NAVIGATING SCALING: MODELLING AND ANALYSING

Transcription

1 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, P. Chainais, B. Lashermes, N. Hohn, S. Roux, P. Borgnat, M.Taqqu, V. Pipiras, R. Riedi Wavelet And Multifractal Analysis, Cargèse, France, July 24.

2 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? []

3 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]

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

5 SCALING? 4 Trafic (WAN) Internet nb connexions Temps (s) Trafic (LAN) Ethernet Densite Spectrale de Puissance temps (s) Trafic (WAN) Internet Log (DSP) Nombre Octets nb connexions Log (Frequence (Hz)) temps (s) [4]

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. 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. [5]

7 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) [6]

8 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), -... [7]

9 MODELLING TOOLS SELF-SIMILARITY: IE X(t + aτ ) X(t) q = C q a qh - Power Laws, - a (for all scales), - q/ie d X (j, k) q <, - A single parameter H - Additive Structure. MULTIFRACTAL: IE X(t + aτ ) X(t) q = C q a ζ(q) - Power Laws, - a < L, (for fine scales only, in the limit a,) - q? - A whole collection of scaling parameter ζ(q) - Multiplicative Structure. BEYOND POWER LAWS IE X(t + aτ ) X(t) q = C q a qh = C q exp(qh ln a) IE X(t + aτ ) X(t) q = C q a ζ(q) = C q exp(ζ(q) ln a) IE X(t + aτ ) X(t) q = = C q exp(ζ(q)n(a)) VISIT PIERRE CHAINAIS S POSTER [8]

10 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 [9]

11 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. []

12 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? []

13 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 ) f (u) = (I ) 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.5-2 temps - 2 temps 2 Temps [2]

14 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 [3]

15 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]

16 MOTHER-WAVELET AND BASIS : WAVELET COEFFICIENTS: CONTINUOUS WT WAVELET TRANSFORMS ψ (u)du =, ψ a,t (u) = a ψ ( u t a ) AND DISCRETE WT T X (a, t) = X, ψ a,t d X (j, k) = T X (a = 2 j, t = 2 j k) SCALE SCALE TIME MODULUS MAXIMA WT TIME AND FAST PYRAMIDAL ALGORITHM WTMM.5.5 log 2 (a) Time [5]

17 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. [6]

18 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, IEX(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 [7]

19 MODELLING TOOL : SELF-SIMILARITY DEFINITION: δ τ X(t) fdd = c H δ τ/c X(t/c), c >, DILATION FACTOR, H > : SELF-SIMILARITY EXPONENT INTERPRETATIONS: - COVARIANCE UNDER DILATION (CHANGE OF SCALE), - 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. [8]

20 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, - AGGREGATION : X at (t) = R t+at X(u)du, VAR X t at (t) a α, Ca +, - INCREMENTS OF SELF.-SIM. PROC. (WITH H > /2) ARE LONG RANGE DEP. (WITH α = 2H ). [9]

21 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]

22 WAVELETS AND LONG RANGE DEPENDENCE H =.5 H =.5 H =.95 Daubechies2 Haar [2]

23 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 jζ q q/ie d X (, k) q <. [22]

24 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 [23]

25 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 [24]

26 2ND ORDER WAVELET-BASED STATISTICAL ANALYSIS FOR SELF -SIMILARITY 5 5 α = 2.57 j y j Octave j [25]

27 2ND ORDER WAVELET-BASED STATISTICAL ANALYSIS FOR LONG RANGE DEPENDENCE α =.55 c f = j y j Octave j [26]

28 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 [27]

29 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. [28]

30 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! [29]

31 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]

32 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 [3]

33 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.? [32]

34 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.? [33]

35 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. [34]

36 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, - FROM DENSITY TO MEASURE AND TO FRACTIONAL BROWNIAN MOTION, - IN MULTIFRACTAL TIME. W, W W W,2 W 2 W 2 W W, WW, NON STATIONARITY, - LOCAL HOLDER EXPONENT, t - MULTIFRACTAL SAMPLE PATHS, MULTIFRACTAL SPECTRUM D(h) - CASCADES, MULTIPLICATIVE STRUCTURE, IE /a R t+aτ q X(u)du t = Cq a ζq, - MULTIPLE EXPONENTS ζ q, FINITE RANGE OF q, - ζ q = log 2 IEW q, NON LINEAR IN q, - FINE SCALES a, a L INTEGRAL SCALE, - NO CHARACTERISTIC SCALE OF TIME BEYOND AN INTEGRAL SCALE. Q r (t) IMPLICATIONS: [35]

37 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. [36]

38 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 [37]

39 TEST FOR THE FINITENESS OF MOMENTS GONÇALVÈS, RIEDI [38]

40 TEST FOR THE FINITENESS OF MOMENTS [39]

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 [4]

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 [42]

44 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 [43]

45 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 [44]

46 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. [45]

47 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. [46]

48 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) [47]

49 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 [48]

50 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 >. [49]

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

52 EXTENSION: 2D MULTIPLICATIVE CASCADE (2/3) theo est theo est ζ(q) 6 D(h) q h [5]

53 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 [52]

54 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. [53]

55 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 ]. [54]

56 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 [55]

57 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, [56]

58 NEGATIVE VALUES OF qs DIFFICULTIES? 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? WEAK HÖLDER EXPONENT VS EXACT HÖLDER EXPONENT 5 5 ζ(q) ζ(q) q SOLUTIONS? MM q [57]

59 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) [58]

60 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 [59]

61 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]

62 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. WARPED INF. DIV. CASCADES : IE d X (j, k) q = C q (2 j ) ζ(q) = C q exp(ζ(q)n(2 j )) VISIT PIERRE CHAINAIS S POSTER. [6]

63 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 darryl fraclab chainais [62]

64 WANT TO TALK TO US? perso.ens-lyon.fr/patrice.abry [63]