Wavelet techniques for p-exponent
multifractal analysis
Stéphane Jaffard
Université Paris Est
based on joint works with
Patrice Abry, Roberto Leonarduzzi, Clothilde Melot,
Stéphane Roux, Marı́a Eugenia Torres, Herwig Wendt
Harmonic Analysis, Probability
and Applications
Orléans University, June 10-13, 2014
Type of data concerned with Multifractal Analysis
70
Jet turbulence Eulerian velocity signal (ChavarriaBaudetCiliberto95)
∆ = 3.2 ms
35
0
Fully developed turbulence
567!689
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Euro vs Dollar (2001-2009)
Everywhere irregular data
0
300
temps (s)
Internet trafic
600
900
Type of data concerned with Multifractal Analysis
70
Jet turbulence Eulerian velocity signal (ChavarriaBaudetCiliberto95)
∆ = 3.2 ms
35
0
Fully developed turbulence
0
300
temps (s)
600
Internet trafic
567!689
!&(
!&"
!&'
!&%
!&$
!&#
!
!!&#
!!&$
!
"!!
#!!!
#"!!
$!!!
)*+,-./01234
$"!!
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Euro vs Dollar (2001-2009)
Everywhere irregular data
Data that share the same statistical properties
should be classified as identical
900
Pointwise exponent
One associates to such data a pointwise regularity exponent h(x)
which describes how the regularity fluctuates from point to point
Pointwise exponent
One associates to such data a pointwise regularity exponent h(x)
which describes how the regularity fluctuates from point to point
Examples :
Let µ be a Probability measure on Rd and x0 ∈ Rd
µ ∈ M α (x0 ) if there exist C > 0 such that
|µ(B(x0 , r ))| ≤ C r α
The local dimension of µ at x0 is hµ (x0 ) = sup{α : µ ∈ M α (x0 )}
Pointwise exponent
One associates to such data a pointwise regularity exponent h(x)
which describes how the regularity fluctuates from point to point
Examples :
Let µ be a Probability measure on Rd and x0 ∈ Rd
µ ∈ M α (x0 ) if there exist C > 0 such that
|µ(B(x0 , r ))| ≤ C r α
The local dimension of µ at x0 is hµ (x0 ) = sup{α : µ ∈ M α (x0 )}
Let f be a locally bounded function Rd → R and x0 ∈ Rd
f ∈ C α (x0 ) if there exist C > 0 and a polynomial P such that
|f (x) − P(x − x0 )| ≤ C|x − x0 |α
The Hölder exponent of f at x0 is hf (x0 ) = sup{α : f ∈ C α (x0 )}
Difficulty to use directly the pointwise regularity
exponent for classification
For classical models, such exponents are extremely erratic :
The Hölder exponent
of “most” Lévy processes
The Local dimension
of multiplicative cascades
Difficulty to use directly the pointwise regularity
exponent for classification
For classical models, such exponents are extremely erratic :
The Hölder exponent
of “most” Lévy processes
The Local dimension
of multiplicative cascades
The function h is
everywhere discontinuous
Difficulty to use directly the pointwise regularity
exponent for classification
For classical models, such exponents are extremely erratic :
The Hölder exponent
of “most” Lévy processes
The Local dimension
of multiplicative cascades
The function h is
everywhere discontinuous
Goal : Recover some information on h(x) from (time or space)
averaged quantities that are numerically computable on a sample
path of the process, or on real-life data
A general framework : Admissible exponents
k1 k1 + 1
Dyadic cubes : λ =
,
2j
2j
kd kd + 1
× ··· ×
,
2j
2j
λj (x0 ) denotes the dyadic cube of scale j that contains x0
Dyadic cubes at scale j : Λj = {λ :
|λ| = 2−j }
A general framework : Admissible exponents
k1 k1 + 1
Dyadic cubes : λ =
,
2j
2j
kd kd + 1
× ··· ×
,
2j
2j
λj (x0 ) denotes the dyadic cube of scale j that contains x0
Dyadic cubes at scale j : Λj = {λ :
|λ| = 2−j }
Definition : A positive sequence (dλ ) is a hierarchical sequence if
0
∃α ∈ R such that if λ0 ⊂ λ then 2−αj dλ0 ≤ 2−αj dλ
!
