Identification of the multiscale fractional Brownian motion with biomechanical applications

0.54 MB·English

by  Jean-Marc Bardet

Identification of the multiscale fractional Brownian
7 motion with biomechanical applications
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J February 2, 2008

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] Jean-Marc BARDET∗ and Pierre BERTRAND∗∗
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. ∗ SAMOS-MATISSE - UMR CNRS 8595, Universit´e Panth´eon-Sorbonne (Paris I), 90 rue de Tolbiac, 75013
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Paris Cedex, France, E-mail: [email protected]
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∗∗ Laboratoire de Math´ematiques - UMR CNRS 6620, Universit´e Blaise Pascal (Clermont-Ferrand II), 24 Av-
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enue des Landais, 63117 Aubi`ere Cedex, France. E-mail: [email protected]
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Abstract : In certain applications, for instance biomechanics, turbulence, finance, or Internet traffic, it seems
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8 suitable to model the data by a generalization of a fractional Brownian motion for which the Hurst parameter H
1
is depending on the frequency as a piece-wise constant function. These processes are called multiscale fractional
0
7 Brownian motions. In this contribution, we provide a statistical study of the multiscale fractional Brownian mo-
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tions. We develop a method based on wavelet analysis. By using this method, we find initially the frequency
/
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changes, then we estimate the different parameters and afterwards we test the goodness-of-fit. Lastly, we give the
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numerical algorithm. Biomechanical data are then studied with these new tools.
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Keywords: Biomechanics;Detection of change;Goodness-of-fittest; FractionalBrownianmotion; Semi-parametric
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X estimation; Wavelet analysis.
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2 Statistic of multi-scale fractional Brownian motion
1 Introduction
Fractional Brownian Motion (F.B.M.) was introduced in 1940 by Kolmogorov as a way to generate Gaussian
”spirals” in a Hilbert space. But the seminal paper of Mandelbrot and Van Ness (1968) emphasizes the rel-
evance of F.B.M. to model natural phenomena: hydrology, finance... Formally, a fractional Brownian motion
B =(B (t), t IR ) could be defined as a real centered Gaussian process with stationary increments such that
H H +
∈
B (0)=0and E B (s) B (t)2 = σ2 t s2H, forallpair(s,t) IR IR whereH ]0,1[andσ >0. This
H H H + +
| − | | − | ∈ × ∈
process is characterized by two parameters : the Hurst index H and the scale parameter σ. We lay the emphasis
on the fact that the same parameter H is linked to different properties of the F.B.M. as the smoothness of the
sample paths, the long range dependence of its increments and the self-similarity.
During the decades 1970’s and 1980’s, the statistical study of F.B.M. was developed, to look at for instance
the historical notes in Samorodnitsky & Taqqu (1994), [30, chap.14] and the references therein. Modelling by a
F.B.M.becamemoreandmorewidespreadduringthelastdecade(trafficInternet,turbulence,imageprocessing...).
Nevertheless, in many applications the real data does not fit exactly F.B.M. Thus, the F.B.M. must be regarded
only as an ideal mathematical model. Therefore, various generalizations of F.B.M. have been proposed these last
years to fill the gap between the mathematical modelling and real data. In one hand, Gaussian processes where
the Hurst parameter H has been replaced by a function depending on the time were studied, see for instance
Peltier and L´evy Vehel (1996), Benassi, Jaffard and Roux (1997), Ayache and L´evy Vehel (1999). However, this
dependence of time implies the loss of the stationarity of the increments. In other hand, non Gaussian processes,
mainly α stable (0 < α < 2) infinite variance processes, were considered, see for example the study of telecom
processes in Pipiras and Taqqu (2002).
