Zaka Ratsimalahelo
email: zaka.ratsimalahelo@univ-fcomte.fr

CANONICAL CORRELATION IN MULTIVARIATE TIME SERIES ANALYSIS

Keywords: Tim e Series, Canonical Correlation, Singular Value Decomposition, Balancing, Innovations models, subspace methods, Instrumental Variables Estimators. JEL classification: C13, C 32. 1. Introduction. l

W e analyze a class o f state space identification algorithms for time series, based on canonical correlation analysis in the ligth o f recent results on stochastic systems theory called « subspace methods ». These can be describe as covariance estimation followed by stochastic realization. The methods offer the major advantage o f converting the nonlinear parameter estimation phase in traditional V A R M A models identification into the solution o f Riccati equation but introduce at the same time some nontrivial mathematical problems related to positivity. The states o f the forward-backward innovations representation have an interpretation: Instrumental Variables estimators.

State space models have several inherent advantages in model specification. The states are formal, minimal sufficient statistics for the past history o f the series, an important consideration in time series. Recently there has been a renewed interest in identification algorithms based on a two steps procedure which in principle can be described as covariance estimation follow ed by stochastic realization, [START_REF] Aoki | State Space Modeling of Time Series[END_REF][START_REF] Aoki | Two complementary representations of multiple time series in state space innovation forms[END_REF]. These algorithms are based on canonical correlation analysis and can naturally accomodate multivariate time series. They offer the major advantage o f converting the nonlinear parameter estimation phase which is necessary in traditional V A R , V M A , and V A R M A models identification into the solution o f an algebraic Riccati equation, a much better undestood problem for which efficient numerical solution techniques are available. On the other hand these methods introduce some very nontrivial mathematical problems related to positivity, Heij et al (1992), [START_REF] Vaccaro | A solution to the positivity problem in the state space approach to modeling vector-valued time series[END_REF], [START_REF] Ratsimalahelo | On Approximate Dynamic Systems[END_REF]. In this paper we shall analyze the above class o f stochastic state-space identification methods in the light o f recent results on stochastic systems theory. Lindquist and Picci (1996), Van Overschee and De M oor (1996).

From the point o f view o f identification in state-space models, there are two possible approaches to determine the parameters o f a model from the finite covariance sequence. One that has been proposed in the literature is do minimal factorization o f a finite block Hankel matrix in balanced form, [START_REF] Aoki | State Space Modeling of Time Series[END_REF], [START_REF] Aoki | State space modeling of multiple time series[END_REF], [START_REF] Aoki | Two complementary representations of multiple time series in state space innovation forms[END_REF], this yields a solution to the minimal partial realization problem which no guarantee a positive extension.

The second approach would be instead to perform positive extension first and then to use a stochastic model reduction procedure on the parameters of the positive extended sequence, Lindquist and Picci (1996). [START_REF] Akaike | Markovian representation of stochastic processes by canonical variables[END_REF] has developed a stochastic realization theory based on the information interface between the past and the future of a time series and the concepts of canonical correlation analysis. Here the pair of canonical vectors with positive canonical correlations form a minimal interface between the past and the future of the process. It is shown that these two canonical vectors are the states of the forward and backward innovations representation (extreme Markovian representations) introduced by [START_REF] Faurre | Stochastic realization algorithm[END_REF] and are also basis vectors of what Akaike calls the forward and backward predictor spaces, respectively. Moreover, the canonical correlations coefficients provide a rational basis for obtaining reduced order model. [START_REF] Desai | A transformation approach to stochastic model reduction[END_REF], [START_REF] Desai | A realization approach to stochastic model reduction[END_REF] extended Akaike's work by introducing the concept of balanced stochastic realizations. Here a forward-backward dual pair is obtained with state covariance matrices being equal and diagonal and these diagonal elements are the canonical correlation coefficients. A forward-backward pair that satisfies these conditions is said to be in balanced form. These balancing conditions are the stochastic counterpart to the deteministic balancing conditions using by [START_REF] Aoki | State Space Modeling of Time Series[END_REF], [START_REF] Aoki | State space modeling of multiple time series[END_REF], [START_REF] Aoki | Two complementary representations of multiple time series in state space innovation forms[END_REF]. Balanced representation state space models have several property: an orderly specification search over a small number of dimensions and a choice of coordinate systems for states that makes sequential model order tests uncorrelated, thus preserving their confidence levels. The balanced truncation is in fact a stochastic model reduction procedure. In all such procedures there must be a guarantee that the reduced degree matrix function is positive real. . [START_REF] Aoki | State Space Modeling of Time Series[END_REF], [START_REF] Aoki | State space modeling of multiple time series[END_REF] [START_REF] Aoki | Two complementary representations of multiple time series in state space innovation forms[END_REF] used an algorithm deterministic realization procedure and hence does not a priori insure that Ricatti equation is positive real even the Toeplitz matrix is positive definite. This paper is organized as follows. In section 2, we present the classical realization approach background. In section 3, the main principles of canonical correlation are outlined. In section 4, useful properties of dynamic systems are summarized, with the canonical correlation concept and balanced stochastic realization, in section 5 we show how approximate stochastic realization.

The classical realization approach.

Let yt e Rm , t = 0, 1, 2,... T be a data sequence that is generated by the following system

x,+i = Axt + u, (1) yt = Cxt + V( (2)
where x, e Rn is an unobserved state vector. A and C are matrices of appropriate dimensions. The vector sequence ut e Rn is a process noise while the vector sequence vt € Rm is a measurement noise, they are both assumed to be zero mean white, with covariance matrix

N / ■ • \ S ' Ut Vr V 1 1 / ,S' Ry
where 5ts is the Kronecker delta function and primes denote the transpose of a vector and matrix. It is assumed that the stochastic process is zero mean stationary, i. e. E(xt) = 0 and E(xtx,) = P. The state covariance matrix P is independent of the time t. This implies that A is a stable matrix.

