Nonparametric estimation of the local Hurst function of multifractional Gaussian processes

(1)

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multifractional Gaussian processes

Jean-Marc Bardet

To cite this version:

Jean-Marc Bardet. Nonparametric estimation of the local Hurst function of multifractional Gaussian

processes. Colloque Franco-Roumain de Mathématiques Appliquées, Aug 2012, Bucarest, Romania.

�hal-00728913�

(2)

Nonparametric estimation of the local Hurst function of

multifractional Gaussian processes

Joint paper with Donatas Surgailis (Lithuania)

Jean-Marc Bardet

bardet@univ-paris1.fr

SAMM, Universit´

e Paris 1

(3)

Outline

1 _Introduction

2 A new nonparametric estimator of the local Hurst function

First definition, assumptions and limit theorems

Second definition and limit theorems

Case of the General multifractional Brownian motion

3 _{Numerical comparison with the quadratic variations estimators}

Definition and limit theorems

Numerical comparisons

(4)

Outline

1 _Introduction

2 A new nonparametric estimator of the local Hurst function

First definition, assumptions and limit theorems

Second definition and limit theorems

Case of the General multifractional Brownian motion

3 _{Numerical comparison with the quadratic variations estimators}

Definition and limit theorems

Numerical comparisons

(5)

An athletic example...

0.5 1 1.5 2 x 104 320 340 360 380 400 420 HR(Msec.) Ath.1 0.5 1 1.5 2 x 104 2.2 2.4 2.6 2.8 3 HR(Hertz) Ath.1 0.5 1 1.5 2 x 104 140 150 160 170 180 HR(BPM) Ath.1 0.5 1 1.5 2 2.5 x 104 140 150 160 170 180 HR(BPM) Ath.2 0.5 1 1.5 2 2.5 x 104 110 120 130 140 150 160 170 HR(BPM) Ath.3 0.5 1 1.5 2 2.5 x 104 130 140 150 160 170 180HR(BPM) Ath.4

(6)

A fact...

0 0.5 1 1.5 2 2.5 x 104 −14000 −12000 −10000 −8000 −6000 −4000 −2000 0 2000 4000 6000 1 1.1 1.2 1.3 1.4 1.5 x 104 −13000 −12000 −11000 −10000 −9000 −8000 −7000 −6000 −5000 −4000 −3000 1.2 1.25 1.3 x 104 −1.25 −1.2 −1.15 −1.1 −1.05 x 104

(7)

The fractional Brownian motion

B

H

₌

_{B

H

t

, t

∈ R}

fractional Brownian motion

(FBM) with H

∈ [0, 1]:

B

H

_{is a Gaussian centered process with stationary increments and}

Var (B

H

t

) = σ

2 |t|

2H

.

Property

A Gaussian process X having stationary increments H-selfsimilar

⇐⇒ X F.B.M. with paremeter H

A trajectory of B

H

_{is a.s.}

_α-H¨

_olderian

_{for any α < H:}

(8)

Two trajectories of FBM

0 100 200 300 400 500 600 700 800 900 1000 −1.5 −1 −0.5 0 0.5 1 f.B;m. (H=0.3) 0 100 200 300 400 500 600 700 800 900 1000 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 f.B.m. (H=0.9)

(9)

Other definitions of FBM

Harmonizable representation:

B

_t

H

= C

1 (H) σ

2 Z

R

e

itξ

_{− 1}

|ξ|

2H+1

W

c

(dξ)

t

∈ R

Temporal representation:

B

_t

H

= C

2 (H) σ

2 Z

R

(t

− u)

H−1/2

+

− (−u)

H−1/2

+

W

(du)

t

_{∈ R}

(10)

Another example

0 1 2 3 4 5 6 7 8 9 x 104 200 400 600 800 1000 1200 1400 1600 Time in seconds Heart interbeat in ms 0 1 2 3 4 5 6 7 x 104 −0.5 0 0.5 1 1.5 2 2.5 x 106

Heartbeats during 24h.

=

⇒

H

depending on t!

(11)

Multifractional Brownian motion

Two first versions:

Harmonizable representation

(Benassi et al., 1997)

:

X

t

= C

1 (H(t)) σ

2 Z

R

e

itξ

_{− 1}

|ξ|

2H(t)+1

W

c

(dξ)

t

∈ R

Temporal representation

(Peltier et L´

evy-V´

ehel, 1995)

:

X

t

= C

2 (H(t)) σ

2 Z

R

(t

_{− u)}

H(t)−1/2

+

− (−u)

H(t)−1/2

+

W

(du)

t

_{∈ R}

(12)

Aims

From an observed trajectory X

1

n

, X

2n

, . . . , X

1 ,

Define a

new non-parametric estimator H(

·)

;

Asymptotic properties

of this estimator.