Here, we are concerned with Gaussian processes having stationary increments and a Hurst index changing with
the frequencies. To our knowledge, these kinds of processes were introduced implicitly in biomechanics by Collins
and de Luca (1993),in finance by Rogers (1997)and Cheridito (2003)and explicitly by Benassiand Deguy (1999)
for image analysis or image synthesis. In any case, the probabilistic properties of these processes have not been
thoroughly established and no rigorous statistical studies have been done. Both Collins and de Luca (1993) and
Benassiand Deguy (1999)proposea modelwith two differentHurst indices correspondingrespectivelyto the high
andthelowfrequenciesseparatedbyonechangepointatthefrequencyω . Theyusethelogvariogramtoestimate
c
thesetwoHurstindices. Indeed,inthiscase,thelogvariogramconsideredasafunctionofthelogarithmofthescale
presents two asymptotic directions with slopes being twice the Hurst index at low (respectively high) frequencies.
Thechangepointω isthenestimatedastheabsciseoftheintersectionofthethesetwostraightlines. Numerically,
c
this method is not robust. Moreover it could not be adapted in the case of more than one change point. Let us
stressthatitisnotaquestionofatheoreticalrefinement,butonethatcorrespondspreciselytothetruesituations.
Indeed,inapplications,weconsideronlyfinitefrequencybands,thereforeweshoulduseastatisticalmethodbased
on the information included in finite frequency bands. Wavelet analysis seems the tool had hoc, when the Fourier
transform of the associated wavelet is compactly supported.
For these reasons, we put forward in Bardet and Bertrand (2003) a model of generalized F.B.M. including the
caseswithmorethanonefrequency changepoint. We calledit(M )multiscale fractionalBrownianmotionwhere
K
K denote the number of frequency change points. More precisely, a (M ) multiscale fractional Brownian motion
K
is aGaussianprocesswithstationaryincrementswhere the HurstparameterH isreplacedby apiecewise constant
function of the frequency ξ H(ξ) in the harmonizable representation, see Formula (3) below. The main proba-
7→
bilistic properties of this model were studied in Bardet and Bertrand (2003). In this work, we treat the statistical
study of the multiscale F.B.M. and we focus on its application to biomechanics.
J.M. Bardet and P. Bertrand 3
The remainder of the paper is organized as follows: in Section 2, we describe the biomechanical data and the
corresponding statistical problem. In section 3, we recall the initial definition of the partial Brownian motion and
its principal probabilistic properties. Then, we show that the variogrammethod is not suitable for the estimation
ofthe variousparametersof a (M )-F.B.M. We then developa statisticalestimationframework,basedonwavelet
K
analysis. We investigate the discretization of the wavelet coefficient and we state a functional Central Limit The-
orem for the empirical wavelet coefficients. In Section 4, we first estimate the different frequency change points
andHurstparameters. Then,we proposeagoodnessoffit testandderiveanestimatorofthe number offrequency
changes. The numerical algorithm is detailed at the end of this section. Finally, in Section 5, the biomechanical
data are studied with the tools developed in Section 4. The proof of the results of Sections 3 and 4 are given in
appendix.
2 The Biomechanical Problem
One of the motivations of this work is to model biomechanicaldata corresponding to the regulationof the upright
position of the human being. By using a force platform, the position of the center of pressure (C.O.P.) during
quiet posturalstance is determined. This positionis usually measuredata frequency of100Hz for the oneminute
period, which yields a data set of 6000 observations. The experimental conditions are formed to the standards of
the Association Franc¸aise de Posturologie (AFP), for instance the feet position (angle and clearance), the open or
closed eyes.
8
6
4
wing Y axis 2
Position follo 0
−2
−4
−6
−6 −4 −2 0 2 4 6 8
Position following X axis
Figure 1 : An example 1 of the trajectory of the C.O.P. during 60s at 100Hz (in mm)
The X axis of the platform corresponds to the fore-aft direction and the Y axis corresponds to the medio-lateral
direction. During the 1970’s, these data were analyzed as a set of points, i.e. without taking into account their
temporal order. During the following decade some studies considered them as a process, and Collins and de Luca
(1993) introduced the use of F.B.M. to model these data. In fact, they used a generalization of F.B.M. More
precisely, let the position X of the C.O.P. be observed at times t =i∆ for i=1,...,N (∆=0.01 s). The study
i i
of Collins and de Luca is based on the empirical variogram
N−δ
1 2
V (δ)= X X (1)
N (N δ) (i+δ)∆− i∆
− i=1
X (cid:0) (cid:1)
1theseexperimentaldatawererealizedbyA.Mouzatandareusedin[13].
