self-similarity, symmetry and anisotropy in the ... · laha & rohatgi (1981), d & pipiras...

Self-similarity, symmetry and anisotropy inthe multivariate and multiparameter

settings

Gustavo Didier

Tulane University

Multifractal Analysis: From Theory to Applications and Back (BIRS)

February 2014

Collaborators

• Patrice Abry (ENS-Lyon)

• Changryong Baek (Sungkyunkwan University)

• Mark Meerschaert (Michigan State)

• Vladas Pipiras (University of North Carolina)

1

Outline

1. introduction

2. harmonizable representations

3. symmetry, exponents, anisotropy

4. wavelet analysis

2

1. Introduction

3

Introduction

why talk about Gaussian monofractals?!. . .

4

Introduction

why talk about Gaussian monofractals?!. . .

. . . because of multidimensionality!

5

Introduction

From a technical standpoint...

difficulty = αAn Analysis + αAlg Algebra + αTop Topology

• typical probability problem: αAn ≈ 1

• multidimensional problems: αAn ≈ αAlg � αTop > 0

6

Introduction

why multidimensional? (applications)

• (range) data is usually multivariate: anomalous diffusion particle

position (X(t)-Y (t)), climatology, Internet data comes in packets or

bytes within bins of certain duration. . .

• (domain) random fields have become a topic of great interest:

image processing (bone radiographic imaging), hydrology (aquifers),

geostatistics. . ., especially in the presence of anisotropy

7

Introduction

why multidimensional? (theory)

• (range): operator self-similar processes (Hudson & Mason (1982),

Laha & Rohatgi (1981), D & Pipiras (2011,2012), etc.)

• (domain): operator scaling measures, scalar s.s. random fields

(Samorodnitsky & Taqqu (1994), Meerschaert & Scheffler (2001),

Xiao (2009), and many, many others)

current stage of this research: carry out the synthesis between the

range and the domain, both in probability and statistics (Li & Xiao

(2011))

8

Introduction

inter-related ideas:

• symmetry and self-similarity/scaling are related in Probability

theory (⇒ multidimensionality )

• anisotropy and intrinsic physical symmetries

• (non)identifiability of the parametrization

identifiability (statistical inference): θ 7→ Pθ is injective

9

Introduction

isotropy means that. . .

{X(Ot)}t∈RmL= {X(t)}t∈Rm, O ∈ O(m)

i.e. every O ∈ O(m) is a (domain) symmetry of X

m = 1: time-reversibility

10

Harmonizable representations

11


def.(o.s.s. vector random fields): {X(t)}t∈Rm satisfies

{X(cEt)}t∈RmL= {cHX(t)}t∈Rm, E ∈M(m), H ∈M(n)

matrix exponentiation cM := exp(M log(c)) =∑∞k=0

(M log(c))k

k!

def.(operator fractional Brownian field (OFBF)): Gaussian, stationary

increment, o.s.s. random field

def.(operator fractional Brownian motion (OFBM)): m = 1, E = 1

12


example: FBM (H characterizes the law up to a constant):

EBH(s)BH(t) =EBH(1)2

2{|t|2H + |s|2H − |t− s|2H}

OFBM: EBH(t)BH(s)∗ + EBH(s)BH(t)∗

= |t|HΓ(1, 1)|t|H∗

+ |s|HΓ(1, 1)|s|H∗− |t− s|HΓ(1, 1)|t− s|H

∗,

Γ(1, 1) = EBH(1)BH(1)∗

It is not true in general that EBH(t)BH(s)∗ = EBH(s)BH(t)∗

13


goal: Fourier domain representations for OFBFs

polar coordinates (Bierme, Meerschaert & Scheffler (2007)): given a

matrix E (min <eig(E) > 0), there exists a norm ‖ · ‖0 such that

Rm 3 x = radialE ∗ spherical = τ(x)El(x) homeomorphically

where S0 = {x : ‖x‖0 = 1} = {x : τ(x) = 1}

14


theorem(Baek, D & Pipiras (2014)): For an OFBF {X(t)}t∈Rwith 0 < min <eig(H) ≤ max<eig(H) < min <eig(E) and density

fX(x),

{X(t)}t∈RmL={∫

R(ei〈t,x〉 − 1)τ(x)−HEf

1/2X (l(x))B(dx)

}t∈Rm

,

• HE = H + tr(E∗)I/2

• B(dx) := B1(dx) + iB2(dx) (complex-valued multivariate

Brownian measure satisfying B(−dx) = B(dx)).

