STOCHASTIC DISCRETE SCALE INVARIANCEAND LAMPERTI TRANSFORMATION

Pierre Borgnat, Patrick Flandrin

Laboratoire de Physique (UMR CNRS 5672)Ecole normale sup´erieure de Lyon

46, allee d’Italie 69364 Lyon Cedex 07, France

Pierre-Olivier Amblard

Laboratoire des Images et des SignauxLIS-UMR CNRS 5083, ENSIEG-BP 46

38402 Saint Martin d’H`eres Cedex, France

ABSTRACT

We define and study stochastic discrete scale invariance (DSI), aproperty which requires invariance by dilation for certain preferredscaling factors only. We prove that the Lamperti transformation,known to map self-similar processes to stationary processes, is animportant tool to study these processes and gives a more generalconnection: in particular between DSI and cyclostationarity. Somegeneral properties of DSI processes are given. Examples of ran-domsequences with DSI are then constructed and illustrated. Weaddress finally the problem of analysis of DSI processes, first usingthe inverse Lamperti transformation to analyse DSI processes bymeans of cyclostationary methods. Second we propose to re-writethese tools directly in a Mellin formalism.

1. DISCRETE SCALE INVARIANCE

Scale invariance, also called self-similarity, is frequentlycalled upon. Its central point is that the signal is scale in-variant if it is equivalent to any of its rescaled versions, upto some amplitude renormalization [1]. More precisely, afunction���� is scale-invariant with exponent� , or�-ss,if for any � � �: ����� � �������

This definition is given here for a deterministic signal.The concept can be extended to stochastic signals when onethinks of the previous equality in a probabilistic way: theequality of the finite-dimensional probability distributions

[1]. We will write�� this equality.

The strict notion of scale invariance, valid for all dila-tion factors above, is in somecases too rigid; the middle-third Cantor set is for example invariant only by dilationsof a factor 3 (or a power of 3). Several weakened versionsof self-similarity have been proposed to enlarge scale in-variance’s relevance and one is of special interest here: it isto require invariance by dilation for certain preferred scal-ing factors only, as it is the case for the Cantor set. This isknown asdiscrete scale invariance (DSI), a concept whichas been stressed upon by Sornette and Saleur [2, 3] as anefficient model in many situations (fracture, DLA, criticalphenomena, earthquakes).

They studied DSI as a property of deterministic signals,and provided general arguments as why should DSI nat-urally occur: classical scenarii involve the existence of acharacteristic scale, the apparition by instability of a pre-ferred scale or more general arguments in non-unitary fieldtheories [4]. They also found ways to estimate the preferredscaling ratio in this context, based on classical spectral anal-ysis (Lomb periodogram).

As far as weknow, this property has not been envisionedfor stochastic processes, a framework which is often fruit-ful to dispose of when dealing with real measurements, as itallows to use statistical signal processing methods. The ex-tension of DSI property to stochastic processes is straigth-forward. We propose the following definition.

A process ������ � � ��� has discrete scale invari-

ance with scaling exponent � and scale � if

������� ������� � � �� � (1)

We will refer to this property as�����-DSI. The equal-ity here is the probabilistic equality. In the following onlywide-sense property will be used (second-order statisticalproperties only).

2. LAMPERTI TRANSFORM : DSI AS AN IMAGEOF CYCLOSTATIONARITY

2.1. Lamperti transformation

A main issue is to find a way to studyboth theoretically andpractically DSI processes. The answer is given by a trans-formation introduced by J. Lamperti in 1962 [5], which isan isometry between self-similar and stationary processes.It will be called the Lamperti transformation and is definedas follows.