4 Statistic of multi-scale fractional Brownian motion
where δ IN∗. For a F.B.M., we have EV (δ) = σ2∆2H δ2H and after plotting the log-log graph of the
N
∈ ×
variogramasafunctionofthe time lag,i.e. (logδ,logV (δ)), a linearregressionprovidesthe slope 2H. Typically,
N
one gets the following type of figure (see Figure 2). It is considered by Collins and de Luca to be a ”F.B.M.” with
two regimes : with slope 2H (short term) and with slope 2H (long term) separated by a critical time lag δ and
0 1 c
these parameters are estimated graphically :
4
3
2
m
Variogra 1
ms of the 0
Logarith−1
−2
−3
−4
0 1 2 3 4 5 6 7 8
Logarithms of the scales
Figure 2 : An example of the log-log graph of the variogramfor the previous trajectories X (-.) and Y (-).
They found H > 0.5, H < 0.5 and a critical time lag δ 1 s. These results were interpreted as corre-
0 1 c
≃
sponding to two different kinds of regulation of the human stance : in the long term H < 0.5 and the process is
1
anti-persistent, in the short term H >0.5 and the process is persistent. This method was employed several times
0
inbiomechanicsunderthevariousexperimentalconditions(openedeyesversusclosedeyes,differentfeetangles,...).
But, a lack of mathematical models and of statistical studies has made impossible to obtain confidence intervals
on the two slopes 2H , 2H and the critical time lag δ .
0 1 c
3 The multiscale fractional Brownian motion and its statistical study
based on wavelet analysis
3.1 Description of the model
AfractionalBrownianmotionB = B (t), t IR ofparameters(H, σ)isarealcenteredGaussianprocesswith
H H
{ ∈ }
stationaryincrements and E B (s) B (t)2 = σ2 t s2H, for all (s,t) IR2 where H ]0,1[ and σ >0. The
H H
| − | | − | ∈ ∈
fractionalBrownianmotion(F.B.M.)hasbeenproposedbyKolmogorov(1940)whodefineditbytheharmonizable
representation :
eitξ 1
B (t)= − W(dξ), for all t IR, (2)
H ξ H+1/2 ∈
ZIR (cid:0)| | (cid:1)
where W(dx) is a Brownian measure and W(dξ) its Fcourier transform (namely for any function f L2(IR)
∈
one has almost surely, f(x)W(dx) = f(ξ)W(dξ), with the convention that f(ξ) = e−iξxf(x)dx when
IR IR IR
c
f L1(IR) L2(IR)). We refer to Samorodnitsky and Taqqu (1994) for the question of the equivalence of the
R R R
∈ b c b
different representations of the F.B.M. From the harmonizable representation, a natural generalization is the
T
multiscalefractionalBrownianmotionwithaHurstindexdepending onthe frequency. Moreprecisely,wedefine :
J.M. Bardet and P. Bertrand 5
Definition 3.1 For K IN, a (M )-multiscale fractional Brownian motion X = X(t),t IR (simplify by
K
∈ { ∈ }
(M )-F.B.M.) is a process such as
K
K ωj+1 (eitξ 1 )
X(t)=2 σ − W(dξ) for all t IR (3)
j=0Zωj j |ξ|Hj+1/2 ∈
X
c
with ω =0<ω < <ω <ω = by convention, σ > and H ]0,1[ for i 0,1, ,K .
0 1 K K+1 i i
··· ∞ ∈ ∈{ ··· }
The (M )-F.B.M. was notably introduced in order to relax the self-similarity property of F.B.M. Indeed, the
K
self-similarityis aformofinvariancewithrespecttochangesoftime scale[27]anditlinksthe behaviortothe high
frequencies with the behavior to the low frequencies. In Bardet and Bertrand(2003), the main properties of these
processes are provided : X is a Gaussian centered process with stationary increments, its trajectories are a.s. of
Ho¨lder regularity α, for every 0 α<H and its increments form a long-memory process (except if the different
K
≤
parameters satisfy a particular relationship, i.e., if its spectral density is a continuous function with 0<H <1/2
i
for i=0,1, ,K).
