < leads to mathematical inconsistencies (spaces o unctions thus deined would not necessarily be included in spaces o distributions). Another advantage o the unction space interpretation o the scaling unction is that it allows to draw relationships with areas o modeling where the a priori assumptions which are made are unction space regularity assumptions. Let us mention two amous examples, which will be urther used as illustrations or the wavelet methods developed in the next section: Bounded Variations (BV) modeling in image processing (c. Section 4.), and bounded quadratic variation hypothesis in inance (c. Sections 4.4 and 6., this latter case being related with the determination o the oscillation scaling unctions (5)).

21 Bounded variations and image processing. A unction belongs to the space BV, i.e., has bounded variation, i its gradient, taken in the sense o distributions, is a inite (signed) measure. A standard assumption in image processing is that real-world images can be modeled as the sum o a unction u BV which models the cartoon part, and another term v which accounts or the noise and texture parts (or instance, the irst u + v model, introduced by Rudin, Osher and Fatemi in 99 ([]) assume that v L, see also [9, 3]). The BV model is motivated by the act that i an image is composed o smooth parts separated by contours, which are piecewise smooth curves, then its gradient will be the sum o a smooth unction (the gradient o the image inside the smooth parts) and Dirac masses along the edges, which are typical inite measures. On the opposite, characteristic unctions o domains with ractal boundaries usually do not belong to BV, see Fig. or an illustration). Thereore, a natural question in order to validate such models is to determine whether an image (or a portion an image) actually belongs to the space BV or not. We will see in Section 4. how this question can be given a sharp answer using a direct extension o Kolmogorov scaling unction. Bounded quadratic variations. Another motivation or unction space modeling is supplied by the inite quadratic variation hypothesis in inance. In contradistinction with the previous image processing example, this hypothesis is not deduced rom a plausible intrinsic property o inancial data, but rather constitutes an ad hoc regularity hypothesis which (considering the actual state o the art in stochastic analysis) is required in order to use the tools o stochastic integration. There exist several slightly dierent ormulations o this hypothesis. One on them, which we consider here, is that the -oscillation is uniormly bounded at all scales. This means that, assumed to be deined on [, ] d, satisies C, j, dj R (, j) = λ Λ j (Os (λ)) C (where R (p, j) was deined by (4)). The deinition o the oscillation scaling unction (5) shows that, i O () > d, then has inite quadratic variation, and i O () < d, then this hypothesis is not valid. This will be urther discussed in Section 4. and illustrated in Section 6. and Fig Wavelets: A natural tool to measure global scale invariance The general considerations developed in Section 3. emphasized the importance o scaling unctions as a data analysis tool. To urther develop this program, reormulations o scaling unctions which are numerically more robust than the Kolomogorov and oscillation scaling unctions introduced so ar are o both practical and theoretical interests. Motivations or introducing wavelet techniques or this purpose were already mentioned in Section.. We now introduce these techniques in a more detailed way, and develop and explain the beneits o rewriting scaling unctions in terms o wavelet coeicients.

22 4. Wavelet bases Orthonormal wavelet bases on R d exist under two slightly dierent orms. First, one can use d unctions ψ (i) with the ollowing properties: The unctions dj/ ψ (i) ( j x k), or i =, d, j Z and k Z d (4) orm an orthonormal basis o L (R d ). Thereore, L, (x) = c i j,k ψ(i) ( j x k), (5) j Z i k Z d where the c i j,k are the wavelet coeicients o, and are given by c i j,k = dj R d (x)ψ (i) ( j x k)dx. (6) Note that, in practice, discrete wavelet transorm coeicients are generally not computed through the integrals (6), but instead using the discrete time convolution ilter bank based pyramidal and recursive algorithm (c. []), reerred to as the Fast Wavelet Transorm (FWT). However, the use o such bases rises several diiculties as soon as one does not have the a priori inormation that L. For instance, i the only assumption available is that is a continuous and bounded unction, then one can still compute wavelet coeicients o using (6) (indeed, these integrals still make sense), however, the series at the right-hand side o (5) may converge to a unction which diers rom. This is due to the act that certain special unctions (which do not belong to L ) have all their wavelet coeicients that vanish, and thereore, such possible components o unctions will always be missed in the reconstruction ormula (5). These special unctions always include polynomials, however, in the case o Daubechies compactly supported wavelets, other ractal-type unctions also have this property, see [67]. This explains why, at the end o Section, we mentioned that the properties o wavelet coeicients o certain processes that we obtained were not characterizations: Indeed inormation on these special unctions, as possible components o the processes, is systematically missed, whatever the inormation on the wavelet coeicients is. However, this drawback can be turned by using a slightly dierent basis, which we now describe. The alternative wavelet bases that will systematically be used rom now on are o the ollowing orm: There exists a unction ϕ(x) and d unctions ψ (i) with the ollowing properties: The unctions ϕ(x k) (or k Z d ) and dj/ ψ (i) ( j x k) ( or k Z d, and j ) (7) orm an orthonormal basis o L (R d ). In practice, we will use Daubechies compactly supported wavelets [56], which can be chosen arbitrarily smooth. Since these unctions orm an orthonormal basis o L, it ollows, as previously, that L, (x) = C k ϕ(x k) + c i j,k ψ(i) ( j x k); (8) k Z d j= k Z d i

23 the c i j,k are still given by (6) and C k = (x)ϕ(x k)dx. R d (9) This choice o basis enables to answer the ollowing important data analysis questions (which was not permitted by the previous choice (4)): In real-world data, the a priori assumption that L does not need to hold (or instance, the sample paths o standard models, such as Brownian motion, do not belong to L (R)). As already mentioned, the sole inormation that the series o coeicients o a unction on an orthonormal basis is square summable, does not imply that L (consider the example o = and the basis (4)). However, or the alternative basis given by (7), the ollowing property now holds: Assume that is a unction in L loc with slow growth, i.e. satisies C, A > : (x) dx C( + R) A, (3) B(,R) where B(x, R) denotes the ball o center x and radius R; then the wavelet expansion (8) o converges to almost everywhere. Additionally, i the wavelet coeicients o are square summable, then L (in contradistinction with what happens when using the basis (4)). Note that the slow growth assumption is a very mild one, since it is satisied by all standard models in signal processing, and actually is necessary, in order to remain in the mathematical setting supplied by tempered distributions. Note that one can even go urther: (6) and (9) make sense even i is not a unction; indeed, i one uses smooth enough wavelets, these ormulas can be interpreted as a duality product between smooth unctions (the wavelets) and tempered distributions. This rises the problem o determining i the series (8) converges in other unctional settings, and it is indeed one o the most remarkable properties o wavelet expansions that it is very oten the case. Wavelet characterizations o unction spaces are detailed in [43]. Let us mention a property which is particularly useul, and shows that convergence properties o wavelet expansions are as good as possible : Suppose that is continuous with slow growth, i.e. that it satisies C, N >, x R d, (x) C( + x ) N ; (3) then the wavelet expansion o converges uniormly on every compact set. However, while the dierences that we pointed out between the two types o wavelet bases are important or reconstruction, they are not or the type o analysis that we will perorm: Indeed, properties will be deduced rom the inspection o asymptotic behaviors o wavelet coeicients at small scale, where both bases (4) and (7) coincide. As an example o the ruitul interplay between the algorithmic structure o wavelet bases and the concept o selsimilarity, we now prove (5). We start by recalling the deinition o equality in law o stochastic processes. Deinition 5 Two vectors o R l : (X, X l ) and (Y, Y l ) share the same law i, or any Borel set A R l, P({X A}) = P({Y A}). 3

24 Two processes X t and Y t share the same law i, l, or any inite set o time points t, t l, the vectors o R l share the same law. (X t, X tl ) and (Y t, Y tl ) We apply this deinition to the two processes X at and a H X t (such as stated in (3)), which are assumed to share the same law. Taking or A the domain between two parallel hyperplanes, this, in turns, implies that any inite linear combinations l α i X ti i= and l α i Y ti share the same law. Thereore, i X t is a selsimilar process o exponent H, then (3) means that t, t l, l α i X ati i= and i= l α i a H X ti share the same law. (3) i= Assume now that or almost every ω Ω, the sample path o X t is Riemann integrable. The coeicient c j,k = j X t ψ( j t k)dt is a limit o the Riemann sums (t i+ t i ) j X ti ψ( j t i k); (33) using (3), it ollows that (33) has the same law as (t i+ t i ) H j X ti ψ( j t i k) i i which, writing u i = t i, can be written (u i+ u i ) H j X ui ψ( j u i k), i which is a Riemann sum o the integral H j X u ψ( j u k)du = H c j,k. Passing to the limit when the largest spacing between sampling points t i (and thereore u i ) tends to, we obtain that c j,k L = H c j,k. The argument that we developed or one coeicient can be reproduced similarly or an arbitrary vector o coeicients, hence (5) holds. 4

25 4. Wavelet Scaling Function We now investigate properties which arise as consequences o the wavelet characterizations o unction spaces. Notations. Compact notations are used or indexing wavelets. Instead o the three indices (i, j, k), dyadic cubes are used: λ (= λ(j, k)) = k j + [, j ) d. In order to keep notations as simple as possible, the index i is dropped in ormulas and notations, with the implicit convention that sums or suprema over indices are also taken on the index i. Accordingly, c λ = c i j,k and ψ λ(x) = ψ (i) ( j x k). The wavelet ψ λ is localized near the cube λ: Since we use Daubechies compactly supported wavelets, C > such that i, j, k, supp (ψ λ ) C λ (where C λ denotes the cube o same center as λ and C times wider). Finally, let Λ j denote the set o dyadic cubes λ which index a wavelet o scale j, i.e. wavelets o the orm ψ λ (x) = ψ (i) ( j x k) (note that Λ j is a subset o the dyadic cubes o size j+ ), and Λ will denote the union o the Λ j or j. Wavelet scaling unction. A key property o wavelet expansions is that many unction spaces have a simple characterization by conditions bearing on wavelet coeicients. This property has a direct consequence on the determination o the Kolmogorov scaling unction. Let S (p, j) = dj c λ p. (34) λ Λ j Classical embeddings between unction spaces imply that, i the wavelets are smooth enough, then the Kolmogorov scaling unction can be re-expressed as (c. [88]) p, η (p) = lim in j + log (S (p, j)) log( j, (35) ) This ormulation, which, again, yields the scaling unction through linear regressions in log-log plots, has several advantages when compared to the earlier version (). First, it allows to extend the Kolmogorov scaling unction to the range < p ; it will be shown in Section 4.4 how to go even urther and deine scaling unctions or negative ps (see also [] or an extension o η (p) to p < ). We will call this extension o the Kolmogorov scaling unction the wavelet scaling unction, but we will keep the same notation. By construction, η (p) is a concave increasing unction, o derivative at most d, see [99]. Wavelet regularity. Let us say a ew words about the required smoothness o wavelets. The rule o thumb is that wavelets should be smoother and have more vanishing moments than the regularity index appearing in the deinition o the unction space. In signal processing, one is not necessarily aware beorehand o the regularity o the data. In practice, one uses smoother and smoother wavelets, and when the results do not vary, this means that smooth enough wavelets are used. (Note that this is reminiscent o the deinition o the Kolmogorov scaling unction where, in theory, one has to use higher and higher order dierences and stop when the results are numerically stabilized.) In the ollowing we will 5

26 never speciy the required smoothness, and always assume that smooth enough wavelets are used (the minimal regularity required being always easy to iner). Note that, in the wavelet setting, one should also draw a dierence between (35), which allows to deine the wavelet scaling unction o any unction, and its use in data analysis, where scaling properties need to hold to enable measurements based on linear regressions in log-log plots. Function space interpretation. Let us examine how the wavelet scaling unction can be used in order to validate the unction space assumptions discussed in Section 3.. For instance, regarding the u+v model, where u BV and v L, the values taken by wavelet scaling unction at p = and p = allow practitioners to determine i data belong to BV or L : I η () >, then BV, and i η () <, then / BV I η () >, then L and i η () <, then / L. Thus wavelet techniques allow to check i the unction space assumptions which are made, e.g., in certain denoising algorithms relying on then u + v model, are valid. Examples o synthetic and natural images are shown in Fig., together with the corresponding measures o η () and η (). The image consisting o a simple discontinuity along a circle and no texture, (i.e., a typical cartoon part o the image in the u+v decomposition) is in BV, while the image o textures or discontinuities existing on a complicated support (such as the Von Koch snowlake) are not. Y. Gousseau and J.-M. Morel were the irst authors to rise the question o inding statistical tests to veriy i natural images belong to BV [77]. Our results conirm that the BV requirement is seldom met; actually, images oten turn out not to be even in L, as illustrated in Fig. (bottom row). 4.3 Uniorm Hölder exponent It has been shown in the previous section that the Kolmogorov scaling unction can be rewritten and extended as a wavelet coeicient based version. It is hence natural to ask whether the same can be perormed or the oscillation scaling unction (5). Because oscillations are deined only i is locally bounded, this condition needs to be checked in practice. This test can be perormed using the uniorm Hölder exponent, which constitutes the subject o this subsection. However, beore entering technicalities, the ollowing general question concerning unction space modeling needs to be addressed. Function space modeling. The issue o determining whether experimental data, acquired by a physical device, can be modeled by a bounded unction, or not, may appear meaningless at irst sight. Indeed, data can be collected only with inite size and resolution, and hence consist o inite, and thus bounded, sequences. For images, a common pitall is to conclude that images are grey-levels, so that they necessarily take values in [, ], and, thereore, by construction, are appropriately modeled by bounded unctions. The issue is even more general regarding the whole strategy o unctional space modeling: Given any 6

27 inite resolution and size sequence o data practically available, they can always be extrapolated by a C unction, that thus belongs to any possible unction space. The resolution o this paradox requires the use o the notion o scale invariance. Let us consider again the example o image modeling and address the issue o deciding whether a digital image can be modeled by a bounded unction or not: It is clear that i the image is analyzed only at its inest scale, then the answer is positive. However, ollowing the point o view that we developed until now, the problem can be reinterpreted in terms o analysis o scaling properties, over a range o available scales, and will thereore amount to determine whether an observed scaling exponent its with those that are compatible with bounded unctions, or not. The uniorm Hölder exponent, which will be considered in this section, answers this question. More generally, numerical results, obtained rom inite resolution and size data, leading to conclude that data belong to certain unction spaces and not to others, should actually be understood in the ollowing way: The unction space regularity thus determined holds under the hypothesis that scaling that are observed on the range o available scales would continue to hold at iner scales i they were available. Uniorm Hölder exponent: Deinition. The mathematical idea behind testing whether data are locally bounded or not is to consider boundedness as a particular case in the whole range o Lipschitz spaces, which corresponds to the regularity exponent s =. Let us be more speciic. The Lipschitz spaces C s sg(r d ) are deined or < s < by the conditions : and C, N, x, y R d, (x) C( + x ) N. C, N, x, y R d, (x) (y) C x y s ( + x + y ) N. I s >, they are then deined by recursion on [s] by the condition: Csg(R s d ) i L and i all its partial derivatives (taken in the sense o distributions) / x i (or i =, d) belong to Csg s (R d ). I s <, then the Csg s spaces are composed o distributions, also deined by recursion on [s] as ollows: Csg(R s d ) i is a inite sum o partial derivatives (in the sense o distributions) o order o elements o Csg s+ (R d ). This allows to deine the Csg s spaces or any s / Z (note that a consistent deinition using the Zygmund classes can also be supplied or s Z, see [43], however we will not need to consider these speciic values in the ollowing). We can now deine the parameter that we will use or determining boundedness. Deinition 6 The uniorm Hölder exponent o a tempered distribution is h min = sup{s : C s sg(r d )}. (36) This deinition does not make any a priori assumption: The uniorm Hölder exponent is deined or any tempered distribution, and it can be positive and negative. Note that we have already met this notion: Proposition can be reinterpreted as stating that, i 7

