John Douglas
email: j.douglas@brgm.fr

Fifty years of ground-motion models

Keywords: ground-motion models, ground motion prediction equations, attenuation relations, seismic hazard assessment, aleatory variability, epistemic uncertainty, ground-motion simulations

. These models are commonly known as: ground-motion models, ground-motion prediction equations (GMPEs) or attenuation relation(ship)s. They are one of the key components of seismic hazard assessment since they predict the median ground motion at a site and its variability given the occurrence of a particular earthquake at a certain distance. A summary of the history of GMPEs and suggestions for future developments are presented here.

Introduction

GMPEs are currently published at the rate of more than one a month and at the last count the total number of equations for the prediction of peak ground acceleration (PGA) was more than 400 [START_REF] Douglas | Ground motion prediction equations 1964-2014[END_REF]. The purpose of this paper is to summarize the current state of the art in the field of GMPEs. The improvements made since the first GMPEs were developed in the 1960s are highlighted in terms of, for example: the increase in the quantity and quality of ground-motion data used; the increase in the complexity of the models; and improvements in the derivations methods. Progress in ground-motion modeling is highlighted based on a meta-analysis of the recent GMPE compendium of Douglas (2014). [START_REF] Douglas | Earthquake ground motion estimation using strong-motion records: A review of equations for the estimation of peak ground acceleration and response spectral ordinates[END_REF] provided a review of empirical GMPEs published until the early 2000s for PGA and linear elastic response spectral ordinates. In the decade since then, GMPE developers have concentrated on: improvements in the estimation of the ground-motion variability (standard deviation, σ) associated with their models and its components, including estimating single-station σ [START_REF] Atkinson | Single-station sigma[END_REF]; a move away from simple regression-based curve fitting; attempts at using non-parametric techniques; the use of much more and better (e.g. in terms of site characterizations) data; attempts at including additional independent parameters to reduce σ; a better appreciation of epistemic uncertainty (multiple models by same developer or by various teams using the same master database); extensions of spectral models to shorter (<0.1s) and longer (>2s) periods using individually-processed records; a more careful consideration of how the models perform at small (M w <5) and large (M w >7) magnitudes; and making the models easier to use within PSHA. In addition, until the early 2000s the vast majority of published GMPEs were for PGA and elastic response spectral ordinates but, as shown by [START_REF] Douglas | Consistency of ground-motion predictions from the past four decades: Peak ground velocity and displacement, Arias intensity and relative significant duration[END_REF], there has been a growing interest in developing models for other intensity measures, e.g. peak ground velocity and displacement, Arias intensity and various duration measures.

Recent developments

Epistemic uncertainty

As shown by Douglas (2010[START_REF] Douglas | Consistency of ground-motion predictions from the past four decades: Peak ground velocity and displacement, Arias intensity and relative significant duration[END_REF] average predicted ground motions for scenarios close to the barycentre of available data (M w ~6, R~20km) have remained roughly constant over the past few decades despite improvements to GMPEs. However, predictions for scenarios closer to the edges of available observations (e.g. Mw>7 and R<10km) demonstrate larger differences. As strong-motion networks become denser the average number of stations that record a given earthquake increases (e.g. singly-recorded events are rarer), which means that model source terms (e.g. style-of-faulting factors) and the inter-event variability (τ) are better constrained in recent GMPEs. Similarly a modern station generally records more earthquakes leading to better estimates of site terms and single-station σ. Site terms are now less biased since fewer stations contribute a large proportion of records to strong-motion databases, although the number of records per station remains highly variable. The reduction of epistemic uncertainty (differences in predictions among models) remains a considerable challenge. It is vital that this uncertainty is not artificially reduced but seismic hazard assessments correctly account for the true uncertainty in ground-motion prediction.

Aleatory variability

Following [START_REF] Douglas | Earthquake ground motion estimation using strong-motion records: A review of equations for the estimation of peak ground acceleration and response spectral ordinates[END_REF], [START_REF] Strasser | Sigma: Issues, insights, and challenges[END_REF] observe that σ associated with GMPEs has shown little or no decrease since the 1970s despite the increasing complexity of models. This fact and the importance of σ on the results of probabilistic seismic hazard assessments, especially at long return periods, has encouraged attempts to increase the complexity of models to account for other effects than simply magnitude, distance and site class. To date these attempts have not led to significant reductions in σ since GMPEs remain simple representations of complex physical phenomena. However, one of the major areas of engineering seismology research in the past decade has been in separating σ into its different components [START_REF] Al Atik | The variability of ground-motion prediction models and its components[END_REF] and using the appropriate components when conducting a hazard assessment (e.g. [START_REF] Walling | Non-ergodic probabilistic seismic hazard analyses[END_REF].