···
3.2 The question of the choice of the estimator
In the remainder of this paper, we suggest a statistical study of such a model based on wavelet analysis. In this
subsection, we explain the reason of this choice.
To begin with, we will describe the statistical framework precisely. Let X = X(t),t IR be a (M )-F.B.M.
+ K
{ ∈ }
defined by (3). We observe one path of the process X on the interval [0,T ] at the discrete times t = i ∆ for
N i N
·
i=1,...,N with T =N ∆ . Therefore,
N N
·
(X(∆ ),X(2∆ ),...,X(N∆ )) is known,
N N N
and we consider the asymptotic N , ∆ 0 and T . We want to estimate the parameters of the
N N
→ ∞ → → ∞
(M )-F.B.M. that are (H ,H ,...,H ), (σ ,σ ,...,σ ) and (ω ,...,ω ).
K 0 1 K 0 1 K 1 K
Even if the model is defined as a parametric one, we prefer to use a semi-parametric statistics based on the
waveletanalysis. Thischoiceisjustifiedbythefollowingreasons. First,thespectraldensityofX isnotcontinuous
in the general case. Thus, one cannot use the classical results on the consistency of the maximum likelihood or
Whittle maximum likelihood estimators for long memory processes (see Fox and Taqqu, 1986, Dahlhaus, 1989 or
Giraitis and Surgailis, 1990). Moreover, this is not a classical time series parametric estimation : indeed, we con-
sider (X(∆ ),X(2∆ ),...,X(N∆ )) instead of (X(1),X(2),...,X(N)) and therefore this is also an estimation
N N N
problemoftheparametersofacontinuousstochasticprocess. Secondly,thefollowingsemi-parametricstatisticsare
morerobustthanaparametriconeifthemodelismisspecified. Considerthe examplewherethefunctionH(ξ)isa
not exactly a piece-wise constantfunction, but instead a constant function on severalintervals and some unknown
function on the other intervals. In this case, a parametric estimator could not work while the semi-parametric
method based on the wavelet analysis will remain efficient.
Another semi-parametric method was developed from the seminal paper of Istas and Lang (1997). This method
of estimation is derived from the variogram and provides good results in the case of F.B.M. (see Bardet, 2000)
or of multifractional F.B.M. (see Benassi et al., 1998). However, one faces difficulties in identifying the model
(M )-F.B.M. with this kind of method. Indeed, one can easily satisfy that for δ >0 :
K
K δωj+1 (1 cosv)
(δ)=E (X(t+δ) X(t))2 = 4 δ2Hjσ2 − dv. (4)
V − j=0 j Zδωj v2Hj+1
X
6 Statistic of multi-scale fractional Brownian motion
The principle of the variogram’smethod ensues from the writing of log (δ) as an affine function of logδ. For a
V
∞ (1 cosv) (cid:16) (cid:17)
(M )-F.B.M., with C(H )= − dv for i=0,1,...,K, two cases could provide such a relation :
K i Z0 v2Hi+1
1. for δ , log (δ) =2H logδ+log 4 σ2 C(H ) +O(δ−2H0);
→∞ V 0· · 0 · 0
(cid:16) (cid:17) (cid:0) (cid:1)
2. for δ 0, log (δ) =2H logδ+log 4 σ2 C(H ) +O(δ2−2HK)
→ V K · · K · K
(cid:16) (cid:17) (cid:0) (cid:1)
(the proofof suchexpansionsis in the proofofLemma A.1). In those cases,if one canshow thatthere is a conver-
gentestimatorV (δ)of (δ), thenalog-logregressionoflog V (δ) ontologδ couldprovideanestimationofthe
N N
V
different parameters. Nevertheless, such a method would ha(cid:16)ve a lo(cid:17)t of drawbacks. On one hand, the estimation
of ”intermediate”parameters(H ) and (σ2) requires veryspecific asymptotic properties between
j 1≤j≤K−1 j 1≤j≤K−1
all the frequency changes (ω ) . This implies a lack of generality of the methods based on the variogram.