15


in terms of the spectral measure. . .

fX(x) = τ(x)−HEfX(l(x))τ(x)−H∗E

(fX(x) is assumed to exist)

16


in terms of the spectral measure. . .

fX(x) = τ(x)−HEfX(l(x))τ(x)−H∗E

(fX(x) is assumed to exist)

m = 1 :

fX(x) =1

x2(x−D+ AA∗x−D

∗+ + x−D− AA∗x−D

∗− )

(fX(x) exists)

17


application(m = 1):

time reversibility: {X(t)}t∈RL= {X(−t)}t∈R

EBH(s)BH(t)∗

=1

2(|t|HΓ(1, 1)|t|H

∗+ |s|HΓ(1, 1)|s|H

∗− |t− s|HΓ(1, 1)|t− s|H

∗)

⇔ {BH(t)}t∈R is time-reversible

(... ⇒ the univariate reasoning is insufficient)

18


application(m = 1):

time reversibility: {X(t)}t∈RL= {X(−t)}t∈R

EBH(s)BH(t)∗

=1

2(|t|HΓ(1, 1)|t|H

∗+ |s|HΓ(1, 1)|s|H

∗− |t− s|HΓ(1, 1)|t− s|H

∗)

⇔ =(AA∗) = 0

in which case

fX(x) =1


∗+ + x−D− AA∗x−D

∗− ) =

1

x2|x|−DAA∗|x|−D

∗

19


application(n = 1):

Bierme, Meerschaert & Scheffler (2007):

• Xψ: OFBF with harmonizable representation based on the filter ψ

• Xϕ: OFBF with a MA representation based on the filter ϕ

ψ, ϕ are operator-homogeneous: ϕ(cEz) = cϕ(z)

20


application(n = 1):

Bierme, Meerschaert & Scheffler (2007): under isotropy, moving

average and harmonizable representations of (m, 1)-OFBFs yield

EXϕ(s)Xϕ(t) =1

2{‖s‖2Hcϕ + ‖t‖2Hcϕ − ‖s− t‖2Hcϕ},

EXψ(s)Xψ(t) =1

2{‖s‖2Hcψ + ‖t‖2Hcψ − ‖s− t‖2Hcψ},

⇒ XϕL= Xψ (up to a constant)

21


application(n = 1):

Bierme, Meerschaert & Scheffler (2007): however, under anisotropy,

we have

EXϕ(s)Xϕ(t) =1

2{τ(s)2Hσ2

l(s) + τ(t)2Hσ2l(t) − τ(s− t)2Hσ2

l(s−t)},

EXψ(s)Xψ(t) =1

2{τ(s)2Hω2

l(s) + τ(t)2Hω2l(t) − τ(s− t)2Hω2

l(s−t)},

open problem: under what conditions on ψ and ϕ do we have

XϕL= Xψ?

22


theorem(Baek, D & Pipiras (2014)) under assumptions, there is a

function ϕ : Rm→ R s.t.

(i) ϕ(t− ·)− ϕ(·) ∈ L2(Rm)

(ii) ϕ is E/(1− tr(E)/2)-homogeneous

(iii) {BE(t)}t∈RmL={∫

Rm ϕ(t− u)− ϕ(−u)B(du)}t∈Rm

where

ϕ(t− u)− ϕ(−u) =

∫Rme−i〈u,x〉(ei〈t,x〉 − 1)a(x)dx

23

Symmetry, exponents, anisotropy

24


non-linear = “not linear”: not very informative. . .

anisotropic: can we do better than say “not isotropic”?. . .

25


isotropy:

{X(Ot)}t∈RmL= {X(t)}t∈Rm, O ∈ O(m)

i.e. every O ∈ O(m) is a (domain) symmetry of X

overarching question: in what ways can the isotropy condition be

violated?

