Multifractal characterization for bivariate data

Transcription

1 Multifractal caracterization for bivariate data R. Leonarduzzi, P. Abry, S. Roux Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Pysique, Lyon, France. H. Wendt IRIT, Univ. Toulouse, CNRS, Toulouse, France. S. Jaffard, S. Seuret LAMA, Univ. Paris-Est, UPEM, UPEC, CNRS, Créteil, France. Abstract Multifractal analysis is a reference tool for te analysis of data based on local regularity, wic as been proven useful in an increasing number of applications. However, in its current formulation, it remains a fundamentally univariate tool, wile being confronted wit multivariate data in an increasing number of applications. Recent contributions ave explored a first multivariate teoretical grounding for multifractal analysis and sown tat it can be effective in capturing and quantifying transient iger-order dependence beyond correlation. Building on tese first fundamental contributions, tis work proposes and studies te use of a quadratic model for te joint multifractal spectrum of bivariate time series. We obtain expressions for te Pearson correlation in terms of te random walk and a multifractal cascade dependence parameters under tis model, provide complete expressions for te multifractal parameters and propose a transformation of tese parameters into natural coordinates tat allows to effectively summarize te information tey convey. Finally, we propose estimators for tese parameters and assess teir statistical performance troug numerical simulations. Te results indicate tat te bivariate multifractal parameter estimates are accurate and effective in quantifying non-linear, iger-order dependencies between time series. I. INTRODUCTION Context: Multifractal analysis. Multifractal analysis is a signal processing tool tat provides a robust caracterization of data in terms of pointwise regularity properties [], [2]. It does so troug an upper-bound L for te so-called multifractal spectrum, wic quantifies geometrically te pointwise regularity fluctuations of data. Suc fluctuations produce, on average, scale-free dynamics, and are tus efficiently modeled and analyzed troug te paradigm of scale-invariance. Multifractal analysis as led to significant successes in many real-world applications in very different contexts [3] [8], and is nowadays establised as a versatile and standard signal processing tool. However, it remains essentially univariate, wic constitutes a major limitation in view of te increasing number of applications involving multivariate data in many domains. Indeed, multifractal analysis is currently conducted independently for eac component in suc cases, ence not accounting for te joint information and cross-dependencies in te data. Surprisingly, attempts to extend it to a multivariate setting remain scarce (see, e.g., [9], [] for notable exceptions in specific applicative contexts. Related works. State-of-te-art tools for multifractal analysis rely on te use of wavelet leaders, defined troug a nonlinear transformation of wavelet coefficients (see Section II-B, and also [] for a discussion and references on alternative formalisms. Very recently, te first cornerstone for a teoretical foundation of multivariate multifractal analysis was laid in [2] (see also [9] for te first istorical work on te topic, in te context of turbulence. Moreover, in te recent contribution [3] a bivariate multifractal random process was defined and studied, following earlier work in [4]. Furter, [3] defined a bivariate wavelet leader multifractal formalism and studied it numerically, yielding first intuitions on wat type of information is actually captured by te bivariate multifractal spectrum and sowing tat multifractal features can effectively capture transient, local dependencies tat cannot be accounted for by te Pearson correlation coefficient. Goals, contributions and outline. Te present contribution aims to build on, complement and go beyond [3] in several ways. First, in