j 1≤j≤K−1
Moreover,concretely, the frequency changes are fixed and one obtains rough approximationinstead of asymptotic
properties. For instance, numerical simulations show that in some cases the log-log plot of the variogramdoes not
exhibitanyintermediatelinearpart. Ontheotherhand,whenthemodelismisspecifiedthevariogrammodelcould
leadto inadequate results. For example the followingpicture givesthe caseofa (M )-F.B.M. where the variogram
2
methodwoulddetectonlyonefrequencychangeandcouldnotpreciselyestimateitsvalue. Finally,thevariogram’s
method could perhaps be applied in the two first previous situations 1. and 2., i.e. for the estimation of (H ,σ2)
0 0
or (H ,σ2 ) with δ will have to be a function of N (number of data). But this choice of function will depend on
K K
the unknown parameters H or H for obtaining central limit theorems for log V (δ) ... (see the same kind of
0 K N
problem in Abry et al., 2002). (cid:16) (cid:17)
15
10
5
d))
V(
g(
o
l
0
−5
−10
−6 −4 −2 0 2 4 6
log(d)
Figure 3: An example of a theoretical variogram for a (M )-f.B.m, with H = 0.9, H = 0.2, H = 0.5, and
2 0 1 2
σ = σ = σ = 5 and ω = 0.05, ω = 0.5 (in solid, the theoretical variogram, in dot-dashed, its theoretical
0 1 2 1 2
asymptotes for δ 0 and δ ).
→ →∞
We deduce from the definition of the model and the previous discussion that a wavelet analysis could be an
interesting semi-parametric method for estimating the parameters of a (M )-F.B.M. Indeed, such a method is
K
based on the change of scales (or frequencies). Therefore, as it is developed below, a wavelet analysis is able to
detect the different spectral domain of self-similarity and then estimate the different parameters of the model.
J.M. Bardet and P. Bertrand 7
3.3 A statistical study based on wavelet analysis
This method has been introduced by Flandrin (1992) and was developed by Abry et al. (2002) and Bardet et al.
(2000). We also use in the following similar results on wavelet analysis for (M )-F.B.M. obtained in Bardet and
K
Bertrand (2003). Let ψ be a wavelet satisfying the following assumption :
Assumption (A1): ψ : IR IR is a ∞ function satisfying :
7→ C
for all m IR, tmψ(t) dt< ;
• ∈ | | ∞
ZIR
its Fourier transform ψ(ξ) is an even function compactly supported on [ β, α] [α,β] with 0<α<β.
• − − ∪
We stress these conditions are sufficiently mild and are satisfied in particular by the Lemari´e-Meyer ”mother”
b
wavelet. The admissibility property, i.e. ψ(t)dt=0, is a consequence of the second one and more generally,for
ZIR
all m IN,
∈
tmψ(t)dt=0. (5)
ZIR
Note that it is not necessary to choose ψ to be a ”mother” wavelet associated to a multiresolution analysis of
IL2(IR). The whole theory can be developed without resorting to this assumption. The choice of ψ is then very
large.
1 t
Let (a,b) IR∗ IR and denote λ = (a,b). Then define the family of functions ψ by ψ (t)= ψ b .
∈ + × λ λ √a a −
(cid:18) (cid:19)
Parametersaandbareso-calledthescaleandtheshiftofthewavelettransform. Letusunderlinethatweconsider
a continuous wavelet transform. Let d (a,b) be the wavelet coefficient of the process X for the scale a and the
X
shift b, with
1 t
d (a,b)= ψ( b)X(t)dt=<ψ ,X > .