For any process�� ���� � � ��, its Lamperti transform������ � � �

�� and its inverse are given by

���� � ��� � ��� �� ��� ��� ��� � � �� � (2)

� ��� ������

���� �� �������� � � �� (3)

The theorem in the paper ofLamperti is that a process� ��� is stationary if and only if its Lamperti transform� ��� is �-ss. The central argument of the derivation is thatthe Lamperti transformation maps a time-shifted process tothe dilated version of the Lamperti transform of the originalprocess. Let

���� �

���� ���������� be the dilation oper-

ator and���� � ��� ��� �� � � the time-shift operator. Theproperty is that

���� ��

� � �����

�� ������ � ���� (4)

Understanding this correspondence between time-shiftand dilation operators, we can propose many variations aroundLamperti’s theorem, relaxing in some way the stationarityfor � and the self-similarity for� . We will only con-sider here the DSI property but some results about differentclasses of processes and their description are proposed in[6]. A useful property is that one can give the (potentiallynonstationary) correlation function of the Lamperti trans-form� of a process� :

� ���������� �� ����� �� � ������� ��� �� �� ��� (5)

In the recent years some results have been obtained for�-ss processes with this transformation. Yazici and Kashyapproposed a general description of wide-sense self-similarprocesses and linear models for�-ss [7]. Burneckiet al.study -stable and�-ss processes with this transform [8].Nuzman and Poor give important results about the predic-tion, the whitening and the interpolation of�-ss processes,mainly applied to the fractional Brownian motion [9]. Fi-nally Vid acs and Virtamo [10] proposed a method of esti-mation of� for a fBm, based on the same idea. All theseauthors use the inverse Lamperti transformation (3) to mapthe question to a stationaryproblem and then use the knownresults for stationary issues in this context. Our objective isto show that nonstationary methods can be adapted in thesame way, especially for DSI.

2.2. DSI and cyclostationarity

A process is called cyclostationary [11] or periodically-cor-related [12, 13], if its correlation function is periodic intime. More precisely, if a period � is given, a process�� ���� � � �� is wide-sense cyclostationary if it satisfiesfor anytimes�� �

� � ��� � � � � � ����

� �� ��� � �� ��� � �� � � �� ���� ���� � (6)

The correlation function�� ��� � � � is then periodic in �of period� and one can decompose�� in a Fourier series

�� ��� �� � �

�������

��������� � (7)

Using the definitions of cyclostationarity and�����-DSI and the correspondance (4), we can state the followingimportant result.

A process �� ���� � � �� is cyclostationary of period� if and only if its Lamperti transform of parameter�:����� � ��� ��� ��� � � ���� is ��� � �-DSI.

This is one possible extension of Lamperti’s theorem,one of importance in our study of DSI. A first consequence,using (5), is that the general form of covariance of�����-DSI processes is naturally expressed on a Mellin basis:

����� ��� � ��������

����

��������� ���� (8)

Note that if the process� is real-valued, a necessarycondition is imposed:������ � ������. TheMellin func-tion ������ ��� in (8) is central in the study of DSI pro-cesses. This is not a surprise: Lamperti transformation mapsthe Fourier basis (invariant up to a phase by time-shift) tothe Mellin basis (invariant up to a phase by dilation and hav-ing also the deterministic DSI property). We stress the factthe Mellin functions are a basis and that they have an asso-ciated transformation whichcan be numerically computed[14].

3. EXAMPLES OF PROCESSES AND SEQUENCESWITH DSI

Continuous-time systems with DSI property are easily con-structed. Applying� to an ARMA(�,�) system,we obtaina generalization of the Euler-Cauchy (EC) system. It is amodel for self-similar processes [7], driven by a multiplica-tive Gaussian noise����, whose correlation is� ����������������� ��. Theprocess���� verifies

����

���� d�

d������ �

�����

������ d�

d������� (9)

In the same manner that a nonstationary ARMA model withperiodic time-varying coefficients is cyclostationary [15],one obtains a DSI model when taking log-periodic time-varying coefficients�� and �� in the (EC) system. Thiswill be not detailed further.

In order to obtain DSI processes in discrete time (ran-dom sequences with self-similarity and log-periodicity), apossibility is to consider a discrete-time system analog to(EC) (�-ss in a certain way), then introduce log-periodicityin the coefficients. We describe two approaches here.