26


For any field X, the symmetry sets are. . .

Gran1 = {A ∈ GL(n,R); {AX(t)}t∈Rm

L= {X(t)}t∈Rm}

Gdom1 = {A ∈ GL(n,R); {X(At)}t∈Rm

L= {X(t)}t∈Rm}

27


For any field X, the symmetry sets are. . .

Gran1 = {A ∈ GL(n,R); {AX(t)}t∈Rm

L= {X(t)}t∈Rm}

Gdom1 = {A ∈ GL(n,R); {X(At)}t∈Rm

L= {X(t)}t∈Rm}

Gdom1 and Gran

1 are compact subgroups of GL(n,R)

⇒ G•1 = WO0W−1, O0 ⊆ O(n)

28


under mild conditions, for any o.s.s. random field X. . .

• Hudson & Mason (1982), Li & Xiao (2011): given E, there exists

H such that. . .

• D, Meerschaert & Pipiras (2014): given H, there exists E such

that. . .

{X(cEt)}t∈RmL= {cHX(t)}t∈Rm

29


under mild conditions, for any o.s.s. random field X. . .

Furthermore, it turns out that. . .

{X(cE+?t)}t∈RmL= {cH+?′X(t)}t∈Rm

question: why should exponents be non-unique in the first place?

30


example: {BH(t)}t∈R is a vector of uncorrelated FBMs with the

same scalar parameter h (H = hI)

For L ∈ so(n), cL ∈ SO(n) since

cL(cL)∗ = cL+L∗

= I and det(cL) = etr(L log(c)) = 1

{BH(ct)}t∈RL= {cHBH(t)}t∈R

L= {cHcLBH(t)}t∈R

L= {cH+LBH(t)}t∈R

⇒ both H and H + L are exponents for BH

31


theorem(D, Meerschaert & Pipiras (2014)): under mild conditions,

for any o.s.s. random field X. . .

• E ran(X) = H + T (Gran1 )

• Edom(X) = E + T (Gdom1 )

T (G•1) ={

limk→∞

Ak − Idk

, some {Ak} ⊆ G1, some 0 6= dk → 0}

32






33






example: OFGN (non-identifiable parametrization):

gYH(x) ∼ x−DAA∗x−D∗, x→ 0

34






example: OFGN (non-identifiable parametrization):

gYH(x) ∼ x−D(Gran1 )(AA∗)(Gran

1 , Gdom1 )x−D(Gran

1 )∗, x→ 0

35


To recap:

• Gran1 = WO0W

−1, E ran(X) = H + T (Gran1 )

• Gdom1 = W ′O′0W ′−1, Edom(X) = E + T (Gdom

1 )

question: domain and range symmetries look quite analogous. Are

they of the same nature, and thus amenable to the same kind of

technique?

36


To recap:

• Gran1 = WO0W

−1, E ran(X) = H + T (Gran1 )

• Gdom1 = W ′O′0W ′−1, Edom(X) = E + T (Gdom

1 )

question: domain and range symmetries look quite analogous. Are

they of the same nature, and thus amenable to the same kind of

technique?

answer: no, they are completely different

37


fact (later): SO(2) cannot be a domain symmetry group

theorem (types): For proper Gaussian random fields in R2,

Gran1∼=...

(i) ... {±I} (cyclic)

(ii) ... {±I,±diag(1,−1)} (dyhedral)

(iii) ... SO(2) (rotational)

(iv) ... O(2) (full)

38


example: blind source separation

X(t) = (X1(t), X2(t))∗: independent, unobservable FGN signals with

parameters d1, d2

P ∈ GL(2,R): mixing matrix

{Y (t)} = {PX(t)}: mixed signal, observable

goal: retrieve X, i.e., estimate P

(hope: P−1Y (t) ≈ X(t))

39


proposition:

(a) d1 < d2⇒ Gran1 = {±I,±diag(1,−1)}

(b) d1 = d2⇒ Gran1 = O(2)

Then

(a) {demixing matrices} = {Q = PO,O ∈ {±I,±diag(1,−1)}}

(b) {demixing matrices} = {Q = PO,O ∈ O(2)}

... ⇒ any estimation procedure estimates P up to a factor O ∈ Gran1

question: consequences for asymptotics? (PP→?)

