NAVIGATING SCALING: MODELLING AND ANALYSING · Wavelet And Multifractal Analysis, Cargese, France,...
Transcript of NAVIGATING SCALING: MODELLING AND ANALYSING · Wavelet And Multifractal Analysis, Cargese, France,...
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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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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 ?
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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.
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IRREGULARITIES, VARIABILITIES
SCALING OR NON STATIONARITIES?
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SCALING ?
temps (s)
Trafic (WAN) Internet
nb c
onne
xion
s
temps (s)
Trafic (WAN) Internet
nb c
onne
xion
s
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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)
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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
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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.
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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
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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
)
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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),- . . .
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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
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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.
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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
?
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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
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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
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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
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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.
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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
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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.
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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).
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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
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WAVELETS AND LONG RANGE DEPENDENCE
(Flandrin)H = 0.15
Haa
rD
aube
chie
s2
H = 0.5 H = 0.95
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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 <∞.
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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
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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
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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
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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
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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
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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.
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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 !
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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
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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)
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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.
• ?
• ?
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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.
• ?
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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.
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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.
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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.
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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
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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
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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
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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
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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
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ESTIMATING THE PARTITION FUNCTION SUPPORT
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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)
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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)
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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.
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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.
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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
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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
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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.
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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
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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
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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
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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.
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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 ].
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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)
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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,
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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 ?
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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)
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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
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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
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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
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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
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