Since ut and vt are zero mean white noise vector sequences independent of xt. We know that E(xtut ) = 0 = E(xtvt). We find the Lyapunov equation for the state covariance matrix P = A P A ' + Q Defining the output covariance matrices Ai = E(yt+iyt). We find for Ao:

Ao = CPC' + R we get Ai = CAmG (3) A-i = (A,)' 1 = 1,2,... where G = APC' + S = E(xt+iyt' )
Given a system of the form (l)-( 2), the spectral density function S(z) of yt is defined by

S (z)= ¿ A tz -' or S(z) = Ao + C(zl -A ) 1 G + G' ( z 'l -A ') ''C' (4
) where P, G and Ao are uniquely determined by

P = A P A ' + Q G = APC' + S Ao = CPC' + R. S(z) is interpreted as an expression of the form S(z) = Z(z) + Z ' (z-') (5)
where

Z(z) = Ao /2 + C(zl -A )'1 G (6)
is the transfer matrix of a linear dynamic system.

The positivity of ( 6) is characterized in terms of the linear matrix inequality

P -APA' G -APC G'-CPA' A 0 -CPC >0, (7)
The following result is well-known Lemme 1. « Kalman-Yakubovich-Popov », [START_REF] Caines | Linear stochastic systems[END_REF], p. 246.

A transfer function Z(z) given ( 6) is positive real with A asymptotically stable if and only if there exist a matrix symmetric P > 0 which satisfies (7).

The identification algorithms based on a two steps procedure consists first to estimate the matrices A, C and G. then the Riccati equation is solved for P.

Let the past and future of yt be defined respectively by

yt' = [yt-i, yt-2, -] yt+ = [yt, yt+i, •••]

Let Span(Yt+) and Span(Y,') be the Hilbert spaces obtained by taking the closed linear span of the random variables in Y t+ and Y t\ respectively. (In general, Span(.) will be the Hilbert space generated by the closed linear span of elements of (.) The inner product on this space is the cross covariance. Orthogonal projection in these Hilbert spaces is equivalent to conditional expectation i.e. x \Y = E(x \Y) denotes the orthogonal projection of x onto span(Y). This projection is given explicitly by x \ Y = E(xY) {E tY Y ') } ' ^ (8) where x and Y are zero-mean random vectors The projection Y t+ \ Y t' defines the state space of a markovian model of the form (l)-(2). Thus this projection is central to any stochastic realization algorithm. From (8), it can be seen that this projection is given in terms of the following matrices formed from the covariance sequence:

Y t+ \ Y,' = H R. lY (9)
where

Y t' \ Y t+ = HR+-'Y+ H = E [Y t+(Y t ) ' ] = A, A 2 A 2 A 3 A 3 a 4 ( 10
) R+ = E [Y t+(Y t+) ' ] = A 0 A,' a 2' A, > o A,' a 2 A, > o ( 11) A 0 Aj A 2 A ,' A 0 A, . . R. = E [Y t-(Yt ) ' ] = A 2 A j A q
H and R are the Hankel and Toeplitz matrices formed from the covariance lags of the output process. Let the full-rank factorization of R+ and R. be R+ = L+L+\ R. = L.L.' where L+ and L. are lower triangular Cholesky factors. From deterministic realization theory there exists an (A, B, C) whose impulse response equals the covariance sequence Aj, j = 1, 2,..; .Such a realization will be called a covariance realization. It is known that a given covariance realization corresponds to a particular factorization of the Hankel matrix as H = OQ'

We saw in Eq(3.) that the covariance matrices have the structure A i = CAmG, for 1 > 0 this indicates that the infinite Hankel matrix H formed from the covariance sequence can be factorized as

A , C CA C A 2 [G AG A 2G . .] H = O Q ' ( 13
)
Thus a covariance Hankel matrix with both row and column dimension not less than p will have rank equal to p.

To illustrate our point we shall first consider the identification procedure proposed in Aoki (1990) [START_REF] Aoki | State space modeling of multiple time series[END_REF]. Given the covariance sequence, factorize the Hankel matrix H by singular value decomposition (SVD) H = U r V ' ( 14) where U'U = I = V 'V Here £ is the square diagonal matrix obtained by listing in decreasing order the nonzero singular values along the diagonal. Let O = U £l/2 and Q. = VX1 / 2 then the Hankel matrix H may be factored as H = 0£T (15) a n d 0 ' 0 = £ = Q 'Q (16) which is balanced in the sense that the observability and constructibility grammians O' O and respectively are both equal to E.

The estimation of the parameters A, C and G are given

A = E •1 / 2 U 'HV2: -m C = H 1 . V 2 1 / 2 G = 2 1 / 2 U 'H*,
where H is the shifted Hankel matrix obtained from H by delecting the first block row and adding a block row at the bottom. Hi» (H*i )is the first block row (column) of H. U' are a singular value generalized inverse of O and G respectively so that V I '1 / 2 and I ' 1 / 2

QQ+ = ( L 0 +0 = ( I -1 / 2 U ')( UX ",/z) = I -1/2 -l/2\ V ' ) ( V I '1 / 2 ) = I.
Modem systems theory has established the balanced representation as the coordinate system of the choice for computational reasons. An implication of this choice of coordinate system is that some of the parameter estimates are « strictly nested ». The principal problem of this procedure is that it is a deterministic realization procedure and hence does not a priori insure that Z(z) is positive real even if the Toeplitz matrix R formed with the data is positive definite. Further, by calculating A, C and G from the n non-zero singular values in En and the orthogonal singular vectors of Un and V n subject to the balanced representation, the estimates have the property referred to as stricly nested, [START_REF] Aoki | State space modeling of multiple time series[END_REF].

Thus the parameters estimates are given by An = E, -^u r H . v , ^ ' m

Cn = H i. V j V 7 2 (17) Gn=zr1 / 2 urH,!
it is a principal subsystem truncation in the sense that

A" = A " C" = C, Gn = Gi ( 18
)
where An Ci, and Gi are sub-matrices of the original balanced state space model, that is,

A = A u A,2 A 2i A 2 2 C = [C ,C 2] G = [ G , G 2] ( 19
)
Further it has been shown by Pemebo and Silverman (1982) that An is stable if A is stable.

Model reduction is done by Hankel norm approximation commencing from the Hankel matrix of the covariances of the output process. In calculating the Hankel norm approximation one issue is to obtain a Hankel matrix as the approximating matrix again which is still a best approximation among the matrices of the rank n, [START_REF] Glover | All optimal Hankel-norm approximations of linear multivariable systems and their L error bounds[END_REF] one obtain a matrix will not be a Hankel matrix in general. Hence it can not in general be used to determine a reduced order system.

