NAVIGATING SCALING: MODELLING AND ANALYSING

P. Abry(1), P. Goncalves(2)

(1) SISYPH, CNRS, (2) INRIA Rhone-Alpes,Ecole Normale Superieure 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, Cargese, France, July 2004.

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Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18

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 ?

[1]

OUTLINE

I. INTUITIONS, MODELS, TOOLS

I.1 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.1 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.1 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]

IRREGULARITIES, VARIABILITIES

SCALING OR NON STATIONARITIES?

[3]

SCALING ?

temps (s)

Trafic (WAN) Internet

nb c

onne

xion

s

temps (s)

Trafic (WAN) Internet

nb c

onne

xion

s

[4]

SCALING ?

0 500 1000 1500 2000 25000

2000

4000

1000 1100 1200 1300 1400 1500 16000

2000

4000

1240 1260 1280 1300 1320 1340 1360 13800

2000

4000

1285 1290 1295 1300 1305 1310 1315 13200

2000

4000

Temps (s)

[5]

SCALING ?

2 2.5 3 3.5 4 4.5 56.5

7

7.5

8

8.5

Log10

(Frequence (Hz))

Trafic (LAN) Ethernet −−− Densite Spectrale de Puissance

Log 10

(DS

P)

−−

− N

ombr

e O

ctet

s

[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.

temps (s)

Trafic (WAN) Internet

nb c

onne

xion

s

temps (s)

nb c

onne

xion

s

0 500 1000 1500 2000 25000

2000

4000

1000 1100 1200 1300 1400 1500 16000

2000

4000

1240 1260 1280 1300 1320 1340 1360 13800

2000

4000

1285 1290 1295 1300 1305 1310 1315 13200

2000

4000

Temps (s)

• ANALYSIS: Rather than for a characteristic scale,look for a relation, a mecanism, a cascade between scales.

[7]

SCALING : OPERATIONAL DEFINITIONS

• MULTIRESOLUTION QUANTITY:TX(a, t) (e.g., Wavelet Coef.).

• POWER LAWS:IE|TX(a, t)|q = cq|a|ζ(q),

1n

∑nk=1 |TX(a, tk)|q = cq|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|TX(a, t)|q = Cq|a|ζ(q) = Cq exp(ζ(q) ln a)IE|TX(a, t)|q = = Cq exp(ζ(q)n(a))

→ VISIT PIERRE CHAINAIS’S POSTER

[8]

UBIQUITY ?

0 1 2 3 4 5 6 7 8 9 100

5000

10000

15000

temps(s)

Trafic (LAN) Ethernet

Nom

bre

Oct

ets

2050 2100 2150 2200 2250 2300 2350 2400 2450 2500450

500

550

600

650

temps (s)

nb c

onne

xion

s

Trafic (WAN) Internet

0 1 2 3 4 5 60

0.005

0.01

0.015

0.02

0.025

0.03

temps(s)

Turbulence de Jet, Rλ ∼ 580

Dis

sipa

tion

0 1 2 3 4 5 60

2

4

6

8

10

12

temps(s)

Turbulence de Jet, Rλ ∼ 580

Vite

sse

(m/s

)

[9]

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),- . . .

[10]

ANALYSING TOOL 1 : AGGREGATION

COMPARE DATA AGAINST A BOX, THEN VARY aTX(a, t) = 1

aT0

∫ t+aT0

tX(u)du

AVERAGE

-1 0 1 2

-0.5

0

0.5

temps

T

1/T

WORKS ONLY FOR POSITIVE TIME SERIES, DENSITY

[11]

ANALYSING TOOL 2 : INCREMENTS

COMPARE DATA AGAINST A DIFFERENCE OF DELTA FUNCTIONS, THEN VARY aTX(a, t) = X(t+ aτ0)−X(t)

DIFFERENCE

-1 0 1 2

-0.5

0

0.5

temps

τ

INCREMENTS OF HIGHER ORDERS OR GENERALISED N -VARIATIONS

− Order 2 : TX(a, t) = −X(t+ 2aτ0) + 2X(t+ aτ0)−X(t),− Order N : TX(a, t) =

∑Np=0(−1)papX(t+ paτ0),

where∑Np=0(−1)pappk ≡ 0, k = 0, . . . , N − 1.

[12]

ANALYSING TOOL: MULTIRESOLUTION ANALYSIS

• MULTIRESOLUTION QUANTITIES:X(t) → TX(a, t) = 〈fa,t|X〉, fa,t(u) = 1

af0(u−ta )

AGGREGATION INCREMENTS ?f0(u) = (β0) f0(u) = (I0) ?

= 1aT0

∫ t+aT0

tX(u)du = X(t+ aτ0)−X(t) ?

