Statistics & Probability Letters 43 (1999) 393–397

Reducing non-stationary stochastic processes to stationarityby a time deformation

Olivier Perrin ∗;1, Rachid Senoussi 2

I.N.R.A., Domaine Saint-Paul, Site Agroparc, 84914 Avignon Cedex 9, France

Received July 1998; received in revised from September 1998

Abstract

A necessary and su�cient condition is given to reduce a non-stationary random process {Z(t): t ∈ T ⊆R} to stationarityvia a bijective di�erentiable time deformation � so that its correlation function r(t; t′) depends only on the di�erence�(t′) − �(t) through a stationary correlation function R: r(t; t′) = R(�(t′) − �(t)). c© 1999 Elsevier Science B.V. Allrights reserved

Keywords: Correlation function; Stationary reducibility; Weak stationarity

1. Introduction

In most applications dealing with non-stationary processes, the �rst step in classical approaches consists inremoving expectation, dividing the residuals by standard deviation and modelling the residuals as stationaryprocesses (AR, MA, ARMA, etc.). The process Z = {Z(t): t ∈ T ⊆R} under study is of the form

Z(t) = �(t) +√�(t)�(t);

where �(t)=EZ(t), �(t)=E(Z(t)−�(t))2 and �(t) a centred and standardised weakly (or strongly) stationaryprocess. The non-stationarity of Z is then understood as non-stationarity of both the �rst-order moment �(t)and the variance �(t).Modelling �(t) as

�(t) = �(�(t));

∗ Correspondence address: Mathematical Statistics, Chalmers University of Technology and G�oteborg University, S-412 96 G�oteborg,Sweden. E-mail: olivier@math.chalmers.se.1 Partially supported by the European Union’s research network “Statistical and Computational Methods for the Analysis of Spatial

Data”, grant ERB-FMRX-CT96-0096.2 E-mail: senoussi@avignon.inra.fr.

0167-7152/99/$ – see front matter c© 1999 Elsevier Science B.V. All rights reservedPII: S0167 -7152(98)00278 -8

394 O. Perrin, R. Senoussi / Statistics & Probability Letters 43 (1999) 393–397

where � is weakly stationary and � is a bijective deformation, or equivalently modelling the correlationfunction r(t; t′) of the process Z as

r(t; t′) = R(�(t′)− �(t)); (1)

where R is a stationary correlation function, is a way to introduce further non-stationarity. This is discussedby Sampson and Guttorp (1992) and is later developed in Meiring (1995) and Perrin (1997).When a correlation function satis�es (1) we call it a stationary reducible correlation function. Unlike the

classical approaches, non-stationarity through second-order moments is thus taken into account and (1) givesopportunity to enlarge the class of models for studying non-stationary processes.According to Proposition 7.1.4 in Samorodnitsky and Taqqu (1994) the correlation function of any H -self-

similar process with parameter H ¿ 0 indexed by T={t: t ¿ 0} satis�es model (1) with �(t)=ln(t). However,not every correlation function can be reduced to a stationarity one.This work appears closely related to the following question of Krivine (in Assouad, 1980):

On a locally compact Abelian group (T;+), how can the positive Hermitian kernels r which sat-isfy r(t; t′) = R(�(t′) − �(t)) be characterised, where � is a transformation of the space T and R apositive-de�nite function?

In this paper we characterise stationary reducible correlation functions under smoothness assumptions. Thepaper is organised as follows. In Section 2 we give some properties of model (1) and propose a characterisationfor smooth stationary reducible correlation functions in the form of a di�erential equation. Further, we proveuniqueness of the deformation � up to a translation and a scale change. Section 3 contains two examples ofstationary reducible correlations and two counterexamples.

2. Properties, characterisation and uniqueness

2.1. Properties of stationary reducibility

So far we have considered correlation functions. However, model (1) can be applied to covariance functionsprovided that the variance of the process is constant. Moreover, when the mean of the process is also constant,model (1) generalises the notion of weak stationarity (or strong stationarity for Gaussian processes); indeed,when the deformation � is the identity function stationarity and reducible stationarity agree.