Further it has to be shown that (18) generates a covariance sequence, i.e. that Zn(z) = C " (z I-A n)-,G"' + ' /2 A0 is positive real. The crucial question of positivity is neglected in [START_REF] Aoki | State Space Modeling of Time Series[END_REF] [START_REF] Aoki | State space modeling of multiple time series[END_REF].

We shall demonstrate next that this reduction procedure indeed produces a positive real and balanced reduced model structure something that has not been available in the literature. Notice that this is a nontrivial and surprising result since it really seems to be too much to hope for that the singular values of the truncated system are exectly to the n first singular values of the original system.

Canonical correlation.

Canonical correlation analysis, introduced for the first time by [START_REF] Hotelling | Relations between two sets of variates[END_REF] essentially consists in computing the principal (canonical) angles between subspaces, a straightforward generalisation of angles between vectors [START_REF] Golub | Matrix computations[END_REF]. Let F and G be subspaces in Rm dim(F) = p dim(G) = q p > q > 1

The principal angles 0i, 02, •••, 0q between F and G are defined recursively by subject to cos(0t ) = max max U' V = U kVk for k = 2 ,..., q

1 Ue F V eG * u = V =1 U 'lij = 0 for i = 1 , k-1 V 'vj = 0 for i = 1,..., k-1
first, two vectors U and V are searched for, between which the angle is minimal (maximal cosine), resulting in Ui, Vi and 0i. Cos(0i) egal 1 if the subspaces intersect. The vectors U = {Uj,..., uq} and V = { Vi, ...,vq} are called principal vectors of the subspace pair (F, G). the idea of using SVD to compute the principal angles and vectors is due to Bjork and [START_REF] Bjorck | Numerical methods for computing angles between subspaces[END_REF]

Canonical correlations and balanced stochastic realization.

The application of the canonical correlation analysis to the realization problem has been pioneered by [START_REF] Akaike | Markovian representation of stochastic processes by canonical variables[END_REF][START_REF] Akaike | Canonical Correlation Analysis of Time Series and Use of an Information Criterion[END_REF], [START_REF] Desai | A transformation approach to stochastic model reduction[END_REF], [START_REF] Desai | A realization approach to stochastic model reduction[END_REF], [START_REF] Aoki | A method for approximate representation of vector valued time series and its relation to two alternatives[END_REF], [START_REF] Aoki | State Space Modeling of Time Series[END_REF]) (1994), Lindquist and Picci (1996), [START_REF] Van Overschee | Subspace identification for linear systems[END_REF] who tied this in with balancing technique. The state is in fact the information interface between past and future. The canonical correlations lead, therefore to a natural distance measure between the past and the future, which in the Gaussian case is nothing else than the Kullback-Leibler mutual information.

The model (l)-( 2) can be converted into a so called forward innovation model and backward innovation model, [START_REF] Aoki | Two complementary representations of multiple time series in state space innovation forms[END_REF]. Forward innovation zt+i = Az, + Ket (20a) yt = Cz, + et (20b) where zt = E(xt/y1 .1') = £2R. 'yt-i (21) This has to be interpreted as follows zt is the optimal prediction of the state xt based on the measurements yt-i'.

e, = yt -E(yt/y ,.i) is called the forward innovation component in the state vector yt, E(et) = 0.

Conclusions.

The purpose of this paper is to analyze a class of state space identification algorithms for time series based on canonical correlation analysis. The usual deterministic arguments based on factorization of a Hankel matrix are not valid for data generating process something that is habitually overlooked in the literature.

We have shown how the IV estimators for the state space models are related to the forward and backward innovations in state space form.

The principal subsystem truncation proposed preserves positivity and balancing then the strict nesting property enjoyed by the original algorithm can be restored to the models constructed by the IV estimators.

State space systems provide a flexible and parsimonious way for time series modelling.

  and summarized in the following theorem: If the columns of Qf and Qg define orthonormal bases for F and G respectively, the principal angles and vectors follows from the SVD of Qf Qg Qf Qg = PSQ' [ U i , . . " Uq] = QfP [Vl,..., Vq] = QgQ S = diag(si,...,sq) cos(0k) = Sk for k = l,...,q Several alternative methods to obtain the required orthonormal bases are conceivable (Q-R decomposition, SVD, generalized SVD, etc ...). If only knowledge of intersection is needed an SVD of the concatenation of two matrices defining the subspaces can suffice: the intersection of the row spaces of two matrices f and G can be calculated from PiF, where [p, pa£ ] -o matrices Pi and P2 being determined through the SVD.

By construction E(ety"t-i) = 0 and E(etzs' ) = 0.

t > s

The matrix of covariance is E(ete,') = Ao-CP.C' The Kalman gain is K = (G -APC' )( A0-CP.C' )'1. P. = E(ztzt' ) is the forward state covariance matrix which can be determined from the forward the algebraic Riccati equation P. = AP.A' + (G -AP.C'X Ao -CP.C')'1 (G -AP.C' ) '

Backward innovation wt.i = A 'wt + Bet (22a) yt = G'wt + £t (22b) where wt = E ( xt/yt+i+) = 0 ,R+'1 yt+i+ (23) et = yt -E(yt/y+,+i) is the backward innovation, E(wt) = 0 It is uncorrelated with wt :E(£,wt) = 0. The matrix of covariance is E (e, et') = ( Ao -G'P+G) The Kalman gain is B = (C' -A ' P+G) ( Ao -G'P+G)'1 Where P+ = E(w,wt' ) is the backward state covariance matrix, this is the solution of algebraic Riccati equation

Following [START_REF] Van Overschee | Subspace identification for linear systems[END_REF] the states of the forward (zt) and the backward (wt) innovations representation are generated from the principal directions between the past and future output data.

Estimates of parameters A, C, G and Ao. Estimate of C. Following [START_REF] Aoki | Two complementary representations of multiple time series in state space innovation forms[END_REF], given z , , the least squares estimate of C is

it is also the IV estimate using z t as instruments. From the definition zt = QR/'y't-i we can form a reduced form for estimate of the covariance matrice of state by P. = QR-_1Q ' . So that

Estimate of G The matrix G is estimated exactly as C in the forward innovation G = (i:y,w t+1-)(E w t+1w t+1 ') -1 where w t is used as an instrument.