BOX, AVERAGE DIFFERENCE ?

-1 0 1 2

-0.5

0

0.5

temps

T

1/T

-1 0 1 2

-0.5

0

0.5

temps

τ

?

[13]

ANALYSING TOOL: MULTIRESOLUTION ANALYSIS

• MULTIRESOLUTION QUANTITIES:X(t) → TX(a, t) = 〈fa,t|X〉, fa,t(u) = 1

af0(u−ta )

• CHOICES FOR MOTHER FUNCTIONS: f0,AGGREGATION INCREMENTS WAVELETS

f0(u) = (β0)∗N f0(u) = (I0)∗N f0(u) = ψ0,N

= 1aT0

∫ t+aT0

tX(u)du = X(t+ aτ0)−X(t) =

∫X(u) 1

|a|ψ0(u−ta ),BOX, AVERAGE DIFFERENCE AVERAGE, DIFFERENCE

-1 0 1 2

-0.5

0

0.5

temps

T

1/T

-1 0 1 2

-0.5

0

0.5

temps

τ

−1 0 1 2

−0.5

0

0.5

Temps

N N N

[14]

WAVELETS AND SCALING: KEY INGREDIENTS

• DILATION OPERATOR, 1|a|ψ0( t

|a|)

-4 -2 0 2 4

-1

-0.5

0

0.5

1

Temps

a = 1

-4 -2 0 2 4

-1

-0.5

0

0.5

1

Temps

a = 2

-4 -2 0 2 4

-1

-0.5

0

0.5

1

Temps

a = 4

• NUMBER OF VANISHING MOMENTS,N ≥ 1,

∫tkψ0(t)dt ≡ 0, k = 0, 1, . . . , N − 1.

−3 −2 −1 0 1 2 3 4−1.5

−1

−0.5

0

0.5

1

1.5

2

temps

Daubechies 2

−3 −2 −1 0 1 2 3 4−1.5

−1

−0.5

0

0.5

1

1.5

2

temps

B Spline Cubique

−3 −2 −1 0 1 2 3 4−1.5

−1

−0.5

0

0.5

1

1.5

2

temps

Daubechies 6

[15]

WAVELET TRANSFORMS

• MOTHER-WAVELET AND ”BASIS”:∫ψ0(u)du = 0, ψa,t(u) = 1

|a|ψ0(u−ta )

• WAVELET COEFFICIENTS: CONTINUOUS WT TX(a, t) = 〈X,ψa,t〉

TIME

SCALE

MODULUS MAXIMA WT AND DISCRETE WTSKELETON : MAXIMA LINES dX(j, k) = TX(a = 2j, t = 2jk)

Time

log2(a)

WTMM

0 100 200 300 400 500 600 700 800 900 1000

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

TIME

SCALE

[16]

OUTLINE

I. INTUITIONS, MODELS, TOOLS

I.1 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.1 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.1 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.

[17]

MOD. TOOL 1: RAND. WALKS AND SELF SIMILARITY

RANDOM WALK: X(t+ τ) = X(t) + δτX(t)︸ ︷︷ ︸Steps or Increments

STATISTICAL PROPERTIES OF THE STEPS:- A1: 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 > −1.

ANOMALIES:⇒ IEX(t)2 = 2D|t|γ,⇒ IEX(t)2 = ∞.

SELF SIMILAR RANDOM WALKS:- B1: Stationary,- B2: Self Similarity

[18]

MODELLING TOOL 1: SELF-SIMILARITY

• DEFINITION: δτX(t)fdd= cHδτ/cX(t/c), ∀c > 0, DILATION FACTOR,

1 > H > 0 : 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+ aτ0)−X(t)|q = Cq|a|qH,- ∀a > 0, ∀c > 0, ∀q /IE|X(t)|q <∞,- A SINGLE SCALING EXPONENT H .- ADDITIVE STRUCTURE,- (CORRELATED) RANDOM WALK, LONG MEMORY, LONG RANGE CORRELATIONS.

[19]

MOD. TOOL 1 (BIS): LONG RANGE DEPENDENCE

• DEFINITIONS :

- LET X BE A 2ND STATIONARY PROCESS WITH,- COVARIANCE : cX(τ) = IEX(t)X(t+ τ)

- SPECTRUM : ΓX(ν)

cX(τ) = cτ |τ |−β, 0 < β < 1, |τ | → +∞ΓX(ν) = cf |ν|−α, 0 < α < 1, |ν| → 0

WITH α = 1− β AND cf = 2(2π) sin((1− γ)π/2)cτ .