28 < H /α <, then h min X H /α. In particular the remark ollowing the proo o Proposition allows to recover the act that, or Bm, h min = H; additionally, it yields that the same result holds or the Rosenblatt process (see [6, 59] or the deinition and the properties o the wavelet expansion o this process). The values taken by the uniorm Hölder exponent have the ollowing interpretation: I h min i h min >, then is a locally bounded unction, <, then is not a locally bounded unction. Typical examples are supplied by Lévy ractional stable processes which already appeared in Section 3.; they satisy h min = H /α, and thereore (as already mentioned) they have nowhere locally bounded stable sample paths i H < /α, and continuous sample paths i H > /α. The importance o this exponent lies in the act that it does not only play the role o a prerequisite (i.e., a parameter whose value has to be positive i one wants to determine the oscillation scaling unction), but also that it turns out to be a very eicient parameter or discrimination. As will be illustrated later (c. Section 6), the classiication o several types o data can actually be based on the determination o their uniorm Hölder exponent. Uniorm Hölder exponent and wavelets. In practice, h min can be derived directly rom the wavelet coeicients o through a simple regression in a log-log plot; indeed, it ollows rom the wavelet characterization o the spaces Csg s that: h min = lim in j + log ( sup c λ λ Λ j log( j ) ). (37) This estimation procedure has been studied in more detail in [97]. The exponent h min can also be derived rom the wavelet scaling unction, [97], h min η (p) = lim = lim p + p p + η (p). Uniorm Hölder exponent and applications. In Section 6, which reviews a number o real-world applications, we will see that h min can be ound either positive or negative depending on the nature o the application. For instance, velocity turbulence data (c. 6. and Fig. 6, middle row, let plot) and price time series in inance (c. 6. and Fig. 7, nd and 3rd row row, second plots and bottom row, irst plot) are ound to always have h min >, while aggregated count Internet traic time series always have h min < (c. 6.4 and Figs. and, top row, let plots). For biomedical applications (c. e.g., etal hear rate variability as in Section 6.3 and Fig. 8, 3rd row), as well as or image processing (c. Section 6.5 and Fig., 4th column), h min can commonly be ound either positive 8

29 or negative. For these last two examples, h min is ound to be a highly relevant parameter or classication purposes (c. Fig. 8, bottom row, and Fig., bottom row, respectively). Functions that are not locally bounded and ractional integration. The examples listed above indicate that it is not uncommon in real-world application to ind that h min <. However, this does not imply that urther analyses which require that h min > (and will be described in the ollowing) are impossible and should be discarded. Indeed, ractional integration oers a one to one way to associate with any distribution (with potentially negative h min ) a unction ( s), whose exponent h min is positive, and thereore ( s) classiication can then be operated on this new unction ( s) rather than on the initial data. We now expose this procedure, both rom a theoretical and a practical point o view. Note that there exist many variants o the deinition o ractional integration, which would however all lead to the same deinition or the exponents used here, see [6] or a discussion o ractional integration, especially in the context o stochastic processes. Deinition 7 Let be a tempered distribution. Fractional integral o order s o, denoted by ( s), is deined as convolution operator (Id ) s/ which, in the Fourier domain, is the multiplication by the unction ( + ξ ) s/. Numerically, ractional integration is intricate to perorm (because o inite size and border eect issues). However, or our purpose here, these diiculties can be circumvented by instead using pseudo-ractional integration, deined as ollows. Deinition 8 Let s >, let ψ λ be a wavelet basis whose elements belong to C r with r > s+; let be a unction, or a distribution, with wavelet coeicients c λ. The pseudo-ractional integral (in the basis ψ λ ) o o order s, denoted by I s (), is the unction whose wavelet coeicients (on the same wavelet basis) are c s λ = sj c λ. This operation is numerically straightorward, and the ollowing result shows that it retains the same pointwise and multiractal properties as exact ractional integration, see [97, 98, 99, 3]. Proposition 4 Let be a tempered distribution; then p >, η ( s)(p) = η I s () = η (p) + sp, s R, = h min I s () = hmin + s. h min s Additionally, the pointwise Hölder exponents satisy s > h min, x R d, h ( s)(x) = h I s ()(x). 9

30 This last property also holds or the leader scaling unctions which will be introduced in Section 4.4 (under the same condition s > h min, so that those quantities are well deined). Thereore, in practice, or multiractal analysis, pseudo-ractional integration is preerred to exact ractional integration. These results point toward a natural strategy in order to perorm the multiractal analysis o a unction which is not locally bounded (or o a distribution): First determine its exponent h min ; then, i h min <, perorm a pseudoractional integration o order s > h min ; it ollows that the uniorm regularity exponent o I s () is positive, and thereore that it is a bounded unction, to which multiractal analysis can be applied. Yet, this strategy rises an important problem: There is no canonical choice or the ractional integration order, and the shape o the wavelet leader scaling unction (see Deinition below) obtained ater ractional integration may depend on this choice. In practice, when multiractal properties o a collection o data have to be studied, one ollows the ollowing strategy: One irst determines the exponent h min o each signal in the database; i some o them have negative h min, one picks an exponent s such that h min +s is positive or all signals. Under these constraints, s should be picked as small as possible, so that data are modiied by a ractional integration o order as small as possible. The rule o thumb practically used is to select s as the smallest multiple o.5 ensuring that h min + s is positive or all signals and to perorm a pseudo-ractional integration o the same order s or all signals. 4.4 Wavelet leaders From oscillations to wavelet leaders. Our purpose is now to obtain a wavelet reormulation o the oscillation scaling unction (5) (in the same way as the wavelet scaling unction is the wavelet reormulation o the Kolmogorov scaling unction). This requires the introduction o wavelet leaders, which are local suprema o wavelet coeicients. Let us start with a heuristic argument showing that they are natural candidates to estimate oscillation. Recall that, i λ denotes a dyadic cube, then 3λ denotes the cube with same center and three times wider. Deinition 9 Let be a locally bounded unction satisying (3). The wavelet leaders o are the coeicients d λ = sup c λ. (38) λ 3λ The act that the supremum in (38) is inite is a consequence o the boundedness assumption or. Thereore, checking that h min > is an absolute prerequisite prior to computing leaders. Let us now sketch the reason why wavelet leaders allow to estimate the oscillation. Consider a wavelet coeicient c λ. Since we use compactly supported wavelets, c λ = dj (x)ψ (i) ( j x k)dx Cλ 3

31 Since wavelets have vanishing integral, c λ = dj ((x) (k j )ψ (i) ( j x k)dx dj Cλ Cλ dj Os (Cλ) = C Os (Cλ). (x) (k j ) ψ (i) ( j x k) dx Cλ ψ (i) ( j x k)dx Consider now a given cube K; this argument works or any wavelet whose support is included in K, so that each wavelet coeicient is bounded by COs (Cλ) COs (K). This explains why wavelet leaders are bounded by the local oscillation. Converse estimates ollow orm the ollowing argument: From (5), we get (x) (y) = j,k c (i) j,k ( ψ (i) ( j x k) ψ (i) ( j y k) ) Let j be such that j x y. The low requency terms (j j ) are estimated using the smoothness o the wavelets, which makes the dierence ψ (i) ( j x k) ψ (i) ( j y k) small. The terms such that j j Aj (where A has to be chosen correctly) are estimated using the bound o the wavelet coeicients supplied by the wavelet leader. Finally, high requency terms (j Aj ) are estimated using the uniorm regularity o. This allows to obtain the required converse estimates. Note that subtle relationships between oscillations and wavelet coeicients have recently been obtained by M. Clausel and her collaborators, see e.g. [54] and reerences therein. Leader scaling unction. The previous heuristic argument motivates the introduction o a new scaling unction, constructed on the model o the wavelet scaling unction, but replacing wavelet coeicients by wavelet leaders. Deinition Let be a locally bounded unction with slow growth (i.e. satisying (3)), and let T (p, j) = dj λ Λ j d λ p. (39) The leader scaling unction ζ (p) is given by p R, ζ (p) = lim in j + log(t (p, j)) log( j ). (4) The relationship between the leader scaling unction and the oscillation scaling unction is very similar to the one mentioned to exist between the wavelet scaling unction and the 3

32 Kolmogorov scaling unction: They coincide when p ; thereore the ormer extends the later or p <, see [93]. Let us discuss two consequences o this result: First, the act that they coincide or p = implies that, i h min >, then the upper box dimension o the graph o can be derived rom the leader scaling unction, see [9]: dim B (Gr()) = max(d, d + ζ ()). Second, when h min >, one can determine whether has bounded quadratic variation or not by inspecting its leader scaling unction or p =, indeed: I ζ () > d, then has bounded quadratic variation, i ζ () < d, then the quadratic variation o is unbounded. Discussion o the condition h min >. One may wonder i the condition h min > is really necessary in order to obtain such results. It is actually the case, and one can not obtain estimates o the quadratic variation (or o any p-variation) using wavelet methods or unctions which are locally bounded but do not have some uniorm smoothness or the ollowing reason: Recall that the Hilbert transorm is deined as the convolution with p.v.(/x) (or alternatively as the Fourier multiplier by the unction sign(ξ)) and consider as an example the sawtooth unction x [x] which is bounded and has discontinuities at the integers. It obviously has bounded p-variation or any p. However, ater applying the Hilbert transorm to it, discontinuities are transormed into logarithmic singularities, and thereore the property o bounded p-variation is lost or any value o p. Since the Hilbert transorm does not modiy the wavelet-based quantities that we have considered, such as h min, ζ (p), η (p), it is clear that wavelet methods can not estimate p-variations o unctions that have discontinuities, and thereore some uniorm regularity assumption is indeed required. In inance, or instance, checking whether h min > is positive or not is o double importance: First, it suggests to reject jump models or price; second, it enables to probe the initeness o quadratic variations, a requested property to validate the use o stochastic integration on such data. This is urther discussed in Section 6. and illustrated in Fig. 7. The bonus o negative ps. The leader scaling unction can also be given a unction space interpretation or p > in the ramework o oscillation spaces, studied in [94, 89]. In particular, embeddings between these spaces and other, more classical, unction spaces (such as Besov spaces) allow to derive an important relationship between the two scaling unctions constructed through wavelet coeicients, see [93]. Proposition 5 Let be a unction satisying h min p >, >. Then η (p) ζ (p). Let p c be the critical exponent deined by the condition η (p c ) = d, then p p c, η (p) = ζ (p). 3

33 An important property o the leader scaling unction is that it is well deined also or p <, although it can no longer receive a unction space interpretation. By well deined, it is meant that it has the ollowing robustness properties i the wavelets belong to the Schwartz class (partial results still hold otherwise), see [98, 99]: ζ (p) is independent o the wavelet basis. ζ (p) is invariant under the addition o a C perturbation. ζ (p) is invariant under a C change o variable. The invariance under a smooth change o variable is a mandatory property or texture classiication: Indeed, it is needed that natural textures can be recognized even ater the deormation produced by setting them on smooth suraces. The possibility o involving negative ps in analysis can turn crucial: A celebrated example is that o hydrodynamic turbulence where two multiplicative cascade models are classically in competition. Using positive ps only, the Kolomogorov and the wavelet scaling unctions computed rom experimental data ail to discriminate between the two models. However, the leader scaling unction, enabling the use o negative ps, permits to show that one model is unambiguously preerred by data. This is discussed in Section 6. and illustrated in Fig. 6 (bottom row plots). 5 Beyond unctional analysis: Multiractal analysis The act that the leader scaling unction extends the analysis to negative ps shows that multiractal analysis allows to go beyond the scope o unctional analysis. Indeed, the scaling unction no longer has a unction space interpretation i p <. Note that this extension has some canonical character, as a consequence o its independence o the chosen wavelet basis, and also because o its robustness properties. The point made in the present section is to show that even more is true by exhibiting a natural interpretation o scaling unctions, which requires both positive and negative values o p. This interpretation is in terms o the pointwise singularities o the unction, and will be supplied by the multiractal ormalism, which we now describe. 5. Multiractal ormalism or oscillation Multiractal spectrum versus multiractal ormalism. From now on, it is assumed that h min >, which implies that is a continuous unction, and thereore that its oscillations are well deined. Recall that the notion o oscillation came up in two occurrences until now: The deinition o the oscillation scaling unction (5), and the characterization o the pointwise Hölder exponent (6). This points towards a possible relationship between both quantities, which can be made explicit via multiractal ormalisms: Speciically, a multiractal ormalism establishes a connection between the scaling unctions and the spectrum o singularities (or multiractal spectrum) o, which is deined as ollows. 33

34 Deinition Let be a locally bounded unction. The spectrum o singularities o is the unction D (h) = dim(e (h)), where we recall that dim denotes the Hausdor dimension, and the E (h) are the isohölder sets E (h) = {x : h (x ) = h}. This deinition justiies the denomination o multiractal unctions: On the theoretical side, one typically considers unctions that have non-empty isohölder sets or h taking all values in an interval [h min, h max ], and thereore one has to deal with potentially an ininite (and even uncountable) number o ractal sets E (h) whose Hausdor dimensions have to be determined. Recall that dim( ) = so that D (h) = i h does not belong to the range o h. We will call support o D the set supp(d ) = {h : D (h) } = {h : E (h) }. (Note that the support o the spectrum needs not be a closed set, in contradistinction with the usual deinition o the support o a unction). The initial ormulation o the multiractal ormalism was due to the physicists G. Parisi and U. Frisch and was based on Kolmogorov scaling unction [5]. It was urther grounded mathematically in [4], see also [8]. Here, the ocus is on a ormalism based on the oscillation scaling unction, which we describe irst since it paves the way towards a more satisactory leader scaling unction ormulation. Oscillation multiractal ormalism. In this paragraph, it is assumed that h (x) takes on values between and, so that (6) is valid. Since pointwise Hölder exponents can thus be derived rom the quantities Os (3λ), it is natural to consider scaling unctions based on these quantities, i.e., the oscillation scaling unction o deined by (5). Let us now show how the spectrum o singularities can be obtained rom the scaling unction. The deinition o the scaling unction roughly means that the quantity R (p, j) deined by (4) satisies R (p, j) Os (p)j. Let us estimate the contribution to R (p, j) o the cubes λ that cover the points o E (h); (6) asserts that they satisy Os (3λ) hj ; since we need about D (h)j such cubes to cover E (h), the corresponding contribution roughly is dj D (h)j hpj = (d D (h)+hp)j. When j +, the dominant contribution stems rom the smallest exponent (the others bring contributions which are exponentially smaller), so that Os (p) = in h (d D (h) + hp). (4) It ollows rom the deinition o the scaling unction Os (p) that it is a concave unction on R. This is in agreement with the act that the right-hand side o (4) necessarily is a concave unction (as an inimum o a amily o linear unctions) no matter whether D (h) is concave or not. However, i the spectrum also is a concave unction, then the Legendre 34