Section II-C te analysis of [3] of te bivariate multifractal spectrum is refined, a quadratic (i.e., second order model for te bivariate multifractal spectrum is proposed, and te exact expression for L under tis model is obtained. Second, in Section II-D, a parametrization of L in natural coordinates is derived, effectively capturing te leading-order information contained in te multifractal spectrum in a small number of easily interpretable quantities. Tird, Section III-C derives an expression for te Pearson correlation coefficient of te model introduced in [3], [4], wic complements te analysis in [3] and provides a natural expansion of te joint dependence structure of te quadratic multifractal model in an (additive self-similar random walk correlation and a (multiplicative multifractal correlation. Finally, Section IV studies te performance of te estimators proposed for all parameters of te model troug Monte Carlo simulations using syntetic multifractal data. II. BIVARIATE MULTIFRACTAL ANALYSIS A. Multifractal spectrum Te goal of multifractal analysis is te quantification of te fluctuations along time of te regularity of a signal or function X(t at position t R, see, e.g., []. Pointwise regularity is usually measured using te Hölder exponent, (t, as follows: X is said to be in C α (t, α, if tere exist a polynomial P t wit deg(p t < α and a constant C > suc tat X(t + a P t (t + a C a α, a, ( (t is defined as te largest α suc tat ( is satisfied, (t sup{α : X C α (t}. (2

2 Te closer (t to, te more irregular X is around t. Let (t ( (t, 2 (t denote te Hölder exponents of te components of te bivariate signal X = (X, X 2. Te bivariate multifractal spectrum D(, 2 of X is defined as te collection of Hausdorff dimensions dim H of te sets of points t R at wic ( (t, 2 (t takes on te values = (, 2 [9], [2], D(, 2 dim H { t : ( (t, 2 (t = (, 2 }. (3 It provides a global, geometrical description of te pointwise regularity properties of te components of X. Its precise sape and widt, and its orientation wit respect to te, 2 axes, quantify information regarding te joint local fluctuation and dependence of te regularity of te components of X. B. Wavelet leader bivariate multifractal formalism Te estimation of te multifractal spectrum (3 cannot be based on its formal definition but requires taking recourse to formulas tat are numerically robust for discrete data, te socalled multifractal formalisms. Te state-of-te-art multifractal formalism is constructed from te multiscale statistics of wavelet leaders [], [2], as been first developed for multivariate data in [2], [3] and is briefly recalled in te following paragraps. Let ψ denote te moter wavelet, an oscillating reference pattern tat is caracterized by its number of vanising moments N ψ, a positive integer defined as ψ C Nψ and n =,..., N ψ, R tk ψ(tdt and R tn ψ ψ(tdt and tat is designed suc tat te collection {ψ j,k (t = 2 j/2 ψ(2 j t k} (j,k Z 2 of its dilated and translated templates forms an ortonormal basis of L 2 (R [5]. Te discrete wavelet transform coefficients d X (j, k of X are defined as d X (j, k = 2 j/2 ψ j,k X, were we ave adered to a L normalization. Ten, te wavelet leaders of X are defined as l X (j, k sup λ 3λ j,k d X (λ, were λ j,k = [k2 j, (k + 2 j denotes te dyadic interval of size 2 j and 3λ j,k stands for te union of λ j,k wit its 2 neigbors [], [2]. It can be sown tat wavelet leaders reproduce Hölder exponents in te limit of fine scales, L X (j, k C2 j(t as 2 j for t = 2 j k. Consequently, n j L X (j, k q L X2 (j, k q2 c q 2 jζ(q,q2, 2 j. (4 n j k= Most importantly, te so-called scaling exponents ζ(q in (4 are tigtly related to D( via teir Legendre transform, te bivariate Legendre spectrum L( = inf( + q, ζ(q, (5 q wic provides an estimate for D( for large classes of processes, see [2]. C. Cumulant expansion of te bivariate Legendre spectrum Using te arguments developed in [6], one can sown tat for a large number of commonly used classes of multifractal processes