• CONSEQUENCES :

-P+∞

A cX(τ)dτ = +∞, A > 0,- NO CHARACTERISTIC SCALE,- AGGREGATION : TX(a, t) = 1

aT0

R t+aT0t

X(u)du,

⇒ VAR TX(a, t) ∼ Caα−1, a→ +∞,- INCREMENTS OF SELF.-SIM. PROC. (WITH H > 1/2)ARE LONG RANGE DEP. (WITH α = 2H − 1).

[20]

WAVELETS AND SELF-SIMILAR PROCESSES WITH

STATIONARY INCREMENTS - SUMMARY(Flandrin et al., Tewfik and Kim)

• P1: {dX(j, k), k ∈ Z} STATIONARY Sequences for each Scale 2j .N ≥ 1

• P2: SELF-SIMILARITY : Dilation{X(t)} d= {cHX(t/c)} ⇒ {dX(0, k)} d= {2−jHdX(j, k)}

- P3 : MARGINAL DIST. Pj(d) = 1β0Pj′( dβ0

), β0 =(

2j′

2j

)H.

• P4 : {dX(j, k)} SHORT RANGE DEPENDENT IF N > H + 1/2.|2jk − 2j

′k′| → +∞, |Cov dX(j, k)dX(j′, l)| ≤ D|2jk − 2j

′k′|2(H−N),

N ≥ 1 and Dilation

[21]

WAVELETS AND LONG RANGE DEPENDENCE

(Flandrin)H = 0.15

Haa

rD

aube

chie

s2

H = 0.5 H = 0.95

[22]

WAVELETS AND SELF-SIMILAR PROCESSES WITH

STATIONARY INCREMENTS - SUMMARY

• P1: {dX(j, k), k ∈ Z} STATIONARY Sequences for each Scale 2j .N ≥ 1

• P2: SELF-SIMILARITY : Dilation{X(t)} d= {cHX(t/c)} ⇒ {dX(0, k)} d= {2−jHdX(j, k)}

- P3 : MARGINAL DIST. Pj(d) = 1β0Pj′( dβ0

), β0 =(

2j′

2j

)H.

• P4 : {dX(j, k)} SHORT RANGE DEPENDENT IF N > H + 1/2.|2jk − 2j

′k′| → +∞, |Cov dX(j, k)dX(j′, k′)| ≤ D|2jk − 2j

′k′|2(H−N),

N ≥ 1 and Dilation

⇒ IDEALISATION : dX(j, k) INDEPENDENT VARIABLES .

⇒ INTERPRETATIONS: X(t) =∑k aX(J, k)ϕJ,k(t)+

∑j=1,...,J,k dX(j, k)ψj,k(t).

⇒ IMPLICATIONS: IE|dX(j, k)|q = IE|dX(0, k)|q2jqH ∀q/IE|dX(0, k)|q <∞.

[23]

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 ν0 and bandwith ∆ν0.

IE|dX(j, k)|2 =∫

ΓX(ν)|Ψ(2jν)|dν' 2−jΓX(2−jν0) within bandwith 2−j∆ν0.

• LET X BE LONG RANGE DEPENDENT :- POWER LAW: ΓX(ν) = cf |ν|−α, 0 < α < 1, |ν| → 0- POWER LAW: IE|dX(j, k)|2 ∼ C2j(α−1), j → +∞,

• {dX(j, k)} SHORT RANGE DEPENDENT IF N > α− 1.|2jk − 2j

′k′| → +∞, |Cov dX(j, k)dX(j′, k′)| ≤ D|2jk − 2j

′k′|α−1−2N ,

N ≥ 1 and Dilation

[24]

2ND ORDER WAVELET STATISTICAL ANALYSIS

Abry, Goncalves, Flandrin

PRINCIPLES:- IDEAS : P1 ⇒ IE|dX(j, k)|2 = C22j2H

⇒ log2 IE|dX(j, k)|2 = j2H + βq,

- PROBLEMS: ESTIMATE IE|dX(j, k)|2 FROM A SINGLE FINITE LENGTH OBSERVATION ?

- SOLUTION : P2 et P3 ⇒ STATISTICAL AVERAGES ⇒ TIME AVERAGES,S2(j) = (1/nj)

∑njk=1 |dX(j, k)|2

LOG-SCALE DIAGRAMS: log2 S2(j) vs log2 2j = j

[25]

2ND ORDER WAVELET-BASED STATISTICAL

ANALYSIS FOR SELF -SIMILARITY

1 2 3 4 5 6 7 8 9 10−35

−30

−25

−20

−15

−10

−5

Octave j

y j

α = 2.57

1 ≤ j ≤ 10

[26]

2ND ORDER WAVELET-BASED STATISTICAL

ANALYSIS FOR LONG RANGE DEPENDENCE

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Octave j

y j

α = 0.55

cf = 4.7

4 ≤ j ≤ 10

[27]

WAVELETS AND 2ND-ORDER SCALING: ESTIMATION

• DYADIC GRID (DISCRETE WAVELET TRANSFORM): aj = 2j, tj,k = k2j,

TIME

SCALE

• STRUCTURE FUNCTION (TIME AVERAGE):Yj = (1

2 log2 S2(2j) = 12 log2(1/nj)

∑njk=1 |dX(j, k)|2

• DEFINITION :Yj versus log2 2j = j,H =

∑j2j=j1

wjYj .