Using the estimate of the covariance matrice of state P+ = O'R+ 'O, we obtain

The matrix A appears both in the forward and backward innovations, hence two alternative estimates are obtained by using zt or wt as instruments. For zt as instrument, we obtain  = ( I z t+lz , ) ( I z tz't) _l where Ezt+i zt' = QR/'iZyt'yt+i'^R.'1 £2' For wt as instrument, we get  = ( I w tw t.,' )(L w tw t' ) "'

Estimate of Ao A0 = T'Eytyt'

Following [START_REF] Akaike | Markovian representation of stochastic processes by canonical variables[END_REF], [START_REF] Desai | A transformation approach to stochastic model reduction[END_REF] and [START_REF] Desai | A realization approach to stochastic model reduction[END_REF] Van Overschee and De Moor (1996) consider a normalized Hankel matrix H* = L'+ H(L'+)' ^ (24) Taking the singular value decomposition of H instead of H H* = UEV' with U'U = I = V 'V Suppose that a finte dimensional realization of order n for yt exist, rank (H*) = n = rank(H) then Z = diag[ai, a2, . , a"], 1 > at > a 2 >... > a"-So that H = (L+U)Z(L.V)' . The singular values become the canonical correlation coefficients i.e. the cosines of the angles between the past and the future of the process y We have

) This is a realization stochastic balanced, its corresponds to a more natural type of balancing corresponding to a Hankel operator describing. The interface between the past and the future of the times series y. Definition: A Balanced stochastic realization (B.S.R) is an innovations representation with state covariance matrix equal to the canonical correlation coefficient matrix Z.

Gel 'fand and Yaglom (1959) have shown that the mutual information between the past and the future is given by I(Y t',Yt+) = -Yi In det (I -Z2) = -Vi Z ln(la?) Since H = OQ.' and this factorization is unique modulo coordinate transformation in state space, we may take O = L+UZ1 / 2 and Q = L_V II/2

Proposition.

In the stochastically balanced basis, an estimate of parameters are

This result is different of the result obtain by [START_REF] Aoki | Two complementary representations of multiple time series in state space innovation forms[END_REF].

5.Approximate stochastic realization.

A stright-forward positive extension of a finite covariance sequence leads to a spectral density S(z) o r, to equivalently, to a positive real matrix function Z(z) = C(zl -A y 'G ' + Vi Ao of a degree n which is often too large. In this section we shall discuss a method of approximating Z, and hence the positive extension, by a positive real function Z, of lower degree, this method is called balanced truncation [START_REF] Desai | A realization approach to stochastic model reduction[END_REF], Lindquist and Picci (1996).

The parameters of a model A, C, G can be chosen in a stochastically balanced form, so that the minimal and maximal solutions, P. and P+, of the algebraic Riccati equation P = A P A ' + (G -APC' X A0-CPC' )'1 (G -APC' ) ' are such that P. = Z = P+, where Z = diag{Ci, a2,..., an} is the matrix of canonical correlation coefficients of the process y. The idea is to replace Z by a rational matrix function Z^C^-An^Gr+ViAo, of lower degree n*, where n* should be chosen, if possible, so that the last n -n* canonical correlation coefficients {<T ".+,, G n*+2,-, a n}, to be disregard, are small. With (A, C, G) in balanced form, (Au, Ci, GO are chosen so that

where Zi = diag {<Ji, Gi,..., <?"♦} and £2 = diag {a n*+i, a n*+2,-, <*n}-The questions to be answered here is whether this procedure leads to a Zi which is positive real and hence corresponds to a spectral density and wheteher {An, Cj, G i } is in stochastically balanced form, i.e. whether the corresponding minimal and maximal solutions, Then the following theorem holds Theorem. Suppose that R = A 0 -CP. C' > 0 and that P+ -P. > 0. Then the balanced truncation (An, Ci, Gi) defines a strictly positive real function for which it is a minimal realization in balanced form. Consequently, S,(z) = Z,(z) + Z ,(z' ) is strictly positive on the unit circle. The proof can be found in Lindquist and Picci (1996).

Further, by calculating A, C and G from the n* non-zero canonical correlation coefficients in X , the estimates have the property referred to as strictly nested. The estimates are called « strictly nested » because the leading principal submatrices of A, C and G contain the exact numerical values of the correponding lower order model. This result is due to the orthogonality of the singular vectors which enter the formulae of the parameter estimates. Thus if additional states are included incorrectly in the model, all coefficients of the model are estimated consistenly and, in fact, are numerically identical to the estimates that would be obtained from the correct specification.


Jean-Marie Huriot

Jacky Perreur Latec

Keywords: minimum transport point, labour cost, agglomération savings, distance, location of public services

Journey to the Centre of Europe (Voyage au centre de FEurope)

Jean-Marie Huriot and Jacky Perreur

De nombreuses institutions locales utilisent, dans de véritables opérations de marketing territorial, une analogie entre centre géographique et centre géométrique du cercle pour valoriser leur localisation. Il s'agit là d'une conception particulière et étroite d'une idée plus riche et plus insaisissable. Le concept de centre est une construction économique et sociale mais il repose sur certains principes consensuels : l'accessibilité, la concentration, l'inégalité, la relativité spatiale et temporelle et la subjectivité compte tenu de sa dimension symbolique.

Définir un centre c'est d'abord borner un sous espace de référence. Représenter le centre économique par le centre d'un cercle nécessite de poser certaines hypothèses que la réalité remet en cause. Le recours répété à cette notion de centre conduit à s'interroger sur le rôle que peut avoir un centre, sur sa dimension symbolique et sur ce que l'on peut faire à partir d'une position centrale. Mais aujourd'hui n'assiste-t-on pas à une dilution du centre ? la centralité ne peut plus se ramener à une simple dichotomie centre-périphérie. Tout lieu peut avoir un certain degré de centralité et l'on est en présence d'une juxtaposition de systèmes partiels de centralité. L'idée de centralité doit coexister avec celle de réseaux hiérarchisés.

Key words

Industriai location, Weber,

Wrong directions

Here are two maps showing the centre of Europe. Each one seems self-evident at first sight. But comparison shows they are absolutely meaningless. It seems that two European regions, although in different locations, are both claiming to be the centre of Europe and so in an enhanced position ! (Maps 1 & 2).