35 transorm in (4) can be inverted (as a consequence o a general result on the duality o convex unctions), which justiies the ollowing deinition. Deinition Let be a unction which satisies h min ormalism i its spectrum o singularities satisies > ; ollows the multiractal D (h) = in p (d Os (p) + hp). (4) Validity o the multiractal ormalism. The derivation exposed above is not a strict mathematical proo, but rather consists o a heuristic explanation o the relation between D (h) and Os (p). The determination o the range o validity o (4), or o any o its variants, is one o the main open mathematical problems in multiractal analysis. Nonetheless, let us emphasize that the cornerstone o this derivation relies heavily on (6), i.e., on the act that the Hölder exponent o a unction can be estimated rom the set o values that its oscillation takes on the enlarged dyadic cubes 3λ. In particular, the variant o this multiractal ormalism based on increments (i.e., on the Kolmogorov scaling unction) has a narrower range o validity, see [5, 98, 99] or a discussion. An important remark is that, in applications, one should not ocus too much on the problem o the validity o the multiractal ormalism. On one side, because one can not compute directly a spectrum o singularities o experimental data, the multiractal ormalism can be checked only or mathematically deined unctions or processes. On other side, the purpose o multiractal analysis oten is to use a spectrum as a tool or classiication or model selection, and, with this respect, using a scaling unction (or it Legendre transorm) is a perectly valid approach, independently o the act that the multiractal ormalism holds or not. However, Section 5.4 below details an example where the multiractal ormalism allows to determine eectively the multiractal spectrum: It can be the case when it is degenerate, i.e., when h (x) is a constant unction (in which case D is supported by only one point). Further comments. The ormulation o the multiractal ormalism given by (4) has the ollowing advantage: The scaling unction is invariant under bi-lipschitz deormations o the unction, which is a natural requirement since the spectrum o singularities possesses this invariance property (as a consequence o the assumption that the range o h is included in (, )). To this Legendre transorm based multiractal ormalism, initially proposed in [5] and thoroughly theoretically grounded in [93, 65], has been opposed an alternative ormulation relying on the large deviation principle (c. e.g., [65] or an early introduction and [8] or a recent version involving oscillations o order ). However, we will not ollow this track or the ollowing reasons: Using oscillations restricts the possible range o Hölder exponents to h [, ]; the method can not be adapted to data that are not locally bounded (which is oten the case in signal and image processing). 35

36 This second drawback, as we saw, can be circumvented easily by a pseudo-ractional integration when using wavelet techniques, and this is the reason why we now turn towards these methods. The limitation to the range h [, ] implied by either the use o increments o order (Kolmogorov scaling unction) or oscillations o order (oscillation scaling unction) has long been recognized and led to the pioneering contributions o A. Arneodo and collaborators promoting the use o wavelet based methods in multiractal analysis (c. e.g., [,, 46, 47] and reerences therein). 5. A ractal interpretation o the leader scaling unction: The leader multiractal ormalism The introduction o wavelet leaders was motivated by the act that they allow to estimate local oscillations, and thereore come as a natural ingredient in a wavelet reormulation o the oscillation scaling unction and o pointwise Hölder regularity. Let x R d, and recall that λ j (x ) denotes the dyadic cube o width j which contains x. I h min >, then the heuristic relationship that we gave in Section 4.4 and which related oscillations and wavelet leaders actually leads to the ollowing result (see [93] and reerences therein): h (x ) = lim in j + ( ) log d λj (x ) log( j. (43) ) The heuristic arguments that were developed above or the derivation o the oscillation multiractal ormalism can be reproduced in the setting o wavelet leaders. They lead to the ollowing reormulation o the multiractal ormalism. Deinition 3 Let be a unction statisying h min > ; the leader spectrum o is deined through a Legendre transorm o the leader scaling unction, i.e. L (h) = in p R (d + hp ζ (p)). The wavelet leader multiractal ormalism holds i h R, D (h) = L (h). (44) This ormalism holds or large classes o models used in signal and image processing, such as Bm, lacunary and random wavelet series [7], or multiplicative cascades. In this wavelet leader setting, the justiication o the derivation o (44) relies heavily on (43). In particular, multiractal ormalisms based on other quantities (such as the wavelet scaling unction) have a narrower range o validity, see [93] or a discussion. The multiractal ormalism does not hold in all generality, nonetheless the ollowing upper bound does, yielding a general relation between the leader scaling unction and the spectrum o singularities. 36

37 Theorem I h min >, then, h R, d (h) L (h). Examples o leader multiractal spectra L (h) computed rom a single realization o various processes and ields are illustrated in Figs., 3 and 4, and compared with the theoretical multiractal spectra D (h). It can be observed that, or all these reerence processes, the leader multiractal spectra computed rom a single realization very satisactorily match the theoretical multiractal spectra D (h), hence suggesting that the multiractal ormalism, L (h) = D (h), holds in all cases. The ollowing relationship holds or most standard models used in multiractal analysis: h min = in(supp(d )). (45) Note however that this relationship is not true in all generality; or instance the chirp ( ) F α,β (x) = x α sin x β (or α and β positive) satisies h min F α,β = α β + and in ( supp ( D Fα,β )) = α. However, we will mention a partial result conirming (45) in Section Local spectrum and homogeneity So ar, little attention has been paid to the region where multiractal analysis is perormed, implicitly assuming that it is conducted on the entire set o data available. However, one may wonder whether multiractal analysis would yield dierent results when applied to dierent domains. The answer is oten negative: Scaling unctions, and spectra o singularities, when measured on a sub-part o the data (say a non-empty open subset, in order to dismiss boundary problems) do not depend on the particular open set which is chosen. When such is the case, the unction analyzed is said to have homogeneous multiractal properties. It is observed that large categories o experimental data have homogeneous multiractal properties: It is or instance the case or ully developed turbulence, where analyses perormed on dierent parts o the signal always yield the same results. However, this might not systematically be the case. A simple cause o non-homogeneity can be that data consist o a juxtaposition o dierent pieces, concatenated together. This situation is commonly met in image processing: Indeed, because o occlusion eects (i.e. objects in the ront o the picture partly mask objects behind), images usually appear as a patchwork o homogeneous textures. In this case, it is reasonable to isolate the dierent components o the data, and perorm multiractal analysis over each piece independently. Scaling unctions computed on the whole image simply result as the minimum o the local scaling unctions, and conversely the global multiractal spectrum consists o the maximum o the local spectra. Note that such cases can 37

38 lead to simple examples o (global) non-validity o the multiractal ormalism. Indeed, the scaling unction o the whole image remains, by construction, a concave unction, whereas the spectrum, as a maximum o concave unction needs no more be concave, even i each component is. Thereore, the multiractal ormalism will ail, since, by construction, the right hand side o (4) always yields concave unctions. However, more intricate situations may exist, where multiractal characteristics can evolve with time (or the location, or images) in a way which is more involved than piecewise constant. One can expect, or instance, that stock market data display dierent statistics during dierent market phases, such as crises or bubbles. Dierent multiractal characteristics could stem rom such dierences. One may also expect that the whole conditions o economy evolve with time, as a consequence o the ever increasing complexity o inancial operations, o new inancial products, and (more subtly) o mathematical models which themselves inluence the way that operators react to the market; all this pleads or possible smooth (or not so smooth) modiications o multiractal characteristics with time. Such possible mechanisms may help to interpret the empirical observations that multiractal properties o market price luctuations, such as those reported in Fig. 7 (bottom row plots) or the hourly Euro-USD rate, are signiicantly changing along time (c. Section 6. or discussion). On the mathematical side, such situations have been barely studied so ar (see a contrario the results concerning heterogeneous ubiquity, by J. Barral, A. Durand and S. Seuret, where sophisticated mathematical tools have been developed to deal with the theoretical challenges o this situation [34, 33, 35, 6]). A typical example, studied in [7], is supplied by Markov processes (i.e. processes with independent increment) which do not have stationary increments (and thus are not Lévy processes). Roughly speaking, such processes have jumps o random amplitude, but each jump brings the process in a new environment, and the local multiractal properties will depend on this environment which is randomly chosen. One thereore obtains a non-homogeneous process, with a random spectrum, and random scaling unctions. To the opposite, homogeneous multiractals have some additional regularity properties. For instance, the pitall mentioned in the previous subsection does not occur: An homogeneous multiractal unction satisies (45). An interesting phenomenon o homogeneity breaking will be discussed in the next subsection: We will consider a homogeneous monohölder stochastic process, whose square is not homogeneous, and no longer monohölder. Note also that a test or the constancy along time o the multiractal properties measured using the leader scaling unction and relying on a non parametric bootstrap procedure was proposed in [8]. 5.4 Mono vs. Multiractality? Disentangling issues. For practical purposes, it is oten o interest to decide whether data are mono- or multiractal. However, despite apparent simplicity, as such the question is ill-posed and can be rephrased in dierent ways:. Are data characterized by a single Hölder exponent that takes the same and unique 38

39 value everywhere or do there exist a variety o Hölder exponents in data?. Are data better described by a selsimilar process (oten implicitly meaning Bm) or by a cascade based process? In this later ormulation, the important underlying question is: Is there an additive (as in random walk and hence selsimilar models) or a multiplicative (as in cascades) structure underlying data and hence revealing signiicant dierences in the nature o the data. 3. Are the scaling exponents (i.e., the scaling unction) measured on data linear in p or not? Relying on the Bm (Gaussian based) intuition, these three dierent ormulations are oten thought to be equivalent. This section aims at discussing some o the subtleties underlying the relations between these three questions. Log-cumulants. Recall that, i they exist, the cumulants o a random variable X are the coeicients o the Taylor expansion o the second characteristic unction o X, i.e. the coeicients c m (i they exist) in the expansions log ( IE ( e px)) = m= c m p m n!. (46) Following [44, 46, 58], the cumulants o the log o the wavelet leaders are now introduced. Let C m (j) denote the m-th order cumulant o the random variables log(d j,k ). Under stationarity and ergodicity assumptions on this sequence, C (j) is the expectation o the log(d j,k ), C (j) their variance, etc. Assuming that the moments o order p o the leader d j,k exist, starting rom the assumption IE(d p j,k ) = IE(dp, ) jζ (p), one obtains that log(ie(d p j,k )) = log(ie(dp, )) + ζ (p) log( j ); using (46), one obtains the ollowing expansion or p close to : log(ie(d p j,k )) = log(ie(ep log d j,k )) = m C m (j) pm m!. Comparing these two expansions, we obtain that the C m (j) are necessarily o the orm and that ζ (p) can be expanded around as C m (j) = C m + c m log( j ), (47) ζ (p) = m c m p m m!. (48) The concavity o ζ implies that c. Note that, even i the moment o order p o d j,k is not inite, the cumulant o order m o log(d j,k ) is likely to be inite and (47) above is 39

40 also likely to hold. Moreover, (47) provides practitioners with a direct way to estimate the c m by means o linear regressions in C m (j) versus log( j ) diagrams [83]. The c m are hence not derived rom an a posteriori expansion perormed rom the estimates o the ζ (p), but instead estimated directly. For the sake o completeness, let us mention that the polynomial expansion o ζ (p) around can be translated, via the Legendre transorm, into an expansion o L (h) around its maximum [8], on condition that c : L (h) = d + c! ( h c c ) + c 3 3! ( ) h 3 c + c 4 + 3c ( ) 3 /c h 4 c +... (49) 4! c Mono vs. Multi-Hölder. The only general mathematical result concerning the validity o the multiractal ormalism is supplied by Theorem, which, in general does not yield the spectrum. However, it does yield an interval in which the support o the spectrum is included: Indeed, let h min = in{h : L (h) } and h max = sup{h : L (h) }. Then, the support o the spectrum D (h) is included in the interval [h min, h max ]. (It can indeed be checked that h min thus deined actually coincides with the uniorm Hölder exponent, deined in Section 4.3.) An important particular case stems rom the situation where h min and h max coincide, in which case the spectrum is supported by the sole point h min = h max, so that only one single Hölder exponent exists in the data, which thereore constitute a monohölder unction satisying c x, h (x) = h min = h max. This puts into light the necessity to use both positive and negative values o p in the estimation: I the leader scaling unction was determined only or positive p, the upper bound supplied by Theorem would yield an increasing unction, and thereore would, at best, yield the increasing part o the spectrum. In particular, h max would not be obtained, hence preventing rom drawing any conclusion with respect to data being monohölder or not. However, testing the theoretical equality h min = h max is practically diicult as the estimation o these quantities turns out to be poor (c. e.g., [83, 85]). Instead, one can estimate the parameter c as deined above in (47). When the estimated c is ound strictly negative, this unambiguously indicates multi-hölder unction. As discussed below, the converse is however not necessarily true: c = does not necessarily imply monohölder unction, as may be incorrectly extrapolated rom the Gaussian Bm case. Let us also note that both approaches based on multiractal analysis can be regarded as non parametric: Testing or mono- or multi-hölder can be achieved without recourse to any parametric setting, in sharp contradistinction with other existing estimators (or example, many estimators evaluate the selsimilarity index H assuming that the data are sample paths o an Bm). Estimated c measured on real-world data are reported in Section 6: Turbulence velocity is well-known as the emblematic real-world example where a non vanishing parameter 4

41 c is met, (c. Section 6., Fig. 6). This is observed to be also the case or the Euro-USD price luctuations (c. Section 6. and Fig. 7, bottom row, let plot) and or etal heart rate variability (c. Section 6.3 Fig. 8). However, Internet traic aggregated count time series are ound to be characterized by c = at coarse scales and weakly departing rom at ine scales (c. Section 6.4 and Figs. and, bottom row, let plots). Selsimilarity? Selsimilar processes are oten obtained as renormalized limits o random walks (see or instance the Bm, or stable Lévy processes). Because o the additive structure underlying their nature, selsimilar models are appealing or describing data. From the examples shown in Section 5., it can even be conjectured that scaling unctions oten are piecewise linear unctions. In particular, combining ergodicity and stationarity properties, the Kolmogorov scaling unction o selsimilar processes with stationary increments should behave linearly in p, as η (p) = ph on an interval o p containing (where H is the selsimilarity parameter). Conjecturing that this property also holds or ζ (p), it ollows that testing or selsimilarity can be ormulated as testing or c =. This has been conducted in a non parametric bootstrap ramework, c. e.g., [8, 83]. Note, however, that selsimilarity with stationary increments together with c = does not imply mono-hölder sample paths (c. the example o Levy motion reported in Section 5. above). Furthermore, c = indicates selsimilarity but does not necessarily imply that data ollow a Bm model. This would in addition require to test or joint Gaussianity o all inite distribution unctions. Non linear point transorm. Let us give some urther insights into the relevance o the question o deciding whether data are mono or multi-hölder. We aim at showing that the notion o monohölder unction is, in some sense, unstable since it can be altered under the action o the most simple smooth non-linear operator. To that end, let us consider Bm B H (t), which is the simplest example o a mono-hölder stochastic process: The Hölder exponent is everywhere equal to h = H. Let us now urther consider its square transorm Y H (t) = (B H (t)). On one hand, at points where the sample path o Bm does not vanish, the action o the mapping x x locally acts as a C dieomorphism, and the pointwise regularity is thereore preserved. On other hand, consider now the (random) set A o points where Bm vanishes. The uniorm modulus o continuity o Bm implies that a.s., or s small enough, sup B H (t + s) B H (t) C s H log(/ s ). t thereore, i B vanishes at t, then B H (t + s) C s H log(/ s ), so that h Y (t) H. The converse estimate ollows rom the act that, or every t, so that, i B H (t) =, then lim sup s lim sup s B H (t + s) B H (t) s H, (B H (t + s)) s H, so that h Y (t) H. Thus, at vanishing points o B H, the action o the square is to shit the Hölder exponent rom h = H to h = H. This set o points has been the subject 4

