ON MULTIVARIATE NON-GAUSSIAN SCALE INVARIANCE: FRACTIONAL LÉVY PROCESSES and WAVELET ESTIMATION B Boniece, Gustavo Didier, Herwig Wendt, Patrice Abry

ON MULTIVARIATE NON-GAUSSIAN SCALE INVARIANCE: FRACTIONAL LÉVY PROCESSES AND WAVELET ESTIMATION B Boniece, Gustavo Didier, Herwig Wendt, Patrice Abry

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B Boniece, Gustavo Didier, Herwig Wendt, Patrice Abry. ON MULTIVARIATE NON-GAUSSIAN SCALE INVARIANCE: FRACTIONAL LÉVY PROCESSES AND WAVELET ESTIMATION. Eu- ropean Signal Processing Conference (EUSIPCO), Sep 2019, A Coruna, Spain. hal-02361748

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ON MULTIVARIATE NON-GAUSSIAN SCALE INVARIANCE: FRACTIONAL LEVY´ PROCESSES AND WAVELET ESTIMATION

B. Cooper Boniece(1), Gustavo Didier(1), Herwig Wendt (2), Patrice Abry(3)

(1) Tulane University, New Orleans, LA, USA (2) IRIT, CNRS (UMR 5505), Universite´ de Toulouse, France (3) Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, Lyon, France

