NAVIGATING SCALING: MODELLING AND ANALYSING

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1 NAVIGATING SCALING: MODELLING AND ANALYSING P. Abry (), P. Gonçalvès (2) () SISYPH, CNRS, (2) INRIA Rhône-Alpes, Ecole Normale Supérieure On Leave at IST-ISR, Lisbon Lyon, France IN COLLABORATIONS WITH : P. Flandrin, D. Veitch, P. Chainais, B. Lashermes, N. Hohn, S. Roux, P. Borgnat, M.Taqqu, V. Pipiras, R. Riedi, S. Jaffard Wavelet And Multifractal Analysis, Cargèse, France, July 24.

2 Report Documentation Page Form Approved OMB No Public reporting burden for the collection of information is estimated to average hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 25 Jefferson Davis Highway, Suite 24, Arlington VA Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.. REPORT DATE 7 JAN REPORT TYPE N/A 3. DATES COVERED - 4. TITLE AND SUBTITLE Navigating Scaling: Modelling And Analysing 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) SISYPH, CNRS, Ecole Normale Sup erieure Lyon, France 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES). SPONSOR/MONITOR S ACRONYM(S) 2. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release, distribution unlimited. SPONSOR/MONITOR S REPORT NUMBER(S) 3. SUPPLEMENTARY NOTES See also ADM75, Wavelets and Multifractal Analysis (WAMA) Workshop held on 9-3 July 24., The original document contains color images. 4. ABSTRACT 5. SUBJECT TERMS 6. SECURITY CLASSIFICATION OF: 7. LIMITATION OF ABSTRACT UU a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified 8. NUMBER OF PAGES 64 9a. NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-8

3 SCALING PHENOMENA? DETECTION: SCALING? WHAT DOES IT MEAN? NON STATIONARITY? IDENTIFICATION: RELEVANT STOCHASTIC MODELS? ESTIMATION: RELEVANT PARAMETER ESTIMATION? SIDE ISSUES: ROBUSTNESS? COMPUTATIONAL COST? REAL TIME? ON LINE? []

4 OUTLINE I. INTUITIONS, MODELS, TOOLS I. INTUITIONS, DEFINITION,APPLICATIONS I.2 STOCHASTIC MODELS: SELF-SIMILARITY VS MULTIFRACTAL I.3 MULTIRESOLUTION TOOLS, AGGREGATION, INCREMENTS I.4 WAVELETS, CONTINUOUS, DISCRETE II. SECOND ORDER ANALYSIS, SELF SIMILARITY AND LONG MEMORY II. RANDOM WAKS, SELF SIMILARITY, LONG MEMORY, II.2 2ND ORDER WAVELET STATISTICAL ANALYSIS, II.3 ESTIMATION, ESTIMATION PERFORMANCE, II.4 ROBUSTNESS AGAINST NON STATIONARITIES, III. HIGHER ORDER ANALYSIS, MULTIFRACTAL PROCESSES III. MULTIPLICATIVE CASCADES, MULTIFRACTAL PROCESSES, III.2 HIGHER ORDER WAVELET STATISTICAL ANALYSIS, III.3 FINITENESS OF MOMENTS, III.4 ESTIMATION, ESTIMATION PERFORMANCE, III.5 NEGATIVE ORDERS, III.6 BEYOND POWER LAWS. [2]

5 IRREGULARITIES, VARIABILITIES SCALING OR NON STATIONARITIES? [3]

6 SCALING? Trafic (WAN) Internet Trafic (WAN) Internet nb connexions nb connexions temps (s) temps (s) [4]

7 SCALING? Temps (s) [5]

8 SCALING? 8.5 Trafic (LAN) Ethernet Densite Spectrale de Puissance Log (DSP) Nombre Octets Log (Frequence (Hz)) [6]

SCALING! DEFINITION : NON PROPERTY: NO CHARACTERISTIC SCALE. NON GAUSSIAN, NON STATIONARY, NON LINEAR EVIDENCE: The whole resembles to its part, the part resembles to the whole.

9 SCALING! DEFINITION : NON PROPERTY: NO CHARACTERISTIC SCALE. NON GAUSSIAN, NON STATIONARY, NON LINEAR EVIDENCE: The whole resembles to its part, the part resembles to the whole. Trafic (WAN) Internet 4 2 nb connexions temps (s) nb connexions temps (s) Temps (s) ANALYSIS: Rather than for a characteristic scale, look for a relation, a mecanism, a cascade between scales. [7]