A direct discretization in time of the (EC) system isgiven by the integration of its evolution between two in-stants. This was proposed in [16] for the first order. Thisnonstationary�-ss system is written as�� � ������ ��� where�� � � � �� and� is a time-decorrelated

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−30

−20

−10

0

10

20

30

40Signal with DSI, H=0.1, λ=1.06

0.2 0.4 0.6 0.8 1−6

−4

−2

0

2

4

6

8

10Signal with DSI, H=0.1, λ=1.06

Fig. 1. Typical realizations of DSI random sequences. On the leftthe model is a (EC) system of order 2 discretized, in cascade witha log-periodic AR(1). On the right it is constructed on fractionaldifference (see text). The length is 5000 points,� � ���, � �

���. The oscillations above the signals are indicative of the log-periodicity of the AR.�� � ���, � � ���, �� � ���� and�� � �.

Gaussian noise with variance� �� �����, when� islarge. The generalization to the discretization of (EC) of or-der� is straightforward. The result is of the form, for thelarge times�

�������� � �����������������

� ���AR��� ������ � � ������ (10)

where� is the backward operator, and AR is an AR model.Such a system with log-periodicity in the coefficient��

and in the AR, or in cascade with a log-periodic AR sys-tem (see for example the AR(1) proposed hereafter, equa-tion 12), will present an approximate DSI property. Thereader can see on the left of figure 1 a realization of such aprocess.

Another class of discrete-time self-similar systems isgiven by models constructed on the fractional difference op-erator. The usual method is to use its moving average repre-sentation written as a binomial expansion. We prefer to usethe discretization proposed in [17], constructed with somegeneralization of the bilinear transformation in order to de-fine a scaling operator for sequences. The fractional differ-ence operator is then a filter��� whose impulse reponseis

��� �

�����

�������� � ������ � �� ��

��� � ������ � � ������������ (11)

This filter is in cascade with a nonstationary AR filterwhose coefficients are log-perodic. For example we maylimit ourselves to the first order (coefficient��), taking carethat the filter is stable at each instant:

�� �

��� �� ��

�� �� �

���

�������� ���

�� ����

� (12)

We propose an example of such a signal fig. 1 on the right.

4. ANALYSIS BY DELAMPERTIZATION

In front of a general class of processes (or random sequencesin the context of numerical processing) which are nonsta-tionary, or of unknown structure, one has to find methods toanalyse those. Given a sequence�� suspected of DSI, thesimplest way of analysis is to find the presumed cyclosta-tionary process associated by applying���.

Generally speaking, classical stationary methods are use-ful to analyse self-similar process after “delampertization”of the signal. This was the essence of papers on�-ss pro-cesses cited before [7, 8, 9]. Nonstationary methods canthen beused tu study classes of processes which have notproper self-similarity, but which have some kind of nonsta-tionarity with regards to dilation - a nonstationarity in scale.DSI is thenonly a first interesting example of a precise kindof nonstationarity in scale.

Before using cyclostationary methods, a practical prob-lem must be considered : how to compute in discrete timethe inverse Lamperti transformation ? First, it needs a nonlinear sampling� � �� of the data (but such is not oftenthe case with real signals), or an interpolation to find thedata with this geometrical sampling, given a signal�� withusual arithmetic sampling: the corresponding sequence��is known for � � ���, with � � � and we want it for� � �, � � �. Figure 2 shows on the left the sequence�constructed from the second process on figure 1.

A second difficulty is that� is a priori unknown. Usingthe transformation of parameter� seems tricky... In factthe tools used thereafter have not been found to be sensitiveto this amplitude effect. The cyclostationary tools are foundunaffected if one uses� � ��� to delampertize the processin place of the real� .