The future

Some recent authors (e.g. [START_REF] Field | OpenSHA: A developing community-modeling environment for seismic hazard analysis[END_REF][START_REF] Atkinson | Integrating advances in ground-motion and seismic-hazard analysis[END_REF] have argued that GMPEs will soon be replaced by numerical simulations of earthquake shaking. Such simulations do provide a much richer representation of the earthquake hazard to engineers (full time-histories rather than simply intensity measures) and they allow source-and site-specific calculations. Although ground-motion simulations show significant advances with the advent of high-performance computing and developments of better procedures (e.g. [START_REF] Douglas | A survey of techniques for predicting earthquake ground motions for engineering purposes[END_REF], GMPEs are likely to remain a key component of hazard assessments for the foreseeable future. This is because: reliable simulations of short-periods (<1s), generally the most important for engineering studies, remain problematic; defining the required input parameters and their variability is difficult and prone to large uncertainties; and ground-motion simulations remain time-consuming and expensive. In consequence, even if it is unlikely that GMPEs will continue to be a key component of hazard assessments for all of the next fifty years they will remain so for the next decade or two.


Laurent Manivel
email: laurent.manivel@math.cnrs.fr

ON FANO MANIFOLDS OF PICARD NUMBER ONE

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The classification of smooth Fano threefolds by Iskhovskih, Mori and Mukai has been one of the highlights of 20th century's algebraic geometry. Many interesting cases from that classification, and especially among the prime Fano manifolds of index one, are obtained by taking suitable sections (mostly linear sections) of certain rational homogeneous spaces, and Mukai wrote a wonderful series of papers about their astonishing geometry (see for example [START_REF] Mukai | Fano 3-folds[END_REF], and the more general reference [START_REF] Iskovskih | Fano varieties[END_REF]).

It is a general fact that rational homogeneous spaces are a rich source of interesting Fano manifolds, obtained as zero-loci of global sections of vector bundles, and especially homogeneous vector bundles of low rank. In dimension four, O. Küchle [START_REF] Küchle | On Fano 4-folds of index 1 and homogeneous vector bundles over Grassmannians[END_REF] began the classification of these Fano manifolds by focusing on Fano fourfolds of index one obtained as subvarieties of Grassmannians, and defined as zero-loci of semisimple homogeneous vector bundles. He obtained a list of fourfolds which, as recently stressed by A. Iliev, are potentially a rich source of nice geometry. In particular some of these varieties have special Hodge structures, that could be relevant in the quest for new hyperkähler manifolds. More precisely, they seem to be good candidates for the ideas of [START_REF] Iliev | Fano manifolds of Calabi-Yau Hodge type[END_REF] to be implemented successfully.

Recently, A. Kuznetsov obtained nice structural results about the Küchle fourfolds whose Picard number is bigger than one. He also observed that among those whose Picard group is cyclic, there are two families with the very same discrete invariants. He asked whether this coincidence could be explained by the possibility that the two types of Fano fourfolds are deformation equivalent [START_REF] Kuznetsov | On Küchle manifolds with Picard number greater than 1[END_REF]Question 1.1].

The main result of this short note is that this is indeed the case, and that much more is true: the two families are in fact the same one! Moreover, this phenomenon happens in arbitrary dimension: there are two families of prime Fano n-folds of index one, that look different at first sight but in fact coincide. The first one is that of (n + 2)-codimensional linear sections of the Grassmannian G(2, n + 3). The second one is that of zero-loci of sections of the twisted quotient bundle on G(2, n + 2). We prove in Theorem 3.1 that these two types of varieties are the Date: January 29th, 2015.

same up to projective equivalence. Meanwhile we provide a few elements about the geometry of these Fano manifolds, that would probably deserve further study.

Acknowledgements. I thank A. Kuznetsov for his interesting comments.

This work has been carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the "Investissements d'Avenir" French Government programme managed by the French National Research Agency.

Two families of Fano manifolds of index one

We will denote by G n the Grassmannian G(2, n + 2) parametrizing planes in a complex vector space V n+2 of dimension n + 2. This is a smooth Fano variety of dimension 2n, Picard number one and index n + 2. The very ample generator of the Picard group defines the Plücker embedding of G n , with respect to which its degree is equal to the Catalan number c n = 1 n+1 2n n . 2.1. Linear sections of Grassmannians. Any smooth linear section X of G n+1 of codimension n + 2 is again a Fano variety, of dimension n, Picard number one and index one. This will be our first family of smooth Fano manifolds. Note that for n = 2 we get a del Pezzo surface of degree five, and for n = 3 a prime Fano threefold of genus eight.