(13)

Outline

1 _Introduction

2 A new nonparametric estimator of the local Hurst function

First definition, assumptions and limit theorems

Second definition and limit theorems

Case of the General multifractional Brownian motion

3 _{Numerical comparison with the quadratic variations estimators}

Definition and limit theorems

Numerical comparisons

(14)

First definition

For t

0 ∈ (0, 1) and α ∈ (0, 1), define:

b

H

_n,α

(IR)

(t

0 ) := Λ

−1

1 2n

1−α

[nt

0

_X

+n

1−α

]

k

=[nt

0

−n

1−α

]

∆

k

n

X

+ ∆

k+1

n

X

∆

k

n

X

+

∆

k+1

n

X

with

∆

k

n

X

= X

k+2 n

− 2X

k+1 n

+ X

kn

Λ(h) = E

h |∆

0

1 B

h

+ ∆

1

1 B

h

|

|∆

0

1 B

h

| + |∆

1

1 B

h

|

i

for h

_{∈ (0, 1)}

= 1 πarccos(−ρ2(h)) + 1 π r 1+ρ2(h) 1−ρ2(h)log 1+ρ2(h)2 with ρ2(h) =−32h +22h+2−7_8−22h+1 .

=

_⇒

Explanation...

(15)

Multifractional Gaussian processes?

Assumptions:

X

= (X

t

)

t

is a centered

Gaussian process

such as:

(A)

κ

There exist

η-H¨

olderian

functions 0 < H(t) < 1 and c(t) > 0 for t

∈ (0, 1)

such that for any 0 < ε < 1/2 and j

_{∈ Z,}

max

[nε]≤k≤[(1−ε)n]

n

κ

Cov ∆

k

n

X

, ∆

k+j

n

X

Cov ∆

k

n

B

H(k/n)

, ∆

k+j

n

B

H(k/n)

− c(

k

n

)

−→

_n→∞

0. (B)

α

There exist C > 0, γ > 1/2 and 0

≤ θ < γ/2 such that for any n ∈ N

∗

and

1 ≤ k, k

′

_{< n}

− q

Cor ∆

k

n

X

, ∆

k

′

n

X

)

≤ C n

(1−α)θ

₍

|k

′

− k| ∧ n

1−α

)

−γ

.

(16)

Limit theorems for multifractional Gaussian processes

Theorem

Under Assumptions

(A)

κ

and

(B)

α

, for all t

0 ∈ (0, 1),

• If

0 < α <

_2(γ−θ)

γ−2θ

,

b

H

_n,α

(IR)

(t

0 )

−→

a.s.

n→∞

H(t

0 ).

• If

κ

_{≥ µ and}

_{3γ−2θ+4γ(η∧2)}

2γ−2θ

_{≤ α <}

2γ−2θ

_3γ−2θ

then for any

ǫ > 0

sup

ǫ<t<1−ǫ

b

H

_n,α

(IR)

(t)

− H(t)

= O

p

(n

−µ

).

• If

κ

_{≥ µ}

1 and

_{3γ−2θ+4γ(η∧2)}

γ−2θ

≤ α <

_3γ−2θ

γ−2θ

then for any

ǫ > 0, δ > 0

sup

ǫ<t<1−ǫ

b

H

n,α

(IR)

(t)

− H(t)

= O(n

−(µ

1

−δ)

₎

_a.s.

(17)

Central Limit theorem for multifractional Gaussian

processes

Theorem

Let Z

= (Z (t))

t∈(0,1)

be a zero-mean Gaussian process satisfying

(A)

κ

and

(B)

α

,

with

α >

1 1+2(η∧2)

,

κ

≥

1−α

2 and

θ = 0. Then for 0 < t

1 <

· · · < t

u

< 1,

√

2n

1−α

_H

b

(IR)

n,α

(t

i

)

− H(t

i

)

1≤i≤u

D

−→

n→∞

W

(IR)

_(t

i

)

1≤i≤u

,

where W

(IR)

(t

i

), i = 1,

· · · , u are inedependent centered Gaussian r.v.’s such as

E[W

(IR)

_(t

i

)]

2 :=

h ∂

∂x

(Λ

2 )

−1

_(Λ

2 (H(t

i

)))

i

2 σ

2 (H(t

i

))

where

σ

2 (H) :=

X

k∈Z

Cov

ψ ∆

0

1 B

H

, ∆

1

1 B

H

, ψ ∆

k

1 B

H

, ∆

k+1

1 B

H

.

(18)

Proof: based on a CLT of triangular arrays of functional of

Gaussian vectors

Theorem (

Bardet et Surgailis, 2012)

Let

(Y

n

(k))

1≤k≤n,n∈N

be a triangular array of standard Gaussian R

ν

-vectors.