10 SCALING : OPERATIONAL DEFINITIONS MULTIRESOLUTION QUANTITY: T X (a, t) (e.g., Wavelet Coef.). POWER LAWS: IE T X (a, t) q = c q a ζ(q), n n k= T X(a, t k ) q = c q a ζ(q), - FOR A RANGE OF SCALES a, - FOR A RANGE OF ORDERS q - SCALING EXPONENTS ζ(q). BEYOND POWER LAWS : WARPED INF. DIV. CASCADES IE T X (a, t) q = C q a ζ(q) = C q exp(ζ(q) ln a) IE T X (a, t) q = = C q exp(ζ(q)n(a)) VISIT PIERRE CHAINAIS S POSTER [8]

11 UBIQUITY? 5 Trafic (LAN) Ethernet 65 Trafic (WAN) Internet 6 Nombre Octets nb connexions temps(s) Turbulence de Jet, R λ temps (s) Turbulence de Jet, R λ Dissipation.5. Vitesse (m/s) temps(s) temps(s) [9]

12 UBIQUITY! - Hydrodynamic Turbulence, - Physiology, Biological Rythms (Heart beat, walk), - Geophysics (Faults Repartition, Earthquakes), - Hydrology (Water Levels), - Statistical Physics (Long Range Interactions), - Thermal Noises (semi-conductors), - Information Flux on Networks, Computer Network Traffic, - Population Repartition (local: cities, global: continent), - Financial Markets (Daily returns, Volatily, Currencies Exchange Rates), -... []

13 ANALYSING TOOL : AGGREGATION COMPARE DATA AGAINST A BOX, THEN VARY a T X (a, t) = at t+at t AVERAGE X(u)du.5 /T -.5 T - 2 temps WORKS ONLY FOR POSITIVE TIME SERIES, DENSITY []

14 ANALYSING TOOL 2 : INCREMENTS COMPARE DATA AGAINST A DIFFERENCE OF DELTA FUNCTIONS, THEN VARY a T X (a, t) = X(t + aτ ) X(t) DIFFERENCE τ - 2 temps INCREMENTS OF HIGHER ORDERS OR GENERALISED N -VARIATIONS Order 2 : T X (a, t) = X(t + 2aτ ) + 2X(t + aτ ) X(t), Order N : T X (a, t) = N p= ( )p a p X(t + paτ ), where N p= ( )p a p p k, k =,..., N. [2]

15 ANALYSING TOOL: MULTIRESOLUTION ANALYSIS MULTIRESOLUTION QUANTITIES: X(t) T X (a, t) = f a,t X, f a,t (u) = a f ( u t a ) AGGREGATION INCREMENTS? f (u) = (β ) f (u) = (I )? = t+at at X(u)du = X(t + aτ t ) X(t)? BOX, AVERAGE DIFFERENCE? /T T -.5 τ - 2 temps - 2 temps? [3]

16 ANALYSING TOOL: MULTIRESOLUTION ANALYSIS MULTIRESOLUTION QUANTITIES: X(t) T X (a, t) = f a,t X, CHOICES FOR MOTHER FUNCTIONS: f, f a,t (u) = a f ( u t a ) AGGREGATION INCREMENTS WAVELETS f (u) = (β ) N f (u) = (I ) N f (u) = ψ,n = t+at at X(u)du = X(t + aτ t ) X(t) = X(u) a ψ ( u t a ), BOX, AVERAGE DIFFERENCE AVERAGE, DIFFERENCE /T T -.5 τ temps temps Temps N N N [4]

17 WAVELETS AND SCALING: KEY INGREDIENTS DILATION OPERATOR, a ψ ( t a ).5 a =.5 a = 2.5 a = Temps Temps NUMBER OF VANISHING MOMENTS, N, t k ψ (t)dt, k =,,..., N Temps 2 Daubechies 2 2 B Spline Cubique 2 Daubechies temps temps temps [5]

18 MOTHER-WAVELET AND BASIS : WAVELET TRANSFORMS ψ (u)du =, ψ a,t (u) = a ψ ( u t a ) WAVELET COEFFICIENTS: CONTINUOUS WT T X (a, t) = X, ψ a,t SCALE TIME MODULUS MAXIMA WT AND DISCRETE WT SKELETON : MAXIMA LINES d X (j, k) = T X (a = 2 j, t = 2 j k) WTMM SCALE.5.5 log 2 (a) Time TIME [6]

19 OUTLINE I. INTUITIONS, MODELS, TOOLS I. INTUITIONS, DEFINITION,APPLICATIONS I.2 STOCHASTIC MODELS: SELF-SIMILARITY VS MULTIFRACTAL I.3 MULTIRESOLUTION TOOLS, AGGREGATION, INCREMENTS I.4 WAVELETS, CONTINUOUS, DISCRETE II. SECOND ORDER ANALYSIS, SELF SIMILARITY AND LONG MEMORY II. RANDOM WAKS, SELF SIMILARITY, LONG MEMORY, II.2 2ND ORDER WAVELET STATISTICAL ANALYSIS, II.3 ESTIMATION, ESTIMATION PERFORMANCE, II.4 ROBUSTNESS AGAINST NON STATIONARITIES, III. HIGHER ORDER ANALYSIS, MULTIFRACTAL PROCESSES III. MULTIPLICATIVE CASCADES, MULTIFRACTAL PROCESSES, III.2 HIGHER ORDER WAVELET STATISTICAL ANALYSIS, III.3 FINITENESS OF MOMENTS, III.4 ESTIMATION, ESTIMATION PERFORMANCE, III.5 NEGATIVE ORDERS, III.6 BEYOND POWER LAWS. [7]