log dλj (x0 )
The exponent h defined by :
h(x0 ) = lim inf
j→+∞
log(2−j )
is called an admissible pointwise exponent
A general framework : Admissible exponents
k1 k1 + 1
Dyadic cubes : λ =
,
2j
2j
kd kd + 1
× ··· ×
,
2j
2j
λj (x0 ) denotes the dyadic cube of scale j that contains x0
Dyadic cubes at scale j : Λj = {λ :
|λ| = 2−j }
Definition : A positive sequence (dλ ) is a hierarchical sequence if
0
∃α ∈ R such that if λ0 ⊂ λ then 2−αj dλ0 ≤ 2−αj dλ
!
log dλj (x0 )
The exponent h defined by :
h(x0 ) = lim inf
j→+∞
log(2−j )
is called an admissible pointwise exponent
Proposition : An admissible exponent h(x) is the liminf of a family
of continuous functions
Multifractal Analysis
Notation : 3λ denote the cube of same center as λ and three times
wider (it is the union of λ and its 3d − 1 immediate neighbours)
Examples :
If µ is a probability measure, then hµ is admissible : Take dλ = µ(3λ)
Multifractal Analysis
Notation : 3λ denote the cube of same center as λ and three times
wider (it is the union of λ and its 3d − 1 immediate neighbours)
Examples :
If µ is a probability measure, then hµ is admissible : Take dλ = µ(3λ)
If f is a locally bounded function and if ∀x, hf (x) < 1 , then hf is
admissible : Take dλ = sup f (x) − inf f (x)
x∈3λ
x∈3λ
Multifractal Analysis
Notation : 3λ denote the cube of same center as λ and three times
wider (it is the union of λ and its 3d − 1 immediate neighbours)
Examples :
If µ is a probability measure, then hµ is admissible : Take dλ = µ(3λ)
If f is a locally bounded function and if ∀x, hf (x) < 1 , then hf is
admissible : Take dλ = sup f (x) − inf f (x)
x∈3λ
x∈3λ
Multifractal spectrum :
The isohölder sets are the sets
EH = {x0 :
h(x0 ) = H}
Multifractal Analysis
Notation : 3λ denote the cube of same center as λ and three times
wider (it is the union of λ and its 3d − 1 immediate neighbours)
Examples :
If µ is a probability measure, then hµ is admissible : Take dλ = µ(3λ)
If f is a locally bounded function and if ∀x, hf (x) < 1 , then hf is
admissible : Take dλ = sup f (x) − inf f (x)
x∈3λ
x∈3λ
Multifractal spectrum :
The isohölder sets are the sets
EH = {x0 :
h(x0 ) = H}
The multifractal associated with the exponent h is
D(H) = dim (EH )
where dim stands for the Hausdorff dimension (dim (∅) = −∞)
Multifractal formalism
The scaling function associated with a hierarchic sequence (dλ ) is
defined by
X
∀q ∈ R,
2−dj
|dλ |q ∼ 2−η(q)j
λ∈Λj
Multifractal formalism
The scaling function associated with a hierarchic sequence (dλ ) is
defined by
X
∀q ∈ R,
2−dj
|dλ |q ∼ 2−η(q)j
λ∈Λj
Stability requirement : Invariant with respect to smooth
deformations
Multifractal formalism
The scaling function associated with a hierarchic sequence (dλ ) is
defined by
X
∀q ∈ R,
2−dj
|dλ |q ∼ 2−η(q)j
λ∈Λj
Stability requirement : Invariant with respect to smooth
deformations
Since η is a concave function, there is no loss of information in rather
considering the Legendre Spectrum
L(H) = inf (d + Hq − η(q))
q∈R
Theorem : Let (dλ ) be an admissible sequence