There is a moduli space for these manifolds, that we can construct as the GIT quotient of an open subset of the Grassmannian G(n + 2, ∧ 2 V * n+3 ) by the reductive group P GL n+3 . The dimension of this moduli space is

N = (n + 1)(n + 2) 2 /2 - (n + 2)(n + 4) = (n + 3)(n 2 -4)/2.
Locally around a point in the moduli space corresponding to a given X, deformations are unobstructed: recall this is the case for any Fano manifold, as a consequence of the Kodaira-Akizuki-Nakano vanishing theorem. The tangent space H 1 (T X) to the local Kuranishi space can be computed fom the normal exact sequence, which yields a long exact sequence of cohomology groups on X:

H 0 (T X) → H 0 (T G n+1|X ) → H 0 (O X (1)) n+2 → H 1 (T X) → H 1 (T G n+1|X ).
Indeed, using the Koszul resolution of O X and Bott's theorem on G n+1 , one checks that

H 1 (T G n+1|X ) = 0 and H 0 (T G n+1|X ) = H 0 (T G n+1 ) = sl n+3 . Moreover H 0 (T X) = 0, or equivalently: Proposition 2.1. For n ≥ 4, any smooth X has a finite automorphism group. Proof. Since X has index one, H 0 (T X) = H 0 (Ω n-1 X (1)
). Taking the (n -1)-th wedge power of the conormal exact sequence, we get that this cohomology group is zero as soon as

H k (X, Ω n-1-k Gn+1|X (1 -k)) = 0 ∀0 ≤ k ≤ n -1.
Using the Koszul resolution of the structure sheaf of X, we get that this vanishing will hold as soon as

H k+ℓ (G n+1 , Ω n-1-k Gn+1 (1 -k -ℓ)) = 0 ∀0 ≤ ℓ ≤ n + 2.
If k +ℓ ≥ 2, this follows from the Kodaira-Akizuki-Nakano vanishing theorem, since

(k + ℓ) + (n -1 -k) = n -1 + ℓ is smaller than the dimension of G n+1 . If k + ℓ ≥ 1, we get H 1 (Ω n-1 Gn+1
) and H 1 (Ω n-2 Gn+1 ), which are both zero for n ≥ 4 since H q (Ω p Gn+1 ) is always zero for p = q. Finally if k + ℓ ≥ 0, we get H 0 (Ω n-1 Gn+1 (1)) = 0 by [6, Theorem 2.3].

Remark. It would be interesting to have closed formulas for the Hodge numbers of linear sections of Grassmannians. In the case we are interested in, the Lefschetz hyperplane theorem gives h p,q (X) = h p,q (G n+1 ) for p + q < n. Hence h p,q (X) = δ p,q p+1 2 under this condition. The case p + q = n is more difficult. It would be enough to compute the holomorphic Euler characteristic of the bundles of p-forms for p ≤ n, which can be done by standard techniques from Schubert calculus but remains computationally hard. Even the topological Euler characteristic seems difficult to compute. If we try to use the Gauss-Bonnet formula e(X) = X c n (T X) we can obtain the Chern class of the tangent bundle from the normal exact sequence. We deduce that if

P n (x, y) = n+1 k=0 p n,k x 2k y 2n+2-2k = x n+2 (1 + x) n+2 (1 + x + y 2 ) n+3 1 -x 2 + 4y 2 2n+2
the degree 2n + 2 part of the Taylor expansion of this rational function, then e(X) = In classical langage, Y defines a congruence of lines in P(V n+2 ) = P n+1 . Recall that the order of such a congruence is defined as the number of lines from Y passing through a general point in P n+1 . Proposition 2.2. The congruence of lines defined by Y has order n + 1.

Proof. The set of lines passing through a point in P n+1 is isomorphic to P n , and the restriction of Q(1) to this projective space is R(1), if R denotes the tautological quotient bundle on P n . The order of the congruence is the number of zeroes of the induced section of R(1) = T P n . For Y general and a general point in P n+1 we get a general vector field on P n , and we deduce that the order is c n (T P n ) = e(P n ) = n+1.

The space of global sections of

Q(1) = Q ⊗ det(U * ) is S n := S 10•••0-1-1 V n+2 = Ker(V n+2 ⊗ ∧ 2 V * n+2 -→ V * n+2
) (the latter morphism being the natural contraction map), as follows from the Borel-Weil theorem. Its dimension is n(n + 2)(n + 3)/2.

We would be tempted to think of our family of Fano manifolds Y , as the quotient of an open subset of P(S n ) by P GL n+2 . This is not correct. Indeed, an unusual phenomenon happens: H 0 (T G n|Y ) is bigger than the sl n+2 we would have expected. In fact,

H 0 (T G n|Y ) = sl n+2 ⊕ V n+2 .
This means that there are more linear isomorphisms beteween these varieties than those coming from P SL n+2 . We will explain the appearance of that extra factor in the next section.

Note that Q(1) = Hom(∧ 2 U, Q), and S n ⊂ Hom(∧ 2 V n+2 , V n+2 ). Hence, for any ω ∈ S n , the zero locus of the associated section of Q( 1) is

Y ω = { a, b ∈ G(2, V n+2 ), ω(a, b) ∈ a, b }.