20 MOD. TOOL : RAND. WALKS AND SELF SIMILARITY RANDOM WALK: X(t + τ) = X(t) + δ τ X(t) }{{} Steps or Increments STATISTICAL PROPERTIES OF THE STEPS: - A: Stationary, - A2: Independent, - A3: Gaussian, Ordinary Random Walk, Ordinary Brownian Motion, IEX(t) 2 = 2D t, Einstein relation, IE X(t) q = 2D t q/2, q >. ANOMALIES: IEX(t) 2 = 2D t γ, IEX(t) 2 =. SELF SIMILAR RANDOM WALKS: - B: Stationary, - B2: Self Similarity [8]

21 MODELLING TOOL : SELF-SIMILARITY DEFINITION: δ τ X(t) fdd = c H δ τ/c X(t/c), c >, DILATION FACTOR, INTERPRETATIONS: - COVARIANCE UNDER DILATION (CHANGE OF SCALE), > H > : SELF-SIMILARITY EXPONENT - THE WHOLE AND THE SUBPART (STATISTICALLY) UNDISTINGUISHABLE, - NO CHARACTERISTIC SCALE OF TIME. IMPLICATIONS: - NON STATIONARITY PROCESS WITH STATIONARY INCREMENTS - IE X(t + aτ ) X(t) q = C q a qh, - a >, c >, q /IE X(t) q <, - A SINGLE SCALING EXPONENT H. - ADDITIVE STRUCTURE, - (CORRELATED) RANDOM WALK, LONG MEMORY, LONG RANGE CORRELATIONS. [9]

22 MOD. TOOL (BIS): LONG RANGE DEPENDENCE DEFINITIONS : - LET X BE A 2ND STATIONARY PROCESS WITH, - COVARIANCE : c X (τ) = IEX(t)X(t + τ) - SPECTRUM : Γ X (ν) c X (τ) = c τ τ β, < β <, τ + Γ X (ν) = c f ν α, < α <, ν WITH α = β AND c f = 2(2π) sin(( γ)π/2)c τ. CONSEQUENCES : - P + A c X(τ)dτ = +, A >, - NO CHARACTERISTIC SCALE, R - AGGREGATION : T X (a, t) = t+at at X(u)du, t VAR T X (a, t) Ca α, a +, - INCREMENTS OF SELF.-SIM. PROC. (WITH H > /2) ARE LONG RANGE DEP. (WITH α = 2H ). [2]

23 WAVELETS AND SELF-SIMILAR PROCESSES WITH STATIONARY INCREMENTS - SUMMARY (Flandrin et al., Tewfik and Kim) P: {d X (j, k), k Z} STATIONARY Sequences for each Scale 2 j. N P2: SELF-SIMILARITY : Dilation {X(t)} = d {c H X(t/c)} {d X (, k)} = d {2 jh d X (j, k)} ( ) H - P3 : MARGINAL DIST. P j (d) = β P j ( d 2 β ), β = j. 2 j P4 : {d X (j, k)} SHORT RANGE DEPENDENT IF N > H + /2. 2 j k 2 j k +, Cov d X (j, k)d X (j, l) D 2 j k 2 j k 2(H N), N and Dilation [2]

24 WAVELETS AND LONG RANGE DEPENDENCE H =.5 (Flandrin) H =.5 H =.95 Daubechies2 Haar [22]

25 WAVELETS AND SELF-SIMILAR PROCESSES WITH STATIONARY INCREMENTS - SUMMARY P: {d X (j, k), k Z} STATIONARY Sequences for each Scale 2 j. N P2: SELF-SIMILARITY : Dilation {X(t)} = d {c H X(t/c)} {d X (, k)} = d {2 jh d X (j, k)} ( ) H - P3 : MARGINAL DIST. P j (d) = β P j ( d 2 β ), β = j. 2 j P4 : {d X (j, k)} SHORT RANGE DEPENDENT IF N > H + /2. 2 j k 2 j k +, Cov d X (j, k)d X (j, k ) D 2 j k 2 j k 2(H N), N and Dilation IDEALISATION : d X (j, k) INDEPENDENT VARIABLES. INTERPRETATIONS: X(t) = k a X(J, k)ϕ J,k (t)+ j=,...,j,k d X(j, k)ψ j,k (t). IMPLICATIONS: IE d X (j, k) q = IE d X (, k) q 2 jqh q/ie d X (, k) q <. [23]