The idea underlying these maps is a highly intuitive one. It is based on analogy between the geographical centre that is sought and the geometric centre of the circle. What could be easier than drawing a circle around the chosen area and deciding that the centre of the area is the centre of the circle ? The circle-centre couple is certainly the most convincing one too. Yet it is a very peculiar and very narrow view of an idea that is both much richer and much more elusive. Quite whether it is meaningful to represent a geographical area by the circle-centre pairing depends on whether the area is homogeneous and isotropic and on whether Euclidean distance can be used, three hypotheses that become invalid when we switch from abstract geometry to the actual economic space and its transport networks.

If we reason in terms of Euclidean distance, the idea of equidistance, which strongly influences our mental picture of the centre, is only found in the circle-centre pairing. In a square, we can only define a centre of symmetry, and in a random figure, like the shape of Europe, there is usually no geometric centre.

When Jules Verne takes us on a Journey to the Centre of the Earth, no readers wonder where the centre might be, no-one is mistaken, even through no-one clearly sees how to get there. But when speaking of the "centre of Europe", it is those who believe they know what they are talking about who most risk being misled, or risk misleading others. The expression "the centre of Europe" is meaningless unless we specify just what of, and how, why and for whom we are looking for the centre. The centre is the result of combining an a priori conception of the causes and forms of spatial structure with technical instrumentation, both components being taken from a vast range of possibilities that we will need to run through.

Map 1: The Saône et Loire region in the centre of Europe

(Source : Conseil Général de Saône et Loire)

The Saone-et-Loire region is highly favoured in terms of geography. The area is right inthe| centre of the map of Europe, has very varied landscape and tremendous natural assets.

Map 2: The Paris region in the centre of Europe (LeMonde, 2.7.1995) Beforehand, we shall lay down a few general principles that seem to validate the emergence of a centre within this domain as a whole. We shall end by questioning whether analysis in terms of centre is currently worthwhile. Our journey shall thus lead us from the emergence of the centre to its dilution.

The emergence of the centre

In a given sub-space, the centre is not at any specific point. First, because the concept of centre is a construction of the mind and this construction can be made in many ways and cover a wide range of real situations. Then, because once we know which real situation we are seeking to apprehend, it is the organisation of the economy and of society which determines the shape and location of this reality: the centre is therefore an intellectual, economic and social construction. The centre, if there is one, emerges at the junction between an attempt to conceptualise it and a process of spatial organisation. It depends as much on advances in the one as on the ups-and-downs of the other.

There is however a minimum consensus about the idea of centre. Whenever the idea of the centre arises in economics or in geography, we must bear in mind five inescapable principles.

The principle of accessibility (p\)

This involves the distances between the different places of the reference area.

In an area formed by a set of points, the centre can first be defined as a point with an advantageous relative position within the set. The centre is then the point that minimizes the maximum distance between it and the other points (e.g. the centre of the Euclidean circle, or the centre of a graph) or the sum of the distances to the other points, or more generally any non-decreasing function of these distances. It is therefore usually defined by a minimum function.

A place is a spatial unit that is non-decomposable by nature or by convention within a geographic space (Beguin & Thisse, 1979). This principle can be applied when a point represents a geographical place 1 and the distance used represents a geographical distance measured in units of distance, time or money -transport cost- [START_REF] Huriot | Distances, espaces et représentations, une revue[END_REF], 1994). The position of the central place then depends entirely on the type of distance considered.

By replacing the minimum function, which determines the centre, by a maximum function, we define the periphery.

Identifying a centre or a periphery by distance alone is usually insufficient or meaningless for correctly identifying an economic or social centre.

The principle of concentration (p2)

The centre is also a place of concentration of population, jobs, the supply of goods and services, wealth, knowledge, information, culture, the capacity for economic, scientific and cultural innovation, but also for the resources for action, power and political, legal and economic decision-making [START_REF] Polydorides | The Concept of Centrality in Urban Form and Structure[END_REF][START_REF] Reynaud | Centre et Périphérie[END_REF].

The centre as a place of concentration is a priori different from the centre of accessibility, defined by its topological position alone. In this way, Paris is far from being the geometric centre of France. There is, however, an a posteriori link: there are often good reasons for concentrating activities in a topologically well-sited place. These two principles of accessibility and concentration can be combined to provide a general integrated measurement of centrality [START_REF] Huriot | Centre et périphérie : fondements et critères, colloque de l'ASRDLF[END_REF].

An important consequence of this principle is that the centre is a favoured place for action and interaction, encounters and confrontations. It is the privileged arena for the interplay of positive (agglomeration economies) or negative (pollution, congestion) externalities.

The theoretical literature in geography and regional economics uses the concepts of central place, centre-city and urban centre or central business district (CBD). All these centres are places of concentration.

The principle of inequality (p3)

The idea of the centre is the source of the dichotomy between what is central and what is not, all of which can be lumped under the heading of periphery. The centre-periphery pairing evokes asymmetry, spatial inequality, sometimes with an ideological connotation.

The centre groups everything that is attractive, prestigious, rich, developed, new, etc. The periphery is defined negatively by comparison with the centre. It appears to be marginal, deserted, abandoned, underdeveloped, devoid of independent decision-making capability, dominated or even colonised and exploited [START_REF] Reynaud | Centre et Périphérie[END_REF].

But the principle of inequality is distinct from the previous two. Dominated places, backwardness and poverty are sometimes encountered within a zone that overall has the characteristics of a centre (in terms of accessibility and/or concentration) of an area. Think of the marginality found in New York, in the heart of Manhattan, the islands of poverty in all cities of international standing, of the less developed areas of the wealthiest and most powerful countries in the world.

The centre maintains with the other places asymmetric relations of complementarity and conflict [START_REF] Lacour | Formes d'urbanisation, mobilité et captivité[END_REF], sources of flow. It is a place that attracts and/or diffuses human, material or immaterial flows in an asymmetrical pattern. It attracts people and activities. It diffuses knowledge, information and decisions. Attraction enhances concentration. Attraction and diffusion are connected with the idea of polarisation or of domination as an asymmetric relationship [START_REF] Perroux | L'économie du XXème siècle, 2ème éd[END_REF].

In Marxist inspired literature, spatial inequality inherent in the centre-periphery dualism is asserted more clearly and more explicitly as domination, or even exploitation. This can be noted in the historical analysis of "world-economies" by Braudel [START_REF] Braudel | Civilisation matérielle, économie et capitalisme, XVème-XVIIIème siècle[END_REF] as in the analysis of the spatial division of labour [START_REF] Aydalot | Dynamique spatiale et développement inégal[END_REF].