Note that this makes sense for any

ω ∈ Hom(∧ 2 V n+2 , V n+2 ) = S n ⊕ V * n+2 ; but the component on V * n+2 is in fact unsignificant, since for any v * ∈ V * n+2 , v * (a, b) = v * (b)a -v * (a)b always belongs to a, b .
More interestingly, the previous description shows that Y ω has a natural rational map to P(V n+2 ), that we denote by Ω. By definition

Ω( a, b ) = [ω(a, b)].
Proposition 2.3. The rational map Ω is a birational isomorphism with a determinantal hypersurface of degree n + 1 in P n+1 .

Proof. Let Z = Ω(Y ). A point [c] of P(V n+2 ) belongs to Z if and only if there exists an independant vector d such that ω(c, d) = 0 belongs to c ⊥ . Otherwise said, the induced map from V n+2 / c → (c ⊥ ) * must not be injective. Note that there is a natural duality between c ⊥ and V n+2 / c . Globally, we can therefore describe Z as the first degeneracy locus of a morphism between vector bundles ω : R(-1) -→ R. This implies that [Z] = c 1 (Hom(R(-1), R)) is n + 1 times the hyperplane class, so Z is a hypersurface of degree n + 1. Moreover the fiber of σ over any point can be identified with the kernel of ω, in particular it is always a linear space, and it reduces to a single point for the general point of Z.

Let Ȳ ⊂ F (1, 2, V n+2 ) be the variety parametrizing incident lines and planes c ⊂ c, d such that ω(c, d) belongs to c . The two projections yield a diagram

Ȳ p1 ւ ց p2 G(2, V n+2 ) ⊃ Y Z ⊂ P(V n+2 )
It is easy to see that for ω generic, the variety Ȳ is the zero-locus of a generic section of a globally generated vector bundle, hence a smooth variety. Moreover its projection to Y is birational. More precisely, this projection is the blow-up of the smooth subvariety S ω defined as

S ω = { a, b ∈ G(2, V n+2 ), ω(a, b) = 0}.
Note that S ω is a Calabi-Yau variety, of codimension two in Y .

The projection to Z fails to be an isomorphism over the locus where ω drops rank. For ω generic, this occurs on a codimension three subvariety C ⊂ Z which is also the singular locus of Z. The projection from Ȳ to Z has fibers over the general points of C which are lines, in particular this projection is a small morphism.

Remark. For n = 2, the surface Y is a del Pezzo surface of degree five and Ȳ is its blow-up at two points. The projection to the cubic surface Z is an isomorphism.

For n = 3, the threefold Y is a prime Fano threefold of genus eight, and Ȳ is obtained by blowing-up the elliptic curve E = S ω . The projection to the quartic determinantal threefold Z contracts 25 lines to the 25 singular points of Z.

Remark. As Kuznetsov points it out, ω ∈ Hom(∧ 2 V n+2 , V n+2 ) might be considered as defining a bracket [a, b] = ω(a, b), although not a Lie bracket in general since the Jacobi identity has no reason to hold. Then Y ω parametrizes the planes in V n+2 on restriction to which the bracket defines a Lie algebra structure: each plane in Y ω is required to be stable under the bracket, and the Jacobi identity automatically holds for dimensional reasons. Moreover the codimension two subvariety S ω can be interpreted as parametrizing the two-dimensional abelian subalgebras.

And their coincidence

Let X be a smooth linear section of G n+1 , defined by an (n + 2)-dimensional space of linear forms H n+2 ⊂ ∧ 2 V * n+3 . We suppose in this section that H n+2 is generic.

Fix a decomposition

V n+3 = V n+2 ⊕ v n+3 , yielding a decomposition ∧ 2 V * n+3 = ∧ 2 V * n+2 ⊕ v * n+3 ∧ V * n+2
, where the linear form v * n+3 has kernel V n+2 . Generically, the projection on the second factor of this decomposition yields an isomorphism H n+2 ≃ V * n+2 . The variety X is thus defined by a monomorphism ω ∈ Hom(V * n+2 , ∧ 2 V * n+2 ) (we use the same notation for ω and its transpose): it is cut out by the space of linear forms defined as the graph

H n+2 = {ω(u) + v * n+3 ∧ u, u ∈ V * n+2 }. Since ω is injective, X does not contain any line passing through [v n+3 ]. Any line in X is of the form a, b + χ(b)v n+3 for some non zero vectors a, b ∈ V n+2 , with the condition that χ(b)u(a) = ω(u)(a, b) ∀u ∈ V * n+2 .
This implies that ω(u)(a, b) = 0 for all u ∈ a ⊥ , while the remaining equation determines χ(b). This means in particular that the linear projection from ∧ 2 V n+3 to ∧ 2 V n+2 , which induces a rational map

G n+1 = G(2, V n+3 ) G n = G(2, V n+2 ) ≃ G(2, V n+3 / v n+3 ),
restricts to a well-defined map from X to the subvariety Y ω of G n , which is moreover injective.

