Assessing cross-dependencies using bivariate multifractal analysis Herwig WENDT 1 , Roberto LEONARDUZZI 2 , Patrice ABRY 2 , St´ ephane ROUX 2 St´ ephane JAFFARD 3 , St´ ephane SEURET 3 1 Univ de Toulouse, IRIT-ENSEEIHT, CNRS, Toulouse, France, [email protected]2 Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Lab de Physique, F-69342 Lyon, France [email protected]3 Univ Paris Est, LAMA, UPEC, CNRS, F-94010, Cr´ eteil, France, [email protected]Abstract — Multifractal analysis has become a reference tool to char- acterize scale-free temporal dynamics in time series. It proved successful in numerous applications very diverse in nature. However, such successes remained restricted to univariate analysis, while many recent applications call for the joint analysis of several components. Surprisingly, multivariate multifractal analysis remained mostly overlooked. The present contribu- tion aims at defining a wavelet-leader-based framework for multivariate multifractal analysis and at studying its properties and estimation per- formance. To better understand what properties of multivariate data are actually captured, a multivariate multifractal model is used as representa- tive paradigm and permits to show that multivariate multifractal analysis puts in evidence transient and local dependencies that are not well quan- tified or even evidenced by the classical Pearson correlation coefficient. Multifractal analysis Multifractal Spectrum – Local Regularity: – Get regularity exponent from function X (t) – Compare X with local polynomial approximation P t – Most common: H¨older exponent h(t) ≥ 0 |X (t + a) - P t (a)| a→0 + ∼ a h(t) – Multifractal Spectrum: – Bivariate signal: X =(X 1 ,X 2 ) – H¨olderexponents: (h 1 (t),h 2 (t)) – Multifractal spectrum: D(h 1 ,h 2 )= dim Hausdorff {t : h 1 (t)= h 1 and h 2 (t)= h 2 } -→ “Quantity”of points with given regularity h 2 D(h 1 ;h 2 ) h 1 – Problem: Can not be computed in practice -→ Use multifractal formalism to estimate Multiresolution quantities – X (t) -→ T X (j, k ) (scale 2 -j , position k = t 2 j ) – Choice 1 : Discrete wavelet transform T X (j, k )= d X (j, k )= d X (λ) -→ Poor performance – Dyadic intervals: λ = λ j,k = (2 -j (k - 1), 2 -j k ] and 3λ j,k = S 1 i=-1 λ j,k +i 3λ k j – Choice 2 : Wavelet leaders (Wendt, Abry & Jaffard, 2007) T X (j, k )= ‘ X (j, k ) = sup λ 0 ∈3λ |c λ 0 | -→ Much better performance Multifractal Formalism: Definition – Goal: provide estimate of D(h 1 ,h 2 ) from wavelet leaders – Easily computable in practice. Steps: 1. Structure functions S : S (q 1 ,q 2 ,j )= 1 n j n j X k =1 L X 1 (j, k ) q 1 L X 2 (j, k ) q 2 2. Scaling function ζ : S (q 1 ,q 2 ,j ) ∼ 2 -jζ (q 1 ,q 2 ) , j →∞ 3. Legendre spectrum L: L(h 1 ,h 2 ) = inf q 1 ,q 2 ( 1+ q 1 h 1 + q 2 h 2 - ζ (q 1 ,q 2 ) ) ≥D(q 1 ,q 2 ). – Upper bound L used as estimate for D Multifractal Formalism: Cumulants – Bivariate cumulants of (ln L X 1 (j, k ), ln L X 2 (j, k )) – C p 1 p 2 (j )= E ln ( L X 1 (j, ·) ) p 1 ln ( L X 2 (j, ·) ) p 2 – Order p 1 + p 2 ≥ 1 – Scale dependence: C p 1 p 2 (j )= c 0 p 1 p 2 + c p 1 p 2 ln 2 j – Polynomial expansion: L(h 1 ,h 2 ) ≈ 1+ c 02 b 2 h 1 - c 10 b 2 + c 20 b 2 h 2 - c 01 b 2 - c 11 b h 1 - c 10 b h 2 - c 01 b where b = c 20 c 02 - c 2 11 ≥ 0 – Information synthesized in second order: 5 parameters – Interpretation: – c 01 ,c 10 : average regularity on each component – c 02 ,c 20 : width of regularity fluctuations on each component – c 11 : leading-order joint regularity fluctuation Practical Estimation – ζ (q 1 ,q 2 ) -→ linear regressions log 2 S (q 1 ,q 2 ,j ) vs log 2 2 j – c p 1 p 2 -→ linear regressions C p 1 p 2 (j ) vs ln 2 j Synthetic Process Bivariate Multifractal Random Walk Synthetic process with