26 WAVELETS AND LONG RANGE DEPENDENCE SPECTRAL ANALYSIS : Let X be a 2nd Order stationary process, Let Ψ be the FT of ψ with central frequency ν and bandwith ν. IE d X (j, k) 2 = Γ X (ν) Ψ(2 j ν) dν 2 j Γ X (2 j ν ) within bandwith 2 j ν. LET X BE LONG RANGE DEPENDENT : - POWER LAW: Γ X (ν) = c f ν α, < α <, ν - POWER LAW: IE d X (j, k) 2 C2 j(α ), j +, {d X (j, k)} SHORT RANGE DEPENDENT IF N > α. 2 j k 2 j k +, Cov d X (j, k)d X (j, k ) D 2 j k 2 j k α 2N, N and Dilation [24]

27 2ND ORDER WAVELET STATISTICAL ANALYSIS Abry, Gonçalvès, Flandrin PRINCIPLES: - IDEAS : P IE d X (j, k) 2 = C 2 2 j2h log 2 IE d X (j, k) 2 = j2h + β q, - PROBLEMS: ESTIMATE IE d X (j, k) 2 FROM A SINGLE FINITE LENGTH OBSERVATION? - SOLUTION : P2 et P3 STATISTICAL AVERAGES TIME AVERAGES, S 2 (j) = (/n j ) n j k= d X(j, k) 2 LOG-SCALE DIAGRAMS: log 2 S 2 (j) vs log 2 2 j = j [25]

28 2ND ORDER WAVELET-BASED STATISTICAL ANALYSIS FOR SELF -SIMILARITY 5 5 α = 2.57 j y j Octave j [26]

29 2ND ORDER WAVELET-BASED STATISTICAL ANALYSIS FOR LONG RANGE DEPENDENCE α =.55 c f = j y j Octave j [27]

30 WAVELETS AND 2ND-ORDER SCALING: ESTIMATION DYADIC GRID (DISCRETE WAVELET TRANSFORM): a j = 2 j, t j,k = k2 j, SCALE TIME STRUCTURE FUNCTION (TIME AVERAGE): Y j = ( 2 log 2 S 2 (2 j ) = 2 log 2(/n j ) n j k= d X(j, k) 2 DEFINITION : Y j versus log 2 2 j = j, Ĥ = j 2 j=j w j Y j. WHERE j jw j, j w j, WITH w j B j B, B B 2 B 2 AND p =,, 2, B p = j jp /a j, a j ARBITRARY NUMBERS. WHAT ARE THE PERFORMANCE OF SUCH AN ESTIMATOR? WHEN APPLIED TO A SELF-SIMILAR. OR LRD PROCESS [28]

31 WAVELETS AND 2ND-ORDER SCALING: ESTIMATION Abry, Gonçalvès, Flandrin, ASSUME: - i)x GAUSSIAN, - ii) IDEALISATION: EXACT INDEPENDENCE. Abry, Veitch BIAS : IE log 2 S 2 (j) = log 2 IES 2 (j) + Γ (n j /2) log 2 (n j /2) }{{} IEĤ = H + 2 j w jg j. g j. VARIANCE: Var Ĥ = 4 j w2 j σ2 j, min Var Ĥ = a j Var log 2 S 2 (j) Var log 2 S 2 (j) C/n j 2 j C/n, VAR ((log Ĥ 2 e) 2 ( ) j w2 j 2j ) /n, ANALYTICAL (APPROXIMATE) CONFIDENCE INTERVAL (DOES NOT DEPEND ON UNKNOWN H). ACTUAL PERFORMANCES : NEGLIGIBLE BIAIS, EXTREMELY CLOSE TO MLE. CONCEPTUAL AND PRACTICAL SIMPLICITY : MATLAB CODE AVAILABLE. [29]

32 WAV. AND 2ND-ORDER SCALING: ROBUSTNESS Superimposed Trends Y (t) = X(t) + T (t) d Y (j, k) = d X (j, k) + d T (j, k) - If T (t) Polynomial of degree P, then d T when N > P, - If T (t) smooth trend, then the d T decrease as N increases y j Octave j y j 8 6 α = time Octave j Vary N! [3]

33 WAV. AND 2ND-ORDER SCALING: ROBUSTNESS Superimposed Trends - Ethernet Data (Veitch, Abry) 6 x 5 # bytes in s bins 4 2 Part I Part II time (s) Logscale Diagram, N=2 y j Full trace : α =.6 Part I : α=.62 part II : α= octave j [3]