40


question: why are Gdom1 , Gran

1 so different?

41


range:

X: Gaussian, W = I, Gran1 ⊆ O(n)

O ∈ Gran1 ⇔ E[OX(s)X(t)∗O∗] = EX(s)X(t)∗, s, t ∈ Rm

⇔ OEX(s)X(t)∗ = EX(s)X(t)∗O, s, t ∈ R

⇒ Gran1 = centralizer of {Γ(s, t)}s,t∈Rm

42


domain(example): E = I ∈M(2,R), H = β − 1 ∈ R

R 3 EX(s)X(t) =

∫R2

(ei〈s,x〉 − 1)(e−i〈t,x〉 − 1)‖x‖−2β1 dx

⇒

{X(At)} L= {X(t)} ⇔ ‖(A∗)−1x‖−2β1 = ‖x‖−2β1 , x ∈ Rm\{0}

⇔∥∥∥(A∗)−1

x

‖x‖1

∥∥∥1

= 1, x ∈ Rm\{0}

⇒ Gdom1 = symmetry group of S‖·‖1 = symmetry group of a rhombus

43


what is the symmetry group of?. . .

44


= Gdom1 (E = I,H = β − 1)

45


• range: A ∈ Gran1 ⇔ AFX(dx)A∗ = FX(dx) (commutativity)

• domain: A ∈ Gdom1 ⇔ FX((A∗)−1dx) = FX(dx) (measure)

46


domain:(Meerschaert & Veeh (1995))

example: SO(2)x = O(2)x, x ∈ Rm ⇒ [SO(2)] = [O(2)]

⇒ SO(2) is not maximal in [O(2)]

47


domain:(Meerschaert & Veeh (1995))

example: SO(2)x = O(2)x, x ∈ Rm ⇒ [SO(2)] = [O(2)]

⇒ SO(2) is not maximal in [O(2)]

answer to the overarching question:

theorem(D, Meerschaert & Pipiras (2014)): the domain symmetry

group Gdom1 of OFBFs is maximal (this fully characterizes all types of

anisotropy)

corollary: SO(2) cannot be a domain symmetry group

48


can we convey a parametric characterization of isotropy?

question:

(m,n)-OFBF: FX(dx) = r−HΛ(dθ)r−H∗r−1dr

isotropy

?⇔

(radial) ∃η > 0 such that E0 = ηI ∈ Edom(X)

49


can we convey a parametric characterization of isotropy?

theorem(D, Meerschaert & Pipiras (2014)):

(m,n)-OFBF: FX(dx) = r−HΛ(dθ)r−H∗r−1dr

isotropy

⇔

(radial) ∃η > 0 such that E0 = ηI ∈ Edom(X)

(spherical) Λ(dθ) = Λ(O∗dθ), O ∈ O(m), S0 = c−10 Sm−1

50


issue: how to incorporate departures from isotropy (“poorer” Gdom1 )

into the analysis of physical systems?

51


illustration: (2, n)-OFBFs. All possible domain symmetry groups are

(i) Cn = {Ok2π/n : k = 1, . . . , n}, n ∈ N (cyclic)

(ii) Dn = {Ok2π/n, Fk2π/n : k = 1, . . . , n}, n ∈ N (dihedral)

(iii) D∗1 = {I, diag(−1, 1)}

(iv) O(2)

This is a full description of anisotropy. . .

52


illustration: (2, n)-OFBFs. All possible domain symmetry groups are

(i) Cn = {Ok2π/n : k = 1, . . . , n}, n ∈ N (cyclic)

(ii) Dn = {Ok2π/n, Fk2π/n : k = 1, . . . , n}, n ∈ N (dihedral)

(iii) D∗1 = {I, diag(−1, 1)}

(iv) O(2)

open issue: many OFBFs fall under one given symmetry group; do

we need to refine our notion of anisotropy (e.g., within symmetry

classes)?