2.2. Characterisation and uniqueness

Let the index space T be either R, {t: t¿0}, {t: t ¿ 0} or an interval [a; b]⊂R and consider for thebijective deformation � the following assumption:(A1) � is continuous and di�erentiable in T as is its inverse:Note that if (�; R) is a solution to (1), then for any �¿ 0 and � ∈ R, (�̃; R̃) with �̃(x) = ��(x) + � and

R̃(u) = R(u=�) is a solution as well. Thus, without loss of generality, we may impose the restriction that forsome �xed point t0 in T

�(t0) = 0 and �′(t0) = 1; (2)

where �′ is the derivative of �. Consequently, when the non-stationary correlation r(t; t′) satisfying (1) andthe deformation � are given, the stationary correlation function R is uniquely determined as

R(u) = r(t0; �−1(u)): (3)

We denote by @ir(t; t′), i = 1; 2, the �rst partial derivatives of r(t; t′). We consider correlation functionsr(t; t′) such that

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(A2) r(t; t′) is continuous and di�erentiable for t 6= t′;(A3) the function u 7→ @1r(u; t0)=@2r(u; t0) is locally Lebesgue integrable in T .

Assumptions (A2) and (A3) are satis�ed by a large class of processes including mth-order iterated integrals ofa Wiener process, non-degenerate fractional Brownian motions (which are a particular case of H -self-similarprocesses) with parameter H ∈ ]0; 1] and more generally processes with independent increments provided thevariance is continuously di�erentiable.It follows from (A1) and (A2) that the stationary correlation function R(u) is continuous and di�erentiable

for u di�erent from 0.Here is the necessary and su�cient condition for stationary reducibility via bijective deformation.

Theorem 2.1. Assume (A1)–(A3). A correlation function r(t; t′) satis�es (1) if and only if almost everywherefor t 6= t′

@1r(t; t′)@1r(t′; t0)@2r(t′; t0)

+ @2r(t; t′)@1r(t; t0)@2r(t; t0)

= 0: (4)

The pair (�; R) in (1) is uniquely determined with � given by

�(t) =−∫ t

t0

@1r(u; t0)@2r(u; t0)

du (5)

and R given by (3) and (5).

Proof. First, we show that (1) is equivalent to

@1r(t; t′)�′(t′) + @2r(t; t′)�′(t) = 0; ∀t 6= t′: (6)

Indeed if (1) holds for t 6= t′@1r(t; t′) = −�′(t)R′(�(t′)− �(t));@2r(t; y′) = �′(t′)R′(�(t′)− �(t));

so that (6) is satis�ed. Conversely assume (6) holds, consider the bijective coordinate change

u = �(t′)− �(t);u′ = �(t′) + �(t)

and set �(u; u′) = r(t(u; u′); t′(u; u′)). Eq. (6) then leads to

@2�(u; u′) = 0:

Thus � does not depend on the second coordinate. Therefore, we may de�ne R(u)=�(u; u′) and (1) follows.Further, under condition (2), we infer from (6) that (5) holds and (4) now follows from (5) and (6).Conversely, if (4) holds, we may de�ne � through (5), and this � satis�es (6). Clearly, � is uniquelydetermined by (5) and R by (3).

3. Examples and counterexamples

3.1. Examples

3.1.1. Fractional Brownian motionsAs already mentioned, non-degenerate fractional Brownian motions are a particular case of H -self-

similar processes with parameter H ∈ ]0; 1]. Their correlation functions are of the form for all t; t′ such

396 O. Perrin, R. Senoussi / Statistics & Probability Letters 43 (1999) 393–397

that tt′ 6= 0

r(t; t′) =(t 2H + t′2H − |t′ − t|2H )

2tH t′H: (7)

As pointed out in the introduction, they satisfy (1) with �(t)= ln(t) (Proposition 7.1.4 in Samorodnitsky andTaqqu, 1994). The corresponding stationary correlation function R is

R(u) = cosh(Hu)− 2(2H−1)(sinh(|u|=2))2H ;where cosh and sinh denote hyperbolic cosine and sine functions.