34 WAV. AND 2ND-ORDER SCALING: ROBUSTNESS Constancy along time of Scaling laws (Veitch, Abry) 35 paug 4 paug Y j Time x 5 (j=7,j2=2) 25.8 Not Rejected α Octave j Block 36 OctExt x 4 6 OctExt Y j Time x 4 (j=7,j2=2) α.5 Rejected Octave j Block [32]

35 SELF-SIMILARITY SELF-SIMILARITY:? IE d X (j, k) q = C q (2 j ) qh - Power Laws, - 2 j (for all scales), - q/ie d X (j, k) q <, - A single parameter H - Additive Structure.? [33]

36 BEYOND SELF-SIMILARITY SELF-SIMILARITY: IE d X (j, k) q = C q (2 j ) qh - Power Laws, - 2 j (for all scales), - q/ie d X (j, k) q <, - A single parameter H - Additive Structure. MULTIFRACTAL IE d X (j, k) q = C q (2 j ) ζ(q) - Power Laws, - 2 j < L, (for fine scales only, in the limit 2 j,) - q? - A whole collection of scaling parameter ζ(q) - Multiplicative Structure.? [34]

37 OUTLINE I. INTUITIONS, MODELS, TOOLS I. INTUITIONS, DEFINITION,APPLICATIONS I.2 STOCHASTIC MODELS: SELF-SIMILARITY VS MULTIFRACTAL I.3 MULTIRESOLUTION TOOLS, AGGREGATION, INCREMENTS I.4 WAVELETS, CONTINUOUS, DISCRETE II. SECOND ORDER ANALYSIS, SELF SIMILARITY AND LONG MEMORY II. RANDOM WAKS, SELF SIMILARITY, LONG MEMORY, II.2 2ND ORDER WAVELET STATISTICAL ANALYSIS, II.3 ESTIMATION, ESTIMATION PERFORMANCE, II.4 ROBUSTNESS AGAINST NON STATIONARITIES, III. HIGHER ORDER ANALYSIS, MULTIFRACTAL PROCESSES III. MULTIPLICATIVE CASCADES, MULTIFRACTAL PROCESSES, III.2 HIGHER ORDER WAVELET STATISTICAL ANALYSIS, III.3 FINITENESS OF MOMENTS, III.4 ESTIMATION, ESTIMATION PERFORMANCE, III.5 NEGATIVE ORDERS, III.6 BEYOND POWER LAWS. [35]

38 MODELLING TOOL 2: MULTIPLICATIVE CASCADES DEFINITION: - SPLIT DYADIC INTERVALS I j,k INTO TWO, - I.I.D. MULTIPLIERS W j,k - Q J (t) = Π {(j,k): j J,t Ij,k }W j,k, W W W 2 W 2 W W W,, W,2 WW, IMPLICATIONS: t - LOCAL HOLDER EXPONENT, - MULTIFRACTAL SAMPLE PATHS, MULTIFRACTAL SPECTRUM D(h) - CASCADES, MULTIPLICATIVE STRUCTURE, Q r (t) P k /a R t k +aτ q X(u)du t = Cq a ζq, FINE SCALES a, k - MULTIPLE EXPONENTS ζ q, - NO CHARACTERISTIC SCALE, - ζ q = log 2 IEW q, NON LINEAR IN q. [36]

39 MODELLING TOOL 2: MULTIPLICATIVE CASCADES YAGLOM, MANDELBROT BARRAL, MANDELBROT SCHMMITT ET AL., BACRY ET AL., CHAINAIS ET AL. MANDELBROT S COMPOUND POISSON INFINITELY DIVISIBLE CASCADE (CMC) CASCADE (CPC) CASCADE (IDC) - IID W, - IID W, - CONTINUOUS INFINITELY - DYADIC GRID, - POINT PROCESS, - DIVISIBLE MEASURE M, r= ( k2 j, 2 j ) r (t i, r i ) C r (t) t t r t Q r (t) = Π W j,k, Π W j,k, exp R dm(t, r ), ϕ(q) = log 2 IEW q, = q( IEW ) + IEW q, = ρ(q) qρ(), A(t) = lim r R t Q r(u)du, FOR A RANGE OF qs, IE A(t + aτ ) A(t) q = c q a q+ϕ(q), RESOLUTION DEPTH < SCALE < INTEGRAL SCALE, a m = r < a < a M = L. [37]

40 MULTIFRACTAL PROCESSES DENSITY: Q r (t) = Π W j,k IE a R t+aτ t Q r (u)du q = cq a ϕ(q), Q r (t) t 9 x 4 MEASURE: A(t) = lim r R t Q r(u)du, IE A(t + aτ ) A(t) q = c q a q+ϕ(q), A r (t) t FRACTIONAL BROWNIAN MOTION IN MULTIFRACTAL TIME: V H (t) = B H (A(t)), IE V H (t + aτ ) V H (t) q = c q a qh+ϕ(qh), V r (t) MULTIFRACTAL RANDOM WALK: Y H (t) = R t Qr (s)db H (s), IE Y H (t + aτ ) Y H (t) q = c q a qh+ϕ(q). X MF Process Time x MATLAB SYNTHESIS ROUTINES : CHAINAIS, ABRY t [38]