For m

_{≥ 1, there exists ρ : N → R such as for 1 ≤ p, q ≤ ν,}

∀(j, k),

EY

n

(p)

(j)Y

n

(q)

(k)

≤ |ρ(j − k)| with

X

j ∈Z

|ρ(j)|

m

_<

∞;

For

τ

_{∈ [0, 1] and J ∈ N}

∗

_{, with}

_(W

τ

(j))

_{j ∈Z}

a stationary Gaussian process

(Y

n

([nτ ] + j))

−J≤j ≤J

D

−→

n→∞

(W

τ

(j))

−J≤j ≤J

;

If ˜

f

k,n

∈ L

2

0 (X) (n

≥ 1, 1 ≤ k ≤ n) with Hermite rank ≥ m and ∃ ˜

φ

τ

, τ

∈ [0, 1]

such as

sup

τ ∈[0,1]

k˜f

[τ n],n

− ˜

φ

τ

k

2 = sup

τ ∈[0,1]

E(˜f

[τ n],n

(X)

− ˜

φ

τ

(X))

2 −→

n→∞

0.

(19)

Theorem

Then with

σ

2 =

Z

1

0 dτ

X

j ∈Z

E ˜

_φ

_τ

_(W

_τ

_{(0)) ˜}

_φ

_τ

_(W

_τ

_(j))

<

_∞,

n

−1/2

n

X

k

=1

˜

f

k

,n

(Y

n

(k))

D

−→

n→∞

N 0, σ

2 _.

(20)

Second definition

For t, α

_{∈ (0, 1) and j = 1, . . . , p,}

b

H

_n,α,j

(IR)

(t

0 ) := Λ

−1

_j

1 2n

1−α

[nt

0

_X

+n

1−α

]

k=[nt

0

−n

1−α

]

∆

k

n

X

+ ∆

k

n

+j

X

|∆

k

n

X

| + |∆

k

+j

n

X

|

,

.

Theorem

Under Assumptions

(A)

κ

and

(B)

α

, for all t

0 ∈ (0, 1),

n

(1−α)/2

H

b

_n,α,j

(IR)

(t

0 )

− H(t

0 )

_1≤j≤p

D

−→

n→∞

N

0 , Γ(H(t

0 ))

(21)

Second definition (end)

Let b

Γ := Γ( b

H

n,α

(IR)

).

A new nonparametric estimator of H(

·) using

Pseudo-Generalized least squares

:

e

H

_n,α

(IR)

(t

0 ) := 11

′

p

bΓ

−1

11 p

−1

11 ′

p

bΓ

−1

H

b

(IR)

n,α,j

(t

0 )

1≤j≤p

.

Theorem

Under Assumptions

(A)

κ

and

(B)

α

, for all t

0 ∈ (0, 1),

n

(1−α)/2

H

e

n,α

(IR)

(t

0 )

− H(t

0 )

D

−→

n→∞

N

0 , 11

′

p

Γ

−1

_(H)11

p

−1

(22)

Definition

(Stoev and Taqqu, 2006)

On pose

Y

(a

+

_,a

−

)

(t) := K (H(t))

Z

R

e

itx

_{− 1}

|x|

H(t)+

1 2

U

(a

+

_,a

−

)

(H(t), x) c

W

(dx),

with for h

∈ (0, 1)

U

(a

+

_,a

−

₎

(h, x)

:=

a

+

e

−i

sign

(x)(h+

1 2

)

π2

+ a

−

e

i

sign

(x)(h+

12

)

π2

(a

+

₎

2 _{+ (a}

−

₎

2 _{− 2a}

+

_a

−

_{sin(π h)}

1/2

.

:=

√

1 _π

if h = 1/2 and a

+

_{= a}

−

_.

(23)

Estimation of H(·)

Theorem

For a trajectory of the process

(Y

(a

+

_,a

−

₎

(t))

_t

,

If

max(0, 1

− 4((η ∧ 2) − H(t))) < α <

1

2 ,

e

H

n,α

(IR)

(t

0 )

a.s.

