Identification of space deformation using linear and superficial quadratic variations

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Abstract

We use linear and superficial quadratic variations to identify a bijective space deformation that makes a non-stationary Gaussian random field stationary.

Introduction

In spatial statistics we are often concerned with non-stationary phenomena. For most applications dealing with a non-stationary Gaussian random field. The first step in classical approaches consists of removing expectation, dividing the residual by the standard deviation and modelling the residual as a stationary process. That is to say, the random field Y={Y(x,y):(x,y)∈G⊆R2} (where G stands for Geographical space) under study is of the formY(x,y)=μ(x,y)+σ(x,y)Z(x,y),where μ(x,y)=EY(x,y), σ(x,y)2=E(Y(x,y)−μ(x,y))2 and Z(x,y) is a reduced (centred and standardised) stationary Gaussian random field. Then the non-stationarity of the random field Y is simply understood as non-stationarity of both the first-order moment μ(x,y) and the standard deviation σ(x,y). Nevertheless, Z can also be non-stationary. In this case, Sampson and Guttorp (1992) propose a transformation of the index space G with a bijective space deformation. Formally, it consists of modelling Z(x,y) asZ(x,y)=δ(Φ(x,y)),where δ is stationary and Φ=(Φ12):G→D (where D stands for Deformed space) is a bijective deformation. Equivalently, the correlation function r of Z isr(x,y,x′,y′)=R(Φ(x′,y′)−Φ(x,y)),where R is a stationary correlation function in R2.

In this paper, we consider a reduced Gaussian random field Z indexed by G=[0,1]2 satisfying , and we suppose that the stationary correlation R is known. Our concern is the functional estimation of the space deformation Φ using a set of suitable linear and superficial quadratic variations of Z.

The quadratic variations are first introduced by Lévy (1940) who shows that if Z is the standard Wiener process on [0,1], then almost surely its quadratic variation on [0,1] converges to 1. Baxter (1956) and further Gladyshev (1961) generalise this result to a large class of Gaussian processes. Guyon and León (1989) introduce the H-variations for stationary Gaussian processes, a generalisation of these quadratic variations. They study the convergence in distribution of the H-variations, suitably normalised.

For Gaussian process Z with stationary increments, Istas and Lang (1997) define general quadratic variations, substituting a general discrete difference operator to the simple difference Z(k/n)−Z((k−1)/n). They use these quadratic variations to estimate the Hölder index of a process.

For non-stationary Gaussian processes, with increments stationary or not, Perrin (1998) gives a general result concerning the functional asymptotic normality of the process of the quadratic variations which corresponds to the linear interpolation of the points (p/n,Vn(p/n)), p=1,2,…,n, with Vn(p/n) the discrete quadratic variations at points p/n. This result is applied to the estimation of a time deformation for non-stationary models of the form Z(x)=δ(Φ(x)),x∈[0,1].

The generalisation of quadratic variations for stationary Gaussian fields is studied in Guyon (1987) and León and Ortega (1989). Another generalisation for non-stationary Gaussian processes and quadratic variations along curves is done in Adler and Pyke (1993). Guyon (1987) shows that some stationary random fields can be identified in mean square sense using different families of variations. Using some of these families allows us, in this paper, to generalise the result of Perrin (1998) to the estimation of a space deformation for non-stationary models of the form , .

The paper is structured as follows. Section 2 sets up notations, assumptions, definitions and describes the superficial and the linear quadratic variations. In Section 3, we study the pointwise mean square (L2) convergence of these quadratic variations. In Section 4, we propose an estimator of Φ which converges in L2 to Φ. This estimator is defined with the help of the superficial and the linear quadratic variations. Finally, Section 5 discusses two extensions of the present work for future research.

Section snippets

Linear and superficial quadratic variations

Let Z={Z(x,y),(x,y)∈[0,1]2} be a real-valued reduced Gaussian random field with correlation r satisfying (2).

For any differentiable function f:(x,y)∈[0,1]2R, we denote using f(p1,p2) the p1,p2-partial derivative of f with respect to x and y. Assume for the stationary correlation function R

(A1)R(u,v) satisfies when u→0 and v→0:
R(u,v)=1−α|u|−β|v|+O(uv) where α>0 and β>0.
(A2)R(2,0)(u,v),R(1,1)(u,v),R(0,2)(u,v) exist and are uniformly bounded in {(u,v):u≠0},{(u,v):uv≠0},{(u,v):v≠0}.
For instance,

Asymptotic properties

In this section, we are concerned with asymptotic properties of the quadratic variations defined in the previous section. Because of the symmetry in the definitions of Hn,λ(x,y) and Vm,λ(x,y) (respectively hn(x) and vm(y)), we focus our attention on Hn,λ(x,y) and hn(x).

Estimator of the space deformation

Using the fact that λ, the geometry of the rectangular partition Πn,m, is a parameter under our control, the superficial quadratic variations, Hn,λ(x,y) and Vm,λ(x,y), for two distinct λ's, together with the linear quadratic variations, hn(x) and vm(y), provide a useful tool for identifying the space deformation Φ in model , . Let us define for all (x,y)∈[0,1]2 and any two distinct values λ1 and λ2 of λ in ]a,b[∩Q+Φ̂1,n(x,y)=λ1Vλ2n,λ2(x,y)−λ2Vλ1n,λ1(x,y)+2(λ1−λ2)hn(x)4α(λ1−λ2)1Hn,λ1(x,0)−λ2H

Discussion

In this section, we wish to point out two developments for improving and extending the present work:

  • So far the identification procedure we have presented is only concerned with stationary correlation functions satisfying (A1). By using the same kind of estimation method through quadratic variations, we would like to obtain identification of the space deformation for other types of correlation structures. More precisely we are currently interested in identifying the space deformation that makes

Acknowledgements

Part of this work was done while the second author was visiting the department of Mathematical Statistics at Chalmers University of Technology and Göteborg University in Sweden. We thank this University for its heartfelt hospitality.

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