WHERE∑j jwj ≡ 1,

∑j wj ≡ 0, WITH wj ≡ B0j−B1

B0B2−B21,

AND p = 0, 1, 2, Bp =∑j jp/aj, aj ARBITRARY NUMBERS.

• WHAT ARE THE PERFORMANCE OF SUCH AN ESTIMATOR ?WHEN APPLIED TO A SELF-SIMILAR. OR LRD PROCESS

[28]

WAVELETS AND 2ND-ORDER SCALING: ESTIMATIONAbry, Goncalves, Flandrin, Abry, Veitch

• ASSUME: - i)X GAUSSIAN,- ii) IDEALISATION: EXACT INDEPENDENCE.

• BIAS : IE log2 S2(j) = log2 IES2(j) + Γ′(nj/2)− log2(nj/2)︸ ︷︷ ︸gj

.

⇒ IEH = H + 12

∑j wjgj.

• VARIANCE: −Var H = 14

∑j w

2jσ

2j ,

− min Var H =⇒ aj ∝ Var log2 S2(j)− Var log2 S2(j) ' C/nj ' 2jC/n,

⇒ VAR H '((log2 e)2(

∑j w

2j2j)

)/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]

WAV. AND 2ND-ORDER SCALING: ROBUSTNESS

Superimposed TrendsY (t) = X(t) + T (t) ⇒ dY (j, k) = dX(j, k) + dT (j, k)

- If T (t) Polynomial of degree P , then dT ≡ 0 when N > P ,- If T (t) smooth trend, then the dT decrease as N increases.

2000 4000 6000 8000 10000 12000 14000 16000−10

−5

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 110

2

4

6

8

10

12

14

16

18

Octave j

y j

2000 4000 6000 8000 10000 12000 14000 16000−10

−5

0

5

10

15

20

25

30

time1 2 3 4 5 6 7 8 9 10 11

0

2

4

6

8

10

12

14

16

18

Octave j

y j α = 0.59

Vary N !

[30]

WAV. AND 2ND-ORDER SCALING: ROBUSTNESS

Superimposed Trends - Ethernet Data (Veitch, Abry)

0 200 400 600 800 1000 1200 1400 16000

2

4

6x 10

5

time (s)

# by

tes

in 1

0s b

ins

Part I

Part II

0 5 10 1520

22

24

26

28

30

octave j

y j

Logscale Diagram, N=2

Full trace : α = 0.60 Part I : α= 0.62 part II : α=0.58

[31]

WAV. AND 2ND-ORDER SCALING: ROBUSTNESS

Constancy along time of Scaling laws (Veitch, Abry)

0 5 10 1520

25

30

35

Octave j

Yj

pAug

0.5 1 1.5 2 2.5

x 105

0

2000

4000

6000

8000

10000

12000

14000

Time

pAug

2 4 6 8 10 12

0

0.2

0.4

0.6

0.8

1

α

Not Rejected

Block

(j1=7,j2=12)

0 2 4 6 8 10 12 1420

22

24

26

28

30

32

34

36

Octave j

Yj

OctExt

2 4 6 8 10 12

x 104

0

1

2

3

4

5

6

x 104

Time

OctExt

2 4 6 8 10 12

−0.5

0

0.5

1

1.5α

Rejected

Block

(j1=7,j2=12)

[32]

SELF-SIMILARITY

• SELF-SIMILARITY:IE|dX(j, k)|q = Cq(2j)qH

- Power Laws,- ∀2j (for all scales),- ∀q/IE|dX(j, k)|q <∞,- A single parameter H- Additive Structure.

• ?

• ?

[33]

BEYOND SELF-SIMILARITY

• SELF-SIMILARITY:IE|dX(j, k)|q = Cq(2j)qH

- Power Laws,- ∀2j (for all scales),- ∀q/IE|dX(j, k)|q <∞,- A single parameter H- Additive Structure.