ABSTRACT provides efficient multiscale representations, leading to the- In the modern world of “Big Data,” dynamic signals are often oretically well-grounded, robust and accurate estimation of multivariate and characterized by joint scale-free dynamics H [2, 5]. However, success has remained mostly restricted (self-similarity) and non-Gaussianity. In this paper, we exam- to univariate analysis. In the modern world of “Big Data,” ine the performance of joint wavelet eigenanalysis estimation a plethora of sensors monitor natural and artificial systems, for the Hurst parameters (scaling exponents) of non-Gaussian generating large data sets in the form of several joint time multivariate processes. We propose a new process called op- series. This creates an ever-increasing demand for adequate erator fractional Levy´ motion (ofLm) as a Levy-type´ model multivariate self-similarity modeling. for non-Gaussian multivariate self-similarity. Based on large Related work. Operator fractional Brownian motion size Monte Carlo simulations of bivariate ofLm with a combi- (ofBm), a generalization of fBm, was recently put forward nation of Gaussian and non-Gaussian marginals, the estima- as a model for multivariate Gaussian self-similarity [6, 7, 8, tion performance for Hurst parameters is shown to be satis- 9]. In particular, it allows multiple correlated fBm coordi- factory over finite samples. nate processes with possibly distinct self-similarity exponents H , m = 1,...,M, occurring in a non-canonical set of co- Index Terms— multivariate self-similarity, non-Gaussian m ordinates (mixing). In [10, 11], a statistical method is devised process, Levy´ processes, wavelets. for jointly estimating the vector of self-similarity exponents H = (H1,...,HM ). Based on multivariate wavelet repre- 1. INTRODUCTION sentations and eigenvalue decompositions, the method was mathematically studied and shown to have satisfactory the- Univariate self-similarity. A signal is called scale invariant oretical and practical performance. when its temporal dynamics lack a characteristic scale. Un- der scale invariance, a continuum of time scales contributes to Furthermore, non-Gaussian behavior is pervasive in a the observed dynamics, and the analyst’s focus is on identify- myriad of natural phenomena (e.g., turbulence, anomalous ing mechanisms that relate the scales, often in the form of the diffusion [11, 12]) and artificial systems (e.g., Internet traf- so-called scaling exponents [1, 2, 3]. Scale invariant dynamic fic [13]). Among non-Gaussian scale invariant models, frac- signals are phenomenologically observed in a wide range of tional Levy´ processes (e.g., [14, 15, 16, 17, 18]) have become physical and engineering systems. In applications, it often popular in physical applications since they make up a very manifests itself in the form of self-similarity. A signal X broad family of second order models displaying fractional co- is called self-similar when its finite-dimensional distributions variance structure [19, 20, 21, 22, 23]. Nevertheless, the mod- (fdd) are invariant with respect to a suitable scaling of time, eling of non-Gaussian, multivariate fractional Levy´ signals, while of great importance in applications, is a research topic i.e., {X(t)} fdd= {aH X(t/a)} , a > 0, where H is the t∈R t∈R that has been relatively little explored (e.g., [24, 25, 26]), and so-named Hurst parameter. The celebrated fractional Brow- the estimation of scaling exponent even less. nian motion (fBm) is the only Gaussian, self-similar process with stationary increments [4]. Estimation of the Hurst pa- Goals, contributions and outline. In this paper, we exam- rameter H is of central interest for signal processing tasks ine the performance of joint wavelet eigenanalysis estimation such as characterization, diagnosis, classification and detec- for non-Gaussian multivariate scaling processes. To this end, tion. It is now well documented that the wavelet transform we define a new class of multivariate non-Gaussian fractional stochastic processes called operator fractional Levy´ motion Work supported by Grant ANR-16-CE33-0020 MultiFracs. G.D. was (ofLm) (Section 2). This class encompasses and generalizes partially supported by the ARO grant W911NF-14-1-0475. G.D.’s long term visits to France were supported by ENS de Lyon, CNRS and the Carol Lavin the Gaussian class of ofBm, while conveniently preserving Bernick faculty grant. the property of covariance (operator) self-similarity. After in- troducing wavelet eigenanalysis estimation (Section 3), per- 3. WAVELET EIGENVALUE-BASED ESTIMATION formance is then assessed with respect to non-Gaussianity by means of Monte Carlo experiments (Section 4). To illustrate In the statistical analysis of self-similar signals, the central the versatility of the method and the effect of non-Gaussian task is to estimate the Hurst exponents H = (H1,H2) from marginal distributions, we allow the unobserved signal to con- a single time series Y . When P is diagonal, (2) suggests tain both non-Gaussian and Gaussian entries. that H1 and H2 can be estimated independently using standard univariate methodologies [28, 29]. However, in the gen- eral framework of nondiagonal mixing (coordinates) matrices ´ 2. OPERATOR FRACTIONAL LEVY MOTION P , univariate estimation does not yield relevant results. For Operator self-similarity. Let H = P diag(H)P −1 be the this reason, we put forward a multivariate wavelet transform- so-named Hurst matrix parameter, where the vector H con- based joint estimation methodology that has been success- tains the Hurst exponents (eigenvalues). For a given process fully applied to the Gaussian case (cf. [10, 11]) . Y = (Y1,...,Yn) with Hurst matrix parameter H, multi- Multivariate wavelet transform. Let ψ0 be a mother variate covariance self-similarity, a key property of ofBm and wavelet, i.e., a reference oscillating pattern (function) with ofLm, reads as good joint time and frequency localization. A mother wavelet ∗ H ∗ H∗ is parametrized by the so-named number of vanishing mo- EY (s)Y (t) = a EY (s/a)Y (t/a) a , (1) ments Nψ, i.e., a positive integer such that ∀n = 0,...,Nψ − H +∞ k k P R n R Nψ for all a > 0, where a := k=0 log (a)H /k!. When the 1, t ψ0(t)dt ≡ 0 and t ψ0(t)dt 6= 0. Let {ψj,k(t) = R R mixing matrix P is diagonal, namely, when we can set P ≡ −j/2 −j 2 ψ0(2 t − k)}(j,k)∈ 2 be the collection of dilated and I, the covariance self-similarity relation (1) takes the simple N translated templates of ψ0 that forms an orthonormal basis of form of component-wise covariance self-similarity relations L2(R). H`+H`0 The multivariate discrete wavelet transform (DWT) of EY`(s)Y`0 (t) = a EY`(s/a)Y`0 (t/a), (2) 0 the multivariate stochastic process {Y (t)}t∈R is defined as for a > 0 and `, ` = 1, 2, . . . , n. j j j j (D(2 , k)) ≡ DY (2 , k) = (DY1 (2 , k),DY2 (2 , k)), ∀k ∈ ∀j ∈ {j , . . . , j } ∀m ∈ {1, 2} D (2j, k) = ´ Z, 1 2 , and : Ym 2.1. Operator fractional Levy motion −j/2 −j h2 ψ0(2 t − k)|Ym(t)i. OfLm is a new and versatile class of non-Gaussian multi- H ,H S(2j) variate fractional processes displaying the same covariance Joint estimation of 1 2. Let denote the empir- structure of ofBm. For clarity of exposition and without ical wavelet spectrum, defined as loss of generality, we construct ofLm in the bivariate case. nj j 1 X j j ∗ N S(2 ) = D(2 , k)D(2 , k) , nj = , (4) Let {L(t) = (L1(t),L2(t))}t∈R be a two-sided symmetric j 2 ∗ nj 2 Levy´ process in R with EL(1)L(1) =: ΣL, |ΣL| < ∞ k=1 0 < H ≤ H < 1 j j j (e.g., [27]). Let 1 2 , and define the where N is the sample size. Let Λ(2 ) = {λ1(2 ), λ2(2 )} (pre-mixed) process X by the component-wise convolution be the eigenvalues of the 2 × 2 matrix S(2j). We de- ˙ X(t) = (gt ∗ L)(t), where g is the diagonal-matrix-valued fine the wavelet eigenvalue regression estimators (H , H ) of D D 1 b1 b2 fractional kernel gt(u) := u −(u−t) , D = diag(H)− I. + + 2 (H1,H2) by means of weighted log-regressions across scales Then, the entrywise processes X , ` = 1, 2, are gener- j j H` 2 1 ≤ a ≤ 2 2 ally correlated fractional Levy´ processes with corresponding   Hurst parameters H ∈ (0, 1) (e.g., [14, 16]). In particular, j2 ` X j 1 Hbm = wj log λm(2 ) 2 − , ∀m = 1, 2. (5) each has stationary increments and its covariance function is  2  2 identical to that of fBm, i.e., j=j1