∀H ∈ R,
D(H) ≤ inf (d + Hq − η(q))
q∈R
Alternative admissible sequences for the Hölder
exponent
Alternative admissible sequences for the Hölder
exponent
A wavelet basis on R is generated
by a smooth, well localized, oscillating
function ψ such that the
j, k ∈ Z
ψ(2j x − k ),
form an orthogonal basis of L2 (R)
∀f ∈ L2 (R),
XX
f (x) =
cj,k ψ(2j x − k )
j∈Z k∈Z
where Z
cj,k = 2j f (x) ψ(2j x − k ) dx
Daubechies Wavelet
Credit to : http ://www.kfs.oeaw.ac.at/content/blogcategory/0/502/lang,8859-1/
Notations for wavelets on R
Dyadic intervals
k k +1
λ= j,
2
2j
Wavelets
ψλ (x) = ψ(2j x − k )
Wavelet coefficients
cλ = 2
j
Z
R
f (x)ψ(2j x − k)dx
Dyadic intervals at scale j
Λj = {λ :
|λ| = 2−j }
Wavelet expansion of f
f (x) =
XX
j
λ∈Λj
cλ ψλ (x)
Wavelets in 2 variables
In 2D, the wavelets used are tensor products :
ψ 1 (x, y ) = ψ(x)ϕ(y)
ψ 2 (x, y ) = ϕ(x)ψ(y )
ψ 3 (x, y) = ψ(x)ψ(y)
Notations
l (l + 1)
k (k + 1)
× j,
λ= j,
2
2j
2
2j
Dyadic squares :
cλ = 22j
Z Z
Wavelet coefficients
f (x, y ) ψ i 2j x − k , 2j y − l dx dy
Wavelet leaders
Let f be a locally bounded function ; the wavelet leaders of f are
dλ = sup |cλ0 |
λ0 ⊂3λ
dλ = supλ’∈ 3 λ |cλ|
c(j, k)
...
2j
j+1
2
2j+2
...
λ’∈ 3 λ
Computation of 2D wavelet leaders
Proposition : Let f be a uniform Hölder function (f ∈ C ε (Rd ) for an
ε > 0). If one uses the wavelet leaders dλ for hierarchical sequence,
then the associated pointwise exponent is the Hölder exponent
Uniform Hölder regularity
How can one check that the data correspond to a locally bounded
function ?
Hölder spaces : Let α ∈ (0, 1) ; f ∈ C α (Rd ) if f ∈ L∞ and
∃C, ∀x, y :
∀α ∈ R,
|f (x) − f (y)| ≤ C · |x − y |α
α
C α (Rd ) = B∞
(Rd )
Uniform Hölder regularity
How can one check that the data correspond to a locally bounded
function ?
Hölder spaces : Let α ∈ (0, 1) ; f ∈ C α (Rd ) if f ∈ L∞ and
∃C, ∀x, y :
∀α ∈ R,
|f (x) − f (y)| ≤ C · |x − y |α
α
C α (Rd ) = B∞
(Rd )
The uniform Hölder exponent of f is
Hfmin = sup{α : f ∈ C α (Rd )}
Uniform Hölder regularity
How can one check that the data correspond to a locally bounded
function ?
Hölder spaces : Let α ∈ (0, 1) ; f ∈ C α (Rd ) if f ∈ L∞ and
∃C, ∀x, y :
∀α ∈ R,
|f (x) − f (y)| ≤ C · |x − y |α
α
C α (Rd ) = B∞
(Rd )
The uniform Hölder exponent of f is
Hfmin = sup{α : f ∈ C α (Rd )}
Numerical computation from the wavelet coefficients
Let
ωj = sup |cλ |
λ∈Λj
then
Hfmin = lim inf
j→+∞
log(ωj )
log(2−j )
Uniform Hölder regularity
How can one check that the data correspond to a locally bounded
function ?
Hölder spaces : Let α ∈ (0, 1) ; f ∈ C α (Rd ) if f ∈ L∞ and
|f (x) − f (y)| ≤ C · |x − y |α
∃C, ∀x, y :
∀α ∈ R,
α
C α (Rd ) = B∞
(Rd )
The uniform Hölder exponent of f is
Hfmin = sup{α : f ∈ C α (Rd )}
Numerical computation from the wavelet coefficients
Let
ωj = sup |cλ |
λ∈Λj
then
Hfmin = lim inf
j→+∞
Hfmin > 0
=⇒ f is continuous
Hfmin < 0
=⇒ f is not locally bounded
log(ωj )
log(2−j )
Is Hmin > 0 fulfilled in applications ?
Is Hmin > 0 fulfilled in applications ?