X √a a − λ L2(IR)
ZIR
If ψ satisfies Assumption (A1) and X is a (M )-F.B.M., the family of wavelet coefficients verifies the following
K
properties (see Bardet and Bertrand, 2003) :
1. for a>0, (d (a,b)) is a stationary centered Gaussian process such as :
X b∈IR
E d2 (a,.) = (a)=a ψ(au)2 ρ−2(u)du. (6)
X I1 | | ·
ZIR
(cid:0) (cid:1)
α β b
2. for all i=0,1, ,K, if the scale a is such as [ , ] [ω ,ω ], then
i i+1
··· a a ⊂
2
ψ(u)
E d2 (a,.) =a2Hi+1 σ2 K (ψ), with K (ψ)= du. (7)
X · i · Hi H (cid:12)u2H+(cid:12)1
ZIR |(cid:12)b| (cid:12)
(cid:0) (cid:1) (cid:12) (cid:12)
Property(7)meansthatthelogarithmofthevarianceofthewaveletcoefficientisanaffinefunctionofthelogarithm
of the scale with slope 2H +1 and intercept logσ2+logK (ψ). This property is the key tool for estimating the
i i Hi
parameters of X. Indeed, if we consider a convergent estimator of log E d2 (a,.) , it provides a linear model
X
in loga and logσ2. Before specifying such an estimator, let us stress that one only observes a discretized path
i (cid:0) (cid:0) (cid:1)(cid:1)
(X(0),X(∆ ),...,X(N∆ )) instead of a continuous-time path.
N N
As a consequence, for a>0 and N IN∗, a natural estimator is the logarithm of the empirical variance of the
∈
wavelet coefficient, that is logI (a) where :
N
1
I (a)= d2 (a,k∆ ), (8)
N D (a) X N
N
| |k∈XDN(a)
with :
8 Statistic of multi-scale fractional Brownian motion
r ]0,1/3[;
• ∈
m =[r(N/a)] and M =[(1 r)(N/a)] where [x] is the integer part of x IR;
N N
• − ∈
D (a)= m ,m +1,...,M and D (a) is the cardinal of the set D (a).
N N N N N N
• { } | |
For 0 < a < a , a functional central limit theorem for (logI (a)) can be established (see a
min max N amin≤a≤amax
similar proof in Bardet and Bertrand, 2003) :
Proposition 3.1 Let X be a (M )-F.B.M., 0<a <a and ψ satisfy Assumption (A1). Then :
K min max
D
N∆ (logI (a) log (a)) (Z(a)) (9)
N N − I1 amin≤a≤amax N−→→∞ amin≤a≤amax
p
with (Z(a)) a centered Gaussian process such as for (a ,a ) [a ,a ]2,
1 2 min max
∈
2
2a a ψ(a ξ)ψ(a ξ)
cov(Z(a ),Z(a ))= 1 2 1 2 e−iuξdξ du. (10)
1 2 (1 2r) (a ) (a ) ρ(ξ)2
− I1 1 I1 2 ZIR ZIR | | !
b b
Then, if we specify the locations of the change points in terms of scales, i.e. frequencies, we obtain the following:
β ω
i+1
Corollary 3.1 Let i 0,1, ,K and assume that . Then,
∈{ ··· } α ≤ ω
i
N∆ logI (1/f)+(2H+1)logf logσ2 logK (ψ)
N N i − i− Hi ωi/α≤f≤ωi+1/β
p (cid:0) D ((cid:1)Z(1/f)) (11)
N−→→∞ ωi/α≤f≤ωi+1/β
ω ω
with the centered Gaussian process (Z(.)) such as for (f ,f ) [ i, i+1]2,
1 2
∈ α β
2
2(f f )2Hi ψ(ξ/f )ψ(ξ/f )
cov(Z(1/f ),Z(1/f ))= 1 2 1 2 e−iuξdξ du. (12)
1 2 (1−2r)KH2i(ψ)ZIR ZIR |ξ|2Hi+1 !
b b
For ∆ small enough, this result shows that all parameters H and σ2 could be estimated by using a linear
N i i
regression of logI (1/f ) versus logf , when the frequencies ω are known. Moreover, this central limit theorem
N j j i
shows that a graph of (logf, logI (1/f)) for f > 0 exhibits different areas of asymptotic linearity : it suggests
N
the procedure of the following section to estimate and test the frequency changes (see for instance figures 4 or 6).