2H` 2H` 2H` 2 For purely Gaussian instances (ofBm), it was shown theoreti- EXH` (t)XH` (s) = {|t| + |s| − |t − s| }σ` /2. (3) cally in [10, 11] that (Hb1, Hb2) are consistent estimators with Note that, if non-Gaussian, the stochastic behavior of the pro- asymptotic joint normality under mild assumptions. Also, cess X is not characterized by its covariance function. Let H` these estimators have very satisfactory performance for finite P be a 2 × 2, real-valued, invertible matrix. We define (bi- sample sizes. Their covariances decrease as a function of the H,L,P variate) ofLm Y by inverse of the sample size and are approximately normal even H,L,P H,L,P for small sample sizes. It was further observed that the vari- {Y1 (t),Y2 (t)}t∈R = P {XH1 (t),XH2 (t)}t∈R ances of (Hb1, Hb2) do not significantly depend on the actual (in short, Y H,L,P = PX). Bivariate ofLm is well- values of (H1,H2). defined only if Γ(2H1 + 1)Γ(2H2 + 1) sin(πH1) sin(πH2) 2 2 2 − ρ Γ(H1 + H2 + 1) sin (π(H1 + H2)/2) > 0, where 4. ESTIMATION PERFORMANCE ASSESSMENT ρ := Corr(XH1 (1),XH2 (1)), thus showing that H and ρ can- not be selected independently (cf. [8]). The process Y H,L,P With a view towards contrasting the tail behavior of ofLm to exhibits the operator covariance self-similarity property (1). its purely Gaussian counterpart, ofBm, we conducted Monte univariate analysis eigenanalysis (multivariate) 30 30 Carlo studies for the case where the non-Gaussian part of L1 2 2 Y1 λ1 225 Y 225 λ is symmetric tempered stable, i.e., the marginals of XH1 dis- 2 2 play distinctly non-Gaussian tail behavior but have finite mo- 220 220 ments of all orders [30, 31]. Such processes are parameter- 215 215 ized by a stability index α ∈ (0, 2) and a tempering param- 210 210 eter γ > 0, and can be constructed by exponentially tem- 25 25 pering the density p(x, t) at time t of an α-stable process: 20 20