3.1.2. A simple di�usion processFor a measurable function f on T = {t: t¿0}, with f(t) 6= 0, locally square integrable with respect to

the Lebesgue measure, and W the standard Wiener process, the process Z(t) =∫ t0 f(s) dW (s) is a centred

Gaussian process with correlation function de�ned for all t; t′ such that tt′ 6= 0

r(t; t′) =

√∫ t∧t′0 f2(u) du√∫ t∨t′0 f2(u) du

:

Calculation of the partial derivatives of r(t; t′) show that (4) is satis�ed for tt′ 6= 0. Thus, it followsfrom Eq. (5) that the deformation � which satis�es (2) with t0 = 0 and that makes Z strongly stationary isde�ned by

�(t) =

∫ 10 f

2(u) duf2(1)

ln

(∫ t0 f

2(u) du∫ 10 f

2(u) du

):

If f(u) = u� with �¿ − 12 , the same deformation �(t) = ln(t) as before is obtained. The corresponding

stationary correlation function R is

R(u) = exp(−(2� + 12

)|u|):

3.2. Counterexamples

3.2.1. A smooth correlation functionConsider a stationary isotropic Gaussian random �eld Z̃(x) indexed by R2 which has the correlation function

r̃(x; y)=exp(−‖y−x‖22) where ‖u‖2 is Euclidean distance in R2. Strolling along the parabolic path t 7→ (t; t2),we consider the process Z(t)= Z̃(t; t2) indexed by R. Its correlation function is r(t; t′)=exp{−(t′− t)2(1+(t+t′)2)} and is in�nite di�erentiable. Suppose there is a continuous and di�erentiable deformation � reducingr. Then with t0 = 0 (4) gives

(1 + 2t(t + t′))(1 + 2t′2) = (1 + 2t′(t + t′))(1 + 2t2); ∀t 6= t′:This equation implies necessarily that t = t′. Therefore, Z(t) is not stationary reducible.

3.2.2. Assouad’s counterexampleThis counterexample constitutes a partial answer of Assouad (1980) to the question of Krivine quoted in

the introduction.

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We endow R3 with the canonical scalar product 〈: ; :〉3. The counterexample is obtained as follows:1. consider six points X1; : : : ; X6 on the unit sphere S2 of R3 that meet the always satis�ed conditions

X1 + X2 = 2X6 cos �; X2 + X3 = 2X4 cos �; X1 + X3 = 2X5 cos �

for any � ∈ ]0; �=3[;2. build the Gram matrix (symmetric and positive de�nite)

(ri; j) = (〈Xi; Xj〉3)16i; j66;3. then construct a continuous (in the mean square sense) Gaussian process Z = {Z(t); t ∈ [0; 1]} with corre-lation r(t; t′) that satis�es for 16i; j66

r((i − 1)=5; (j − 1)=5) = ri; j;4. �nally Perrin (1997) proves that there is no bijective continuous deformation � of R that makes Z stationary.

Acknowledgements

This work was done while O. Perrin was visiting the department of Mathematical Statistics at ChalmersUniversity of Technology and G�oteborg University in Sweden. We thank this University for its heartfelthospitality.

References

Assouad, P., 1980. Plongements isom�etriques dans L1: aspect analytique. S�eminaire d’Initiation �a l’Analyse 14, 1–23.Meiring, W., 1995. Estimation of heterogeneous space–time covariance. Ph.D. Thesis, University of Washington, Seattle.Perrin, O., 1997. Mod�ele de covariance d’un processus non-stationnaire par d�eformation de l’espace et statistique. Ph.D. Thesis, Universit�ede Paris I Panth�eon-Sorbonne, Paris.

Samorodnitsky, G., Taqqu, M.S., 1994. Stable Non-Gaussian Random Processes. Chapman & Hall, London.Sampson, P.D., Guttorp, P., 1992. Nonparametric estimation of nonstationary spatial covariance structure. J. Amer. Statist. Assoc. 87,108–119.