42 o investigations by probabilists (c. the lourishing literature on the local time either o Brownian motion, or o Bm); in particular, it is known to be a ractal set o dimension H, c [45]. It ollows that Y H (t) is a bi-hölder process, whose spectrum is given by i h = H, D YH (h) = H i h = H, (5) elsewhere. Because the multiractal ormalism yields the Legendre transorm, i.e., the concave hull, o the multiractal spectrum, the leader based estimated multiractal spectrum would suggest practitioners to (wrongly) conclude that Y H (t) is a ully multiractal unction whose spectrum is supported by the interval [H, H]. This is illustrated in Fig 5 where it can be observed that the leader based measured multiractal spectrum yields a very precise estimation o the exact concave hull o D YH (h). Again, note that this is made possible only because negative values o p can be used in the leader ramework. Further, the analysis o the y = x point transorm o Bm raises another disturbing issue. The multiractal properties o Bm are homogeneous (i.e., the spectrum measured rom a restriction o the sample path to any interval (a, b) on the real line is the same as that corresponding to the whole one). This is no longer true or (B H ) : the spectrum measured on a restricted interval (a, b) will vary depending on whether the interval includes or not a point where B H vanishes. This example shows that the multiractal properties are not preserved under simple non linear transormations. 6 Real-World Applications Reviewing the literature dedicated to real-world applications reveals that the concepts o scaling, scale invariance, selsimilarity, ractal geometry and multiractal analysis have been and continue to be used to analyze data in numerous contexts o very dierent natures, ranging rom natural phenomena physics (hydrodynamic turbulence [76, 4, 44], astrophysics and stellar plasmas [74], statistical physics [5, 4], roughness o suraces [5]), biology (human heart rate variabilities [86, 87,, 9, 7], MRI [53, 8], physiological signals or images), geology (ault repartition, [73]) to human activities computer network traic [5], texture image analysis [5], population geographical repartition [57, 74], social behaviors or inancial markets [4, 57]. B. Mandelbrot himsel signiicantly contributed to the studies o many dierent applications, two o the most prominent possibly being hydrodynamic turbulence, the scientiic ield that actually gave birth to the essence o multiractal, (c. e.g., [4, 3]) and inance (c. e.g., [,, 43, 35]). As a tribute to B. Mandelbrot s outstanding research work, a small number o applications rom widely dierent origins have been selected to be revisited. Their presentations are not intended as state-o-the-art reviews, but rather as illustrations, both o the theoretical arguments developed in the previous sections and o the beneits obtained rom using multiractal analysis in these applications. 4

43 6. Hydrodynamic Turbulence In the early s, Richardson, combining empirical observations with the analysis o Navier- Stokes equations, that mostly govern luid lows, ormulated a heuristic description o the apparently random movements observed in hydrodynamic turbulence lows, in terms o a cascade o energy: Between a coarse scale (where energy is injected by an external orcing) and a ine scale (where energy is dissipated by viscous riction), no characteristic scale can be identiied. This seminal contribution paved the road to tremendous amounts o works involving the concept o scaling to model turbulence lows (c. [76, 44] or ancient and modern reviews, respectively). In 94, Kolmogorov proposed a non explicit version o Bm to model velocity (which implies that c = H, and c m or m ) [6]. However, a general consensus (a rare situation worth mentioning or the ield o turbulence) amongst practitioners conducting real experiments rejected Bm as a valid model because the empirically observed scaling exponents (Kolomogorov scaling unction) did not behave linearly in p, or p. Framed in seminal contributions due to A. Yaglom [86] and supported by physical arguments developed by Kolmogorov (and his collaborator A. Obukhov) in 96 [8, 49], the energy transer rom coarse to ine scales has been modeled via split/multiply iterative random procedures that account or the physical intuitions beyond the vorticity stretching mechanisms implied by the Navier-Stokes equation. In 974, B. Mandelbrot studied these various declinations o cascades and their properties [4], paving the way or their interpretation as a collection o singularities and hence to the irst ormulation o the multiractal ormalism by G. Parisi and U. Frisch, based on the Kolmogorov scaling unction [5]. This work triggered enourmous amounts o analysis o experimental and real-world turbulence data, aiming at measuring as precisely as possible scaling exponents (i.e., the scaling unction). The earliest contributions relied on the Kolmogorov scaling unction (c. e.g. [6, 4, 45, 47, 5, 58]), and later work initiated by A. Arneodo, E. Bacry and J.-F. Muzy relied on the use o wavelets (c. e.g. [58, 4, 4, 46, 47, 83]). Fig. 6 (top plot) shows an example o a turbulence longitudinal Eulerian velocity signal, measured with hot-wire anemometry techniques. It has been collected on a jet turbulence experiment [5], at average Reynolds number 58, and has been made available to us by C. Baudet and collaborators (LEGI, Université Joseph Fourier, INPG, CNRS, Grenoble, France), who are grateully acknowledged. Middle row plots depict its scaling property in the so-called inertial range, a priori deined rom the physical characteristics o the low conditions: Let, the wavelet leader based log T (j, q) versus log j (or p = 3), and h min on the right. In the bottom row the measured wavelet leader scaling unction and multiractal spectrum are plotted, clearly indicating multiractality. While the generic multiractality o turbulence data (understood as a strict departure rom linearity in p o the scaling unction) has never been called into question by any o these studies, a major open issue remains: Which precise cascade model better describes turbulence? This question is not purely theoretical, since the precise ingredients entering the cascade that best match data may reveal physical mechanisms o importance or turbulence. Two such models are emblematic o this issue: In 96, Obukhov and Kolmogorov [8, 49] proposed a model mostly based on a law o large numbers argument and reerred to as the 43

44 log-normal multiractal model. It predicts that ζ (p) = c p + c p / and hence that c m or m 3. In 994, She and Lévêque [7] proposed an alternative construction based on the key ingredient that energy dissipation gradients must remain inite within turbulence lows. It is reerred to as the log-poisson model and yields a multiractal process with all non zero c m s. Fig. 6 (bottom plots) compares the scaling unction (let) and multiractal spectrum (right) measured rom data (solid black), together with bootstrap based conidence intervals (c. [83] or details) against those predicted by the log-normal (solid red) and log-poisson (dashed blue) model. Together with the act that the measured c 3 can not be distinguished rom (c 3 =. ±.), these plots clearly indicate that the log-normal model is preerred by turbulence data. This has been studied systematically and on larger data sets in [4, 83] and unambiguously conirms earlier indings reported in [4, 58]. 6. Finance Time Series: Euro-USD rate luctuations Modeling stock market prices received considerable eorts and attention starting with the seminal early contribution o Bachelier in 9 who, in his thesis, based his description o stock market on Brownian motion, i.e., on an ordinary Gaussian random walk [8]. Essentially, this model implies that daily returns, i.e., increments o (or dierences between) prices taken along two consecutive days, X(t) = P (t) P (t ), consist o independent, identically distributed Gaussian zero-mean random variables. This hence leaves little room to earn money rom stock market investments by pure statistical model based prediction. However, it has long been observed and/or expected that stock market prices signiicantly depart rom the Brownian model: First, returns may exhibit statistics that signiicantly depart rom Gaussian distributions. Second, it is oten observed that volatility, a quantity essentially based on the squared (or absolute) value o the returns, display some orm o dependency. Notably, in their celebrated contribution, Granger and Joyeux indicated that volatility is well characterized by long memory, or long term dependence [78, 79], a model that is closely related with selsimilarity and Bm. This motivated substantial eorts to model stock market returns with the GARCH models amily or FARIMA alternatives (c. e.g., [36, 6, 78]). B. Mandelbrot signiicantly contributed to the analysis o the statistical properties o stock market returns by perorming an iconoclastic analysis o commonly used inance models, emphasizing heavy tails, as opposed to Gaussian distributions (the Noah eect), and long memory, as opposed to independence (the Joseph eect), c. [33] or a review, and several seminal contributions o B. Mandelbrot; see also e.g., [, ]. In particular, this called into question the Black and Scholes model strongly relying on Gaussianity, and led B. Mandelbrot and collaborators to explicitly propose that stock market prices may have multiractal properties in [7] in contradistinction with most Itô-martingale based models used so ar. B. Mandelbrot s work paved the way towards numerous eorts to assess multiractality empirically in stock market prices and exploit it (c. e.g., [,, 4, 4, 57]). Almost one century ater Bachelier, B. Mandelbrot and collaborators proposed a new model or stock market prices: Bm in multiractal time [43, 35], a process that explicitly 44

45 incorporates multiractal properties by a multiractal time jittering, obtained by a multiplicative cascade construction. In this model, a new generation o cascades is used, based on Compound Poisson distributions, ollowing again an original idea o B. Mandelbrot [9, 3, 3]. Let F denote the distribution unction o the multiplicative cascade µ on [, ]: F (t) = µ([, t)). Then, Bm in multiractal time is deined as X t = B H (F (t)), (5) where B H stands or Bm with selsimilarity parameter H (which is independent o µ). Note that, as a consequence o (), the spectrum o X is deduced rom the spectrum o F by a simple dilation along the h axis D X (h) = D F (h/h); see the results o R. Riedi and S. Seuret [65, 7] or additional discussions concerning multiractal subordination, i.e. understanding multiractal properties o composed processes). Numerous rich and interesting declinations o such constructions o cascades and multiractal processes have been proposed since (c. e.g., [9,,, 3, 4, 4, 48, 49]). Note that the idea o multiractal subordination (i.e., composition by a multiractal change o time) is extremely powerul, and can be applied in many settings. Let us mention just one example: In [], B. Mandelbrot and one o the authors perormed the multiractal analysis o the Polya unction (a amous example o continuous space-illing unction) by simply remarking that it can be actorized as the composition o the repartition unction o a simple dyadic cascade with a monohölder unction. As a tribute to B. Mandelbrot s contributions, though the authors have not personally contributed to the ield, the multiractal properties o the Foreign exchange Euro-USD price luctuations are now studied. Fig. 7 (top row) shows the hourly Euro-USD quotations (and hourly returns) since the birth o Euro (Data Courtesy Vivienne Investissement, Lyon, France, / The return (or increment) time series clearly shows large luctuations that are not consistent with a constant along time Gaussian distribution. Scaling properties analyzed here are ound to hold or scales ranging rom the day to the month, and hence correspond to long term characteristics, as opposed to intraday analysis, not conducted here. Multiractal spectra measured over one year long time series (second and third rows, let plots), using the wavelet leader multiractal ormalism, show that the Euro-USD quotations exhibit multiractal spectra that are in agreement with the Bm in multiractal time model, yet indicate that the multiractal properties may vary along the years. To urther analyze these variations, scaling attributes are measured in one year long time windows (with a 3-month sliding time shit) and reported (together with conidence intervals) in Fig. 7 (bottom row). Interestingly, it is ound that h min is permanently above : h min >, which clearly rules out the use o jump processes to model ForEx time series. Moreover, because h min >, the initeness o quadratic variations can be evaluated using the leader scaling unction at p =, ζ () (as in (39), c. Section 4.4). Fig. 7 (bottom row, nd plot) shows that ζ () luctuates and oten remains below the inite quadratic variations limit o : ζ () <. This result validates the hypothesis o the initeness o quadratic variations, underlying the use o Itô-martingale processes as models 45

46 to describe the Euro-USD variations, at least when long term variations are analyzed, as is the case here; this does not exclude that shorter term (or intra-day) variations might show dierent properties. In Fig. 7 (bottom row, 3rd and 4th plots), parameters c, c are plotted as unctions o time. These plots urthermore indicate that though a Bm in multiractal time-like process provides a valid model to describe the price luctuations, the multiractal properties may vary along time. Two dierent regimes are suggested by these plots: Beore 7, with a c.45 <.5 and c. ; and ater 8, with c.5 and c.. Recalling that the ordinary Brownian motion (proposed by Bachelier) would correspond to c =.5 and c =, it may be concluded that the (long term) luctuations o the Euro- USD quotations show less structure or the later period than or the ormer, hence making predictions or investments more diicult (see also discussion in Section 5.3 on homogeneous multiractal properties). Assuming that the Bm in multiractal time model studied by L. Calvet, A. Fisher and B. Mandelbrot is the correct description o prices, let us now show how the parameter H o the corresponding Bm can be inerred rom the observed data using the wavelet scaling unction (the heuristic argument that ollows can easily be turned into a mathematical proo). Let us irst return to the deinition o the Kolmogorov scaling unction (8). Approximating the integral by a Riemann sum, and taking h = j, one obtains X t+h X t p dt j k X k+ j X k j p = j k B H ( F ( k + j )) ( ( )) k p B H F j which, using the act that B H (t) B H (s) t s H, is o the order o magnitude o j k ( ) ( ) k + k ph F j F j. In particular, or p = /H, using the act that F is continuous and increasing, we obtain ( ) ( ) j k + k F j F j = ( ( ) ( )) k + k j F j F j = j k Since this quantity is, by deinition, o the order o magnitude o ( j ) η (p), we obtain that the exponent H o the Bm used in this model (c. [43]) satisies ( ) η =. (5) H Using the act that the Kolmogorov and the wavelet scaling unction coincide in the range o exponents involved here, (5) allows to recover the selsimilarity exponent o the Bm in this model rom the wavelet scaling unction. In the example considered in Fig. 7, one obtains that, in practice, the parameter H thus estimated is very close to c. k 46

47 6.3 Fetal Heart Rate Variability Heart rate variability (HRV), which reers to the short time luctuations (within the minute) o heart rate, has long been considered as a powerul tool to characterize the health status o a given subject: In a nutshell, the more variability, the healthier the subject is (c. e.g., the seminal contribution []). To characterize variability in heart rate time series, spectral analysis has long been used to measure the so called LF/HF balance, i.e., the ratio o energies measured in requency bands attached to the sympathic and parasympathic activities o the autonomous central nervous system []. Yet, it has long been observed that heart rate time series display ractal properties and that the corresponding ractal exponent could be used as a non invasive tool or non healthy status detection (c. []). This opened the way to numerous analyses aiming at describing HRV data with Bmlike models and at correctly estimating the corresponding selsimilarity parameter H; later, multiractal analysis has naturally been considered as an analysis tool potentially improving the characterization o HRV scaling properties (c. e.g., [86, 87, ]). Recently, multiractal analysis has been involved in the detection o per partum etal asphyxia. Obstetricians are monitoring etal-hrv during the delivery process to detect potential etal asphyxia. Using a set o analysis criteria (the international FIGO rules), obstetricians measure the health status o the etus and decide on operative delivery or not. A major diiculty they are acing is the high number o False Positive (etuses diagnosed as unhealthy during the delivery process but a posteriori evaluated as perectly healthy), and hence o operative delivery that might have been avoided. This constitutes an important public health issue, as operative deliveries are accompanied by a signiicant number o serious posterior complications or both newborns and their mothers. Fig. 8 (top row) shows examples o heart rate time series (in Beats-per-Minute) or a healthy subject, correctly diagnosed as such by the obstetrician (let) (True Negative), or a healthy subject, incorrectly diagnosed as suering rom per partum asphyxia (False Positive), and or a etus actually suering orm per partum asphyixia and diagnosed as such (True Positive). Data Courtesy M. Doret, Obstetrics Dept. o the University Hospital Femme-Mère-Enant, Lyon, France, [9, 6]. As shown in Fig. 8 (second and third rows), or all three classes (True Negative, True Positive, False Positive), HRV exhibits scaling properties or scales ranging rom the second to the minute. The ourth row in Fig. 8 shows multiractal spectra, estimated rom 5 min. long heart rate time series using wavelet leaders, which suggests that the healthier the subject, the larger its variability in the sense that its c parameter is smaller (spectrum shited to the let). To conirm such an observation, multiractal parameters are estimated within 5 min. long sliding windows (with a 5% time overlap) or the last hour beore delivery. The 5 min. window duration is chosen ater the obstetrician s request that asphyxia should be detected as early as possible. Multiractal parameters are then averaged within the three classes (each containing 5 patients careully selected as representative by obstetricians). A Wilcoxon ranksum test is conducted or each time window between the True Positive and False Positive classes. The results reported in Fig. 8 (bottom row) clearly indicate a signiicant increase o c and h min (hence a signiicant decrease in HRV) or the Unhealthy True Positives compared to the Healthy True Negatives and False Positives. The p-value o the Wilcoxon ranksum test 47