−γ|x| 2-5 2-5 pγ (x, t) ∝ e p(x, t) [31]; increasingly small tempering 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 parameter γ corresponds to increasingly heavier-tailed behav- octave octave ior. For comparison, we take L2 to be Gaussian (i.e., XH2 is Fig. 1. Monte Carlo wavelet log-scaling plots. Aver- an fBm), so the resulting bivariate ofLm Y H,L,P contains a age curves for component-wise (univariate) wavelet scal- combination (sum) of both Gaussian and non-Gaussian com- ing plots (left) and wavelet eigenvalue scaling plots j j ponents. log λ1(2 ), log λ2(2 ) (right) over R = 1000 runs for the ofLm with γ = 10−5. The error bars represent approxi- 4.1. Monte-Carlo simulation mate 95% confidence intervals; the green dashed reference lines are the theoretical slopes associated with H1 and H2. In all instances, we generated sets of independent ofLm paths The multivariate analysis manages to identify the presence of 15 at the sample size n = 2 , based on a combination of frac- a smaller Hurst exponent H1 < H2 as well as that of dis- tional Levy´ process [16] and fBm with corresponding Hurst tinctly non-Gaussian/heavier-tailed behavior (larger standard exponents H1 = 0.35 and H2 = 0.75, respectively. The error around the λ1 log-scaling curve). Both phenomena are tempered stable part of L1 was chosen with α = 1 and missed by univariate-like analysis. different γ for each individual study (described below); the Gaussian part of L1 was chosen as the same as L2. Con- is overborne by the dominant exponent H2 > H1. It detects sequently, ΣL is non-diagonal and XH1 and XH2 are cor- R neither the presence of the smaller Hurst exponent H1 nor of related, i.e., EXH1 (t)XH2 (s) =c gH1,t(x)gH2,s(x)dx, R non-Gaussian tail behavior. By contrast, multivariate analysis where g (x) := (u − x)H`−1/2 − (−x)H`−1/2. The H`,u + + reveals the presence of both exponents H and H as well as fBm was simulated via circulant matrix embedding (e.g., 1 2 that of non-Gaussian tail behavior. In all results from multi- [32]). The symmetric tempered stable marginals were sim- variate analysis, the effect of the presence of non-Gaussianity ulated via an accept-reject procedure for the tempered stable turned out more strongly in the scaling behavior of the smaller random variables and an adaptation of the FFT-based algo- eigenvalue λ (2j). In other words, not only is λ (2j) driven rithm described in [33] for the tempered case. In the nota- 1 1 by the smaller Hurst exponent H , as typical in wavelet eige- tion of [33], the mesh size was chosen m = 8 with kernel 1 nanalysis, but also the non-Gaussianity of the pre-mixed com- cutoff parameter M = 215. The mixing matrix was set to T T ponent XH1 , associated with H1, was mostly absorbed by P = ((1, 0) ; (1, 1) ). For this choice, it can be shown that j H,L,P λ1(2 ). Accordingly, the scaling of the larger eigenvalue Y1 is a sum of Gaussian and non-Gaussian components, λ (2j) is, over large scales, driven by H and similar to H,L,P 2 2 and that Y2 is Gaussian. that for the wavelet eigenanalysis of (the purely Gaussian) Computational studies were carried out as follows. In ofBm. By construction, the wavelet (second order) covariance all instances, Daubechies wavelets with Nψ = 2 vanishing structure of ofLm is identical to that of an ofBm. However, moments were used. In Figure 1, R = 1000 independent the confidence intervals associated with the scaling curves of j ofLm paths were generated, and both component-wise (uni- λ1(2 ) behave differently compared to the purely Gaussian variate) wavelet regression and wavelet eigenvalue regression case, since such intervals involve fourth order properties of were conducted over the octave range (j1, j2) = (4, log2 n − the underlying process. In particular, whereas wavelet scal- Nψ − 2) = (4, 11). In Figure 2, the wavelet eigenvalue re- ing curves display increased variability for larger octaves in gression was carried out over (j1, j2) = (4, 11) for seven the purely Gaussian case, it is a striking feature of ofLm that −7 −6 −1 j sets of R = 1000 paths with γ ∈ {10 , 10 ,..., 10 }, confidence intervals associated with λ1(2 ) are shown in our and the estimates from these paths were used in Figure 3 for studies to be approximately constant as a function of j for γ ∈ {10−6,..., 10−3}. In Figure 4, two sets of R = 1000 moderate j. Other choices of P yielded qualitatively simi- ofLm paths were generated with γ ∈ {10−5, 10−2} and were lar results; for most choices of P , the large-variance behav- analyzed using the octave range (j1, j2) = (3 + k, 7 + k) for ior associated with non-Gaussianity generally displays in both k = 0, 1,..., 5, and another set of R = 1000 ofBm paths eigenvalues. was generated and analyzed over the same octave ranges for In more detail, Figure 2 shows that the wavelet eigenvalue comparison. regression bias remains approximately fixed as a function of Performance with respect to non-Gaussianity. Figure 1 γ, and is comparable to that observed for ofBm instances. shows that univariate-like (component-wise) wavelet analysis However, the estimation performance degrades for small γ bias std err √MSE 0.03 0.08 0.1

0.02 0.07 j 0.08 the associated wavelet matrices S(2 ) (cf. Figure 1), as with 0.01 0.06

0 ofBm. In addition, when γ is large (lighter tails), the per- 0.05 0.06 -0.01 0.04 formance of wavelet eigenvalue regression estimation is close 0.04 -0.02 0.03 λ1 to that for an ofBm, resulting in the same choice of optimal -0.03 0.02 λ2 0.02 λ , ofBm -0.04 0.01 1 (j1, j2). When γ is small (heavier tails), the performance de- λ2, ofBm -0.05 0 0 10-7 10-6 10-5 10-4 10-3 10-2 10-1 10-7 10-6 10-5 10-4 10-3 10-2 10-1 10-7 10-6 10-5 10-4 10-3 10-2 10-1 grades in standard error. However, the resulting standard error γ γ γ curve associated with Hb1 is approximately a translation of the curve corresponding to the ofBm case. This leads to reduced Fig. 2. Estimation performance√ over different tail behav- √ ior. Bias, standard error, and MSE as the tempering pa- MSE performance, but also to the same choice of optimal rameter γ increases (i.e., increasingly lighter tails). Compared (j1, j2) as compared to ofBm.