• MULTIFRACTAL

IE|dX(j, k)|q = Cq(2j)ζ(q)

- Power Laws,- ∀2j < L, (for fine scales only, in the limit 2j → 0,)- ∀q?- A whole collection of scaling parameter ζ(q)- Multiplicative Structure.

• ?

[34]

OUTLINE

I. INTUITIONS, MODELS, TOOLS

I.1 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.1 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.1 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]

MODELLING TOOL 2: MULTIPLICATIVE CASCADES

• DEFINITION:- SPLIT DYADIC INTERVALS Ij,k INTO TWO,- I.I.D. MULTIPLIERS Wj,k

- QJ(t) = Π{(j,k):1≤j≤J,t∈Ij,k}Wj,k,

1

1

W

W

W

WW

W W

1

1

1,2

2

1,1

1,1

W2

W W1,21

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

t

Q r(t)

• IMPLICATIONS:- LOCAL HOLDER EXPONENT,- MULTIFRACTAL SAMPLE PATHS, MULTIFRACTAL SPECTRUM D(h)

- CASCADES, MULTIPLICATIVE STRUCTURE,-

Pk

“1/a

R tk+aτ0tk

X(u)du”q

= Cq|a|ζq , FINE SCALES a→ 0,- MULTIPLE EXPONENTS ζq,- NO CHARACTERISTIC SCALE,- ζq = − log2 IEW q, NON LINEAR IN q.

[36]

MODELLING TOOL 2: MULTIPLICATIVE CASCADESYAGLOM, 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 ,

...2−2

2−1

r=1

t

( k2j, 2j )

0

1

t

r

(ti, r

i)

0

1

t

r

Cr(t)

Qr(t) = Π Wj,k, Π Wj,k, expRdM(t′, r′),

ϕ(q) = − log2 IEW q, = −q(1− IEW ) + 1− IEW q, = ρ(q)− qρ(1),

A(t) = limr→0

R t

0Qr(u)du,

FOR A RANGE OF qS, IE|A(t+ aτ0)− A(t)|q = cq|a|q+ϕ(q),RESOLUTION DEPTH < SCALE < INTEGRAL SCALE, am = r < a < aM = L.

[37]

MULTIFRACTAL PROCESSES

DENSITY: Qr(t) = Π Wj,k

IE“

1a

R t+aτ0t

Qr(u)du”q

= cqaϕ(q),

0 1 2 3 4 5 6 7 8 9 10 110

1

2

3

4

5

6

7

t

Qr(t

)

MEASURE: A(t) = limr→0

R t

0Qr(u)du,

IE|A(t+ aτ0)− A(t)|q = cq|a|q+ϕ(q),

0 1 2 3 4 5 6 7 8 9 10 110

1

2

3

4

5

6

7

8

9x 10

4

t

Ar(t

)

FRACTIONAL BROWNIAN MOTIONIN MULTIFRACTAL TIME:

VH(t) = BH(A(t)),IE|VH(t+ aτ0)− VH(t)|q = cq|a|qH+ϕ(qH),

0 1 2 3 4 5 6 7 8 9 10 11-100

-50

0

50

100

150

200

250

300

t

Vr(t

)

MULTIFRACTAL RANDOM WALK:YH(t) =

R tQr(s)dBH(s),

IE|YH(t+ aτ0)− YH(t)|q = cq|a|qH+ϕ(q).0 1 2 3 4 5 6 7

x 104

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time

X

MF Process

MATLAB SYNTHESIS ROUTINES : CHAINAIS, ABRY

[38]

HIGHER-ORDER WAVELET STATISTICAL ANALYSIS

PRINCIPLES :- IDEAS : P1 ⇒ IE|dX(j, k)|q = IE|dX(0, k)|q2jζq

⇒ log2 IE|dX(j, k)|q = jζq + βq,

- PROBLEMS: ESTIMATE IE|dX(j, k)|q FROM A SINGLE FINITE LENGTH OBSERVATION ?

- SOLUTION : P2 et P3 ⇒ STATISTICAL AVERAGES ⇒ TIME AVERAGES,Sq(j) = (1/nj)

Pnjk=1 |dX(j, k)|q

LOG-SCALE DIAGRAMS: log2 Sq(j) vs log2 2j = j

[39]

LOGSCALE DIAGRAMS - MULTIFRACTAL PROC.