The inverse mapping can be described as follows. Note that ω(a, b) is the vector c ∈ V n+2 defined by the identity u

(c) = ω(u)(a, b) for all u ∈ V * n+2 . The equations defining X reduce to the condition that c = χ(b)a. The line a, b + χ(b)v n+3 is thus represented by [a ∧ (b + χ(b)v n+3 )] = [a ∧ b + c ∧ v n+3 ] = [a ∧ b + ω(a, b) ∧ v n+3 ].
This concludes the proof of our main result: Theorem 3.1. X ⊂ G n+1 and Y ω ⊂ G n are projectively equivalent.

Remark. In order to identify X with the subvariety Y ω of G n , we started from a decomposition V n+3 = V n+2 ⊕ v n+3 . Note that if we choose another hyperplane V n+2 , or equivalently if we change the defining linear form v * n+3 into v * n+3 -e * for some e * ∈ V * n+2 , then ω is changed into the morphism from V * n+2 to ∧ 2 V * n+2 that sends u to ω(u) + e * ∧ u. In particular the class of ω in S n is not affected.

On the contrary, changing the line v n+3 has a non trivial effect on the class of ω (that could easily be expressed explicitely), but does not affect Y = Y ω up to projective equivalence. This is precisely what explains the extra factor V n+2 inside H 0 (T G n|Y ).

Remark. The Calabi-Yau two codimensional subvariety S of Y can be seen directly in X as the intersection of G(2, V n+2 ) ⊂ G(2, V n+3 ) with the linear space that defines X. Of course there is a whole family of such Calabi-Yau's in Y , parametrized by an open subset of the projective space of hyperplanes in V n+3 . In particular, for n = 2 we get a four dimensional family of K3 surfaces covering Y .

Question. As we have seen, our two families of Fano manifolds of index one coincide generically. An intriguing question is to decide whether they coincide stricto sensu: can any smooth member of each family be described as a member of the second family? A negative answer would be particularly interesting, as a new example of the pathological behaviors of the moduli spaces of Fano varieties.

2 . 2 .

 22 n+1 k=0 p n,k c k . We can deduce the missing Betti number b n = b n (X) for X of small dimension n: b 2 = 5, b 3 = 10, b 4 = 69, b 5 = 380, b 6 = 2321, b 7 = 9442. Zero loci of the twisted quotient bundle.Recall that the Grassmannian G n is endowed with two natural vector bundles, the tautological rank two bundle U , and the quotient bundle Q, which has rank n. Moreover det(U * ) = det(Q) = O Gn (1), the very ample generator of the Picard group. The quotient bundle Q is generated by global sections, and the zero locus of a non zero section is just a projective space. More interesting are the zero loci of global sections of the twisted quotient bundle Q(1). Since det(Q(1)) = O Gn (n + 1), the zero locus Y of a general global section of Q(1) is a smooth Fano manifold of dimension n and index one. This is our second family of such manifolds.


Marc Gaudry
email: marc.gaudry@umontreal.ca

Emile Quinet
email: quinet@enpc.fr

Émile Quinet

Correlation within SNCF administrative regions among track segment maintenance

Keywords: C61, D61, D92, H54, R41, R48 rail infrastructure, current maintenance cost, rail track degradation, Box-Cox transformation, directed autocorrelation, spatial autocorrelation, SNCF, French rail network, network data by segment, SNCF administrative regions, regional differences rail infrastructure, current maintenance cost, rail track degradation, Box-Cox transformation, directed autocorrelation, spatial autocorrelation, SNCF, French rail network, network data by segment, SNCF administrative regions, regional differences

Table of contents

 2. BCT forms and elasticities of maintenance cost m with respect to 8 regressors ................... 8 Table 3. SNCF regions: neighbors (nb), segments (sg) in the sample and t-statistic of dummies ...... 9 Table 4. Four geographic partitions of matrix R 1 into pairs of complementary matrices .................. 11 Table 5. Testing geographic and sociological decompositions of matrix R 1 ..................................... 12 Table 6. Family structure in France and the Liberal/Authoritarian partition of SNCF regions ......... 13

The context: European rail infrastructure maintenance cost modeling 1

Models of infrastructure maintenance costs estimated from rail track segment data, linking such costs to traffic and technical characteristics of ways, are not to be confused with models of total rail cost (including train service provision) estimated from total expenditures of railway firms.