Pkt Count Traffic ! Japon to US ! Backbone Link ! 2005/09/22
80
Internet Traffic
70
60
50
40
30
20
10
0
0
2
4
6
8
10
Time (min)
12
14
16
18
20
hmin LogScale
5
!0.46412
4
wtype 0
3
j:[3!10]
log2 sup [d(j,.)]
2
1
0
Hmin = −0.46
!1
!2
!3
0
5
10
j
15
Heartbeat Intervals
RR Interval ! human ! patient
1200
1150
1100
1050
RRI (ms)
1000
950
900
850
800
750
700
0
1
2
3
4
5
Time (min)
h
min
6
7
8
9
10
LogScale
7
!0.5547
6.5
wtype 0
6
j:[2!6]
log2 sup [d(j,.)]
5.5
5
4.5
Hmin = −0.55
4
3.5
3
1
1.5
2
2.5
3
3.5
j
4
4.5
5
5.5
6
−1
hmin = −0.18079
−2
−3
−4
0
2
4
6
8
−1
h
min
= −0.23042
−2
−3
−4
−5
0
2
4
6
8
Pointwise regularity with negative exponents ?
Pointwise Hölder regularity : f ∈ C α (x0 ) if
|f (x) − P(x − x0 )| ≤ C|x − x0 |α
If α < 0, this definition implies that, outside of x0 , f is locally bounded
Pointwise regularity with negative exponents ?
Pointwise Hölder regularity : f ∈ C α (x0 ) if
|f (x) − P(x − x0 )| ≤ C|x − x0 |α
If α < 0, this definition implies that, outside of x0 , f is locally bounded
=⇒ it could be used only to define isolated singularities of negative
exponent
Pointwise regularity with negative exponents ?
Pointwise Hölder regularity : f ∈ C α (x0 ) if
|f (x) − P(x − x0 )| ≤ C|x − x0 |α
If α < 0, this definition implies that, outside of x0 , f is locally bounded
=⇒ it could be used only to define isolated singularities of negative
exponent
Which definition of pointwise regularity would allow for
negative exponents ?
Pointwise regularity with negative exponents ?
Pointwise Hölder regularity : f ∈ C α (x0 ) if
|f (x) − P(x − x0 )| ≤ C|x − x0 |α
If α < 0, this definition implies that, outside of x0 , f is locally bounded
=⇒ it could be used only to define isolated singularities of negative
exponent
Which definition of pointwise regularity would allow for
negative exponents ?
Clue : The definition of pointwise Hölder regularity can be rewritten
f ∈ C α (x0 ) ⇐⇒ sup |f (x) − P(x − x0 )| ≤ Cr α
B(x0 ,r )
Pointwise regularity with negative exponents ?
Pointwise Hölder regularity : f ∈ C α (x0 ) if
|f (x) − P(x − x0 )| ≤ C|x − x0 |α
If α < 0, this definition implies that, outside of x0 , f is locally bounded
=⇒ it could be used only to define isolated singularities of negative
exponent
Which definition of pointwise regularity would allow for
negative exponents ?
Clue : The definition of pointwise Hölder regularity can be rewritten
f ∈ C α (x0 ) ⇐⇒ sup |f (x) − P(x − x0 )| ≤ Cr α
B(x0 ,r )
Definition (Calderón and Zygmund) : Let f ∈ Lp (Rd ) ; f ∈ Tαp (x0 ) if
there exists a polynomial P such that for r small enough,
!1/p
Z
1
p
|f (x) − P(x − x0 )| dx
≤ Cr α
r d B(x0 ,r )
The p-exponent
Definition : Let f ∈ Lp (Rd ) ; f ∈ Tαp (x0 ) if there exists a polynomial P
such that for r small enough,
!1/p
Z
1
|f (x) − P(x − x0 )|p dx
≤ Cr α
r d B(x0 ,r )
The p-exponent
Definition : Let f ∈ Lp (Rd ) ; f ∈ Tαp (x0 ) if there exists a polynomial P
such that for r small enough,
!1/p
Z
1
|f (x) − P(x − x0 )|p dx
≤ Cr α
r d B(x0 ,r )
The p-exponent of f at x0 is hp (x0 ) = sup{α : f ∈ Tαp (x0 )}
The p-spectrum of f is d p (H)
= dim ({x0 : hp (x0 ) = H})
The p-exponent
Definition : Let f ∈ Lp (Rd ) ; f ∈ Tαp (x0 ) if there exists a polynomial P
such that for r small enough,
!1/p
Z
1
|f (x) − P(x − x0 )|p dx
≤ Cr α
r d B(x0 ,r )
The p-exponent of f at x0 is hp (x0 ) = sup{α : f ∈ Tαp (x0 )}
The p-spectrum of f is d p (H)
= dim ({x0 : hp (x0 ) = H})
Remarks :
I
The case p = +∞ corresponds to pointwise Hölder regularity
I
The normalization is chosen so that a cusp |x − x0 |α has the
same p-exponent for all p : hp (x0 ) = α (as long as α ≥ −d/p)
How can one check that the data belong to Lp ?