0 1 2 3 4 5 6 7 8 9 10 11-100

-50

0

50

100

150

200

250

300

t

Vr(t

)

0 5 10 15−15

−10

−5

Octave

(log 2 S

q(j))/

q

q=0.5

0 5 10 15−16

−14

−12

−10

−8

−6

−4

Octave

(log 2 S

q(j))/

q

q=1

0 5 10 15−14

−12

−10

−8

−6

−4

Octave

(log 2 S

q(j))/

q

q=2

0 5 10 15−14

−12

−10

−8

−6

−4

Octave

(log 2 S

q(j))/

q

q=3

0 5 10 15−14

−12

−10

−8

−6

−4

Octave

(log 2 S

q(j))/

q

q=4

0 5 10 15−14

−12

−10

−8

−6

−4

−2

Octave

(log 2 S

q(j))/

q

q=8

0 2 4 6 8 10 12 14−16

−14

−12

−10

−8

−6

−4

−2

Octave

(log 2 S

q(j))/

q

q=0.5 1 2 3 4 8

[40]

WAV. AND HIGHER-ORDER SCALING: ESTIMATION

• DYADIC GRID (DISCRETE WAVELET TRANSFORM): aj = 2j, tj,k = k2j,

TIME

SCALE

• STRUCTURE FUNCTIONS (TIME AVERAGE):Sq(j) = (1/nj)

Pnjk=1 |dX(j, k)|q

• DEFINITION:Yj,q,n = log2 Sn(2

j, q; f0) VERSUS log2 2j = j,

ζ(q, n) =Pj2

j=j1wj,qYj,q,n .

NON WEIGTHED: aj = cste

• WHAT ARE THE PERFORMANCE OF SUCH ESTIMATORS ?WHEN APPLIED TO MULTIFRACTAL PROCESSES

[41]

TEST FOR THE FINITENESS OF MOMENTS

GONCALVES, RIEDI

THEOREM :LET X BE A RV WITH CHARACTERISTIC FUNCTION χ(s) := IE exp{isX}.

IF H<χ := sup{α > 0 : |<χ(s)− Pα(s)| ≤ C|s|α},IS THE LOCAL HOLDER REGULARITY OF <χ AT THE ORIGIN, THEN

IE|X|q < +∞∀q ≤ q+c AND H<χ ≤ q+

c ≤ bH<χc+ 1.

ESTIMATOR :{Xk}k=1,...,n, n I.I.D RVS, SET

W (a) := n−1

nXk=1

Ψ(a.Xk)

,WITH Ψ A REAL AND SEMI-DEFINITE

FOURIER TRANSFORM OF A

SUFFICIENTLY REGULAR WAVELET ψ.THEN

H<χ = lim supa→0+

log |W (a)|log a

.−60 −50 −40 −30 −20 −10 0 10

−60

−50

−40

−30

−20

−10

0

Lower scale bound

Upper scale bound

slope = Nψ

slope = α slope = −γ

[42]

ESTIMATING THE PARTITION FUNCTION SUPPORT

[43]

METHODOLOGY

• NUMERICAL SYNTHESIS OF PROCESSES:− ACCUMULATE nbreal NUMERICAL REPLICATIONS WITH LENGTH n SAMPLES.

• APPLY SCALING EXPONENTS ESTIMATORS:− COMPUTE ζ(q, n)(l) FOR EACH REPLICATION,− AVERAGE OVER REPL. TO OBTAIN THE STATISTICAL PERFORMANCE OF ζ(q, n)

• ASYMPTOTIC BEHAVIOURS:− THE CASCADE DEPTH INCREASES FOR A GIVEN NUMBER OF INTEGRAL SCALES.− ... ,

0

1

t

r

(ti, r

i)

[44]

METHODOLOGY

• NUMERICAL SYNTHESIS OF PROCESSES:− ACCUMULATE nbreal NUMERICAL REPLICATIONS WITH LENGTH n SAMPLES.

• APPLY SCALING EXPONENTS ESTIMATORS:− COMPUTE ζ(q, n)(l) FOR EACH REPLICATION,− AVERAGE OVER REPL. TO OBTAIN THE STATISTICAL PERFORMANCE OF ζ(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,

0

1

t

r

(ti, r

i)

0

1

t

r

(ti, r

i)

[45]

LINEARISATION EFFECT: ζ(q)LASHERMES, ABRY, CHAINAIS

CPC Qr EI(1) CPC VH EIII(3)

−18 −12 −6 0 6 12 18−8

−7

−6

−5

−4

−3

−2

−1

0

1

q

ζ(q)

0 6 12 18−5

0

5

10

qζ(

q)

q > qo, ζ(q, n) = αo + βoq, qo, αo, βo ARE RV.

[46]

LINEARISATION EFFECT: LEGENDRE TRANSFORM

D(h) = d+ MINq(qh− ζ(q)), (d EUCLIDIEN DIMENSION OF SPACE).