The latter arose in the 1980's and 1990's in a context where the Trans-Log form (Christensen et al., 1971a[START_REF] Christensen | Transcendental Logarithmic Production Frontiers[END_REF] was progressively adopted as a standard tool of analysis of transport firm costs across all modes: representative rail examples include Caves et al. (1980aCaves et al. ( , 1980b[START_REF] Caves | Network effects and the measurement of returns to scale and density for U.S. railroads[END_REF] in the USA and [START_REF] Borger | Hedonic versus homogeneous output specifications of railroad technology: Belgian railroads 1950-1986[END_REF][START_REF] Borger | Estimating a Multiple-Output Generalized Box-Cox Cost Function: Cost Structure and Productivity Growth in Belgian Railroad Operations[END_REF] in Belgium. The former, sometimes separately explaining track renewal costs by track segment in addition to maintenance costs, arose specifically in Europe in the 2000's: early studies include [START_REF] Idström | Reforming the Finnish railway infrastructure chargean econometric analysis of the marginal costs of rail infrastructure use[END_REF], [START_REF] Johansson | An Economic Analysis of Track Maintenance Costs[END_REF][START_REF] Gaudry | The Robust Gravity Form in Transport and Trade Models[END_REF] and [START_REF] Gaudry | Rail track wear-and-tear costs by traffic class in France[END_REF]. Such studies are still few, European, and very rare for other modes: [START_REF] Ben-Akiva | Modelling infrastructure performance and user costs[END_REF], who studied road maintenance costs with Brazilian highway section data, provide a notable exception. We found no example of applications to airport runways or taxiways.

The model of the French rail network maintenance costs for the year 2007 used here to study spatial correlation of its residuals actually belongs to a second wave of rail maintenance cost models. First wave models do not impose complex behavioral conditions on the estimated cost function other than short-run cost minimization. Second wave models are more ambitious in this respect and assume on the part of the infrastructure firm some joint optimization of maintenance and regeneration expenditures: these stronger assumptions have implications for both the contents and the shape of the maintenance cost function.

It is on a function of this latter type that we will perform spatial and more general tests of "directed" correlation pointing both to unexplained similarities within administrative regions and to dissimilarities across such regions of the nation-wide SNCF firm. As the interpretation of extant region-specific correlation among residuals is no less uncertain than that of serial correlation, our analysis will only speculate on the causes of uncovered correlations.

Background of the model selected for tests of spatial correlation

First wave rail track segment models

In the first wave of studies based on rail track segment data, a behavioral assumption of short run cost minimization by infrastructure owners is typically made and the cost function is estimated with an eye for its optimal pricing implications, especially in countries where the marginal cost pricing doctrine of the European Union Commission is applied. A representative function might be:

(1-A) 00 ( ( , ), ( , ), ( )) tt m f K h e q n w S h  ,
where, for a given year t, the maintenance cost per unit of track segment m is a function of its technical characteristics K, current traffic q and latent deterioration indicators S(h). A particularly complete set of recent models within this stream is found in the numerous country reports of the Cost Allocation of Transport INfrastructure cost (CATRIN) Consortium, summarized in [START_REF] Wheat | CATRIN (Cost Allocation of TRansport INfrastructure cost), Deliverable 8 -Rail Cost Allocation for Europe[END_REF], and in related contributions (e.g. [START_REF] Andersson | Estimating the marginal cost of Railways Track Renewals using corner solution models[END_REF].

Note that Eq. (1-A) contains many sub-sets of explanatory variables found in these various country studies: K is split between planned reference quality or "standing" 0 h (such as maximum allowed design speed v) and state characteristics e 0 (such as segment length; circuitousness; number of tracks, switches and tunnels; electrification; centralized automated traffic control; etc.); also, current traffic q is split between the number of currently circulating trains (n) and their weight (w); finally, current deterioration performance indicators S(h) include many possible other factors (such as age of rails and of sleepers; proportion of wood sleepers; etc.). We have found no rail paper using an actual measure of track performance or condition like the Longitudinal Deformation index appearing in the model of interest below: the h in question in (1-A) are only the latent determinants of unobserved track condition S, not direct measures of it as found below.

As the index t in (1-A) refers in practice to very few consecutive years of available data, typically one and at most three, the brevity of large track segment panels makes it possible, if not necessary, for the above early and for later authors to abstract entirely from input prices: they are then assumed constant in space (and during the very short time periods available) for the given national infrastructure provider studied, for instance Swedish in [START_REF] Andersson | Marginal railway infrastructure cost estimates in the presence of unobserved effects[END_REF], Swiss in [START_REF] Marti | Case study 1.2E: Track Maintenance Costs in Switzerland[END_REF] and Austrian in Link ( 2009)all three of which use CES (Log-Log) specifications for () f  , as most early authors did.