48 is observed to be below 5% (c. red in Fig. 8 (bottom row)) oten 3 min earlier than the time the decision to operate the delivery was actually taken. These promising results are reported in details in [9, 6] and will be urther investigated. It is worth noting that or heart rate time series (as or many other types o human rhythms signals, c. e.g., MRI applications [53, 8]), h min can be ound either positive or negative depending on the subject, or its health status. This is consistent with the act that practitioners, in applications, noticed that they oten had to switch rom Bm to Gn (or conversely) to model data and understood this as a puzzling inconsistency (see or instance discussions in [8]). To some extent, the general ramework developed here alleviates such inconsistency by introducing h min as one o the various parameters that can be used to analyze scaling without a priori assumptions on the data. In the analysis o etal HRV described above, a ractional integration o order / (larger than the absolute value o the most negative h min encountered in the whole database) has been applied to enable the use o the wavelet leader multiractal ormalism in a consistent manner on the whole set o data. 6.4 Aggregated Internet Traic Internet traic engineering currently is a challenging task involving a variety o issues ranging rom economy development planning and security assessment to statistical data characterization. The earliest versions o statistical models used or Internet traic modeling were stemming rom the experience gained on the previous large communication network: Telephone communications. B. Mandelbrot himsel made very early contributions to such modeling (c. e.g., [37]), and this certainly was a major source o inspiration or the later developments he made in the theory o ractal stochastic processes. However, such telephone communications models were all based on the assumption that there exist one or several characteristic scales o time in data, while it has soon been recognized that Internet traic is well characterized by long memory and sel similarity (c. [8, 5, 6, 53, 54] or seminal articles). This later raised the question o multiractality o Internet traic (c. e.g., [, 69, 84, 66]) that we briely illustrate here. Any activity perormed on the net ( ing, web browsing, Video-On-Demand, data downloading, IP telephony,... ) amounts to establishing a connection between hosts (serverclients, peer to peer,... ), consisting o computers located all over the world, according to an agreed-on protocol, and exchanging a (very) large number o elementary quanta o inormation, the Internet Protocol packets (in short, IP Pkts). Collecting Internet traic data hence naturally consists o recording the IP Pkt time stamp and header (IP source, IP destination, Port Source, Port Destination). Obviously, analyzing the marked point process resulting o the tremendous number o IP Pkts exchanged is beyond easible, and it is classical to instead monitor so-called aggregated (or count) time series: Practitioners select a given aggregation-level, whose choice depends on the application, and construct either the Pkt or Byte time series that counts the number o IP Pkts whose timestamp alls within [k, (k + ) ), or the corresponding volume in Byte. An example o such Pkt time series is plotted in Fig. 9 (let). It consists o traic rom the MAWI repository, made available by the Japanese WIDE research program [4]. The 48

49 MAWI repository consists o data that have been collected everyday or 5min at pm (Tokyo time) since January on major backbone links (Mbps or Gbps) connecting Japan and the United States (c. [39, 5, 59]. The let plot o the same igure displays the wavelet coeicient based log (S (, j)) vs. log ( j ) = j (as in (34)) and clearly shows that, or Internet traic, there exist two ranges o scales or which scaling can be identiied: Coarse scales, above s, and ine scales, below s. For each o those ranges o scales, the wavelet leader based multiractal ormalism is applied. For the coarse scales, Fig. clearly indicates on a 5min long set o data aggregated at ms that the leader scaling unction is linear, and that the multiractal spectrum collapses on a single point, hence suggesting monoractality or that multiractality is at best very weak. To urther investigate this point, the parameters c and c are estimated on sliding 5min long windows or a trace collected during 4 hours (on Sept., nd, 5), or scales ranging rom s to min. Results are reported or the last 8 hours or simplicity and show, irst, that the estimated parameters are remarkably constant along the day (while the changes in amplitude o the time series may be drastic) and, second, that c. This indicates no multiractality at coarse scale and, because c H.9, a signiicant long memory in the data. These experimental results corroborate those reported in [84] and are consistent with the main model due to Taqqu et al. (c. [76]) relating selsimilarity and heavy-tails o the web ile size distribution, see also [, 83, 8]). For the ine scales, Fig. shows on the same 5min long set o data aggregated at ms that the leader scaling unction is linear or p around, hence indicating a very weak multiractality or no multiractality at all. Parameters c and c, estimated on sliding 5min long windows or the 4 hour trace, again turn out to be constant along time, with c.55±.5 and c.7±.5. This hence validates that ine scales o aggregated traic show at best little multiractality. Furthermore, estimations o the selsimilarity parameters (not shown here) being close to.5 also indicate little correlation. These analyses tend to show that ine scales o aggregated traic are characterized by the quasi-absence o dependence structure. Moreover, the estimated parameters c and c remain remarkably constant along time or the whole day and across the various days analyzed, a highly unexpected result or Internet traic, which is oten described as widely varying, and which is continuously subject to numerous occurrences o anomalies and attacks. This is essentially because data have been pre-processed using a random projection and a median procedure that robustiy estimation against anomalous behaviors (c. [39, 59]). In practice, the eect o this procedure is to eliminate abnormal components in the traic. The results reported here can hence be associated with the properties o regular and legitimate background Internet traic, corresponding to the usual Internet user behaviors. Furthermore, this quasi-absence o dependence structure at ine scales is consistent with the Cluster Point Process modeling o Internet traic aggregated count time series proposed in [84], which implies that ine scales are actually not possessing strictly scaling properties. However, it also shows that the statistical properties o the ine scales are satisactorily approximated by scaling behaviours. Thereore, the act that regular or legitimate traic does not show multiractality does not imply however that multiractal parameters like c and c can not be 49

50 used or traic monitoring, such as anomaly detection or instance. Indeed, the occurrence o anomalies is likely to change the dependence structure o the ine scale statistics (as shown in e.g., [59]) and consequently the empirically estimated parameters c and c, which may thus prove very useul, even i scaling properties are measured that actually consist o approximations o the true statistical properties o the ine scales. Finally, let us mention that or aggregated count time series h min <, as shown on the top let plots o Figs. and, hence a ractional integration o order is systematically applied prior to perorming the wavelet leader multiractal ormalism. 6.5 Texture classiication: Vincent Van Gogh meets Benoît Mandelbrot Applications described so ar all implied the multiractal analysis o D signals only. However, B. Mandelbrot always advocated that ractal analysis should be applied to images, to characterize the roughness o textures (c. e.g., [5, 6]). The wavelet coeicient based analysis o selsimilarity analysis has been readily and immediately extended to D ields and images, thanks to the straightorward ormulation o the D discrete wavelet transorm (c. e.g., []). The estimation o the corresponding H parameter has then been used to characterize textures in images, oten in the context o biomedical applications (c. [38, 97]). The extension o multiractal ormalisms to images turned out to be signiicantly more intricate. Direct extension o the Kolmogorov scaling unction were used to analyze rainall repartitions by D. Schertzer and S. Lovejoy, see [69]. A second attempt based on the modulus maxima o the coeicients o the D continuous wavelet transorm was proposed by A. Arneodo and his collaborators; it employed in medical [3] or geophysical applications [5]. The wavelet leader ormalism recently proposed and based on a D discrete wavelet transorm enabled a signiicant step orward toward the use o multiractal analysis in higher dimensions. Its potential beneits have been assessed in [84], and it has recently been used to characterize texture in paintings [6]. The paintings that Van Gogh perormed while he was living in France are grouped into two periods by art experts and art historians: The Paris period, ending early 888, and the later Provence period. While art experts agree on the classiication o most o Van Gogh s paintings, some are still not clearly attributed to either period. To perorm such classiication, art experts oten make use o material or stylometric eatures such as the density o brush strokes, their size or scale, the thickness o contour lines, the layers, the chemicals o the colors,.... Aiming at promoting the investigation o computer-based image processing procedures to assist art experts in their judgement, the Image Processing or Art Investigation research program (c. in collaboration with the Van Gogh and Kröller-Müller Museums (The Netherlands), made available to research teams two sets o (partial and low resolution digitized versions o) 8 Van Gogh s paintings each rom the Paris and Provence period, and 3 painting whose period remains undecided (nomenclature below corresponds to the Van Gogh museum catalog). The digitized copies are available through the usual RGB (Red, Blue, Green) channels and converted into the HSV (Hue, Saturation, Value) channels, hence enabling to process the gray-level light intensity inormation and the color inormation. 5

51 Three examples, corresponding to the Paris period, the Provence period and unclassiied paintings, respectively, are shown in Fig. (let column) together with examples o homogenous patches selected or analysis (second column). The D wavelet leader based multiractal ormalism has been applied, or each selected patch, independently to the six RGB and HSV channels. Examples o obtained multiractal spectra are shown in Fig. (right column) and suggest, or example, that 65 is closer to 475 than to 45 and hence is likely to belong to the Provence period. For each patch o the paintings and each channel, multiractal attributes h min, η (), c,... are estimated. Fig. (bottom row) shows D projections o various pairs o such attributes, which all indicate that on average the Paris paintings (blue clusters) are globally more regular than the Provence paintings (red clusters). Also, the Saturation and Red channels are ound to be the most discriminant. This suggests that both a color and an intensity inormation (saturation and Red) are useul or discrimination. Interestingly, it has been reported by art experts that saturation is a key eature to distinguish between the Paris and Provence periods in Van Gogh s paintings. Finally, these D projections suggest that 65 and 386 belong to the lower let red cluster and hence to the Provence period, while conclusion is less straightorward or 57. Such results are let to the appreciation o art experts. These analyses are detailed in [6]. 7 Conclusion A key issue or the uture developments o multiractal analysis is to disentangle (at least) our dierent acceptations that can be associated with the key word multiractal, and that are oten misleadingly conused one with another, sometimes leading to potential misunderstandings and erroneous interpretations. Theoretical multiractal analysis and local regularity. From the theoretical or mathematical side, multiractal analysis is deeply tied with the analysis o the luctuations along time (or space) o the local regularity o the unction or data o interest, these luctuations being oten coined as irregularity. Usually, this local regularity is measured in terms o Hölder exponents h (x). The global description o the irregularity o is gathered in the multiractal spectrum D (h), which consists o the Hausdor dimensions o the geometrical iso-hölder sets. For mathematics, multiractal is hence all about Hölder exponent and (ractal, or box or Haussdor) dimensions o geometrical sets. Though listed irst here, the mathematical ormulation o multiractal analysis theory can be considered as the most recent piece o the history o multiractal, as it really started in the 9s only, see [4], as a consequence o the seminal article o G. Parisi and U. Frisch [5]. Note however that precursors can be ound, or instance in articles dealing with slow and ast points o Brownian motion: Recall that this speciic process displays exceptional points where the modulus o continuity is slightly smaller (resp. bigger) than a.e.; the corresponding sets are random ractals. Following the seminal contribution o J.-P. Kahane, their dimensions have been precisely determined, respectively by S. Orey and S. J. Taylor or ast points, see [5], and by E. Perkins or slow points, see [56]. 5

52 Multiractal ormalism and scale invariance. From the more practical perspective o real-world data analysis, scaling unctions, consisting o time (or space) averages o the p- th power o scale dependent quantities (such as increments, oscillations, wavelet coeicients or wavelet leaders) also oten reerred to, in statistical physics, as partition or structure unctions constitute the core aspect o the notion o multiractal. Importantly, practical multiractal analysis amounts in essence to assuming that scaling unctions behave as power laws with respect to the analysis scales, in the range o scales available in the data. The description o these power laws naturally leads to the notion o scaling exponents. Equivalently, these scaling exponents are termed scaling unctions, with the practical and oten implicit request that the limit underlying its deinition actually exists (c. (8) versus (9)). These power law behaviors or scale invariance, or scaling o scale dependent quantities, and the corresponding scaling exponents (or scaling unctions), were at the heart o the intuition that B. Mandelbrot developed, connecting scaling properties to (ractal) dimensions, to selsimilarity and/or to luctuations o local regularity, and relating scaling to sel similar random walks or multiplicative cascades. The multiractal ormalism (in reerence to the Thermodynamic ormalism) thereore consists o the mathematical developments that relate scale invariance and scaling unctions to local regularity and multiractal spectra. In some sense, the multiractal spectrum constitutes the link between ormal mathematics and physics or data analysis. Multiractal processes. Multiractal processes do not constitute a well-deined class. In essence, this notion reers to processes (or unctions) that can be ruitully used to model scale invariance properties in data. Following the seminal distinctions proposed by B. Mandelbrot, they can be thought o as alling into two major classes: Selsimilar random walks and multiplicative cascades (or martingales). The irst one, now very classical ater the seminal review o B. Mandelbrot on Bm [38], pertains to the class o additive models: A large step (or increment) can be obtained as the sum o many dierent small steps, yet sharing the same statistical properties. This class is hence deeply associated with selsimilarity. Oten, this class is conused with its emblematic but restrictive representative Bm, the only Gaussian selsimilar process with stationary increments. The Gaussian nature o this process and its ully parametrized ormulation lead to the ormulation o various parametric estimation o its selsimilarity exponent H. Such parametric estimation are sometimes incorrectly considered as a multiractal analysis o data. This is inaccurate since multiractal analysis aims at measuring much richer properties o data than the sole selsimilarity and, above all, does not rely on the assumption that data are selsimilar or Gaussian. Multiractal analysis thus has a much broader scope and can also be applied to limits o other random walks, such as Lévy processes (and, more generally, jump processes), see [6] or overviews on the mathematical side. The second class, multiplicative cascades, is usually considered as that o archetypal multiractal processes, in the sense that their (scaling unctions and) multiractal spectra can be computed theoretically and ound to oten consist o continuous, smooth bell-shaped unctions. Thereore, or such processes, multiractal analysis unveils inormation that 5

53 can not easily be obtained with other analysis tools (as opposed to the previous class o selsimilar random walks). Furthermore, as mentioned earlier, and again ollowing one o B. Mandelbrot s intuitions, this particular class can be extended by combinations (or subordinations) o selsimilar process with multiplicative cascades (or instance Bm in multiractal time mentioned above, or Lévy processes in multiractal time, as studied by J. Barral and S. Seuret [33, 34]). It is however clear that these examples do not even give a vague idea o the tremendous variety o all possible multiractal unctions. A ew mathematical results actually comort this idea (and allow to transorm it into a precise mathematical statement): Generically, most unctions in a given unction space are multiractal and satisy the multiractal ormalism, see [75, 9] or precise statements. Thereore, multiractality is not a rare property, and is certainly not restricted to the ew examples which have been investigated up to now. The classical opposition between monoractal versus multiractal processes, oten used in practical data analysis, is not well grounded and somehow irrelevant. Conusion stems rom the act that Gaussian Bm is characterized by a single Hölder parameter h (x) = H and is hence monohölder (its spectrum D (h) is supported by a single point). But selsimilarity does not in general imply this monohölder property. Instead, the classiication o data might be by opposing additive to multiplicative structure. Indeed, the physics (or the biology or else) underlying data production may signiicantly dier depending on whether it is driven by additive mechanisms, or by multiplicative ones. In that respect, multiractal analysis may help as the multiractal spectra o selsimilar random walks generally dier rom those o multiplicative constructions. However, in opposition to what is oten used in applications, multiractal analysis per se, understood as the measurement o the scaling unction and multiractal spectrum, is not suicient or proving that there exist any additive or multiplicative structures underlying the data. Multiractal analysis does not aim at stating deinite answers to uzzy questions such as: Are data ollowing a cascade model or not? but, instead, it provides quantitative answers to more restricted questions such as: Is h min >? Is ζ () >? In turns, these precise measurements can contribute to help practitioners to ormulate hypotheses regarding data, or to classiy them. Practical multiractal analysis: signal and image processing. As stated in the previous paragraph, practical multiractal analysis amounts to measuring rom data multiractal attributes such as η (p), h min, ζ (p), c m, L (h). These quantities can be computed rom data without assuming any a priori data model. In other words, data to which multiractal analysis is applied need not stem rom any exact selsimilar random walk or multiplicative construction. Multiractal analysis is hence, by nature, a non parametric analysis tool that can be applied to any type o data, be they (multi)ractal or not. Whether L (h) can be related with D (h) or not does not prevent practitioners to use these multiractal attributes as tools or classiication, diagnosis, detection, or else. This is notably the case in image processing, where it is very unlikely that databases containing images o widely dierent natures can be associated with speciic construction models (such as random walks or cascades). Furthermore, in practice, one question is always central and was already raised by B. 53