CPC Qr EI(1) CPC VH EIII(3)

−0.4 −0.2 0 0.2 0.4

−0.2

0

0.2

0.4

0.6

0.8

1

h

D(h

)

0.5 0.7 0.9

−0.2

0

0.2

0.4

0.6

0.8

1

hD

(h)

ACCUMULATION POINTS : Do(ho), WITH Do = d− αo, ho = βo,Do, ho ARE RV.

[47]

LIN. EFFECT: ASYMPTOTIC BEHAVIOURS

• GIVEN RESOLUTION, INCREASING NUMBER OF INTEGRAL SCALES, q0 h0 D0

8 9 10 11 12 13 14 15 16 170

5

10

15

20

25

30

log2(n)

E q

o

8 9 10 11 12 13 14 15 16 17−0.4

0

0.4

0.8

1.2

log2(n)E

ho

8 9 10 11 12 13 14 15 16 17

0.2

0.6

1

1.4

1.8

log2(n)

E D

o

• GIVEN NUMBER OF INTEGRAL SCALES, INCREASING RESOLUTION, q0 h0 D0

10 11 12 13 14 15 160

5

10

15

20

25

30

log2(n)

E q

o

10 11 12 13 14 15 16−0.4

0

0.4

0.8

1.2

log2(n)

E h

o

10 11 12 13 14 15 16

0.2

0.6

1

1.4

1.8

log2(n)E

Do

[48]

LINEARISATION EFFECT: CONJECTURE

• CRITICAL POINTS:8><>:

D±∗ = 0,

D(h±∗ ) = 0,

h±∗ = (dζ(q)/dq)q=q±∗

.

• RESULTS:

EI :

8><>:ζ(q, n) = d−D−

o + h−o q → d−D−∗ + h−∗ q, q ≤ q−∗ ,

ζ(q, n) → ζ(q), q−∗ ≤ q ≤ q+∗ ,

ζ(q, n) = d−D+o + h+

o q → d−D+∗ + h+

∗ q, q+∗ ≤ q.

EII&III :

(ζ(q, n) → ζ(q), 0 < q ≤ q+

∗ ,

ζ(q, n) = d−D+o + h+

o q → d−D+∗ + h+

∗ q, q+∗ ≤ q.

• ILLUSTRATION:

0 6 12 18

0

5

10

q

ζ(q)

theoest

[49]

LINEARISATION EFFECT: COMMENTS

WHEN DOES THE LINEARISATION EFFECT EXIST ?− FOR ALL TYPES OF CASCADES: CMC, CPC, IDC,− FOR ALL TYPES OF PROCESSES: Qr, A, VH, YH ,− FOR ALL NUMBERS OF VANISHING MOMENTS: N ≥ 1,− FOR ALL MRA-BASED ESTIMATORS: WAVELETS, INCREMENTS, AGGREGATION,− CAN BE WORKED OUT FOR q < 0,− EXTENDS TO DIMENSION HIGHER THAN d > 1.

[50]

EXTENSION: STANDARD WT VERSUS WTMM (1/3).

0 5 10 150

1

2

3

4

5

6

7

8

q

Est

imat

ed ζ

q

Standard vs WTMM

TheoCWTWTMMcWTMM

[51]

EXTENSION: 2D MULTIPLICATIVE CASCADE (2/3).

020

4060

0

20

40

600

5

10

15

20

−12 −8 −4 0 4 8 12 16 20−14

−12

−10

−8

−6

−4

−2

0

2

q

ζ(q)

theoest

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2

0

0.5

1

1.5

2

h

D(h

)

theoest

[52]

EXTENSION: 3D MULTIPLICATIVE CASCADE (3/3).

3D CMC (LOG NORMAL), EI(1) COMPARED TO A 2D SLICE.

−20 −15 −10 −5 0 5 10 15 20

−18

−16

−14

−12

−10

−8

−6

−4

−2

0

q

ζ(q)

theo2d3d

−1 −0.5 0 0.5 1 1.5−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

h

D(h

)

theo2d3d

[53]

LINEARISATION EFFECT: COMMENTS

WHEN DOES THE LINEARISATION EFFECT EXIST ?− FOR ALL TYPES OF CASCADES: CMC, CPC, IDC,− FOR ALL TYPES OF PROCESSES: Qr, A, VH, YH ,− FOR ALL NUMBERS OF VANISHING MOMENTS: N ≥ 1,− FOR ALL MRA-BASED ESTIMATORS: WAVELETS, INCREMENTS, AGGREGATION,− CAN BE WORKED OUT FOR q < 0,− EXTENDS TO DIMENSION HIGHER THAN d > 1.