The wavelet scaling function is informally defined by
X
∀p > 0
2−dj
|cλ |p ∼ 2−ζf (p)j
λ∈Λj
How can one check that the data belong to Lp ?
The wavelet scaling function is informally defined by
X
∀p > 0
2−dj
|cλ |p ∼ 2−ζf (p)j
λ∈Λj
Besov spaces : Let p > 0 ; f ∈ Bps,∞ (Rd ) if
∃C, ∀j :
2−dj
X
λ∈Λj
|cλ |p ≤ C · 2−spj
How can one check that the data belong to Lp ?
The wavelet scaling function is informally defined by
X
∀p > 0
2−dj
|cλ |p ∼ 2−ζf (p)j
λ∈Λj
Besov spaces : Let p > 0 ; f ∈ Bps,∞ (Rd ) if
2−dj
∃C, ∀j :
λ∈Λj

j→+∞
|cλ |p ≤ C · 2−spj

log 2−dj
ζf (p) = lim inf
X
X
λ∈Λj
|cλ |p 
log(2−j )
n
o
= p · sup s : f ∈ Bps,∞ (Rd )
How can one check that the data belong to Lp ?
The wavelet scaling function is informally defined by
X
∀p > 0
2−dj
|cλ |p ∼ 2−ζf (p)j
λ∈Λj
Besov spaces : Let p > 0 ; f ∈ Bps,∞ (Rd ) if
2−dj
∃C, ∀j :
λ∈Λj

j→+∞
I
I
|cλ |p ≤ C · 2−spj

log 2−dj
ζf (p) = lim inf
X
X
λ∈Λj
|cλ |p 
log(2−j )
If ζf (p) > 0, then f ∈ Lp
If ζf (p) < 0, then f ∈
/ Lp
n
o
= p · sup s : f ∈ Bps,∞ (Rd )
Wavelet scaling functions of synthetic images
Wavelet scaling function ζf (p) :
X
2−2j
|cλ |p ∼ 2−ζf (p) j
λ∈Λj
Van Koch snowflake : ζf (p) = 2−
!4
!10
1
0.5
!f(1)=1.03
!12
!14
!4
!6
!8
Scale (s)
2
4
8
16 32 64 128 256
Koch snowflake + polynomial
log2 Sf(1,j)
.
0
1.5
1
p
1
2
3
!f(1)=0.75
Scale (s)
2
4
8
16 32 64 128 256
0
4
5
Koch snowflake + polynomial
0.5
!10
!12
disc + polynomial
!f(p)
!8
1.5
disc + polynomial
log2 Sf(1,j)
!6
log 4
∼ 0.74
log 3
!f(p)
Disk : ζf (p) = 1
p
Properties of p-exponents
Gives a mathematical framework to the notion of negative regularity
exponents
The p-exponent satisfies : hp (x0 ) ≥ − dp
Properties of p-exponents
Gives a mathematical framework to the notion of negative regularity
exponents
The p-exponent satisfies : hp (x0 ) ≥ − dp
p-exponents may differ :
Theorem : Let f be an L1 function, and x0 ∈ Rd . Let
p0 = sup{p : f ∈ Lploc (Rd ) in a neighborhood of x0 }
The function p → hp (x0 ) is defined on [1, p0 ) and possesses the
following properties :
1. It takes values in [−d/p, ∞]
2. It is a decreasing function of p.
3. The function r → h1/r (x0 ) is concave.
Furthermore, Conditions 1 to 3 are optimal.