On this last point of functional form, CATRIN mandated simple Box-Cox (SBC) tests, consisting in using a single Box-Cox transformation (BCT), defined in (2-D) below, on the dependent and independent variables of a CES reference model: all resulting national model results so obtained (e.g. [START_REF] Andersson | CATRIN (Cost Allocation of TRansport INfrastructure cost), Deliverable 8, Rail Cost Allocation for Europe -Annex 1A, Marginal Costs of Railway Infrastructure Wear and Tear for Freight and Passenger Trains in Sweden[END_REF][START_REF] Link | CATRIN (Cost Allocation of TRansport INfrastructure cost), Deliverable 8, Rail Cost Allocation for Europe -Annex 1C -Marginal costs of rail maintenance in Austria[END_REF][START_REF] Marti | CATRIN (Cost Allocation of TRansport INfrastructure cost) Deliverable 8, Rail Cost Allocation for Europe -Annex 1B. Track maintenance and renewal costs in Switzerland[END_REF] proved far superior to the references ones.

But the most extensive Box-Cox tests carried out within CATRIN showed that, although the popular Trans-Log function was not superior to the CES function for () f  , CES-inspired generalizations named Unrestricted-Generalized Box-Cox (U-GBC) -and even the single-BCT Restricted (R-GBC) specification inspired by [START_REF] Berndt | Parametric Productivity Measurement and Choice among Flexible Functional Forms[END_REF] 2 were, statistically speaking, "infinitely" superior to both CES and Trans-Log forms. These new forms [START_REF] Gaudry | CATRIN (Cost Allocation of TRansport INfrastructure cost), Deliverable 8, Rail Cost Allocation for Europe -Annex 1Di-Track Maintenance Costs in France[END_REF][START_REF] Gaudry | Track Wear and Tear Cost by Traffic Class: Functional Form, Zero-Output Levels and Marginal Cost Pricing Recovery on the French Rail Network[END_REF], already tried in the 2003 round of tests by the same authors, were:

(2-A)   () ( ) ( ) 0 1 1 1 , k r i r j r k k ij i j k i j y X X X i j                   R-GBC   () () () 0 1 1 1 , ij y k k r i r j r k k ij i j k i j y X X X i j                    U-GBC
where ij  because BCT are invariant to a power transformation (Gaudry & Laferrière, 1989) and, in consequence, the BCT of a squared variable is not distinct from that of the simply transformed variable. As many as 13 BCT were used in various model runs, including some with five  k specific to train types demonstrating the existence of differences in track damages across train types3 .

Remembering that prices are absent from these cross-sectional models, the U-GBC and R-GBC tests in fact showed that the Trans-Log is a very poor model of physical interaction between track and traffic because this interaction is demonstrably not Log-Log. They also showed, by systematically studying the impact of total traffic disaggregation among train types that, if certain precautions were taken4 , the BCT was the best tool to test the forms of variables that include some zero observations (e.g. not all types of trains use all track segments), and was in fact superior, in theory and practice, to both CES (Log-Log) and Trans-Log specifications in handling zero values.

A particular second wave track segment model

Economic specification of a three-phase model. If the assumption of short-run cost minimization is replaced by that of joint optimization of maintenance and renewal, the cost function is enriched by new terms: in particular, the cumulative value of traffic Q (in terms of number of trains N or their weight W)5 and t, time elapsed since the last regeneration, now matter in principle.

The specifics of the particular source model considered for our tests [START_REF] Gaudry | Infrastructure maintenance, regeneration and service quality economics: A rail example[END_REF], from which we will borrow heavily in this derived paper, also include some dynamic control variables c, functions of the firm's track quality target Sc, because deviations of actual track conditions S from their target levels bring forth corrections of maintenance expenditures. Noting that the latent condition indicators S(h) of (1-A) are replaced by actual condition S and target condition Sc, the doubly enriched cost function may now be written:

(1-B) 00 ( ( , ), ( , ), ( , ), , ( , ))

tt m f K h e q n w Q N W t c S Sc  .
Indeed, the actual theoretical model at hand in fact defines three maintenance phases. In the first and third phases, i.e. after a segment has been renewed and before it is to be renewed, maintenance is nil or very low and limited to surveillance; but during the central cruising phase, maintenance is performed in such a way as to follow the target track quality or condition path. This means that, for the estimation of determinants of maintenance, only this central phase is relevant: one then has to remove from the available sample observations belonging to unrepresentative first and third phases.

Econometric specification with and without a ∩-shaped expenditure time profile. This is effected in a first step by estimating with the full sample, within a given multivariate U-GBC specification, the inverse U-shaped6 time profile of maintenance expenditures associated with one particular regressor, cumulative traffic Q (here the cumulative weight of trains W since the last track segment regeneration):

(

2-A)     12 12 () QQ t Q t Q t Q W W W   .
In the second step, very small and very large values of observations of this variable are progressively removed until the cost function is monotonic in terms of W because the two BCT powers assigned to the same variable W converge to a unique value and (2-A) becomes monotonic.