The principle of relativity (p4)

The centre is a relative reality, in space and time.

It is misleading to speak of the centre. Each level of the spatial scale can have its centre (district, city, region, nation, etc.). Each urban function can have its centre. Do we not speak of leisure centres, shopping centres, administrative centres ? These places may be spatially separate, or even remote. Everyday language reflects this relativity clearly. Even so, these different functional centres are often found together or very close to each other. It is the specific role of the city to act as multifunctional centre.

Even for a specific function, an area seldom has a single centre. There is usually one (or several) main centre(s) and centres of lesser importance.

The integration of several centres of unequal importance is the first step from the centreperiphery dichotomy towards a continuous scale of centrality: each place of a set E is characterised by a degree of centrality, as in the theory of central places.

A centre changes and moves. A place may lose its central properties (desertification) or recover them (rehabilitation). A centre may be the scene of different activities as relative capacities for bidding for occupation of ground space change. A centre can be created from scratch, deliberately, as for example with the building of shopping centres. Thus a centre may be the result of spontaneous development, self-organisation or a decision from outside.

The principle of subjectivity (ps)

Implicitly, when we speak of a centre, we imagine it surrounded by a circular area, because its geometry is more evocative, more directly representable than complex spatial structures.

The centre is enhanced and enhancing, objectively and subjectively. The mental picture of it has a high value of historical, cultural or economic prestige, as a place of power and responsibility, for the emotional attachment it elicits. This enhancement is especially related to symbolic or mythical representations.

The centre-periphery pairing has a very rich symbolic content (see Huriot, 1994aHuriot, , 1994b)). The centre-periphery pairing is a symbol of hierarchy, control, domination or even transcendence.

In this sense, being at the centre gives the feeling of being better located. Too bad if the centre corresponds to a criterion with only a distant link with the one we are really trying to optimise.

The realm of imagination has an important role, which means that talk about the centre, even if it aims to be scientific and objective, is never entirely neutral.

Principles of centrality versus principles of spatial analysis

Although they were determined independently, these five principles seem to fit in perfectly with the five principles of spatial analysis dear to Jean Paelinck (e.g. [START_REF] Paelinck | Eléments d'analyse économique spatiale[END_REF], viz.:

Pi Spatial models should be based on spatial interdependence. Pi These relations of spatial interdependence are probably asymmetrical.

P'i

• • • Pi • • P3 • • PA • • Ps

The centre of what ?

Of course, first of all we must define a sub-space of which to seek the centre. This is not a trivial point, since the two dimensional space of the earth's surface has no centre (thus on map 3, Turkey appears at the centre of the world only as a consequence of an entirely arbitrary representation of the world).

Nor is it as simple as might be thought, and the problem is not solved by saying we are looking for the centre of Europe. The first, commonplace, difficulty is that Europe has no single hard and fast definition. Even if we restrict ourselves to the EU, the border is constantly expanding. To avoid multiplying such difficulties, we shall assume that the area of Europe is defined unambiguously at a given instant. If we use the principle of accessibility, combined or not with that of concentration, Europe must be reduced again to a finite set of points (inset 1). How can this set be defined ? We might use the set of European regions, each being marked by its capital, or its centre ... which only puts off the problem. We might use only the cities whose population exceeds a certain level, which means we lose part of the information about the spatial distribution of population and we do not necessarily dispose of a meaningful centre from the point of view of European economics: e.g. the influence of international functions is not related exclusively to the size of cities (Equipe [START_REF] Paris | Le poids économique des villes dans le système urbain européen[END_REF]. The different ways of defining the set of places inevitably leads to different locations of the centre.

Map 3 : Turkey, centre of the world ! (LeMonde, 2.21.1995) La Turquie n'est pas le centre du monde ...elle y est située, tout simplement.

Inset 1: Discrete space

To define a centre, we may need to reduce the study area to a finite number of points, or to reduce the number of places considered for convenience of computation. Aggregations must therefore be made, which are never neutral. The effects of aggregation on spatial models have been investigated by numerous authors [START_REF] Openshaw | The Modifiable Areal Unit Problem[END_REF][START_REF] Fotheringham | The Modifiable Areal Unit Problem in Multivariate Statistical Analysis[END_REF]Ruhigira, 1994). [START_REF] Arbia | Spatial Data Configuration in Statistical Analysis of Regional Economics and Related Problems[END_REF] and [START_REF] Putman | Effects of Spatial System Design on Spatial Interaction Models, 1 : the Spatial System Definition Problem[END_REF] claim that variation in results stems from two sources: (1) variation in the number of elementary units (places) used for collecting information (regardless of the shape and dimensions of such places) and (2) variation due to the procedure which produced the shape and dimensions of the places. Errors caused by aggregation may have a sizeable impact on how the centre is determined.

The centre -how ?

Starting with a discrete area (inset 1), there are several techniques for determining a centre. Inset 2 quotes some, with their main properties.

Inset 2: Determining a centre from bivariate statistics

Let us suppose the area under consideration is composed of a finite number of places, elementary spatial units, on which a distribution of a variable is defined. Several types of centre can be determined, displaying different properties.

The centre of gravity. This is an arithmetic mean. It is the point that minimizes the sum of the squares of weighted distances to all other points. The main limitation with it is that it may lie outside the study area if the area is not convex. Knowing this centre provides no socio economic indication about the place where it is located, which may be completely deserted. Its position is very sensitive to any variation in spatial distribution. Because the squares of distances are used, extreme locations have a relatively important weighting.

The median centre. This minimizes the sum of distances to all other points. Its position is only independent of the direction of the axes if the distance is Euclidean. It involves the first two limitations of the centre of gravity. It is less affected by extreme locations. Its insensitivity to certain changes in position limits its ability to account for temporal changes.

The modal centre. This is the areal unit where the density of the study variable is maximum. There may be several non-contiguous modal centres. By definition, the modal centre is always located within the study area and provides information on the areal unit where it is located. But it is insensitive to movements in distribution until a greater concentration appears. By definition again, it only includes one value of the distribution and therefore overlooks a sizeable amount of information. Choice of the modal centre may differ with the study variable. Values of the variables are sometimes distributed by classes. For "international urban functions", Paris and London usually appear in the modal class.

p The centre, harmonic mean. This is the solution of min -where Pi is the value i of the study variable in place /, P -]T P t et is the distance from unit / to unit j, or, in an equivalent way, the solution of max^^/d^ where the maximized expression is Stewart's J i location potential if Pi is a population. It is located in the study area and is not very sensitive to population movements. It often coincides with the modal centre, but unlike the modal centre, it includes all available information. Isopotential curves are often associated with it.