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 < 0 < 1 < q+

c , q − 1 + ϕ(q) = 0,- q−c < q−∗ < 0 < 1 < q+

∗ < q+c ,

WHAT THE LINEARISATION EFFECT MIGHT BE:− MULTIPLICATIVE MARTINGALES ?− OSSIANDER, WAYMIRE 00, KAHANE, PEYRIERE 75, BARRAL, MANDELBROT 02.

[54]

LINEARISATION EFFECT: PICTURE

• TWO POWER-LAWS, TWO FUNCTIONS OF q:

− BARE CASCADE: IEQr(t)q= r

ϕ(q), q ∈ R.

− DRESSED CASCADE:

IETQ0(t, a; β0)

q = cq|a|ζ(q), q ∈ [q−c , q+c ],

IETQ0(t, a; β0)

q = ∞, ELSE,

ffWITH:

ζ(q) = 1 + qh−∗ , q ∈ [q−c , q−∗ ],

ζ(q) = ϕ(q), q ∈ [q−∗ , q+∗ ],

ζ(q) = 1 + 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]

LINEARISATION EFFECT: SKETCHED VIEWS

q−c

q+c

q−*

q+*0 1−1

q

q

q

Moments

E Aτ(t)q =

Estimated ζ(q,n)

Estimated ζ(q,n)

EI

EII & EIII

ζ(q)1+qh−*

1+qh+*

ζ(q) 1+qh+*

∞ ∞< ∞ < ∞cq |τ |ζ(q)

[56]

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 [am, aM ]

− 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 ζ(q, n),− DISCRIMINATION BETWEEN MONOFRACTAL AND MULTIFRACTAL,

[57]

NEGATIVE VALUES OF qS

DIFFICULTIES ?− FINITENESS ? Sq(j) = (1/nj)

Pnjk=1 |dX(j, k)|q <∞?

− NUMERICAL INSTABILITY ? dX(j, k) ' 0 → |dX(j, k)|q = ∞− THEORY ? FULL MULTIFRACTAL SPECTRUM

−15 −10 −5 0 5 10 15−15

−10

−5

0

q

ζ(q)

−15 −10 −5 0 5 10 15−15

−10

−5

0

5

10

q

ζ(q)

SOLUTIONS ?

[58]

NEGATIVE VALUES OF qS - SOLUTION 1

AGGREGATION: TX(a, t) = 1aT0

R t+aT0t

X(u)du

−15 −10 −5 0 5 10 15−15

−10

−5

0

q

ζ(q)

APPLIES ONLY TO POSITIVE DATA (MEASURE)

[59]

NEGATIVE VALUES OF qS - SOLUTION 2

WT MODULUS MAXIMA (ARNEODO ET AL.)

LX(a, tk) = SUPa′<a|TX(a′, tk(a′))|

0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5WTMM

log 2(a

)

Time

log 2(a

)

Skeleton of the Wavelet Transform

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

−15 −10 −5 0 5 10 15−15

−10

−5

0

5

10

q

ζ(q)

COMPUTATIONALLY EXPENSIVE

[60]

NEGATIVE VALUES OF qS - SOLUTION 3

WAVELET LEADERS: (JAFFARD ET AL.)

dX(j, k) → LX(j, k) = SUPj′<jdX(j′, 2−j′)

−15 −10 −5 0 5 10 15−15

−10

−5

0

5

10

q

ζ(q)

COMPUTATIONALLY EFFICIENT AND EXCELLENT STATISTICAL PERFORMANCE

[61]

BEYOND POWER LAWS

• SELF-SIMILARITY:IE|dX(j, k)|q = Cq(2

j)qH = Cq exp(qH ln 2j)

- POWER LAWS,- ∀2j (FOR ALL SCALES),- ∀q/IE|dX(j, k)|q <∞,- A SINGLE PARAMETER H

- ADDITIVE STRUCTURE.• MULTIFRACTAL

IE|dX(j, k)|q = Cq(2j)ζ(q) = Cq exp(ζ(q) ln 2j)

- POWER LAWS,- ∀2j < L, (FOR FINE SCALES ONLY, IN THE LIMIT 2j → 0,)- ∀q?- A WHOLE COLLECTION OF SCALING PARAMETER ζ(q)

- MULTIPLICATIVE STRUCTURE.

• BEYOND POWER LAWS : WARPED INF. DIV. CASCADES

IE|dX(j, k)|q = Cq(2j)qH = Cq exp(qH ln 2j)

IE|dX(j, k)|q = Cq(2j)ζ(q) = Cq exp(ζ(q) ln 2j)

IE|dX(j, k)|q = = Cq exp(ζ(q)n(2j))

→ VISIT PIERRE CHAINAIS’S POSTER

[62]

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— www.cubinlab.ee.mu.oz.au/ ∼darryl— fraclab— www.isima.fr/ ∼chainais

[63]