53

Wavelet analysis

54

Wavelet analysis

How about multidimensional inference? Some references:

• scalar fields: Abry, Clausel, Jaffard, Roux and Vedel (2013); Lim,

Meerschaert, and Scheffler (2014)

• vector processes: Becker-Kern and Pap (2008); Amblard and

Coeurjolly (2011)

our focus: vector processes from a wavelet perspective

55

Wavelet analysis

wavelet coefficients/transform: a ∈ N, k ∈ Z

Da,k = a−1∫Rψ(a−1t− k)X(t)dt =

(dp(a, k)

)p=1,...,n

where ψ is a wavelet with compact support (Daubechies)

56

Wavelet analysis

• when X(t) = OFBM

EDa,kD∗a,k = aHED1,0D

∗1,0a

H∗

⇒ a wavelet regression method can be developed

• caveat: the Fourier and wavelet spectra are not equivalent. They

do not blow up entrywise according to the same power law (up to a

known constant). Equivalence holds under time reversibility

57

Wavelet analysis

assuming time reversibility. . .

• when H is non-diagonal, as a→∞ there is a competition between

power laws. A wavelet regression only captures the highest power law

• an extrapolation of the univariate framework yields that “scaling

behavior is characterized by a power law at the origin of the spectrum”

However. . .

58

Wavelet analysis

why non-diagonal scaling? (H is non-diagonal)

• blind source separation

• cointegration (Robinson (2008)): Y (t) = PX(t), where X is a

long-memory vector

• how many others?. . .

under investigation: how to deal with non-diagonal scaling from a

wavelet perspective

59

Summary

• multidimensionality can technically change the problem

• current goal: to develop the synthesis of domain and range

multidimensionality

• integral representations: a powerful paradigm for the analysis of

operator fractional fields

• interrelated ideas: symmetry groups, anisotropy, operator scaling,

exponents

60

Integral representations (Bierme et al. (2007))

harmonizable representation:

Xψ(t) = <(∫

Rm(ei〈t,x〉 − 1)ψ(x)−q−H/2W2(dx)

)ψ : [0,∞)m → R is a continuous, E∗-homogeneous function such

that ψ(x) 6= 0 when x 6= 0

61

Integral representations (Bierme et al. (2007))

moving average representation:

Xϕ(t) = <(∫

Rm(ϕ(t− u)H−q/2 − ϕ(−u)H−q/2)W2(dx)

)ϕ : [0,∞)m→ R is a E-homogeneous, (β,E)-admissible function

• (β,E)-admissibility: x 6= 0 ⇒ ϕ(x) > 0, and there exists C > 0

such that for any 0 < A < B

τ(x) ≤ 1⇒ |ϕ(x+ y)− ϕ(y)| ≤ Cτ(x)β

62

Wavelet analysis

(D = H − (1/2)I)

Fourier spectrum:

fX(x) =1


∗+ + x−D− AA∗x−D

∗− )

wavelet spectrum:

w(a) := EDa,kD∗a,k = aH

(∫ ∞0

|ψ(x)|2

x2x−D+ Re(AA∗)x−D

∗+

)aH∗

63

Wavelet analysis

problem: the wavelet and Fourier spectra are not equivalent

example:

H = diag(h1, h2), AA∗ =( 1 i

−i 1

)Fourier spectrum:

x−D+ AA∗x−D∗

+ =( x−2d1+ ix

−(d1+d2)+

−ix−(d1+d2)+ x−2d2+

)wavelet spectrum (diagonal!):