wit scaling exponents ζ(q, te cumulants C pp 2 (j of te 2-variable vector of logaritm of leaders (ln L X (j, k, ln L X2 (j, k at scale 2 j take te form [3] C pp 2 (j = c p p 2 + j c pp 2 ln 2, p + p 2 (6 and te coefficients c pp 2 are related to te ζ(q, q 2 as ζ(q, q 2 = c pp 2 q p qp2 2 /(p! p 2!. (7 p,p 2 : p +p 2 Furter, from (6, estimators for c pp 2 can be defined as linear regressions of sample cumulants Ĉp p 2 (j: ĉ pp 2 j 2 j=j w j Ĉ pp 2 (j/ ln 2, (8 over a range of scales j (j, j 2, were w j are linear regression weigts. By truncating te sum in (7 to te leading order terms p, p 2 : p + p 2 2, we can gain insigt into te information provided by te sape of te bivariate spectra L(. Te quadratic approximation ζ(q, q 2 c q +c q 2 + c 2 2 q2 + c2 2 q2 + c q q 2 yields te expression L(, 2 + c 2b 2 ( 2 c + c ( 2b 2 c b 2 b ( ( c 2 c c b b b 2, (9 were b c 2 c 2 c 2, sowing tat - te position of te maximum of te bivariate spectrum is given by m = (c, c - c 2 and c 2 quantify te widts of te fluctuations independently for eac component and - c yields a leading order joint caracterization of te regularity fluctuations of bot components. As an extreme ( c 2 case, wen c =, L( = + c2 2 c 2 + ( 2 c 2 2 c 2 c 2 (i.e., it equals te sum of te univariate spectra up to a constant and te regularity fluctuations of te components are independent (in consistency wit te generic properties of multivariate L( proven in [2]. D. Bivariate multifractal parameters in natural coordinates Inspection of te expression (9 leads to conclude tat te level sets and te support of L( are rotated and translated ellipses in te (, 2 plane. Terefore, te natural parameters for L( are given by its center m, and by te rotation angle θ and te major and minor alf-axes α and α 2 of its support. Expressions for tese quantities are obtained by straigt-forward but tedious calculations, and are given by θ = ( 2 arctan 2c, ( c 2 c 2 c 2 c 2 c 2 α = 2 (c 2 c c 2 c, ( 2 c 2 α 2 = 2 c 2 c 2 c 2 + (c 2 c c 2 c 2 c 2, (2

3 respectively. From tese expressions, it can be seen tat te linear eccentricity of te support ɛ α α 2 = ( γ c2 c 2 γ c 2 c 2, 2 (3 were γ (c 2 c c 2, increases wit c, as expected. Moreover, in te special case c 2 = c 2 (i.e., marginal spectra of equal widt, θ = ±45, and α,2 = 2 c 2 ± c ; in oter words, te spectrum remarkably flips between diagonal and anti-diagonal orientation as c canges sign. Finally, estimators ˆθ, ˆα, ˆα 2 and ˆɛ can be readily defined by replacing c pp 2 wit estimates (8 in (-2. III. BIVARIATE MULTIFRACTAL MODEL PROCESS A. Definition of bivariate multifractal random walk Multifractal random walks (MRW were originally proposed as realistic models for multifractal data [7]. Teir construction is based on te increments of fractional Brownian motion (fbm, te reference Gaussian self-similar process [8], wose variance is modulated using an independent process wose properties mimic tose of Mandelbrot cascades, and ence impart teir multifractality to te MRW [4], [7]. Building on te unpublised work [4], a bivariate extension of MRW was proposed in [3], wic we denoted bmrw and briefly summarize next. Te construction of bmrw requires two pairs of stocastic processes: First, a pair of increments of fbm, ( G (t, G 2 (t, wic is determined by two self-similarity parameters, H and H 2, and a point covariance Σ ss. Its correlation coefficient is ereafter referred to as ρ ss. Tese processes can be constructed as a specific case of te operator fractional Brownian motion framework developed in [9]. Second, a pair of Gaussian processes ( ω (t, ω 2 (t wit prescribed covariance function Σ mf, wit entries given by ( {Σ mf } ij (k, l = ρ mf (i, jλ i λ j log T k l +, i =, 2 (4 for k l T and oterwise, were T is an arbitrary integral scale. To simplify notations, we consider ρ mf (, = ρ mf (2, 2 =, and ρ mf (, 2 = ρ mf. Bot pairs of processes are numerically syntesized as described in [2]. Finally, eac component i =, 2 of bmrw is defined as X i (t = B. Multifractal