−→

n→∞

H(t

0 )

If

max

n

1 1 + 2(η

_{∧ 2)}

, 1

− 4 η ∧ 2 − H(t

0 )

o

< α < 1,

n

(1−α)/2

H

e

_n,α

(IR)

(t

0 )

− H(t

0 )

D

−→

n→∞

N

0 , 11

′

p

Γ

−1

(H)11

p

−1

(24)

Outline

1 _Introduction

2 A new nonparametric estimator of the local Hurst function

First definition, assumptions and limit theorems

Second definition and limit theorems

Case of the General multifractional Brownian motion

3 _{Numerical comparison with the quadratic variations estimators}

Definition and limit theorems

Numerical comparisons

(25)

Generalized quadratic variations estimator of H(·)

Define (see

Istas and Lang, 1994, Benassi et al., 1998, Coeurjolly, 2005

):

b

H

_n,α

(QV )

(t

0 ) :=

1

2 A

′

A

′

_A

log

1 2n

1−α

[nt

0

_X

+n

1−α

]

k=[nt

0

−n

1−α

]

X

k+2j n

− 2X

k+j n

+ X

k n

2 ′

1≤j≤p

with A := log i

−

1 p

P

p

j=1

log j

1≤i≤p

∈ R

p

_.

Theorem

(26)

0 500 1000 1500 2000 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 0 500 1000 1500 2000 −1 −0.5 0 0.5 1 1.5

Figure:

Examples of MBM trajectories (up, H ∈ C

η−

_{with η = 0.6, down}

(27)

0 0.2 0.4 0.6 0.8 1 0.4 0.5 0.6 0.7 0.8 0.9 1 t H(t) alpha=0.3 0 0.2 0.4 0.6 0.8 1 0.4 0.5 0.6 0.7 0.8 0.9 1 t H(t) alpha=0.4 0 0.2 0.4 0.6 0.8 1 0.4 0.5 0.6 0.7 0.8 0.9 1 t H(t) alpha=0.3 0 0.2 0.4 0.6 0.8 1 0.4 0.5 0.6 0.7 0.8 0.9 1 t H(t) alpha=0.4

Figure:

Estimation of H

4

(t) = 0.1 + 0.8(1 − t) sin

2

(10t) for n = 6000, α = 0.3 (left) and

(28)

H1(t) = 0.6 α 0.2 0.3 0.4 0.5

n= 2000 p_p[MISEfor bHn,α(QV ) 0.041 0.051 0.069 0.096

[MISEfor bHn,α(IR) 0.061 0.077 0.106 0.145

n= 6000 p_p[MISEfor eHn,α(QV ) 0.025 0.033 0.050 0.074

[MISEfor eHn,α(IR) 0.037 0.049 0.076 0.115

H2(t) = 0.1 + 0.8t α 0.2 0.3 0.4 0.5

n= 2000 p[MISEfor eHn,α(QV ) 0.170 0.073 0.072 0.096

p

[MISEfor eHn,α(IR) 0.059 0.071 0.098 0.135

n= 6000

p

[MISEfor eHn,α(QV ) 0.114 0.044 0.048 0.070

p

[MISEfor eHn,α(IR) 0.036 0.046 0.069 0.103

H3(t) = 0.5 + 0.4 sin(5t) α 0.2 0.3 0.4 0.5

[MISEfor eHn,α(IR) 0.093 0.071 0.091 0.124

[MISEfor eHn,α(IR) 0.057 0.047 0.065 0.097

H4(t) = 0.1 + 0.8(1 − t) sin2 (10t) α 0.2 0.3 0.4 0.5

n= 2000

p

[MISEfor eHn,α(QV ) 0.320 0.164 0.117 0.112

p

[MISEfor eH_n,α(IR) 0.165 0.098 0.091 0.112

[MISEfor eH_n,α(IR) 0.148 0.062 0.067 0.091

Table:

Estimators e

H

n,α(QV )

et e

H

n,α(IR)

when H(·) is a C

∞

(29)

H ∈ C

1.5

_{(0, 1)}

α

0.2

0.3

0.4

0.5 n

= 2000

p

_p

\

MISE

for e

H

n,α(QV )

0.261

0.112

0.085

0.098 \

MISE

for e

H

n,α(IR)

0.098

0.077

0.093

0.128 n

= 6000

p

_p

\

MISE

for e

H

n,α(QV )

0.164

0.066

0.053

0.070 \

MISE

for e

H

n,α(IR)

0.054

0.047

0.066

0.098 H ∈ C

0.6

_{(0, 1)}

_α

_0.2

_0.3

_0.4

_0.5

n

= 2000

p

_p

\

MISE

for e

H

n,α(QV )

0.140

0.086

0.081

0.094 \

MISE

for e

H

n,α(IR)

0.088

0.078

0.096

0.135 n

= 6000

p

\

MISE

for e

H

n,α(QV )

0.130

0.067

0.056

0.071 p

\

MISE

for e

H

n,α(IR)

0.066

0.052

0.067

0.103

(30)

Figures

0 0.2 0.4 0.6 0.8 1 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.7 H L(H) 3 4 7 4 4 4 5 4 1 4 2 4 6 4 H 0.34 0.36 0.38 0.42