w(a) = diag(a2h1

∫ ∞0

|ψ(x)|2

x2x−2d1dx, a2h2

∫ ∞0

|ψ(x)|2

x2x−2d2dx

)64

Wavelet analysis

theorem (spectral equivalence): under time reversibility,

diagonalizable H with real roots, the spectra are equivalent. For

a fixed entry (m1,m2), either

• both fm1,m2(·) and wm1,m2(·) are identically zero; or

• for some −1 < δ < 1,

fm1,m2(x) ∼ cF |x|−δ, x→ 0

wm1,m2(a) ∼ cW |a|δ+1, a→∞, cF , cW 6= 0

65

Wavelet analysis

interest: estimate the entry-wise powers “δ” based on the behavior

of w(a) as a→∞

simplifying assumption: entry-wise scaling, i.e.,

(A) For m1,m2 = 1, . . . , n, there exists ηm1,m2 ∈ (0, 2) such that

Edm1(a, 0)dm2(a, 0) = aηm1,m2Edm1(1, 0)dm2(1, 0),

66

Wavelet analysis

interest: estimate the entry-wise powers “δ” based on the behavior

of w(a) as a→∞

simplifying assumption: entry-wise scaling, i.e.,

(A) For m1,m2 = 1, . . . , n, there exists ηm1,m2 ∈ (0, 2) such that

Edm1(a, 0)dm2(a, 0) = aηm1,m2Edm1(1, 0)dm2(1, 0),

central idea behind the wavelet regression: take logs

log|Edm1(a, 0)dm2(a, 0)| = ηm1,m2log|a|+ log|Edm1(1, 0)dm2(1, 0)|

67

Wavelet analysis

regression equations (n = 2):

j = 1, . . . ,m (# scales), k = 1, . . . , Nj (# terms per scale)

log∣∣∣Ed21(aj, k)

∣∣∣log∣∣∣Ed22(aj, k)

∣∣∣log∣∣∣Ed1(aj, k)d2(aj, k)

∣∣∣

=

η1 log ajη2 log ajη1,2 log aj

+

log∣∣∣Ed21(1, k)

∣∣∣log∣∣∣Ed22(1, k)

∣∣∣log∣∣∣Ed1(1, k)d2(1, k)

∣∣∣

68

Wavelet analysis

regression equations (n = 2): (η1, η2, η1,2)least squares

j = 1, . . . ,m (# scales), k = 1, . . . , Nj (# terms per scale)

log∣∣∣ 1Nj

∑Nj−1k=0 d21(aj, k)

∣∣∣log∣∣∣ 1Nj

∑Nj−1k=0 d22(aj, k)

∣∣∣log∣∣∣ 1Nj

∑Nj−1k=0 d1(aj, k)d2(aj, k)

∣∣∣

=

η1 log ajη2 log ajη1,2 log aj

+

log∣∣∣Ed21(1, k)

∣∣∣log∣∣∣Ed22(1, k)

∣∣∣log∣∣∣Ed1(1, k)d2(1, k)

∣∣∣

+ stoch. error

69

Wavelet analysis

example: H = diag(h1, h2)

Then

EDaj,0D∗aj,0

=

(a2h1j Ed21(1, 0) ah1+h2j Ed1(1, 0)d2(1, 0)

ah1+h2j Ed1(1, 0)d2(1, 0) a2h2j Ed22(1, 0)

)

⇒ for entry (1,1), j = 1, . . . ,m (scales)

log( 1

Nj

Nj−1∑k=0

d21(aj, k))A= 2h1 log aj+log(Ed21(1, 0))+

√a(N)

NN(0, ν1j )

⇒ by least squares, h1

70

Wavelet analysis

9 10 11 12 13 14 15 16 17 18 19−0.1

0

0.1Bias

alp

haX

9 10 11 12 13 14 15 16 17 18 19−0.1

0

0.1

alp

haY

9 10 11 12 13 14 15 16 17 18 19−0.5

0

0.5

alp

haX

Y

9 10 11 12 13 14 15 16 17 18 19−0.5

0

0.5

S

log2(samplesize)

Figure 1: Bias vs sample size (d1 = −0.2, d2 = 0.3)

71

Wavelet analysis

−0.1 0 0.1 0.2 0.30

5

10

15alphaXY − N = 2048

−0.1 0 0.1 0.2 0.30

5

10

15alphaXY − N = 524288

Figure 2: Asymptotic normality: MC (solid black lines) vs best

Gaussian fits (dashed red lines), d1 = −0.2, d2 = 0.3

72

self-similarity, symmetry and anisotropy in the ... · laha & rohatgi (1981), d & pipiras...

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