bias, shifting range std err, shifting range √MSE, shifting range to ofBm (dashed reference lines), the presence of heavier- 0.2 0.2 0.06 −5 λ1, γ = 10 0.18 −2 tailed noise has little effect on the bias. However, for standard 0.04 λ1, γ = 10 λ , ofBm 0.15 0.16 1 0.02 −5 error, performance degrades signiﬁcantly for smaller γ. λ2, γ = 10 −2 0 0.14 λ2, γ = 10 λ2, ofBm 0.5 0.5 0.5 0.5 -0.02 0.1 0.12

-0.04 0.1 c −6 c −5 c −4 c −3 0.4 H1, γ = 10 0.4 H1, γ = 10 0.4 H1, γ = 10 0.4 H1, γ = 10 -0.06 0.05 0.08

0.3 0.3 0.3 0.3 -0.08 0.06

-0.1 0 0.04 0.2 0.2 0.2 0.2 (3,7) (4,8) (5,9) (6,10) (7,11) (8,12) (3,7) (4,8) (5,9) (6,10) (7,11) (8,12) (3,7) (4,8) (5,9) (6,10) (7,11) (8,12) (j1, j2) (j1, j2) (j1, j2)

0.1 0.1 0.1 0.1 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 −5 0.9 0.9 0.9 0.9 Fig. 4. Choice of regression range over small (10√ ) and 0.85 c −6 0.85 c −5 0.85 c −4 0.85 c −3 −2 H2, γ = 10 H2, γ = 10 H2, γ = 10 H2, γ = 10 large (10 ) values of γ. Bias, standard error, and MSE 0.8 0.8 0.8 0.8 as the regression range [j1, j2] shifts from [3, 7] to [8, 12]. 0.75 0.75 0.75 0.75

0.7 0.7 0.7 0.7 For comparison, the dashed lines represent results for purely

0.65 0.65 0.65 0.65 Gaussian (ofBm) instances. When γ is small (heavier tails), -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 j the increased variability in the wavelet eigenvalues λ1(2 ) Fig. 3. Non-Gaussianity of estimates. Q-Q plots for Hb`, gives rise to increased variability in the estimates H ; the op- −6 −3 b1 ` = 1, 2 for γ ∈ {10 ,..., 10 }. The distribution of timal choice of scales is the same for both heavy and lighter- Hb2 is approximately Gaussian for all γ. In contrast, as the tailed cases. Levy´ noise tail behavior departs further from Gaussianity, Hb1 appears increasingly non-Gaussian, i.e., the asymptotically 5. CONCLUSION Gaussian behavior of the estimator Hb1 requires larger sam- In this work, we propose a new model for non-Gaussian scal- ple sizes to manifest itself. ing dynamics called operator fractional Levy´ motion (ofLm). Given the distributions of the driving noise components, due to the presence of heavier tails. Interestingly, among the ofLm is parametrized by a vector of Hurst (scaling exponent) subclass of ofLms with tempered stable driving Levy´ noise, parameters and a coordinates (mixing) matrix P . We further all studies conducted suggest that the estimation performance construct a wavelet eigenanalysis-based method for the iden- degrades only in standard error as a function of the tail be- tification of ofLm. For bivariate instances based on a com- havior of the driving process. This indicates that the wavelet bination of Gaussian and non-Gaussian marginals, large size eigenvalue regression method is bias-robust with respect to Monte Carlo studies demonstrate that the estimation of Hurst non-Gaussianity. parameters is satisfactory. We further characterize the effect 15 In Figure 3, at the sample size n = 2 , asymp- of non-Gaussian marginals on wavelet-based estimation as totic Gaussianity is achieved for Hb2 over all instances γ ∈ a function of the strength of the distribution tail tempering −6 −3 {10 ,..., 10 }. However, in the presence of heavier- parameter. In particular, unlike univariate-type analysis, we tailed behavior, Hb1 departs further from Gaussianity due to show that the proposed multivariate analysis method is able j the increased variability of the eigenvalue curve λ1(2 ), indi- to identify the presence of multiple Hurst exponents as well cating that larger sample sizes are required to achieve Gaus- as that of non-Gaussian noise. sian behavior for Hb1 when γ is small. Future work includes mathematically establishing the Performance with respect to scale choices. 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