41 HIGHER-ORDER WAVELET STATISTICAL ANALYSIS PRINCIPLES : - IDEAS : P IE d X (j, k) q = IE d X (, k) q 2 jζ q log 2 IE d X (j, k) q = jζ q + β q, - PROBLEMS: ESTIMATE IE d X (j, k) q FROM A SINGLE FINITE LENGTH OBSERVATION? - SOLUTION : P2 et P3 STATISTICAL AVERAGES TIME AVERAGES, S q (j) = (/n j ) Pn j k= d X(j, k) q LOG-SCALE DIAGRAMS: log 2 S q (j) vs log 2 2 j = j [39]

42 LOGSCALE DIAGRAMS - MULTIFRACTAL PROC V r (t) q= Octave q=3 (log 2 S q (j))/q (log 2 S q (j))/q Octave q= (log 2 S q (j))/q Octave q=4 (log 2 S q (j))/q Octave q= Octave q= Octave t (log 2 S q (j))/q q= Octave [4]

43 WAV. AND HIGHER-ORDER SCALING: ESTIMATION DYADIC GRID (DISCRETE WAVELET TRANSFORM): a j = 2 j, t j,k = k2 j, SCALE STRUCTURE FUNCTIONS (TIME AVERAGE): S q (j) = (/n j ) Pn j k= d X(j, k) q TIME DEFINITION: Y j,q,n = log 2 S n (2 j, q; f ) VERSUS log 2 2 j = j, ˆζ(q, n) = P j 2 j=j w j,q Y j,q,n. NON WEIGTHED: a j = cste WHAT ARE THE PERFORMANCE OF SUCH ESTIMATORS? WHEN APPLIED TO MULTIFRACTAL PROCESSES [4]

44 TEST FOR THE FINITENESS OF MOMENTS THEOREM : GONÇALVÈS, RIEDI LET X BE A RV WITH CHARACTERISTIC FUNCTION χ(s) := IE exp{isx}. IF H Rχ := sup{α > : Rχ(s) P α (s) C s α }, IS THE LOCAL HÖLDER REGULARITY OF Rχ AT THE ORIGIN, THEN IE X q < + q q + c AND H Rχ q + c H Rχ +. ESTIMATOR : {X k } k=,...,n, n I.I.D RVS, SET nx W (a) := n Ψ(a.X k ) slope = α slope = γ, k= WITH Ψ A REAL AND SEMI-DEFINITE FOURIER TRANSFORM OF A SUFFICIENTLY REGULAR WAVELET ψ. THEN Lower scale bound Upper scale bound H Rχ = lim sup a + log W (a). log a 5 slope = N ψ [42]

45 ESTIMATING THE PARTITION FUNCTION SUPPORT [43]

46 METHODOLOGY NUMERICAL SYNTHESIS OF PROCESSES: ACCUMULATE nbreal NUMERICAL REPLICATIONS WITH LENGTH n SAMPLES. APPLY SCALING EXPONENTS ESTIMATORS: COMPUTE ˆζ(q, n) (l) FOR EACH REPLICATION, AVERAGE OVER REPL. TO OBTAIN THE STATISTICAL PERFORMANCE OF ˆζ(q, n) ASYMPTOTIC BEHAVIOURS: THE CASCADE DEPTH INCREASES FOR A GIVEN NUMBER OF INTEGRAL SCALES...., (t i, r i ) r t [44]

47 METHODOLOGY NUMERICAL SYNTHESIS OF PROCESSES: ACCUMULATE nbreal NUMERICAL REPLICATIONS WITH LENGTH n SAMPLES. APPLY SCALING EXPONENTS ESTIMATORS: COMPUTE ˆζ(q, n) (l) FOR EACH REPLICATION, AVERAGE OVER REPL. TO OBTAIN THE STATISTICAL PERFORMANCE OF ˆζ(q, n) ASYMPTOTIC BEHAVIOURS: THE CASCADE DEPTH INCREASES FOR A GIVEN NUMBER OF INTEGRAL SCALES. THE NUMBER OF INTEGRAL SCALES INCREASES FOR A GIVEN CASCADE DEPTH, (t i, r i ) (t i, r i ) r r t t [45]

48 LINEARISATION EFFECT: ˆζ(q) LASHERMES, ABRY, CHAINAIS CPC Q r EI() CPC V H EIII(3) ζ(q) ζ(q) q q q > q o, ˆζ(q, n) = α o + β o q, q o, α o, β o ARE RV. [46]