3.4 The discretization problem
Intheapplications,weonlyobserveafinitetime series(X(0),X(∆ ), ,X((N 1) ∆ ))andwemustderived
N N
··· − ×
the empiricalwaveletcoefficients fromthis time series. Since the processX hasalmostacontinuouspathbutwith
a regularity α < 1 almost surely, we should use the Riemann sum. Thus, for (a,b) IR∗ IR we define the
X ∈ + ×
empirical wavelet coefficient by
N−1
∆ p∆
N N
e (a,b)= ψ( b) X(p∆ ) (13)
X N
√a a − ×
p=0
X
and the discretized estimator by
1
J (a)= e2 (a,k∆ ). (14)
N D (a) X N
N
| |k∈XDN(a)
We also define for every k D (a) the error
N
∈
ε (a,k)=e (a,k∆ ) d (a,k∆ ). (15)
N X N X N
−
Now, it is possible to provide the functional central limit theorem for (logJ (a)) computed from
N amin≤a≤amax
(X(0),X(∆ ), ,X(N∆ )):
N N
···
J.M. Bardet and P. Bertrand 9
Theorem 3.1 Under assumptions of Proposition 3.1 and with ∆ such as N∆ and N(∆ )2 0 when
N N N
→ ∞ →
N . Then, with the same process Z than in (9),
→∞
D
N∆ (logJ (a) log (a)) (Z(a)) . (16)
N N − I1 amin≤a≤amax N−→→∞ amin≤a≤amax
p
β ω
i+1
As a particular case, for i 0,1, ,K and if , then
∈{ ··· } α ≤ ω
i
N∆ logJ (1/f)+(2H+1)logf logσ2 logK (ψ)
N N i − i− Hi ωi/α≤f≤ωi+1/β
p (cid:0) D ((cid:1)Z(1/f)) . (17)
N−→→∞ ωi/α≤f≤ωi+1/β
The convergence rate of the central limit theorem (16) is √N∆ . Thus, the discretization problem implies that
N
the maximum convergence rate is o(N1/4) from the previous conditions on ∆ .
N
4 Identification of the parameters
First, let us describe the method on a heuristic level. From Proposition 3.1, Formula (17), we have
logJ (1/f)= (2H +1) log(f)+log σ2 +log(K (ψ))+ε(N), (18)
N − i × i Hi f
(cid:0) (cid:1)
for the frequencies f which satisfy the condition
log(ω ) log(α) log(f) log(ω ) log(β). (19)
i i+1
− ≤ ≤ −
Moreoverwehave(N∆ )1/2 ε(N) D (Z(1/f )) .Formula(18)andcondition(19)meanthatfor
N fj 1≤j≤m N−→→∞ j 1≤j≤m
log(f) [log(ω ) log(α),log(cid:16)(ω (cid:17)) log(β)], wehavealinearregressionof logJ (1/f) onto log(f) withslope
i i+1 N
∈ − −
(2H +1) and intercept
i
−
logσ2 + logK (ψ) and for log(f) [log(ω ) log(α),log(ω ) log(β)] a linear regression with slope
i Hi ∈ i+1 − i+2 −
(2H +1) and intercept logσ2 +logK (ψ). This is a problem of detection of abrupt change on the
− i+1 i+1 Hi+1
parametersofa linearregression,but witha transitionzone for log(f) ]log(ω ) log(β), log(ω ) log(α)[.
i+1 i+1
∈ − −
β
Remark 4.1 Condition (19) implies that ω > ω . Therefore we could only detect the frequency changes
i+1 i
α ×
sufficiently spaced. For instance, if we choose the Lemari´e-Meyer wavelet, we get β/α = 4 which leads to the
condition ω >4 ω .
i+1 i
×
In this section,we describe the estimationof the parametersanda goodnessof fit test. Both ofthem are basedon
the following assumption :
Assumption (B ) : The process X is a (M )-multiscale fractional Brownian motion. This process is
K K
characterized by the parameters Ω∗, H∗ and σ∗ where Ω∗ = (ω∗, ,ω∗ ) with H∗ = (H∗,H∗,...,H∗) and
1 ··· K 0 1 K
σ∗ =(σ∗,σ∗,...,σ∗ ). Moreover the following conditions are fulfilled
0 1 K
β
ω∗ > ω∗ for i=1, ,K 1;
• i+1 α × i ··· −
2 2
min H∗ H∗ + σ∗ σ∗ >0 and
• 0≤i≤(K−1) i+1− i i+1− i
(cid:26)(cid:16) (cid:17) (cid:16) (cid:17) (cid:27)
there exists a compact set ]0,1[ ]0, [such as (H∗,σ∗) for all i=0,1, ,K.