bivariate multifractal behavior – Definition – Use two pairs of stochastic processes – Pair 1: bivariate fractional Gaussian noise G 1 (t),G 2 (t) * Self-similarity parameters: H 1 ,H 2 * Covariance matrix: Σ ss = 1 ρ ss ρ ss 1 – Pair 2: Gaussian processes ω 1 (t),ω 2 (t) * Multifractality parameters: λ 1 ,λ 2 * Covariance function: Σ mf such that {Σ mf } ij (k,l )= ρ mf (i, j )λ i λ j log N |k - l | +1 ,i =1, 2 where ρ mf = 1 ρ mf ρ mf 1 -→ Logarithmic covariance to induce multifractality – G i (t),ω i (t) synthesized following (Helgason, Pipiras & Abry, 2011) – Final process: X i (t)= t X k =1 G i (k )e ω i (k ) , i =1, 2 – Properties – Correlation coefficient of final process: ρ bMRW = ρ ss · f (ρ mf ,λ 1 ,λ 2 ) -→ Can have ρ bMRW =0 with ρ mf 6=0 ! – Cumulants: * c 10 = H 1 + λ 2 1 /2 and c 01 = H 2 + λ 2 2 /2 * c 20 = -λ 2 1 and c 02 = -λ 2 2 * c 11 = -ρ mf λ 1 λ 2 Results Estimation performance: univariate -1 -0.5 0 0.5 1 0.6 0.605 0.61 0.615 0.62 0.625 ˆ c 10 ρ mf -1 -0.5 0 0.5 1 -0.025 -0.02 -0.015 -0.01 ˆ c 20 ρ mf – Performance independent of correlation – Performance independent of multifractal dependence Estimation performance: bivariate -1 -0.5 0 0.5 1 -0.04 -0.02 0 0.02 0.04 ˆ c 11 ρ ss =0.5 ρ mf -1 -0.5 0 0.5 1 -1.5 -1 -0.5 0 0.5 1 1.5 ˆ ρ mf ρ ss =0.5 ρ mf -1 -0.5 0 0.5 1 0 0.002 0.004 0.006 0.008 0.01 rms ˆ c 11 ρ mf -1 -0.5 0 0.5 1 0 0.05 0.1 0.15 0.2 0.25 0.3 rms ˆ ρ mf ρ mf ρ ss =0 ρ ss =0.5 ρ ss =0.9 – Excellent estimation performance – Estimation performance largely independent of ρ ss and ρ mf – Relevant and robust estimates for bivariate parameters c 11 and ρ mf Higher-order dependence – Set ρ ss =0 = ⇒ ρ bMRW =0 -1 -0.5 0 0.5 1 × 10 -3 -5 0 5 ˆ ρ bMRW ρ ss =0 ρ mf -1 -0.5 0 0.5 1 -1.5 -1 -0.5 0 0.5 1 1.5 ˆ ρ mf ρ ss =0 ρ mf – Measured Pearson correlation indeed null (left) – Estimated ˆ ρ mf 6=0 = ⇒ dependence beyond correlation Multifractal spectra – Orientation and eccentricity of support: higher-order dependence – ρ ss =0,ρ mf =0 = ⇒ : uncorrelated, independent – ρ ss =0,ρ mf > 0 = ⇒ : uncorrelated, positive dependence – ρ ss =0,ρ mf < 0 = ⇒ : uncorrelated, negative dependence References –S. Jaffard, S. Seuret, H. Wendt, R. Leonarduzzi, S. Roux and P. Abry, “Multivariate multifractal analysis,” Applied and Computational Harmonic Analysis, 2018, in press. https://doi.org/10.1016/j.acha.2018.01.004 –H. Wendt, P. Abry, and S. Jaffard, “Bootstrap for empirical multifractal analysis,” IEEE Signal Processing Magazine, vol. 24, no. 4, pp. 38–48, 2007 –H. Helgason, V. Pipiras, and P. Abry, “Fast and exact synthesis of stationary mul- tivariate Gaussian time series using circulant embedding,” Signal Processiong, vol. 91, no. 5, pp. 1123–1133, 2011. ICASSP 2018 – Calgary – Canada
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EXTENDING MULTIFRACTAL ANALYSIS TO NEGATIVE REGULARITY:P-EXPONENTS AND P-LEADERS
Roberto Leonarduzzi^, Herwig Wendt_, Stephane Ja↵ard`, Stephane G. Rouxa, Marıa E. Torres^, Patrice Abrya
Summary. Multifractal analysis characterizes data by describing globally and geometrically the fluctua-tions of local regularity, commonly measured by means of the Holder exponent. A major limitation is that itapplies only to locally bounded functions or signals, i.e., to signals with positive regularity. In this work, wepropose to characterize local regularity with a new quantity, the p-exponent, that permits negative regularity,a widely observed property in real-world data. Relations to Holder exponents are detailed and a p-leadermultifractal formalism for p-exponents is devised and shown at work on synthetic multifractal processes.Even when Holder and p-exponents coincide, the p-leader formalism yields better estimation performance.