54 Mandelbrot: What multiresolution quantity should one start with? Increments, oscillations, wavelet coeicients, wavelet leaders? In this respect, the present contribution aims at providing clear and general answers. First, increments and oscillations only address the restricted case where h (t) <, t; while wavelet based quantities enable to bypass that restriction by selecting smooth enough wavelets with a large enough number o vanishing moments. Thereore, wavelet coeicients generalize increments and the wavelet leaders constitute the generalized counterpart o oscillations. Second, wavelet coeicients and wavelet leaders should not be opposed. Wavelet coeicients enable to measure inormation in data such as the uniorm Hölder exponent and a global selsimilarity type scaling exponent; they should hence always be applied irst to data. Wavelet leaders should be applied next, when relevant (h min > ) and when seeking or additional and more reined analysis o scaling properties in data, via a ull collection o scaling exponents ζ (p), or equivalently c m, or equivalently L (h). Multiractal toolbox. The practical solutions proposed here in terms o discrete wavelet transorm coeicients and leaders are believed to be one o the methods enabling the best practical achievements in terms o combining irm mathematical grounding, satisactory estimation perormance, low implementation and computational costs, robustness and versatility. It can be accompanied with non parametric bootstrap procedures, providing conidence intervals or estimates and hypothesis test procedures. Technical details are reported in [83] and Matlab toolboxes implementing these tools are made publicly available by the authors. Reerences [] P. Abry, R. Baraniuk, P. Flandrin, R. Riedi, and D. Veitch. Multiscale network traic analysis, modeling, and inerence using wavelets, multiractals, and cascades. IEEE Signal Processing Magazine, 3(9):8 46, May. 48 [] P. Abry, P. Borgnat, F. Ricciato, A. Scherrer, and D. Veitch. Revisiting an old riend: On the observability o the relation between long range dependence and heavy tail. Telecom. Syst., Special Issue on Traic Modeling, Its Computations and Applications, 43(3-4):47 65, [3] P. Abry, P. Gonçalvès, and P. Flandrin. Wavelets, spectrum estimation and / processes, chapter 3. Springer-Verlag, New York, 995. Wavelets and Statistics, Lecture Notes in Statistics. [4] P. Abry, P. Gonçalvès, and J. Lévy-Véhel, editors. Lois d échelle, ractale et ondelettes, Vol. &. Hermès Scientiic Publications,. 5 [5] P. Abry, S. Jaard, S.G. Roux, B. Vedel, and H. Wendt. Wavelet decomposition o measures: Application to multiractal analysis o images. In Unexploded ordnance detection and mitigation, J. Byrnes ed. Springer, NATO Science or peace and security, Series B, pages,

55 [6] P. Abry, S. Jaard, and H. Wendt. When Van Gogh meets Mandelbrot: Multiractal classiication o painting s texture. preprint,. 5, 5 [7] P. Abry, B. Pesquet-Popescu, and M.S. Taqqu. Wavelet based estimators or sel similar α-stable processes. In Int. Con. on Signal Proc., 6th World Computer Congress, Beijing, China, August. 6 [8] P. Abry and D. Veitch. Wavelet analysis o long-range dependent traic. IEEE Trans. Ino. Theory, 44(): 5, 998., 48 [9] P. Abry, H. Wendt, S. Jaard, H. Helgason, P. Goncalvès, E. Pereira, C. Gharib, P. Gaucherand, and M. Doret. Methodology or multiractal analysis o heart rate variability: From l/h ratio to wavelet leaders. In 3nd Annual International Conerence o the IEEE Engineering in Medicine and Biology, Buenos Aires, Argentina,. 47, 74 [] S. Akselrod, D. Gordon, F.A. Ubel, D.C. Shannon, A.C. Berger, and R.J. Cohen. Power spectrum analysis o heart rate luctuation: a quantitative probe o beat-tobeat cardiovascular control. Science, 3(454):, [] A. Arneodo, B. Audit, N. Decoster, J.-F. Muzy, and C. Vaillant. Wavelet-based multiractal ormalism: applications to dna sequences, satellite images o the cloud structure and stock market data. The Science o Disasters; A. Bunde, J. Kropp, H.J. Schellnhuber, Eds. (Springer), pages 7,. 36 [] A. Arneodo, E. Bacry, and J.F. Muzy. The thermodynamics o ractals revisited with wavelets. Physica A, 3(-):3 75, [3] A. Arneodo, N. Decoster, P. Kestener, and S.G. Roux. A wavelet-based method or multiractal image analysis: rom theoretical concepts to experimental applications. In P.W. Hawkes, B. Kazan, and T. Mulvey, editors, Advances in Imaging and Electron Physics, volume 6, pages 98. Academic Press, 3. 5 [4] A. Arneodo, S. Manneville, and J.F. Muzy. Towards log-normal statistics in high Reynolds number turbulence. Eur. Phys. J. B, :9, , 44 [5] A. Arneodo, S.G. Roux, and N. Decoster. A wavelet-based method or multiractal analysis o rough suraces: applications to high-resolution satellite images o cloud structure. Experimental Chaos, AIP Conerence Proceeding, 6:8,. 4, 5 [6] A. Arneodo et al. Structure unctions in turbulence, in various low conigurations, at Reynolds number between 3 and 5, using extended sel-similarity. Europhys. Lett., 34:4 46, [7] J.M. Aubry and S. Jaard. Random wavelet series. Communications In Mathematical Physics, 7(3):483 54,

56 [8] L. Bachelier. Théorie de la spéculation. Annales de l École Normale Supérieure, 3(7), [9] E. Bacry, J. Delour, and J.F. Muzy. Multiractal random walk. Phys. Rev. E, 64():63,. 8, 45 [] E. Bacry, L. Duvernet, and J.F. Muzy. Continuous-time skewed multiractal processes as a model or inancial returns. International Journal o Theoretical and Applied Finance,. 44, 45 [] E. Bacry, A. Kozhemyak, and J.F. Muzy. Continuous cascade models or asset returns. Journal o Economic Dynamics and Control, 3():56 99, 8. 4, 44, 45 [] E. Bacry, A. Kozhemyak, and J.F. Muzy. Multiractal models or asset prices. Encyclopedia o quantitative inance, Wiley,. 4 [3] E. Bacry and J.F. Muzy. Multiractal stationary random measures and multiractal random walks with log-ininitely divisible scaling laws. Phys. Rev. E, 66,. 8, 45 [4] E. Bacry and J.F. Muzy. Log-ininitely divisible multiractal processes. Commun. Math. Phys., 36: , 3. 8, 45 [5] V. Bareikis and R. Katilius, editors. Proceedings o the 3th International Conerence on Noise in Physical Systems and / Fluctuations, Palanga, Lithuania. World Scientiic, [6] J. Barral, J. Berestycki, J. Bertoin, A. Fan, B. Haas, S. Jaard, G. Miermont, and J. Peyrière. Quelques interactions entre analyse, probabilités et ractals. S.M.F., Panoramas et synthèses 3,. 8, 5 [7] J. Barral, N. Fournier, S. Jaard, and S. Seuret. A pure jump Markov process with a random singularity spectrum. Annals o Probability, 38(5):94 946,., 38 [8] J. Barral and P. Gonçalves. On the estimation o the large deviations spectrum. Journal o Statistical Physics, 44(6):56 83,. 35 [9] J. Barral, X.O. Jin, and B. Mandelbrot. Uniorm convergence or complex [,]- martingales. Annals o Applied Probability, (4):5 8,. 45 [3] J. Barral and B. Mandelbrot. Multiractal products o cylindrical pulses. Probability Theory and Related Fields, 4(3):49 43,. 8, 45 [3] J. Barral and B. Mandelbrot. Multiplicative products o cylindrical pulses. Probab. Theory Rel., 4:49 43,. 45 [3] J. Barral and S. Peyrière. Mandelbrot cascades abulous ate. (in this volume),. 8 56

57 [33] J. Barral and S. Seuret. The singularity spectrum o Lévy processes in multiractal time. Adv. Math., 4(): , 7. 8, 38, 53 [34] J. Barral and S. Seuret. On multiractality and time subordination or continuous unctions. Adv. Math., (3): , 9. 8, 38, 53 [35] J. Barral and S. Seuret. A localized Jarnik-Besicovich theorem. Adv. Math., 6(4):39 35,. 38 [36] J. Beran. Statistics or Long-Memory Processes. Chapman & Hall, 994., 44 [37] J.M. Berger and B. Mandelbrot. New model or the clustering o errors on telephone circuits. IBM Journal o Research and Development, 7(3):4 36, [38] H. Biermé and F. Richard. Statistical test or anisotropy or ractional Brownian textures. Applications to ull ield digital mammography. J. Math. Imaging Vision, 36(3):7 4,. 5 [39] P. Borgnat, G. Dewaele, K. Fukuda, P. Abry, and K. Cho. Seven years and one day: Sketching the evolution o internet traic. In INFOCOM9, 9. 48, 49, 76, 77 [4] G. Brown, G. Michon, and J. Peyrière. On the multiractal analysis o measures. Journal o Statistical Physics, 66(3-4):775 79, , 5 [4] L. Calvet and A. Fisher. Multiractality in assets returns: theory and evidence. Review o Economics and Statistics, LXXXIV(84):38 46,. 44 [4] L. Calvet and A. Fisher. Multiractal volatility: Theory, orecasting and pricing. Academic Press, San Diego, CA, 8. 4, 44, 45 [43] L. Calvet, A. Fisher, and B. Mandelbrot. The multiractal model o asset returns. Cowles Foundation Discussion Papers: 64, , 44, 46 [44] B. Castaing. The temperature o turbulent lows. J. Phys. II, 6:5 4, [45] B. Castaing, Y. Gagne, and E. Hopinger. Velocity probability density unctions o high Reynolds number turbulence. Physica D, 46:77 9, [46] B. Castaing, Y. Gagne, and M. Marchand. Log-similarity or turbulent lows. Physica D, 68(3-4):387 4, [47] B. Castaing, Y. Gagne, and M. Marchand. Log-similarity or turbulent lows. Physica D, 68:387 4, [48] P. Chainais, R. Riedi, and P. Abry. On non scale invariant ininitely divisible cascades. IEEE Trans. Ino. Theory, 5(3), March 5. 8, 45 [49] P. Chainais, R. Riedi, and P. Abry. Warped ininitely divisible cascades: beyond scale invariance. Traitement du Signal, (), 5. 8, 45 57

58 [5] G.R. Chavarria, C. Baudet, and S. Ciliberto. Hierarchy o the energy dissipation moments in ully developed turbulence. Phys. Rev. Lett., 74: , , 7 [5] A. Chhabra et al. Direct determination o the singularity spectrum and its application to ully developed turbulence. Phys. rev. A, 4:37, [5] K. Cho, K. Mitsuya, and A. Kato. Traic data repository at the wide project. In USENIX FREENIX Track,. 48 [53] P. Ciuciu, P. Abry, C. Rabrait, and H. Wendt. Log wavelet leaders cumulant based multiractal analysis o evi mri time series: evidence o scaling in ongoing and evoked brain activity. IEEE J. o Selected Topics in Signal Proces., (6):99 943, 9. 4, 48 [54] M. Clausel-Lesourd and S. Nicolay. A wavelet characterization or the upper global holder index. Preprint,. 3 [55] S. Cohen and J. Istas. Fractional ields and applications. (preprint),. 5, 7 [56] I. Daubechies. Orthonormal bases o compactly supported wavelets. Comm. Pure and App. Math., 4:99 996, 988. [57] A. Dauphiné. Géographie ractale. Hermès, Lavoisier, Paris, France,. 4 [58] J. Delour, J.F. Muzy, and A. Arneodo. Intermittency o d velocity spatial proiles in turbulence: A magnitude cumulant analysis. The Euro. Phys. Jour. B, 3:43 48,. 39, 43, 44 [59] G. Dewaele, K. Fukuda, P. Borgnat, P. Abry, and K. Cho. Extracting hidden anomalies using sketch and non Gaussian multiresolution statistical detection procedures. In SIGCOMM 7 Workshop LSAD - ACM SIGCOMM 7 Workshop on Large-Scale Attack Deense (LSAD), Kyoto, Japan, 7. 48, 49 [6] I. Dittman and C.W.J. Granger. Properties o nonlinear transormations o ractionally integrated processes. Journal o Econometrics, :3 33,. 44 [6] M. Doret, H. Helgason, P. Abry, P. Gonçalvès, C. Gharib, and P. Gaucherand. Multiractal analysis o etal heart rate variability in etuses with and without severe acidosis during labor. Am. J. Perinatol.,. To Appear. 47, 74 [6] A. Durand. Random wavelet series based on a tree-indexed Markov chain. Comm. Math. Phys., 83():45 477, [63] A. Durand and S. Jaard. Multiractal analysis o Lévy ields. Proba. Theo. Related Fields (to appear),. 7 [64] G.A. Edgar (ed.). Classics on Fractals. Cambridge University Press,

59 [65] P. Embrechts and M. Maejima. Selsimilar Processes. Princeton University Press, Series in Applied Mathematics,. 5, 7 [66] K. Falconer. Fractal Geometry. Wiley, 99. [67] K. Falconer. Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, West Sussex, England, , 8 [68] K. Falconer. Techniques in Fractal Geometry. John Wiley & Sons, West Sussex, England, , 8 [69] A. Feldmann, A.C. Gilbert, and W. Willinger. Data networks as cascades: Explaining the multiractal nature o internet wan traic. In SIGCOMM 98, pages 4 55, [7] A. Fisher, L. Calvet, and B. Mandelbrot. Multiractality o Deutsche Mark / US dollar exchange rate. Cowles Foundation Discussion Papers: 65, [7] P. Flandrin. On the spectrum o ractional Brownian motions. IEEE Trans. Ino. Theory, IT-35():97 99, , [7] P. Flandrin. Wavelet analysis and synthesis o ractional Brownian motions. IEEE Trans. Ino. Theory, 38:9 97, 99. 6, [73] E. Fououla-Georgiou and P. Kumar, editors. Wavelets in Geophysics. Academic Press, San Diego, Cali., [74] P. Frankhauser. L approche ractale : un nouvel outil dans l analyse spatiale des agglomerations urbaines. Population, 4:5 4, [75] A. Fraysse and S. Jaard. How smooth is almost every unction in a sobolev space? Revista Matematica Iberoamericana, ():663 68, [76] U. Frisch. Turbulence, the Legacy o A.N. Kolmogorov. Addison-Wesley, , 8, 4, 43 [77] Y. Gousseau and J.-M. Morel. Are natural images o bounded variation? SIAM J. Math. Anal., 33: ,. 6 [78] C.W.J. Granger. Long memory relationships and the aggregation o dynamic models. J. o Econometrics, 4: 38, [79] C.W.J. Granger and R. Joyeux. An introduction to long memory time series models and ractional dierencing. Journal o Time Series Analysis, :5 9, [8] T.C. Halsey, M.H. Jensen, L.P. Kadano, I. Procaccia, and B.I. Shraiman. Fractal measures and their singularities - the characterization o strange sets. Phys. Rev. A, 33():4 5,