We tried the applicability of these ideas on synthetic se-quences. As an example of a classical cyclostationary tool,weimplemented the methods proposed in [18]. In a nutshellthe algorithm to estimate a time-smoothed cyclic cross pe-riodogram is as follows. First the signal is decomposed in� segmentsof length in order to average on these parts.A filtered and decomposed version is computed, where! isa data tapering window:

��� ��� "� �

����������

!���� ��� �������� �� (13)

Then the spectral components��� ��� �� are correlated atfrequencies" � #��� and" � #��� by a multiplier followedby a low-pass filter$:

%��� �&� "� ���

��� ��� " �#��� �� �� ��� " �

#���$�& � ���

This is an estimate of the spectral cross correlation. Theusual spectrum is distributed on the main diagonal#� � �

6.5 7 7.5 8 8.5−6

−4

−2

0

2

4

6

8

10Delampertzation of a DSI signal

−0.2 0 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Cyclic Marginal for a delampertized DSI process

↓ ↓

Fig. 2. On the left is shown the cyclostationary sequence afterusing��� on the signal plotted on the right of fig. 1. The marginalin cyclic frequency is represented on the right. The main peak onthe center is the total energy of the signal. The two symmetricpeaks (pointed on by arrows) are an indication of cyclostationarityand situated tofrequencies���� ��.

and for cyclostationary sequences it presents non-zero dis-tributions on#� � ���� (and eventually on higher har-monics). The marginal in cyclic frequency of this spectrumhas then sharp peaks on��� where� � � for DSI andgives a reliable estimation of�. See figure 2 the result ofthis procedure for the synthetic model described before.

5. TOWARD MELLIN-BASED TOOLS

Another way of thinking might be fecund to analyse DSIprocesses. We can formulate directly the methods in a Mellinformalism, with no geometrical resampling. That is to saythat we oper a “lampertization” of the tools where the firstway proposed to “delampertize” the signal studied.

By direct interpolation we have few details for the shorttimes (in fact we can’t reconstitute� ' �) and we ignoremany details in the long times (taking one point amongmany). To obtain statistical relevance, one has to have ahuge number of points in the original data to make someprocessing. The avantage, remarked in [8, 10], is that thereare fewer points in� , then�� after geometrical resamplingand this keeps the computational cost low.

Whenone does not dispose of a large number of points,using a geometric sampling loose much information on thesignal. As the Fourier transform of a process is related tothe Mellin transform of the process transformed by�, manymethods for cyclostationary processes can be written withMellin transformation and used on processes with DSI. Forself-similar signals (� � �), estimators constructed in thisway were given in [19] and can be adapted to take into ac-count an exponent� and DSI.

6. REFERENCES

[1] G. Samorodnitsky and M. Taqqu,Stable Non-Gaussian Ran-dom Processes, Chapman&Hall, 1994.

[2] D. Sornette, “Discrete scale invariance and complex dimen-sions,” Physics Reports, vol. 297, pp. 239–270, 1998.

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[7] B. Yazici and R. L. Kashyap, “A class of second-order sta-tionary self-similar processes for��� phenomena,” IEEETrans. on Signal Proc., vol. 45, no. 2, pp. 396–410, 1997.

[8] K. Burnecki, M. Maejima, and A. Weron, “The Lampertitransformation for self-similar processes,”Yokohama Math.J., vol. 44, pp. 25–42, 1997.

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[10] A. Vidacs and J. Virtamo, “ML estimation of the parametersof fBm traffic with geometrical sampling,” inIFIP TC6, Int.Conf. on Broadband communications ’99. Nov. 1999, Hong-Kong.

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[16] E. Noret and M. Guglielmi, “Mod´elisation et synth`ese d’uneclasse de signaux auto-similaires et `a memoire longue,” inProc. Conf. Delft (NL) : Fractals in Engineering. 2000, pp.301–315, INRIA.

[17] W. Zhao and R. Rao, “On modeling self-similar random pro-cesses in discrete-time,” inProc. IEEE Time-Frequency andTime-Scale, Oct. 1998, pp. 333–336.

[18] R. Roberts, W. Brown, and H. Loomis, “Computationallyefficient algorithms for cyclic spectral analysis,”IEEE SPMagazine, pp. 38–49, Apr. 1991.

[19] H. L. Gray and N. F. Zhang, “On a class of nonstationaryprocesses,”Journal of Time Series Analysis, vol. 9,no. 2, pp.133–154, 1988.