properties t G i (ke ωi(k. (5 k= As detailed in [3], following [4], [7], te bivariate scaling exponents of bmrw are conjectured to take te form (7, wit and c = H +λ 2 /2, c = H 2 +λ 2 2/2, c 2 = λ 2, c 2 = λ 2 2, and c = ρ mf λ λ 2. Moreover, c pp 2, (p, p 2 suc tat p + p 2 3; terefore, te second order approximation developed in Section II-C (in particular, (9 is exact for bmrw Ĉ (j ρ ss = ρ mf =.4 c =.7 ĉ =.6 j 5 5 ĉ ρ ss = est teo ρ mf Fig.. Log-scale diagram Ĉ (j and estimation performance for c (from left to rigt. C. Correlation and dependence Te dependence between te components of bmrw is clearly controlled by te correlation parameters ρ ss and ρ mf of te self-similar random walk and multifractal cascade components entering its construction, respectively; we terefore identify ere expressions for tese parameters involving quantities tat can be readily measured from data. An estimator for ρ mf can be defined as [3] ˆρ mf ĉ / ĉ 2 ĉ 2. (6 To identify te parameter ρ ss, we first derive te expression for te Pearson correlation ρ bmrw of te increments of te components of bmrw. Te increments are, for eac k, te product of mutually independent Gaussian and log-normal random variables, see (5; using elementary expressions for te product, expectation and variance of log-normal random variables, we terefore ave E[ X i (k] = E[G i (k] E[e ωi(k ] = Var[ X i (k] = σ 2 i (e λ2 i log(t e λ2 i log(t + σ 2 i e λ2 i log(t = σ 2 i e 2λ2 i log(t E[ X (k X 2 (k] = E[G (kg 2 (k] E[e ω(k+ω2(k ] and tus = ρ ss σ σ 2 e 2 (λ2 +λ2 2 +2ρ mf λ λ 2 log(t, ρ bmrw = E[ X (k X 2 (k] E[ X (k]e[ X 2 (k] Var[ X (k]var[ X 2 (k] = ρ ssσ σ 2 e 2 (λ2 +λ2 2 +2ρ mf λ λ 2 log(t σ σ 2 e (λ2 +λ2 2 log(t = ρ ss e (ρ mf λ λ 2 2 (λ2 +λ22 log(t. (7 Consequently, ρ bmrw = ρ ss f(ρ mf, λ, λ 2 takes te form of a product of te correlation coefficient ρ ss of te random walk components G i (k and a nonlinear function in te parameters of te multifractal components e ωi(k. Notably, tis implies tat te Pearson correlation coefficient can equal zero (in case ρ ss = even wen te data are actually igly dependent (ρ mf. Tus, te bivariate multifractal spectrum can be seen as capturing transient local dependencies beyond second order correlations, see [3] and Section IV. Furter, in view of te model described in Section III, te parameters ρ ss and ρ mf constitute natural expansion

4 .5 ˆρ bmrw ρss =.9 ρss =.5 ρss = est ρss =.5 teo -.5 ρ mf ρss =.5 ρ mf ˆρ mf ρ ss =.5 ˆρ ss ρss =.9 ρss =.5 ρss = - ρ mf - ρ mf ˆρ mf ρ ss =.5 Fig. 2. Model and estimation for dependence parameters. Pearson correlation coefficient ρ bmrw (top left, self-similar random walk correlation ρ ss (top rigt, and multifractal correlation ρ mf (bottom row, for several ρ ss ˆα ρ ss =.3 ρ mf ρ mf ˆǫ ρ ss = ˆθ (deg ρ ss = ˆα 2 ρ ss =.3 ρ mf ρ mf Fig. 3. Model and estimation for natural bivariate multifractal parameters. Semi-axes α (top left and α 2 (top rigt, eccentricity ɛ (bottom left and orientation θ (bottom rigt. coefficients for te joint dependence of X. Upon substitution of c = ρ mf λ λ 2, c 2 = λ 2, c 2 = λ 2 2, we can define a natural estimator for ρ ss as ˆρ ss ˆρ bmrw e (ĉ 2 (ĉ2+ĉ2 log(t. (8 IV. ESTIMATION PERFORMANCE ASSESSMENT A. Monte Carlo simulations and parameter setting Estimation performance is analyzed troug Monte Carlo simulations over independent copies of bmrw of sample size n = 2 8. Te parameters of te process are set to (H, H 2 = (.65,.75, (λ, λ 2 = (.3,.6, and several values for ρ ss and ρ mf are considered. Te integral scale in (4 is set to T = n. Wavelet analysis is conducted wit a Daubecies least asymmetric wavelet, wit N ψ = 3. B. Bivariate multifractality parameter c Te estimation performance for univariate parameters c, c, c 2, and c 2 remains unaltered in te multivariate setting, and as been documented elsewere (see, e.g., [2]. Here we focus on te multivariate parameter c. Fig. (left sows tat estimates of te quantities C (j beave as a clean, linear function of octaves j as