60 [8] B. He. The temporal structures and unctional signiicance o scal-ree brain activity. Neuron, 66: ,. 4, 48 [8] D. Hernad, M. Mouillard, and D. Strauss-Kahn. Du bon usage de R/S. Revue de Statistique Appliquée, 6(4):6 79, 978. [83] N. Hohn, D. Veitch, and P. Abry. Cluster processes: A natural language or network traic. IEEE Transactions On Signal Processing, 5:9 44, [84] N. Hohn, D. Veitch, and P. Abry. Multiractality in tcp/ip traic: the case against. Computer Network Journal, 48:93 33, 5. 48, 49 [85] H.E. Hurst. Long-term storage capacity o reservoirs. Transactions o the American Society o Civil Engineering, 6:77 799, 95. [86] P.C. Ivanov. Scale-invariant aspects o cardiac dynamics. IEEE Eng. In Med. and Biol. Mag., 6(6):33 37, 7. 4, 47 [87] P.C. Ivanov, L.A. Nunes Amaral, A.L. Goldberger, S. Havlin, M.G. Rosenblum, Z.R. Struzik, and H.E. Stanley. Multiractality in human heartbeat dynamics. Nature, 399:46 465, , 47 [88] S. Jaard. Multiractal ormalism or unctions. SIAM J. o Math. Anal., 8(4): , [89] S. Jaard. Oscillation spaces: Properties and applications to ractal and multiractal unctions. Journal o Mathematical Physics, 39(8):49 44, [9] S. Jaard. Sur la dimension de boîte des graphes. Compt. Rend. Acad. Scien., 36:555 56, [9] S. Jaard. The multiractal nature o Lévy processes. Proba. Theo. Related Fields, 4():7 7, [9] S. Jaard. On the risch-parisi conjecture. Journal de Mathématiques Pures et Appliquées, 79(6):55 55,. 53 [93] S. Jaard. Wavelet techniques in multiractal analysis. In M. Lapidus and M. van Frankenhuijsen, editors, Fractal Geometry and Applications: A Jubilee o Benoît Mandelbrot, Proc. Symp. Pure Math., volume 7(), pages 9 5. AMS, 4. 3, 35, 36 [94] S. Jaard. Beyond Besov spaces, part : Oscillation spaces. Constructive Approximation, ():9 6, 5. 3 [95] S. Jaard. Pointwise regularity associated with unction spaces and multiractal analysis. Banach Center Pub. Vol. 7 Approximation and Probability, T. Figiel and A. Kamont, Eds., pages 93,

61 [96] S. Jaard. Wavelet techniques or pointwise regularity. Ann. Fac. Sci. Toul., 5():3 33, 6. 5 [97] S. Jaard, P. Abry, and S.G. Roux. Function spaces vs. scaling unctions: tools or image classiication. Mathematical Image processing (Springer Proceedings in Mathematics) M. Bergounioux ed., 5: 39,. 5, 8, 9, 5 [98] S. Jaard, P. Abry, S.G. Roux, B. Vedel, and H. Wendt. The contribution o wavelets in multiractal analysis, pages Series in contemporary applied mathematics. World scientiic publishing,. 9, 33, 35 [99] S. Jaard, B. Lashermes, and P. Abry. Wavelet leaders in multiractal analysis. In T. Qian, M.I. Vai, and X. Yuesheng, editors, Wavelet Analysis and Applications, pages 9 64, Basel, Switzerland, 6. Birkhäuser Verlag., 5, 9, 33, 35 [] S. Jaard, B. Lashermes, and P. Abry. Wavelet leaders in multiractal analysis. In Wavelet Analysis and Applications, T. Qian, M.I. Vai, X. Yuesheng, Eds., pages 9 64, Basel, Switzerland, 6. Birkhäuser Verlag. 5 [] S. Jaard and B. Mandelbrot. Peano-polya motion, when time is intrinsic or binomial (uniorm or multiractal). The Mathematical Intelligencer, 9(4): 6, [] S. Jaard and C. Melot. Wavelet analysis o ractal boundaries. Communications In Mathematical Physics, 58(3):53 565, 5. 5 [3] J.-P. Kahane and J. Peyrière. Sur certaines martingales de Mandelbrot. Adv. Math., :3 45, [4] M. Keshner. / noise. Proc. IEEE, 7(3): 8, [5] A.N. Kolmogorov. The Wiener spiral and some other interesting curves in Hilbert space (russian),. Dokl. Akad. Nauk SSSR, 6:():58, [6] A.N. Kolmogorov. a) dissipation o energy in the locally isotropic turbulence. b) the local structure o turbulence in incompressible viscous luid or very large Reynolds number. c) on degeneration o isotropic turbulence in an incompressible viscous liquid. In S.K. Friedlander and L. Topper, editors, Turbulence, Classic papers on statistical theory, pages 5 6. Interscience publishers, [7] A.N. Kolmogorov. The local structure o turbulence in incompressible viscous luid or very large Reynolds numbers. Comptes Rendus De L Academie Des Sciences De L Urss, 3:3 35, [8] A.N. Kolmogorov. A reinement o previous hypotheses concerning the local structure o turbulence in a viscous incompressible luid at high Reynolds number. J. Fluid Mech., 3:8 85, 96. 7, 43 [9] N. Kôno and M. Maejima. Selsimilar stable processes with stationary increments. Tokyo J. Math., 4:93,

62 [] K. Kotani, K. Takamasu, L. Saonov, and Y. Yamamoto. Multiractal heart rate dynamics in human cardiovascular model. Proceedings o SPIE, 5:34 347, 3. 4, 47 [] S. Osher L. Rudin and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D, 6:59 68, 99. [] B. Lashermes, P. Abry, and P. Chainais. New insight in the estimation o scaling exponents. Int. J. on Wavelets, Multiresolution and Inormation Processing, (4), 4. [3] B. Lashermes, S. Jaard, and P. Abry. Wavelet leader based multiractal analysis. 5 Ieee International Conerence On Acoustics, Speech, and Signal Processing, Vols -5, pages 6 64, 5. 9 [4] B. Lashermes, S.G. Roux, P. Abry, and S. Jaard. Comprehensive multiractal analysis o turbulent velocity using the wavelet leaders. European Physical Journal B, 6(): 5, 8. 43, 44 [5] W.E. Leland, M.S. Taqqu, W. Willinger, and D.V. Wilson. On the sel-similar nature o ethernet traic (extended version). ACM/IEEE Transactions on Networking, (): 5, February [6] W.E. Leland, M.S. Taqqu, W.Willinger, and D.V. Wilson. On the sel-similar nature o ethernet traic. In SIGCOMM 93: Conerence proceedings on Communications architectures, protocols and applications, pages 83 93, New York, NY, USA, 993. ACM Press. 48 [7] L.S. Liebovitch and A.T. Todorov. Invited editorial on ractal dynamics o human gait: stability o long-range correlations in stride interval luctuations. J. Appl. Physiol., pages , [8] P. Loiseau, P. Gonçalves, G. Dewaele, P. Borgnat, P. Abry, and P.V. Primet. Investigating sel-similarity and heavy-tailed distributions on a large scale experimental acility. IEEE/ACM Transactions on Networking, 8(4):6 74,. 49 [9] S.B. Lowen and M.C. Teich. Fractal-Based Point Processes. Wiley, Hoboken, NJ, 5. 4 [] S. Mallat. A Wavelet Tour o Signal Processing. Academic Press, San Diego, CA, 998., 5 [] B. Mandelbrot. The variation o certain speculative price. The Journal o Business, 36(4):394 49, [] B. Mandelbrot. Limitations o eiciency and o martingale models. The review o econometrics and statistics, 53():5 36,

63 [3] B. Mandelbrot. Possible reinement o the lognormal hypothesis concerning the distribution o energy dissipation in intermittent turbulence, volume, pages Springer, La Jolla, Caliornia, 97. Statistical Models and Turbulence, Lecture Notes in Physics. 8 [4] B. Mandelbrot. Intermittent turbulence in sel-similar cascades: divergence o high moments and dimension o the carrier. J. Fluid Mech., 6:33 358, , 4, 43 [5] B. Mandelbrot. Geometry o homogeneous scalar turbulence: iso-surace ractal dimensions 5/ and 8/3. J. Fluid Mech., 7():4 46, [6] B. Mandelbrot. Poisson approximation o the multi-temporal Brownian unctions and generalizations. Comptes Rendus de l Académie des Sciences (Paris), 8(A):75 78, [7] B. Mandelbrot. Fractals: Form, Chance and Dimension. W. H. Freeman and Company, USA, [8] B. Mandelbrot. The Fractal Geometry o Nature. New York, 98. 3, 4 [9] B. Mandelbrot. Diagonally sel-aine ractal cartoons. part : Mass, box, and gap ractal dimensions, local or global. Fractals in Physics, pages 3 5, , [3] B. Mandelbrot. Diagonally sel-aine ractal cartoons. part : length and area anomalies. Fractals in Physics, pages 7, , [3] B. Mandelbrot. Sel-ainity and ractal dimension. Physica Scripta, 3:57 6, [3] B. Mandelbrot. Limit lognormal multiractal measures. In E.A. Gotsman et al., editor, Frontiers o Physics, Proc. Landau Memorial Con., Tel Aviv, 988, pages 39 34, Oxord, 99. Pergamon Press. 8, 4 [33] B. Mandelbrot. Fractals and scaling in inance. Selected Works o Benoit B. Mandelbrot. Springer-Verlag, New York, 997. Discontinuity, concentration, risk, Selecta Volume E, With a oreword by R.E. Gomory. 44 [34] B. Mandelbrot. Multiractals and / noise. Selected Works o Benoit B. Mandelbrot. Springer-Verlag, New York, 998. Wild Sel-Ainity in Physics. [35] B. Mandelbrot. A multiractal walk down Wall Street. Sci. Am., 8():7 73, , 4, 44 [36] B. Mandelbrot. Gaussian Sel-Ainity and Fractals. Selected Works o Benoit B. Mandelbrot. Springer-Verlag, New York,. 5 [37] B. Mandelbrot. Fractals and Chaos. Selected Works o Benoit B. Mandelbrot. Springer-Verlag, New York,

64 [38] B. Mandelbrot and J.W. van Ness. Fractional Brownian motion, ractional noises and applications. SIAM Reviews, :4 437, ,, 5 [39] B. Mandelbrot and W. Wallis. Noah, Joseph, and operational hydrology. Water Resources Research, 4(5):99 98, , [4] B. Mandelbrot and W. Wallis. Robustness o the rescaled range R/S in the measurement o noncyclic long run statistical dependence. Water Resources Research, 5(5): , 969. [4] MAWI Traic Archive. WIDE project. 48, 75 [4] C. Meneveau. Analysis o turbulence in the orthonormal wavelet representation. J. Fluid Mech., 3:469, [43] Y. Meyer. Ondelettes et Opérateurs. Hermann, Paris, 99. English translation, Wavelets and operators, Cambridge University Press, 99. 3, 7 [44] A.S. Monin and A.M. Yaglom. Statistical Fluid Mechanics. M.I.T. Press, Cambridge, MA, , 43 [45] D. Monrad and L.D. Pitt. Local nondeterminism and Hausdor dimension. Progress in Probability and Statistics, Seminar on Stochastic Processes 986, (E. Cinlar, K.L. Chung, R.K. Getoor, Eds.), Birkhäuser, Boston, page 6389, [46] J.F. Muzy, E. Bacry, and A. Arneodo. Wavelets and multiractal ormalism or singular signals: application to turbulence data. Phys. Rev Lett., 67: , , 43 [47] J.F. Muzy, E. Bacry, and A. Arneodo. The multiractal ormalism revisited with wavelets. Int. J. o Biurcation and Chaos, 4:45 3, , 43 [48] E.A. Novikov. Ininitely divisible distributions in turbulence. Phys. Rev. E, 5:R333 R335, [49] A.M. Obukhov. Some speciic eatures o atmospheric turbulence. J. Fluid Mech., 3:77 8, 96. 7, 43 [5] S. Orey and S. J. Taylor. How oten on a Brownian path does the law o iterated logarithm ail? Proc. London Math. Soc, 8(3):74 9, [5] G. Parisi and U. Frisch. Fully developed turbulence and intermittency. In M. Ghil, R. Benzi, and G. Parisi, editors, Turbulence and Predictability in geophysical Fluid Dynamics and Climate Dynamics, Proc. o Int. School, page 84, Amsterdam, 985. North-Holland. 34, 35, 43, 5 [5] K. Park and W. Willinger. Sel-similar network traic: An overview. In K. Park and W. Willinger, editors, Sel-Similar Network Traic and Perormance Evaluation, pages 38. Wiley,. 4 64

65 [53] K. Park and W. Willinger. Sel-similar network traic: An overview. In K. Park and W. Willinger, editors, Sel-Similar Network Traic and Perormance Evaluation, pages 38, New York,. Wiley. 48 [54] V. Paxson and S. Floyd. Wide area traic: The ailure o Poisson modeling. IEEE TON, 4(3):9 3, [55] H.O. Peitgen and D. Saupe, editors. The science o ractal images. Springer-Verlag, 988. [56] E. Perkins. On the Hausdor dimension o the Brownian slow points. Z. Wahrsch. Verw. Gebiete, 64(3): , [57] E. Peters. Fractal market analysis. Wiley, NYC, USA, 994., 4, 44 [58] S. Pietropinto, C. Poulain, C. Baudet, B. Castaing, B. Chabaud, Y. Gagne, B. Hebral, Y. Ladam, P. Lebrun, O. Pirotte, and P. Roche. Superconducting instrumentation or high Reynolds turbulence experiments with low temperature gaseous helium. Physica C, 386:5 56, [59] V. Pipiras. Wavelet-type expansion o the rosenblatt process. J. Four. Anal. Appli., :599? 634, 4. 8 [6] V. Pipiras and P. Abry. Wavelet-based synthesis o the rosenblatt process. Sig. Proc., 86:36 339, 6. 8 [6] V. Pipiras and M. Taqqu. Multiractal processes. In P. Doukhan, G. Oppenheim, and M.S. Taqqu, editors, Fractional calculus and its connexions to ractional Brownian motion, pages 65. Birkhäuser, 3. 9 [6] V. Pipiras, M.S. Taqqu, and P. Abry. Can continuous-time stationary stable processes have discrete linear representation? Statistics & Probability Letters, 64:47 57, 3. 6 [63] V. Pipiras, M.S. Taqqu, and P. Abry. Bounds or the covariance o unctions o ininite variance stable random variables with applications to central limit theorems and wavelet-based estimation. Bernoulli, 3(4):9 3, 7. 6 [64] L.F. Richardson. Weather prediction by numerical process. Cambridge University Press, 9. 8 [65] R.H. Riedi. Multiractal processes. In P. Doukhan, G. Oppenheim, and M.S. Taqqu, editors, Theory and applications o long range dependence, pages Birkhäuser, 3., 35, 45 [66] R.H. Riedi, M.S. Crouse, V.J. Ribeiro, and R.G. Baraniuk. A multiractal wavelet model with application to network traic. IEEE Trans. Ino. Theory, 45:99 8,

66 [67] P.-G. Lemarié Rieusset. Distributions dont tous les coeicients d ondelettes sont nuls. Comptes Rendus Acad. Sci., 38:83 86, 994. [68] G. Samorodnitsky and M. Taqqu. Stable non-gaussian random processes. Chapman and Hall, New York, , 7, 8 [69] D. Schertzer and S. Lovejoy. Physically based rain and cloud modeling by anisotropic, multiplicative turbulent cascades. J. Geophys. Res., 9: , [7] M. Schroeder. Fractals, Chaos, Power Laws: Minutes rom an Ininite Paradise. W.H. Freeman, 99. [7] S. Seuret. On multiractality and time subordination or continuous unctions. Adv. Math., (3): , [7] Z.S. She and E. Lévêque. Universal scaling laws in ully developed turbulence. Phys. Rev. Letters, 7: , , 44 [73] N. R. Shieh and Y. Xiao. Hausdor and packing dimensions o the images o random ields. Bernoulli, 6 (4):96? 95,. 8, 9 [74] J.-L. Starck, F. Murtagh, and A. Bijaoui. Image Processing and Data Analysis: The Multiscale Approach. Cambridge University Press, Cambridge, [75] S. Stoev, V. Pipiras, and M.S. Taqqu. Estimation o the sel-similarity parameter in linear ractional stable motion. Signal Processing, 8():873 9,. 6 [76] M.S. Taqqu, W. Willinger, and R. Sherman. Proo o a undamental result in selsimilar traic modeling. SIGCOMM Comput. Commun. Rev., 7():5 3, [77] A.H. Tewik and M. Kim. Correlation structure o the discrete wavelet coeicients o ractional Brownian motions. IEEE Trans. Ino. Theory, IT-38():94 99, 99. [78] G. Teyssière and P. Abry. Wavelet analysis o nonlinear long range dependent processes. Applications to inancial time series. In Long Memory in Econometrics, G. Teyssière and A. P. Kirman, Eds. Springer, Berlin, [79] D. Veitch and P. Abry. A wavelet-based joint estimator o the parameters o longrange dependence. IEEE Trans. Ino. Theory, 45(3): , 999. [8] D. Veitch and P. Abry. A statistical test or the time constancy o scaling exponents. IEEE Trans. Signal Proces., 49():35 334,. [8] H. Wendt. Contributions o Wavelet Leaders and Bootstrap to Multiractal Analysis: Images, Estimation Perormance, Dependence Structure and Vanishing Moments. Conidence Intervals and Hypothesis Tests. PhD thesis, Ecole Normale Suprieure de Lyon,