49 LINEARISATION EFFECT: LEGENDRE TRANSFORM D(h) = d + MIN q (qh ζ(q)), (d EUCLIDIEN DIMENSION OF SPACE). CPC Q r EI() CPC V H EIII(3) D(h).4 D(h) h h ACCUMULATION POINTS : D o (h o ), WITH D o = d α o, h o = β o, D o, h o ARE RV. [47]

50 LIN. EFFECT: ASYMPTOTIC BEHAVIOURS GIVEN RESOLUTION, INCREASING NUMBER OF INTEGRAL SCALES, q E q o 5 E h o.4 E D o log2(n) log2(n) log2(n) GIVEN NUMBER OF INTEGRAL SCALES, INCREASING RESOLUTION, q E q o 5 E h o.4 E D o log2(n) log2(n) log2(n) [48]

51 LINEARISATION EFFECT: CONJECTURE CRITICAL POINTS: RESULTS: EI : 8 >< >: 8 >< >: D ± =, D(h ± ) =, h ± = (dζ(q)/dq) q=q ±. ˆζ(q, n) = d D o + h o q d D + h q, q q, ˆζ(q, n) ζ(q), q q q+, ˆζ(q, n) = d D + o + h+ o q d D+ + h+ q, q+ q. EII&III : ( ˆζ(q, n) ζ(q), < q q +, ˆζ(q, n) = d D + o + h+ o q d D+ + h+ q, q+ q. ILLUSTRATION: theo est ζ(q) q [49]

52 LINEARISATION EFFECT: COMMENTS WHEN DOES THE LINEARISATION EFFECT EXIST? FOR ALL TYPES OF CASCADES: CMC, CPC, IDC, FOR ALL TYPES OF PROCESSES: Q r, A, V H, Y H, FOR ALL NUMBERS OF VANISHING MOMENTS: N, FOR ALL MRA-BASED ESTIMATORS: WAVELETS, INCREMENTS, AGGREGATION, CAN BE WORKED OUT FOR q <, EXTENDS TO DIMENSION HIGHER THAN d >. [5]

53 EXTENSION: STANDARD WT VERSUS WTMM (/3). 8 Standard vs WTMM 7 6 Theo CWT WTMM cwtmm Estimated ζ q q [5]

54 EXTENSION: 2D MULTIPLICATIVE CASCADE (2/3) theo est theo est ζ(q) 6 D(h) q h [52]

55 EXTENSION: 3D MULTIPLICATIVE CASCADE (3/3). 3D CMC (LOG NORMAL), EI() COMPARED TO A 2D SLICE. 2 theo 2d 3d.5 theo 2d 3d ζ(q) D(h) q h [53]

56 LINEARISATION EFFECT: COMMENTS WHEN DOES THE LINEARISATION EFFECT EXIST? FOR ALL TYPES OF CASCADES: CMC, CPC, IDC, FOR ALL TYPES OF PROCESSES: Q r, A, V H, Y H, FOR ALL NUMBERS OF VANISHING MOMENTS: N, FOR ALL MRA-BASED ESTIMATORS: WAVELETS, INCREMENTS, AGGREGATION, CAN BE WORKED OUT FOR q <, EXTENDS TO DIMENSION HIGHER THAN d >. WHAT THE LINEARISATION EFFECT IS NOT: A LOW PERFORMANCE ESTIMATION EFFECT. A FINITE SIZE EFFECT : THE CRITICAL PARAMETERS DO NOT DEPEND ON n, BE IT THE NUMBER OF INTEGRAL SCALES, OR THE DEPTH (OR RESOLUTION) OF THE CASCADES. A FINITENESS OF MOMENTS EFFECT, - q c < < < q+ c, q + ϕ(q) =, - q c < q < < < q+ < q+ c, WHAT THE LINEARISATION EFFECT MIGHT BE: MULTIPLICATIVE MARTINGALES? OSSIANDER, WAYMIRE, KAHANE, PEYRIÈRE 75, BARRAL, MANDELBROT 2. [54]

57 LINEARISATION EFFECT: PICTURE TWO POWER-LAWS, TWO FUNCTIONS OF q: BARE CASCADE: IEQ r (t) q = r ϕ(q), q R. DRESSED CASCADE: IET Q (t, a; β ) q = c q a ζ(q), q [q c, q+ c ], IET Q (t, a; β ) q =, ELSE, ff WITH: ζ(q) = + qh, q [q c, q ], ζ(q) = ϕ(q), q [q, q+ ], ζ(q) = + qh +, q [q+, q+ c ]. 9 = ; CONFUSION BETWEEN ϕ(q) AND ζ(q): MULTIPLICATIVE CASCADE: ϕ(q), q R, SCALING EXPONENTS: ζ(q), q [q c, q+ c ]. [55]