When do p-exponents coincide ?
Notation : hp,γ (x0 ) denotes the p-exponent of the fractional integral of
f of order s at x0
When do p-exponents coincide ?
Notation : hp,γ (x0 ) denotes the p-exponent of the fractional integral of
f of order s at x0
Definition : f is a cusp of exponent h at x0 if ∃p, γ > 0 such that
I
hp (x0 ) = h
I
hp,γ (x0 ) = h + γ
When do p-exponents coincide ?
Notation : hp,γ (x0 ) denotes the p-exponent of the fractional integral of
f of order s at x0
Definition : f is a cusp of exponent h at x0 if ∃p, γ > 0 such that
I
hp (x0 ) = h
I
hp,γ (x0 ) = h + γ
Theorem : This notion is independent of p and γ
Types of pointwise singularities (Yves Meyer)
Typical pointwise singularities :
Cusps : f (x) − f (x0 ) = |x − x0 |H
Types of pointwise singularities (Yves Meyer)
Typical pointwise singularities :
Cusps : f (x) − f (x0 ) = |x − x0 |H
After one integration :
f (−1) (x) − f (−1) (x0 ) ∼
1
H |x
− x0 |H+1
Types of pointwise singularities (Yves Meyer)
Typical pointwise singularities :
Cusps : f (x) − f (x0 ) = |x − x0 |H
After one integration :
f (−1) (x) − f (−1) (x0 ) ∼
1
H |x
− x0 |H+1
H
Oscillating singularity : f (x) − f (x0 ) = |x − x0 | sin
After one integration :
f (−1) (x) − f (−1) (x0 ) =
|x − x0 |H+(1+β)
cos
β
1
|x − x0 |β
1
|x − x0 |β
+ ···
Types of pointwise singularities (Yves Meyer)
Typical pointwise singularities :
Cusps : f (x) − f (x0 ) = |x − x0 |H
After one integration :
f (−1) (x) − f (−1) (x0 ) ∼
1
H |x
− x0 |H+1
H
Oscillating singularity : f (x) − f (x0 ) = |x − x0 | sin
After one integration :
f (−1) (x) − f (−1) (x0 ) =
|x − x0 |H+(1+β)
cos
β
1
|x − x0 |β
More generally, after a fractional integration of order s,
I
If f has a cusp at x0 , then hI s f (x0 ) = hf (x0 ) + s
I
If f has an oscillating singularity at x0 , then
hI s f (x0 ) = hf (x0 ) + (1 + β)s
1
|x − x0 |β
+ ···
Further classification
Two types of oscillating singularities
:
“Full singularities” : |x − x0 |H sin
1
|x − x0 |β
(p-exponents coincide)
“Skinny singularities ” |x − x0 |H 1Eγ where
[1 1
1
Eγ =
, + γ
for γ > 1
n n n
(p-exponents differ)
Further classification
Two types of oscillating singularities
:
“Full singularities” : |x − x0 |H sin
1
|x − x0 |β
(p-exponents coincide)
“Skinny singularities ” |x − x0 |H 1Eγ where
[1 1
1
Eγ =
, + γ
for γ > 1
n n n
(p-exponents differ)
Characterization
oscillating singularities : F (x) = |x − x0 |H g
where g is
indefinitely oscillating
1
|x − x0 |β
+ r (x)
Further classification
Two types of oscillating singularities
:
“Full singularities” : |x − x0 |H sin
1
|x − x0 |β
(p-exponents coincide)
“Skinny singularities ” |x − x0 |H 1Eγ where
[1 1
1
Eγ =
, + γ
for γ > 1
n n n
(p-exponents differ)
Characterization
oscillating singularities : F (x) = |x − x0 |H g
where g is
indefinitely oscillating
1
|x − x0 |β
For “full singularities” g is “large at infinity”
For “skinny singularities ” g is “small at infinity”
+ r (x)
Admissible sequences for the p-exponent
Definition : Let f : Rd → R be locally in Lp ; the p-leaders of f are
!1/p
dλp
=
X
λ0 ⊂3λ
p d(j−j 0 )
|cλ0 | 2
where j 0 is the scale associated with the subcube λ0 included in 3λ
0
(i.e. λ0 has width 2−j ).