postulated by (6, allowing te estimation of parameter ĉ by linear regression. Furter, Fig. (rigt clearly indicates te excellent performance of te estimates ĉ for all values of ρ mf, wit negligible bias and variance independent of ρ mf. C. Correlation and dependence Figure (top left sows tat te Pearson correlation ρ bmrw is a nonlinear function of te multifractal correlation ρ mf and tigtly follows te predicted values (7. Moreover, and notably, ρ bmrw = wen ρ ss even if ρ mf, illustrating tat ρ mf measures a type of dependence to wic te second-order Pearson correlation is totally blind. Furter, Fig. 2 (top rigt sows tat estimates ˆρ ss remain constant wit ρ mf, and precisely recover te true values, wit negligible bias. Interestingly, te variances of ˆρ ss and ˆρ bmrw decrease wit increasing ρ mf and decreasing ρ ss ; a precise modeling of tis penomenon is left for future work. Finally, Fig. 2 (bottom row sows estimation performance of ˆρ mf, already reported in [3]. Here, te performance is moreover sown to be independent of ρ ss ; furter, te results indicate tat it is largely unaffected by te true value of ρ mf. D. Second order multifractal analysis in natural coordinates Fig. 3 illustrates te estimation performance for te natural parameters θ, α,2 and ɛ, defined in (-3, for te quadratic bivariate multifractal model. It sows tat te estimates for θ, α,2 and ɛ overall closely reproduce te teoretical values. In particular, estimates for te rotation angle θ (i.e., te orientation of te multifractal spectra in te (, 2 plane are found to be igly accurate. Te estimates for bot te minor and major alf-axes, α, α 2 are affected by a sligt but systematic negative bias, for all ρ mf. However, tis bias as no negative effect on te estimates for te linear eccentricity ɛ, wic are very satisfactory. Similar results are obtained for oter values of ρ ss and not sown for space reasons (ere, ρ ss. Overall, tis leads to conclude tat wile te scale of te quadratic multifractal model is sligtly underestimated, estimates for its sape and orientation are igly accurate. Tis is furter investigated in Fig. 4, wic illustrates te level sets of te Legendre spectrum L (middle row and its estimate ˆL (bottom row, for ρ ss = and several values of ρ mf, togeter wit te major and minor alf axes of te second order model (blue dased lines and teir estimates (red lines. Te results reveal an excellent agreement between te estimates and teir true values in all cases. Moreover, tey indicate tat, despite te sligt negative bias of ˆα and ˆα 2, tese natural parameters of te second order model provide on

5 ACKNOWLEDGMENT Work supported by ANR-6-CE33-2 MultiFracs, France..2 L(, ˆL(, L(, ˆL(, L(, ˆL(, Fig. 4. Legendre spectra. Sample pats (top row, true Legendre spectra L (middle row and estimates ˆL (bottom row, for tree bmrw wit ρ ss = and ρ mf =.4 (left column, ρ mf = (middle column, and ρ mf =.8 (rigt column. Te true and estimated minor and major axes are sown in blue-dased and red-solid lines, respectively. average more accurate estimates of te size of te support of te multifractal spectrum (ence, te domain of joint regularity fluctuations tan te unconstrained Legendre spectra L, wic are furter srank as compared to teory. It is wort pointing out tat for all te examples considered in Fig. 4 (top row, ρ ss =. Tus, ρ bmrw = and Pearson correlation is unable to distinguis tose time series. However, teir multifractal spectra L clearly capture a form of igerorder statistical dependence beyond Pearson correlation, and fully caracterize tese processes (see also [3]. Tis information is conveniently summarized in te natural parameters θ and α,2, wic caracterize te orientation and strengt of joint regularity fluctuations. V. CONCLUSION In tis work, te identification and estimation of secondorder joint multifractal properties for bivariate processes was considered. Expressions for second order parameters in natural coordinates were derived, and sown to provide an intuitive and