58 LINEARISATION EFFECT: SKETCHED VIEWS Moments < c τ ζ(q) q < q E A τ (t) q = q c q * q + * q + c Estimated ζ(q,n) +qh EI * ζ(q) +qh + * q Estimated ζ(q,n) EII & EIII ζ(q) +qh + * q [56]

59 LINEARISATION EFFECT: IMPACTS AND IMPORTANCE CONSEQUENCES: RECAST THE USUAL GOALS : ESTIMATE THE INTEGRAL SCALE AND THE RESOLUTION OF THE CASCADE, I.E., FIND A SCALING RANGE [a m, a M ] ESTIMATE THE CRITICAL PARAMETERS D ±, h±, q±, ESTIMATE THE ζ(q) FOR q [q, q+ ], VISIT B. LASHERMES S POSTER. IMPORTANCE OF THE LINEARISATION EFFECT: DISCRIMINATION OF MF MODELS BASED ON ˆζ(q, n), DISCRIMINATION BETWEEN MONOFRACTAL AND MULTIFRACTAL, [57]

60 DIFFICULTIES? NEGATIVE VALUES OF qs FINITENESS? S q (j) = (/n j ) Pn j k= d X(j, k) q <? NUMERICAL INSTABILITY? d X (j, k) d X (j, k) q = THEORY? FULL MULTIFRACTAL SPECTRUM 5 5 ζ(q) ζ(q) q q SOLUTIONS? [58]

61 NEGATIVE VALUES OF qs - SOLUTION R AGGREGATION: T X (a, t) = t+at at X(u)du t 5 ζ(q) q APPLIES ONLY TO POSITIVE DATA (MEASURE) [59]

62 NEGATIVE VALUES OF qs - SOLUTION 2 WT MODULUS MAXIMA (ARNEODO ET AL.) 5 WTMM L X (a, t k ) = SUP a <a T X (a, t k (a )) 4 log 2 (a) ζ(q) Skeleton of the Wavelet Transform 5 log 2 (a) Time q COMPUTATIONALLY EXPENSIVE [6]

63 NEGATIVE VALUES OF qs - SOLUTION 3 WAVELET LEADERS: (JAFFARD ET AL.) d X (j, k) L X (j, k) = SUP j <j d X (j, 2 j ) 5 ζ(q) q COMPUTATIONALLY EFFICIENT AND EXCELLENT STATISTICAL PERFORMANCE [6]

64 BEYOND POWER LAWS SELF-SIMILARITY: MULTIFRACTAL IE d X (j, k) q = C q (2 j ) qh = C q exp(qh ln 2 j ) - POWER LAWS, - 2 j (FOR ALL SCALES), - q/ie d X (j, k) q <, - A SINGLE PARAMETER H - ADDITIVE STRUCTURE. IE d X (j, k) q = C q (2 j ) ζ(q) = C q exp(ζ(q) ln 2 j ) - POWER LAWS, - 2 j < L, (FOR FINE SCALES ONLY, IN THE LIMIT 2 j,) - q? - A WHOLE COLLECTION OF SCALING PARAMETER ζ(q) - MULTIPLICATIVE STRUCTURE. BEYOND POWER LAWS : WARPED INF. DIV. CASCADES IE d X (j, k) q = C q (2 j ) qh = C q exp(qh ln 2 j ) IE d X (j, k) q = C q (2 j ) ζ(q) = C q exp(ζ(q) ln 2 j ) IE d X (j, k) q = = C q exp(ζ(q)n(2 j )) VISIT PIERRE CHAINAIS S POSTER [62]

65 CONCLUSIONS AND REFERENCES ANALYSING SCALING IN DATA? THINK WAVELET EFFICIENCY, PRACTICAL AND CONCEPTUAL ADEQUATION AND SIMPLICITY, ROBUSTNESS AGAINST NON STATIONARITIES, EASY TO USE, LOW COAST, REAL TIME ON LINE. MODELLING SCALING IN DATA? THINK SELF SIMILARITY VERSUS MULTIPLICATIVE CASCADES, AND POSSIBLY ADD LONG MEMORY. ALSO SCALING MAY NOT BE POWER LAWS REFERENCES AND RESOURCES, VISIT : perso.ens-lyon.fr/patrice.abry inrialpes.fr/is2/ pgoncalv darryl fraclab chainais [63]

NAVIGATING SCALING: MODELLING AND ANALYSING

NAVIGATING SCALING: MODELLING AND ANALYSING P. Abry (), P. Gonçalvès (2) () SISYPH, CNRS, (2) INRIA, Ecole Normale Supérieure Lyon, France Grenoble, France IN COLLABORATIONS WITH : P. Flandrin, D. Veitch,