Other centres. Other methods of computation are available: the geometric mean centre, for which the logarithm of distance reduces the role of remote places, the centre which minimizes the variance of the distribution, or again the minimax centre which minimizes the greatest distance from any place to that place, and which is used for example in the theory of location of public services.

Like any indicator of centrality in statistics, the centre provides no information about the dispersion of the study variable. The use of non Euclidean distances changes the relative positions of places [START_REF] Huriot | Distances, espaces et représentations, une revue[END_REF], as does the inclusion of movements over a network.

Representing the economic centre by the centre of a circle, as we have said, supposes homogeneity, isotropy and Euclidean distance. Reality differs from this ideal picture for three series of reasons. Places, in addition to their different locations, are not all alike. They contain variable densities of attributes. Movement takes place along networks (transport is not ubiquitous). The important thing is often not as much the length of the journey as its cost or duration. These reasons partly overlap. Each underlies a choice for representation of the centre.

Non-homogeneity

Having precisely defined the set of places of which we want to find the centre, we must choose which factor we are seeking: accessibility, concentration or a combination of the two ? In the latter two cases, which are the most common, what attribute or group of attributes of places are to be given priority ? These attributes may be the population size, the value of production, the importance of command functions, political or administrative influence, cultural importance, etc. Location of the centre can be very sensitive to such choices. For example, if we consider concentration alone and select as the centre the mode of spatial distribution, it can be seen that Paris is the centre of Europe for the number of registered offices of major European businesses, but is dominated by London for the number of registered offices of subsidiaries of these businesses (PARIS, 1993, 38-39).

Non-Euclidean

There is no homothety between the length, cost and duration of journeys. Functional distances cannot be represented by a Euclidean distance. We know that the choice of a mathematical function representing the relation of distance is decisive for the properties of the space represented [START_REF] Huriot | Distances, espaces et représentations, une revue[END_REF].

Non-isotropy

Journeys are invariably constrained by specific transport networks which introduce distortions compared with the hypothesis of isotropy. Determining the centre depends on the characteristics of the network or group of networks chosen to evaluate accessibility. All these characteristics are sources of distortion.

The nature of the network is related to the mode of transport chosen, the type of link desired and the speed desired. Rail or road travel is chosen ; depending on the size of the places of departure and destination and the speed required, travellers will take ordinary trains or high speed trains, roads or motorways. We need only look how a map where distances represent access times by high speed train is deformed (by anamorphosis) to understand the consequences of non-isotropy (map4).

Accessibility of the network and the connection between networks also play an important role in determining the centre. Some networks are only accessible at a very limited number of places like airports, stations and toll gates. This is the cause of the famous "tunnel effect" which eliminates from a whole network of relations a number of places through which high speed trains or motorways pass with no possibility of stopping. Thus it may be much faster to go from A to B than from A to a, even though the first journey is longer than the second. Different networks related to the same overall means of transport can then be more or less well connected, which can cause breaks and irregularities in the relationship between the length and the duration of journeys.

Map 4: TGV201S [START_REF] Cauvin | Une accessibilité renouvelée[END_REF] The shape of the network itself can create or reproduce a centrality which may be partly fictitious. Thus the radial form around Paris of the French rail network means many journeys must cross Paris, which reinforces the centrality-accessibility of Paris, which in turn reinforces its centrality-concentration. The radial picture is besides partly misleading as Paris at the centre is in part a simple compulsory point of transit. It is a special demonstration of the "hubs and spokes" model or the radial network, of which the principle can be applied for rationalising freight transport: it is more economical to link n places to a central place (n links) than to link all the places to each other (n 2 -n links). In such a conception of a network, the centre (hub) may be located in a place remote from any concentration of population and activity, as is often the case for lorry transport. But it may also coincide with a major city (Paris, in the previous example, or the large American cities for air transport in North America).

Congestion of the network, by its geographical and temporal variability, is another factor of non homothety between length and time.

The centre -why ?

For what or for whom is it useful to determine the centre ? What are we trying to understand through knowledge of the centre? What are we trying to say ? What are we trying to do?

Understanding

The search for the centre first of all has an intellectual interest that involves a whole series of questions on the role of a centre in an area like Europe, and more generally on the meaning

Doing

Knowledge of the centre, or rhetoric about the centre, leads to action whether warranted or not.

From the viewpoint of the private sector decision-maker, the business leader, a centre seems a priori to be a good location with all the chances of maximizing possibilities for interaction on the spot (centre-concentration) and with the outside (centre-accessibility). From the point of view of the job-seeker, the centre offers the guarantee of greater employment because of the wider range of opportunities. Likewise, it provides those supplying jobs with a better match between employees and jobs. All that makes for "fully comprehensive insurance" centres [START_REF] Veltz | D'une géographie des coûts à une géographie de l'organisation[END_REF]Sallez & Verot, 1993). Such behaviour relative to the centre reproduces and amplifies centrality.

From the standpoint of the public decision-maker, the question is one of regional development and management of regional inequalities. Should they be maintained or reduced ? The notion of balance is a slippery one here. Spatial equilibrium can of course accommodate inequalities. Which inequalities are desirable ? Should we oppose hyper-centralisation of the French type ? If so, how ? The policy of compulsory relocation of some institutions, even if it may serve as a signal, remains modest compared with the hierarchy that is to be overturned. A few measures cannot change a system that has been rooted in geography and in minds for centuries, especially as the systems tends to be self-perpetuating, even self-amplifying.