versatile description for te iger-order dependencies of te data. Crucially, tese parameters distinguis dependencies tat te usual Person correlation cannot identify. Moreover, an expression for te joint dependence structure was provided, enabling a factorization into an (additive self-similar randomwalk correlation and a (multiplicative multifractal correlation. Estimators for te associated (multifractal and correlation parameters ave been defined, and teir performance was assessed on syntetic data and sown to be igly satisfactory. Tese developments open new and promising perspectives for te analysis of real-world multivariate data, including applications in neuroscience, wic are currently being explored. REFERENCES [] S. Jaffard, Wavelet tecniques in multifractal analysis, in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, M. Lapidus and M. van Frankenuijsen, Eds., Proc. Symposia in Pure Matematics, vol. 72(2. AMS, 24, pp [2] H. Wendt, P. Abry, and S. Jaffard, Bootstrap for empirical multifractal analysis, IEEE Signal Proc. Mag., vol. 24, no. 4, pp , 27. [3] K. Kiyono, Z. R. Struzik, N. Aoyagi, and Y. Yamamoto, Multiscale probability density function analysis: non-gaussian and scale-invariant fluctuations of ealty uman eart rate, IEEE Trans. Biomed. Eng., vol. 53, no., pp. 95 2, Jan. 26. [4] B. B. Mandelbrot, Intermittent turbulence in self-similar cascades: divergence of ig moments and dimension of te carrier, J. Fluid Mec., vol. 62, pp , 974. [5] P. Ciuciu, G. Varoquaux, P. Abry, S. Sadagiani, and A. Kleinscmidt, Scale-free and multifractal time dynamics of fmri signals during rest and task, Front. Pysiol., vol. 3, Jun. 22. [6] M. Doret, H. Helgason, P. Abry, P. Gonçalvès, C. Garib, and P. Gaucerand, Multifractal analysis of fetal eart rate variability in fetuses wit and witout severe acidosis during labor, Am. J. Perinatol., vol. 28, no. 4, pp , 2. [7] R. Fontugne, P. Abry, K. Fukuda, D. Veitc, K. Co, P. Borgnat, and H. Wendt, Scaling in internet traffic: a 4 year and 3 day longitudinal study, wit multiscale analyses and random projections, IEEE/ACM T. Networking, vol. 25, no. 4, 27. [8] A. Joansen and D. Sornette, Finite-time singularity in te dynamics of te world population, economic and financial indices, Pysica A, vol. 294, pp , 2. [9] C. Meneveau, K. Sreenivasan, P. Kailasnat, and M. Fan, Joint multifractal measures - teory and applications to turbulence, Pys. Rev. A, vol. 4, no. 2, pp , 99. [] T. Lux, Higer dimensional multifractal processes: A GMM approac, Journal of Business and Economic Statistics, vol. 26, pp. 94 2, 27. [] S. Jaffard, P. Abry, and H. Wendt, Irregularities and Scaling in Signal and Image Processing: Multifractal Analysis. Singapore: World scientific publising, 25, pp [2] S. Jaffard, S. Seuret, H. Wendt, R. Leonarduzzi, S. Roux, and P. Abry, Multivariate multifractal analysis, Appl. Comp. Harm. Anal., 28, in press. [3] H. Wendt, R. Leonarduzzi, P. Abry, S. Roux, S. Jaffard, and S. Seuret, Assessing cross-dependencies using bivariate multifractal analysis, in IEEE Int. Conf. Acoust., Speec, and Signal Proces. (ICASSP, Calgary, Canada, April 28. [4] E. Bacry, J. Delour, and J. F. Muzy, A multivariate multifractal model for return fluctuations, arxiv preprint cond-mat/926, 2. [5] S. Mallat, A Wavelet Tour of Signal Processing. San Diego, CA: Academic Press, 998. [6] B. Castaing, Y. Gagne, and M. Marcand, Log-similarity for turbulent flows, Pysica D, vol. 68, no. 3-4, pp , 993. [7] E. Bacry, J. Delour, and J.-F. Muzy, Multifractal random walk, Pys. Rev. E, vol. 64: 263, 2. [8] G. Samorodnitsky and M. Taqqu, Stable non-gaussian random processes. New York: Capman and Hall, 994. [9] G. Didier and V. Pipiras, Integral representations and properties of operator fractional Brownian motions, Bernoulli, vol. 7, no., pp. 33, 2. [2] H. Helgason, V. Pipiras, and P. Abry, Fast and exact syntesis of stationary multivariate Gaussian time series using circulant embedding, Signal Proc., vol. 9, no. 5, pp , 2.