– R.F. Leonarduzzi, P. Abry, S. Ja↵ard, S.G. Roux, M.E. Torres, H. Wendt, p-exponentand p-Leaders for Multifractal Analysis, 2014, in preparation.
– P. Abry, S. Ja↵ard, and H. Wendt. A bridge between geometric measure theory andsignal processing: Multifractal analysis, in The Abel Symposium 2012, SpringerSeries Abel Symposia, Vol. 9, 2014, to appear.
– S. Ja↵ard, P. Abry, and H. Wendt, Irregularities and Scaling in Signal and Im-age Processing: Multifractal Analysis, in Benoit Mandelbrot: A Life in ManyDimensions, M. Frame, ed., World scientific publishing, 2014, to appear.
Assessing cross-dependencies using bivariatemultifractal analysis
1 Univ de Toulouse, IRIT-ENSEEIHT, CNRS, Toulouse, France, [email protected] Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Lab de Physique, F-69342 Lyon, France
Summary. Multifractal analysis characterizes data by describing globally and geometrically the fluctua-tions of local regularity, commonly measured by means of the Holder exponent. A major limitation is that itapplies only to locally bounded functions or signals, i.e., to signals with positive regularity. In this work, wepropose to characterize local regularity with a new quantity, the p-exponent, that permits negative regularity,a widely observed property in real-world data. Relations to Holder exponents are detailed and a p-leadermultifractal formalism for p-exponents is devised and shown at work on synthetic multifractal processes.Even when Holder and p-exponents coincide, the p-leader formalism yields better estimation performance.
– R.F. Leonarduzzi, P. Abry, S. Ja↵ard, S.G. Roux, M.E. Torres, H. Wendt, p-exponentand p-Leaders for Multifractal Analysis, 2014, in preparation.
– P. Abry, S. Ja↵ard, and H. Wendt. A bridge between geometric measure theory andsignal processing: Multifractal analysis, in The Abel Symposium 2012, SpringerSeries Abel Symposia, Vol. 9, 2014, to appear.
– S. Ja↵ard, P. Abry, and H. Wendt, Irregularities and Scaling in Signal and Im-age Processing: Multifractal Analysis, in Benoit Mandelbrot: A Life in ManyDimensions, M. Frame, ed., World scientific publishing, 2014, to appear.
Abstract — Multifractal analysis has become a reference tool to char-acterize scale-free temporal dynamics in time series. It proved successfulin numerous applications very diverse in nature. However, such successesremained restricted to univariate analysis, while many recent applicationscall for the joint analysis of several components. Surprisingly, multivariatemultifractal analysis remained mostly overlooked. The present contribu-tion aims at defining a wavelet-leader-based framework for multivariatemultifractal analysis and at studying its properties and estimation per-formance. To better understand what properties of multivariate data areactually captured, a multivariate multifractal model is used as representa-tive paradigm and permits to show that multivariate multifractal analysisputs in evidence transient and local dependencies that are not well quan-tified or even evidenced by the classical Pearson correlation coefficient.
Multifractal analysis
Multifractal Spectrum– Local Regularity:
– Get regularity exponent from function X(t)
– Compare X with local polynomial approximation Pt– Most common: Holder exponent h(t) ≥ 0
References– S. Jaffard, S. Seuret, H. Wendt, R. Leonarduzzi, S. Roux and P. Abry, “Multivariate
multifractal analysis,” Applied and Computational Harmonic Analysis, 2018, in press.https://doi.org/10.1016/j.acha.2018.01.004
– H. Wendt, P. Abry, and S. Jaffard, “Bootstrap for empirical multifractal analysis,”IEEE Signal Processing Magazine, vol. 24, no. 4, pp. 38–48, 2007
– H. Helgason, V. Pipiras, and P. Abry, “Fast and exact synthesis of stationary mul-tivariate Gaussian time series using circulant embedding,” Signal Processiong, vol.91, no. 5, pp. 1123–1133, 2011.