Combating overall centrality that is judged excessive involves creating or favouring centres at a smaller geographical scale. Then we can use the fiction of geometric centrality or ready access to encourage and develop a concentration of activity. That is the objective of the designers of maps 1 and 2 in section one. of inequality that is covered by the hierarchy that underlies the idea of centrality. Inequality is the unequal spatial distribution of activities and of power over these activities. The question of the formation of centres comes back to that of the origin of agglomerations of people and activities and that of interaction between places where people live and work. This is therefore the first problem of spatial theory, which must explain the heterogeneity of space. There are several paths available. We can suppose an unequal endowment in natural factors, or the existence of nodes of communication, i.e. an a priori central position (in terms of accessibility) on a network, which supposes the problem is partly resolved. If we reject this attitude, we evoke the important role of agglomeration externalities, or proximity effects in the optimal organisation of inter-individual and inter-firm relations which are not all of the market type.

Saying

What do we mean when we establish that a place is a centre ? What message do we want to get over, apart from the simple knowledge of existing spatial inequality? Beyond knowledge, there is a symbolic factor. The place raised to the rank of centre is enhanced. From inequality, we make a value judgement from more to better. It is easy to see the meaning that this hierarchy can take on in a society that is beset by rankings and records. This symbolic dimension, which appeals to the imagination, inevitably appeals to rhetoric. The desire to persuade may jeopardise the serious scientific endeavour of seeking out the centre. The choice of approach to the centre or of a method for determining it may then be justified only by what we want to say. The problem is back to front: given a place, how can we show it is central ? Maps 1 and 2 in section one are a naive illustration of this procedure.

Dilution of the centre

Nowadays, the centre is all or nothing. Michel Serre refers to urban diffusion over the entire planet: "the world is becoming a single city ...", which he names Newtown, "an invisible city, whose centre is everywhere and edge nowhere" [START_REF] Serre | La légende des anges[END_REF]. Do the density of links and the proximity of concentrations in Europe remove all value from the idea of centre ? There are several reasons for contesting the value of the classical idea of centre.

• Centrality is a diffuse phenomenon which cannot be reduced to a centre-periphery dichotomy, where the periphery is the collection of all places that are not the centre. Each place in a given area has a certain concentration of population and activities, and is more or less well situated compared with other places. Thus we move from the idea of centre to that of degree of centrality, and from a dichotomy to a hierarchy, a quasi-continuum, with places being ranked as more or less central. The structure of an area is then characterised not only by the location and characteristics of its centre, but by the form of the hierarchy of centrality its exhibits.

A good illustration of this phenomenon is the difficulty in visually interpreting the concentration of activities in Europe, from map 5, made by R. Brunet, who ranks towns with more than 200,000 inhabitants by an indicator combining criteria of international influence. R. Brunet sees a backbone arise, which he calls by the surrealist name of "blue banana". Apart from the fact that the "banana" oddly leaves aside one or two towns that are at the top of the rank order (Paris), the shape of the European zone of high concentration only looks like a banana for certain geographic methods of representation, and we could equally see in it a "magic circle" ! This difficulty appears even more clearly if we compare this map with map 6 based on the principle of iso-potential lines. The potentials are calculated from the gross domestic product of the regions and distances evaluated by average road transport costs between regions (Keeble, Owens & Thomson, 1982). The greatest potential appears in the Rhineland-Palatinate region and generally in the "golden triangle" whose corners are Stuttgart, Hamburg and Lille. Other peaks occur in Paris and less so in London and Berlin. On this basis it can be shown that successive widening of the European Community has not disrupted the golden triangle but has sometimes increased potential disparities (enlargement to 12 members).

Map 5 : International urban functions and the "blue banana 9 '

(Le Moniteur, juin 1989)

• Europe is a patchwork of partial systems of centrality which have very different characteristics. The "Parisian" model exhibits a very marked hierarchy with the strong domination of a large centre over a not very dense territory. This structure occurs in France, but also around London, Copenhagen, Vienna, Madrid, Dublin and Athens. The "Rhine" model displays much less inequality between places and links towns of almost equal size, as in the Rhineland, Flanders and to a lesser extent in North-East Italy [START_REF] Paris | Le poids économique des villes dans le système urbain européen[END_REF]. It is difficult, even rash, to try to synthesise these different structures into a single hierarchy of centrality.

• Europe is covered by hierarchical networks which mean there is not much point looking for a central place. Rather than a Christallerian type hierarchy of places, we should perhaps concentrate on a hierarchy of networks. Large cities spaced throughout Europe exhibit both a strong concentration and high geographical proximity because of rapid transport services of the high speed train or airline type. Such networks of cities dominate the secondary networks of the smallest cities, also spread throughout Europe, and less well linked to each other than to cities of the first type, notably because of the tunnel effect described above. We thus see appearing a series of spatially superposed networks. If there is a centre, it is relative to the highest placed network in the hierarchy and covers all the territory. R. Carnagni distinguishes three networks: one of world-status cities, one of specialised national cities and one of specialised regional cities (Carnagni, 1993).

Map 6: Regional economic potential in 12-nation EC [START_REF] Keeble | Regional Accessibility and Economie Potential in the European Community[END_REF]

Conclusion

The image of the centre is widely used by regional leaders for its symbolic content. They endeavour to convince their partners that their region occupies a preferential location relative to a value-enhancing economic area within the world economy. Europe is widely viewed as a symbol of drive, progress, the future, etc. However, regional leaders fail to question the relevance of the idea of centre on which their reasoning is based. Quite apart from the difficulties of definition developed in this paper, this simple concept related to the centreperiphery dichotomy seems ever less relevant in a area as complex as Europe.

The idea of attributing a rank order to places independently of each other, in the Christallerian way, has perhaps become meaningless. The centre becomes a set of places that is superposed and coexists with the periphery. The idea of the centre may persist, but void of its traditional geographical meaning. The important thing, rather than focusing on the idea of changing centres is to seek out the changes in the idea of the centre.

Table 1 : from Paelinck's principles to centrality principles Pi Pi ^3 PA Ps Pl

 1 By the principle of allotopy, phenomena located in one place are often explained by causes located elsewhere. P4 Ex post interaction is different from ex ante interaction. P5 Two-dimensional containing space should appear explicitly. The following links can be drawn between the two series of principles, illustrated by table in the definition of the area whose centre is sought. It is guaranteed specifically by the way accessibility is defined in principle p\.Principle ps brings out an imaginary dimension we think is intimately related to spatial representations, but which does not necessarily occur in every spatial analysis.

	1:
	Pi is achieved through p\ and /73 ;
	Pi is satisfied by pi and /73 ;
	P3 is guaranteed by pi, pi and /73 ;
	P4 is part of the temporal aspect of /74 ;
	P5 is inherent