67 [8] H. Wendt and P. Abry. Bootstrap tests or the time constancy o multiractal attributes. In Proc. IEEE Int. Con. Acoust., Speech, and Signal Proc. (ICASSP), Las Vegas, USA, 8. 38, 4 [83] H. Wendt, P. Abry, and S. Jaard. Bootstrap or empirical multiractal analysis. IEEE Signal Processing Mag., 4(4):38 48, 7., 4, 4, 43, 44, 54, 68, 69, 7 [84] H. Wendt, P. Abry, S. Jaard, H. Ji, and Z. Shen. Wavelet leader multiractal analysis or texture classiication. In Proc. IEEE Int. Con. Image Proc. (ICIP), Cairo, Egypt, 9. 5 [85] H. Wendt, S.G. Roux, P. Abry, and S. Jaard. Wavelet leaders and bootstrap or multiractal analysis o images. Signal Proces., 89: 4, 9. 4 [86] A.M. Yaglom. Eect o luctuations in energy dissipation rate on the orm o turbulence characteristics in the inertial subrange. Dokl. Akad. Nauk. SSR, 66:49 5, , 8, 43 67

68 !.5!()=.3!!4!4!6!8 Scale (s) Koch snowlake + polynomial log S(,j). disc + polynomial p Koch snowlake + polynomial!(p)!8.5 disc + polynomial log S(,j)!6!(p)!4.5!! ()=.75!5!6! disc + F.B.M.! Snow Scale (s) 4 8 6! Trees p 3 4! !.5! ()=.3! Scale (s) 5 Trees! log S(,j) p.8 Snow!()=.35 3 disc + F.B.M..5 log S(,j) Scale (s) 3!()=.73!8. p.5!(p)!4 4!(p)!3 Scale (s)!(p)! log S(,j) !.5 p Figure : Functional space assumptions in image processing. Let column: Indicator unctions o a Figure : row) Image Dierent images (synthetic,polynomial 3 top rows, and disc (top and oprocessing. the Koch snowlake (nd row) with superimposed trend, andnatural, o a disc bottom rows) together with log-log plots o their wavelet coeicients based S (, with F.B.M. texture (3rd row, H =.7); natural images (bottom rows). Corresponding S (j, p =j))(as deined (34)) (middle column) and their wavelet scaling estimated unctionsrom η (p) (as deined (centerincolumn) and wavelet scaling unctions η (p) (right column), the images in the in (35)) Fromunction the wavelet scaling unction, veryunctions accurate let (right column.column). For the indicator o the Koch snowlake and owe theobtain disc, thethe scaling estimate D by = η.5 or the ractal dimension o the Von Koch snowlake (given are given (p)η=() log(4) log(3).74, and η (p) =, respectively. Thereore, the latter is in B.V., by while D = the log(4)/log(3).6). Furthermore, recover that the image simple disk Koch snowlake is not because η () <.we With an added texture, the latteroisthe not any longer in B.V., butisremains L and sincenot η () >. Both natural images not in B.V., the image o discontinuity just ininbv smoother since η () &,are while the Vonyet Koch snowlake row) is ound too be disk in Land, while the image o trees (bottom row)natural is not in images Lp. andsnow the (4th superimposition Bm texture are not. Both are not in BV, and the one at bottom row is not even in L. 68

69 .5 F.B.M. H=.7.5 Levy motion!=.43.5!.5.5! Time (s) Time (s)! ! F.B.M. F.B.M. c p =[.7,,] c =.697 (.) c =.4 (.5).8 c = (.) 3! (p).6 D.4!. p H!4! Levy motion Levy motion ! (p) D.4.!.5 p H!! Figure : Selsimilar processes. Top Figure row: : sample paths o Bm with H =.7 (let) and o stable Lévy motion with H =.7 (right). Bottom row: estimated wavelet leader scaling unctions and Legendre spectra. Both processes have leader scaling unctions that behave linearly in p or p close to : ζ (p) = ph or Bm, ζ (p) = Hp (i p is close to ) or stable Lévy motion. These sel similar processes with stationary increments are thus characterized by c =. While Bm shows a linear in p leader scaling unction and is mono- Hölder, stable Lévy motion is multiractal and its leader scaling unction is only piecewise linear (c. discussion in Section 5.4). Conidence intervals are constructed based on the non parametric bootstrap procedure described in [83]. 69

70 log Normal Multiplicative Cascade log Poisson Multiplicative Cascade.5 F.B.M. in MF CPC time Time (s) Time (s) Time (s)! LN Cascade LP Cascade F.B.M. MF CPC! (p)!!4 p!6!5 5.8 LN Cascade c p =[.7,.6,] c =.68 (.45) c =.57 (.43) c 3 =. (.38) 4! (p)!!4!6.8!4! LP Cascade p c p =[.7,.6,.] c =.7 (.35) c =.65 (.6) c 3 =. (.6) ! (p)! p!4!5 5.8 F.B.M. MF CPC c p =[.7,.6,.] c =.78 (.6) c =.67 (.7) c 3 =.3 (.7).6 D.4.6 D.4.6 D.4... H H H Figure 3: Figure 3: D multiractal processes. Top row: sample paths o log-normal cascade (let), log-poisson cascade (middle), and a Bm in multiractal time (as deined in (5)) (right). nd and 3rd rows show the superimposition o the theoretical (blue) and estimated (black) corresponding leader scaling unctions and multiractal spectra. Theory and estimation are strikingly in agreement both or positive and negative ps and across the entire multiractal spectra (increasing and decreasing parts). Conidence intervals are obtained by a non parametric bootstrap procedure described in [83]. 3 7

71 4 F.B.M. H=.7 log Normal Multiplicative Cascade log Poisson Multiplicative Cascade 5 5! (p)! (p)! (p)!!5!5 p!4!5 5.5 F.B.M. c p =[.7,,] c =.7 (.3) c = (.4) c 3 =. (.) p!!!5 5!.5 LN Cascade c p =[.7,.,] c =.78 (.9) c =.4 (.9) c 3 =.3 (.5).5 p!5 5 5 LP Cascade c p =[.7,.6,.] c =.664 (.5) c =.55 (.3) c 3 =.4 (.3) D D D H H H Figure : Figure 4: D multiractal processes. Top row: D realizations o Bm (let), log-normal cascade (middle), log-poisson cascade (right). nd and 3rd rows show the superimposition o the theoretical (blue) and estimated (black) corresponding leader scaling unctions and multiractal spectra. As in the D case (c. Figs. and 3), theory and estimates are strikingly in agreement both or positive and negative ps and across the entire multiractal spectra (increasing and decreasing parts). Conidence interval are constructed using a non parametric bootstrap procedure described in [83]. 7

72 .5 FBM H=.7.8 FBM square H=.7.6.4!.5. Time (s)! FBM H=.7.8 Time (s) FBM square H= D D.4.4. H H Figure 5: Non linear transorm. FigureTop : FBM row: sample carre ig path o Bm, with H =.7 (let) and its squared transorm (right). Bottom row: estimated wavelet leader based Legendre spectra. Bm is a mono-hölder process and its estimated Legendre spectrum collapses on the theoretical spectrum D(h) = or h = H. Its squared non linear point transorm, however, is a bi-hölder process, with theoretical multiractal spectrum given in (5): The estimated Legendre spectrum perectly reproduces its corresponding concave hull. 7

73 Jet turbulence Eulerian velocity Time (s) Jet turbulence Jet turbulence!5! log T (3,j)!5!4 H min =.8! log Scale (samples) log Scale (samples)!5! ! Jet turbulence! (p)! h min D Jet turbulence c =.88 (.8) c =. (.) c 3 =. (.)!4!6 p!5 5. H Figure 6: Hydrodynamic Turbulence. Top: longitudinal Eulerian turbulence velocity Figure : c=.3; c=.3 signal (hot-wire anemometry, Reynolds number: 5, Courtesy C. Baudet et al. [5])). Middle: log-log plot or the wavelet leader based T (3, j) (as in (39)) and or the wavelet coeicients based h min (as in (37)). Bottom: Wavelet leader scaling unction (let) and multiractal spectrum (right) measured rom a single run o data (black solid lines), compared to the log-normal (red solid lines) and log-poisson (dashed blue lines) models. Data clearly select the log-normal model. 73

74 .8 EUR USD.3 EUR USD.6. Price.4. Hourly Return... Price Time (Year) EUR USD h min EUR USD log S(j,) Time (Year) 5 5 EUR USD 63 64! () = D.6.4. EUR USD Time (Day) Scale (Hour) Scale (Hour).5.5 h EUR USD EUR USD EUR USD 66 67! () = EUR USD Price h min 7 8 log S(j,) 5 D Time (Day) Scale (Hour) Scale (Hour).5.5 h EUR USD.65 EUR USD.4 EUR USD.5.6. H min.4.3! () C.55.5 C Figure 7: Euro-USD. Top: Euro-USD rate luctuations or years (starting January st, ) (let) and hourly returns (right). Data Courtesy Vivienne Investissement, Lyon, France. One year long time series (June 3 to June 4, nd row; June 6 to June 7, 3rd row), with log-log plots or the wavelet coeicients based h min (c. (37)), loglog plots or the wavelet leader based T (, j) (given by (39)), and multiractal spectra, measured using the wavelet leader multiractal ormalism. Bottom row: Estimation in one year long time windows (with a 3-month sliding shit) o h min, η (), c, c (together with conidence intervals). These plots show that while h min >, ζ () oten remains below the inite quadratic variations limit o. Parameter c, c variations indicate that the multiractal properties o the Euro-USD price luctuations vary signiicantly along time. 74

75 8 BpM 6 4 Time (min) 8 5 BpM 5 8 BpM 8 BpM Time beore birth (min) Time beore birth (min) BpM BpM log T (,j) 5 log T (,j) 5 log T (,j)!5!5!5 Scale (s) Scale (s) Scale (s)!!! BpM BpM BpM 4 3 h min 4 3 h min 4 3 h min H =.9 min H =. min H =.35 min Scale (s) Scale (s) Scale (s) BpM c =.96 (.) BpM c =. (.6) BpM c =.4 (.4) c =.5 (.) c =.3 (.3) c =.4 (.) D.6.4 D.6.4 D. H BpM. H BpM. H h min c!.3.5 Time beore birth (min) Time beore birth (min) Figure 8: Fetal-ECG. Top: Figure Examples : last 5o min cardiac TN / requency FP / TP time series (in Beats-per- Minute) or etuses during delivery process - last 5 min. beore delivery, or a healthy subject, correctly diagnosed as such by the obstetrician (let) (True Negative), or a healthy subject, incorrectly diagnosed as suering rom per partum asphyxia (middle) (False Positive), or a etus actually suering rom per partum asphyixia and diagnosed as such (True Positive). Data Courtesy M. Doret, Obstetrics Dept. o the University Hospital Femme- Mère-Enant, Lyon, France. Second Row: log-log plots or the wavelet leader based T (, j) (c. (39)). Third Row: log-log plots or the wavelet coeicients based h min (as in (37)). Fourth Row: Multiractal spectra, measured using the wavelet leader multiractal ormalism. Bottom Row: Parameters h min (let) and c (right) estimated over 5 min. long sliding windows (with a 7.5min overlap) over the last hour beore delivery, values are averaged over the classes o 5 patients each: True Positive (dashed blue line with o ), False Positive (dashed black line with squares, True Positive (solid red line with ). The red correspond to windows where a Wilcoxon ranksum test between True Positive and True Negative gives a p value below 5%. This analysis shows that unhealthy subjects (True Positive) systematically display less HRV as compared to healthy ones (True Positive and 75 False Positive) (c. [9, 6]).

76 Time (s) Wavelet Spectrum log Scale (in s) Figure 9: Internet Traic: Aggregated time series and scaling. Let: Aggregrated time series (Courtesy MAWI research project, [4]); Right: log-log plot or the wavelet coeicient based S (, j) (as in (34)). It clearly displays two independent scaling ranges, below and above the second. 76

77 3 B595 tj 6 t 6 B595 tj 6 t 5 4 log S(j,) 5 5 h min Scale (s) B595 tj 6 t Scale (s) B595 tj 6 t.8 ζ(q) D q. B59 tj.t MF 6 Wav. Coe Wav. leader.5.5 h.5 B59 tj.t MF 6 c.9.8 c Time (h) Time (h) Figure : Internet Traic: Coarse scales. Top: log-log plots or the wavelet leader based T (, j) (c. (39)) (let) and or the wavelet coeicients based h min (as in (37)) (right). Middle: Wavelet leader scaling unction (let) and multiractal spectrum (right) measured rom 5-min long data. Bottom: Evolution along time o parameters c (let) and c (right), 77 measured rom 5-min long sliding windows or 4 consecutive hours (9 hours shown here; c. [39]), suggesting that scaling properties are constant along time and clearly validating that, at coarse scales, i.e., above s, aggregated Internet traic are selsimilar with long memory (c H.9) and show no multiractality (c ).

78 3 B595 tj 6 t B595 tj 6 t 5 5 log S(j,) 5 h min Scale (s) B595 tj 6 t 6 4 ζ(q) Scale (s) B595 tj 6 t.8 D q.8.7 c B59 j.t MF 4 7 Wav. Coe Wav. leader.5.5 h.. c.4.6 B59 j.t MF Time (h) Time (h) Figure : Internet Traic: Fine Scales. Top: log-log plots or the wavelet leader based T (, j) (as in (39)) (let) and or the wavelet coeicients based h min (given by (37)) (right). Middle: Wavelet leader scaling unction (let) and multiractal spectrum (right) measured rom 5-min long data. Bottom: Evolution along time o parameters c (let) and c (right), measured rom 5-min long sliding windows or 4 consecutive hours (9 hours shown here). (c. [39]), also suggest that scaling properties are constant along time and indicate that, at ine scales (i.e., below s), aggregated Internet traic are characterized by 78 c H.6 and c.5 ±.5, revealing at best a very weak multiractality or no multiractality at all.

79 Paris period Period questioned Provence period !3.5!4!4.5!5!5.5!6 log S (,j) 45! ()=.5!6.5 Scale (mm)! !4.5!5!5.5!6 log S (,j) 65! ()=.3 Scale (mm)! !4!4.5!5!5.5!6 log S (,j) 475! ()=.4 Scale (mm)! Wavelet analysis: parameter space D projections h min R h S 358 min h R469 min !.5!!.5!3 45 H min =.3 Scale (mm)! !.5!!.5!3 65 H min =. Scale (mm)! !.6!.8!!.! ! 358 () S H min =.7!.6 Scale (mm)! h min R D.5 45 H D.5 65 H D c S H.5.5 Figure : Van Figure Gogh s 7: R Paintings. (j = [, 5]) [Paris Let columns: QUESTIONED Representative Provence] examples o paintings o the Paris (top), Provence (bottom) and unknown (middle) periods, with selected homogeneous patches (second column), Courtesy Image Processing or Art Investigation research program, c. and the Van Gogh and Kröller-Müller Museums (The Netherlands). Third column: log-log plots or the wavelet coeicient based S (, j) (as in (34)). Fourth column: h min. Right column: Wavelet leader based multiractal spectra, suggesting that the questioned painting has multiractal properties close to that o the Provence period example. Bottom row: D projections o multiractal attributes, indicating that paintings rom the Paris period (blue) are overall globally more regular than paintings rom the Provence period (red)