• K⊂ × ∞ i i ∈K ···
10 Statistic of multi-scale fractional Brownian motion
4.1 Estimation of the parameters
Let X be a (M )-F.B.M. satisfying the assumption (B ) with K a known integer number. We observe one path
K K
of the process at N discrete times, that (X(0),X(∆ ), ,X(N∆ )). Let [f , f ], with 0 < f < f ,
N N min max min max
···
be the chosen frequency band (see section 5, for an example). We discretize a (slightly modified) frequency band
and compute the wavelet coefficients at the frequencies (f ) where
k 0≤k≤aN
f f β 1/aN
f = min (q )k for k=0, ,a , q = max and a =N∆ .
k N N N N N
β ··· f α
(cid:18) min (cid:19)
For notational convenience , we assume here that N∆ is an integer number. By definition, we havef =f /β
N 0 min
andf =f /α,then,usingthewaveletcoefficientsatthefrequencies(f ) ,wecoulddetectallfrequency
aN max k 0≤k≤aN
changes (ω∗) included in the band ]f ,f [. To simplify the notations, we use the following assumption :
i min max
Assumption (C) : ω∗ ]f ,f [ for all i=1,...,K.
i ∈ min max
In this framework, the estimation of the different parameters of X becomes a problem of linear regression with
a known number of changes; thus, we follow the same method as in Bai (1994), Bai and Perron (1998), Lavielle
(1999) or Lavielle and Moulines (2000) and define the estimated parameters (T(N),Λ(N)) as the couple of vectors
which minimize the quadratic criterion :
b b
K+1 tj+1−τN
Q(N)(T,Λ)= Y X λ 2,and thus
i i j
| − |
Xj=0 i=X1+tj
(T(N),Λ(N))=Argmin Q(N)(T,Λ); T (N),Λ
∈AK ∈BK
n o
with
b b
Y =log(J (1/f )), X =(logf ,1) for i=0, ,a ;
i N i i i N
• ···
log(β/α)
τ = , where [x] is the integer part of x.
N
• logq
(cid:20) N (cid:21)
(N)
T =(t ,t , ,t ) where
• 0 1 ··· K+1 ∈AK
(N)= (t , ,t ) INK+2;t =0,t =a +τ ,t t >τ for j =0, ,K ;
AK 0 ··· K+1 ∈ 0 K+1 N N j+1− j N ···
(cid:8) (cid:9)
(2H +1)
• Λ=(λ0,··· ,λK)∈BK where λj =(cid:18) logσ−j2+lojgKHj(ψ) (cid:19) and then
= (λ , ,λ ) with (H ,σ2) for all j 0,1, ,K .
BK 0 ··· K j j ∈K ∈{ ··· }
n o
The integer τ correspondsto the number of frequencies in the transitionzones andlogf =logf +log(β/α).
N i+τN i
Obviously, for j = 0, ,K, the vector λ(N) provides the estimators H(N) of H∗ and σ(N) of σ∗ by the relation
··· j j j j j
(2H(N)+1)
λj(N) = log (σj(N−))2bj+logKHbj(N)(ψ)b. For a given T ∈ A(KN), eacbh λ(jN) is obtainebd from a linear regression
obf (Y ) onto (X(cid:16)) for i=(cid:17) t +1, ,t τ . Thus, with T =(t ) b obtained from the minimization in T
i i j j+1 N j 0≤j≤K+1
b ··· −
of Q(N)(T,Λ), we define the different estimators of the change frequencies as
b b
b tb(jN)
ω(N) =αf =α fmin fmax β aN for j =1, ,K. (20)
j bt(jN) · β (cid:18)fmin α(cid:19) ···
We have the following convbergence :