Admissible sequences for the p-exponent
Definition : Let f : Rd → R be locally in Lp ; the p-leaders of f are
!1/p
dλp
=
X
λ0 ⊂3λ
p d(j−j 0 )
|cλ0 | 2
where j 0 is the scale associated with the subcube λ0 included in 3λ
0
(i.e. λ0 has width 2−j ).
Theorem : (C. Melot)
If ηf (p) > 0, then hp is the admissible exponent associated with the
sequence dλp

log dλpj (x0 )

hp (x0 ) = lim inf 
j→+∞
log(2−j )

p-leaders and negative regularity
_=ï0.1
Cusp
.
x0
hp (x0 ) = −0.1
_=ï0.1
`=0.3
Chirp
.
x0
leaders : ĥ(x0 ) = 0.00
hp (x0 ) = −0.1
leaders : ĥ(x0 ) = 0.00
3.5
ï0.5
h=ï0.1
Cusp ïï leaders (p=inf)
h
3
=0
est
ï1
h=ï0.1
Chirp ïï leaders (p=inf)
h
=0
est
2.5
ï1.5
2
1.5
ï2
1
ï2.5
ï3
0.5
j
2
4
6
8
10
12
14
0
j
2
4
6
8
10
12
14
p-leaders : ĥ(x0 ) = −0.08
p-leaders : ĥ(x0 ) = −0.10
ï0.5
3.5
h=ï0.1
Cusp ïï pïleaders (p=2)
h
=ï0.08
est
ï1
3
h=ï0.1
Chirp ïï pïleaders (p=2)
h
=ï0.1
est
2.5
ï1.5
2
1.5
ï2
1
ï2.5
ï3
0.5
j
2
4
6
8
10
12
14
0
j
2
4
6
8
10
12
14
p-Multifractal Formalism
The p-scaling function is defined informally by
X p
∀q ∈ R,
2−dj
|dλ |q ∼ 2−ηp (q)j
λ∈Λj
p-Multifractal Formalism
The p-scaling function is defined informally by
X p
∀q ∈ R,
2−dj
|dλ |q ∼ 2−ηp (q)j
λ∈Λj


log 2−dj
ηp (q) = lim inf
j→+∞
X
λ∈Λj
|dλp |q 
log(2−j )
p-Multifractal Formalism
The p-scaling function is defined informally by
X p
∀q ∈ R,
2−dj
|dλ |q ∼ 2−ηp (q)j
λ∈Λj


log 2−dj
ηp (q) = lim inf
j→+∞
X
λ∈Λj
|dλp |q 
log(2−j )
Stability properties :
I
Invariant with respect to deformations
I
independent of the wavelet basis
The p-Legendre Spectrum is
Lp (H) = inf (d + Hq − ηp (q))
q∈R
√
MRW (H = 0.72, λ = 0.08)
(min)
p0 = +∞ hX
=0
D(h)
(min)
p0 = 10 hX
p0 = 2.5 hX
D(h)
0
h
0.5
1
−0.5
= −0.3
D(h)
h
−0.5
(min)
= −0.2
0
h
0.5
1
−0.5
0
0.5
1
√
MRW (H = 0.72, λ = 0.08)
(min)
p0 = +∞ hX
=0
D(h) ⌧ Px (p=
−0.5
8 4 2 1)
p = +∞, 8, 4, 2, 1
(min)
p0 = 10 hX
D(h)
p0 = 2.5 hX
h
0.5
1 −0.5
= −0.3
D(h)
h
0
(min)
= −0.2
0
h
0.5
1 −0.5
0
0.5
1
√
MRW (H = 0.72, λ = 0.08)
(min)
p0 = +∞ hX
=0
D(h) ⌧ Px (p=
−0.5
8 4 2 1)
p = +∞
(min)
p0 = 10 hX
D(h) ⌧ Px (p=
h
0
(min)
= −0.2
)
p0 = 2.5 hX
D(h)
h
0.5
1 −0.5
0
= −0.3
h
0.5
1 −0.5
0
0.5
1
√
MRW (H = 0.72, λ = 0.08)
(min)
p0 = +∞ hX
=0
D(h) ⌧ Px (p=
−0.5
8 4 2 1)
p = +∞, 8
(min)
p0 = 10 hX
D(h) ⌧ Px (p=
h
0
(min)
= −0.2
8)
p0 = 2.5 hX
D(h)
h
0.5
1 −0.5
0
= −0.3
h
0.5
1 −0.5
0
0.5
1
√
MRW (H = 0.72, λ = 0.08)
(min)
p0 = +∞ hX
=0
D(h) ⌧ Px (p=
−0.5
8 4 2 1)
p = +∞, 8, 4
(min)
p0 = 10 hX
D(h) ⌧ Px (p=
h
0
(min)
= −0.2
8 4)
p0 = 2.5 hX
D(h)
h
0.5
1 −0.5
0
= −0.3
h
0.5
1 −0.5
0
0.5
1
√
MRW (H = 0.72, λ = 0.08)
(min)
p0 = +∞ hX
=0
D(h) ⌧ Px (p=
−0.5
8 4 2 1)
p = +∞, 8, 4, 2, 1
(min)
p0 = 10 hX
D(h) ⌧ Px (p=
h
0
(min)
= −0.2
8 4 2 1)
p0 = 2.5 hX
D(h)
h
0.5
1 −0.5
0
= −0.3
h
0.5
1 −0.5
0
0.5
1
√
MRW (H = 0.72, λ = 0.08)
(min)
p0 = +∞ hX
=0
D(h) ⌧ Px (p=
−0.5
8 4 2 1)
p = +∞
(min)
p0 = 10 hX
D(h) ⌧ Px (p=
h
0
(min)
= −0.2
8 4 2 1)
p0 = 2.5 hX
D(h) ⌧ Px (p=
h
0.5
1 −0.5
0
= −0.3
)
h
0.5
1 −0.5
0
0.5
1
√
MRW (H = 0.72, λ = 0.08)
(min)
p0 = +∞ hX
=0
D(h) ⌧ Px (p=
−0.5
8 4 2 1)
p = +∞, 8
(min)
p0 = 10 hX
D(h) ⌧ Px (p=
h
0
(min)
= −0.2
8 4 2 1)
p0 = 2.5 hX
D(h) ⌧ Px (p=
h
0.5
1 −0.5
0
= −0.3
8)
h
0.5
1 −0.5
0
0.5
1
√
MRW (H = 0.72, λ = 0.08)
(min)
p0 = +∞ hX
=0
D(h) ⌧ Px (p=
−0.5
8 4 2 1)
p = +∞, 8, 4
(min)
p0 = 10 hX
D(h) ⌧ Px (p=
h
0
(min)
= −0.2
8 4 2 1)
p0 = 2.5 hX
D(h) ⌧ Px (p=
h
0.5
1 −0.5
0
= −0.3
8 4)
h
0.5
1 −0.5
0
0.5
1
√
MRW (H = 0.72, λ = 0.08)
(min)
p0 = +∞ hX
=0
D(h) ⌧ Px (p=
−0.5
8 4 2 1)
p = +∞, 8, 4, 2
(min)
p0 = 10 hX
D(h) ⌧ Px (p=
h
0
(min)
= −0.2
8 4 2 1)
p0 = 2.5 hX
D(h) ⌧ Px (p=
h
0.5
1 −0.5
0
= −0.3
8 4 2)
h
0.5
1 −0.5
0
0.5
1
√
MRW (H = 0.72, λ = 0.08)
(min)
p0 = +∞ hX
=0
D(h) ⌧ Px (p=
−0.5
8 4 2 1)
p = +∞, 8, 4, 2, 1
(min)
p0 = 10 hX
D(h) ⌧ Px (p=
h
0
(min)
= −0.2
8 4 2 1)
p0 = 2.5 hX
D(h) ⌧ Px (p=
h
0.5
1 −0.5
0
= −0.3
8 4 2 1)
h
0.5
1 −0.5
0
0.5
1
Advantages and drawbacks of multifractal analysis
based on the p-exponent
Advantages and drawbacks of multifractal analysis
based on the p-exponent
I
Allows to deal with larger collections of data
I
The estimation is not based on a unique extremal value, but on
an l p average ⇒ better statistical properties
I
Systematic bias in the estimation of p-leaders
Thank you for your attention !