With the doubly censored sample, we then tested a number of U-GBC specifications containing interactions between cumulative traffic and some measure of maximum allowed speed, for both total traffic and traffic by train type, but none of the interactions proved significant under the new specification enriched by cumulative traffic (W) and control variables (S, S*). In the absence of interactions and of non monotonic regressors, these U-GBC specifications reduce to a standard multivariate BCT generalization of the CES which may now be written for our spatial correlation tests with an explicit residual error v t associated with each of the T segments in the sample:

(2-B) () () 0 y k t k kt t k y X v        (2-C)   2 , 1 1 nT t tn n t n v r v w        
where, in the first equation duly containing an intercept (Schlesselman, 1971) to guarantee the invariance of form parameters to changes in the units of measurement of the X k , the Box-Cox transformation (BCT) applicable to any strictly positive 7 variable Var v is commonly defined without Tukey's shift parameter as:

(

2-D) () [( ) 1] , 0, ln ( ) , 0, v v v v v v v v Var Var Var             
and where, in the second equation, ,tn r denotes the typical element of matrix R , a row or column- normalized square (T by T) Boolean matrix R expressing hypothesis concerning the presence of correlation among designated residuals, which may then behave as substitutes ( 0   ) or complements ( 0   ) to the residual of segment t. And rewriting (2-C) in matrix notation eases the explicitation of matrix R , namely

(2-E) 2 1 v R v w      , with 1 (1 ) , (0 1) , R I R R           
where the new proximity parameter  measures the relative influence of "near" and "distant" neighbors of R , as soon discussed below.

Directed autocorrelation. Matrix R in (2-E) results from three steps. In the first, a square matrix R is defined with typical element ,tn r =1, to express an hypothesized correlation between any two residuals v t and v n (and ,tn r =0 otherwise). It is often called a contiguity matrix because, historically, spatial hypotheses were first used to define it; but clearly any criterionspatial, temporal or socioeconomic -may specify a pattern of interdependence among residuals: the researcher's directed choice 8 is not limited by the natural order of the data.

In the second step, it has long been the practice [START_REF] Ord | Estimation methods for models of spatial interaction[END_REF] to row or column normalize this matrix and [START_REF] Bolduc | On the estimation of models with generalized SAR(1) processes on the residuals of a regression[END_REF] has shown that the resulting normalized matrix R guarantees a convex likelihood function over the stable unit interval of  . This Boolean formulation is « idiot-proof » in the sense that, based only on zeroes and ones, it avoids the complications of lack of invariance of estimates to changes in units of measurement that arise when the elements of R l are functions of continuous variables, such as distance or income (Bolduc et al., 1989, p. 369). But something might then be needed to compensate for the discreteness of R , as many distributed phenomena are likely to be smooth and their representation by a few « all-or-nothing » Boolean slices insufficient: the solution resides in taking due account of the whole set of near and distant "neighbor" slices, as long done with distributed lags in time-series analysis.

Contiguous and distributed "spatial" lags. This is precisely the idea of the [START_REF] Blum | From Correlation to Distributed Contiguities: a Family of AR-C-D Autocorrelation Processes[END_REF][START_REF] Blum | From Correlation to Distributed Contiguities: a Family of AR-C-D Autocorrelation Processes[END_REF] approach where powers of R l generate a sequence of contiguity matrices which define degrees of neighborliness or proximity ( 2 R , ... , c R ,...), with 2 R denoting neighbors of neighbors, and c R higher powers.

Under the assumption that the impact of these close and distant neighbors decreases geometrically with «distance» c, as in [START_REF] Koyck | Distributed Lags and Investment Analysis[END_REF] distributed lags of time-series (whence the name 7 The dummy variable and 0-replacement methods available to transform variables that contain some 0 are discussed and used in Gaudry & Quinet (2010). Here 0 traffic values are replaced by 0,00001, unless stated otherwisefor instance in the detection of Phase C segments where the dummy variable method is used. 8 For an example of directed choice involving correlation of residuals across socio-economic groups, see [START_REF] Gaudry | An Example of Correlation Among Residuals in Directly Ordered Data[END_REF]. For distributed processes in transport and trade flow models, see [START_REF] Gaudry | The Robust Gravity Form in Transport and Trade Models[END_REF].

Autoregressive Contiguous Distributed (AR-C-D) for this analogue process), one obtains

1 (1 ) R I R R       
, where the new proximity parameter  allows for endogenization of the relative importance of near and distant effects. If  =1, Rl = R l , in which case only the adjacent neighbors have an impact on the correlations among the residuals selected by the "residue impact criterion" expressed in R , exactly as in the classical (Ord, 1975; Cliff & Ord, 1981) case. By contrast, as  →0, the near effect is reduced to a minimum in favor of the distant effect. In this R-Koyck formulation, the parameter  therefore generally weighs the relative importance of near and distant effects, i.e. the sharpness or slope of the decline.

Using the L-2.1 algorithm [START_REF] Tran | LEVEL: The L-2.1 estimation procedures for BC-DAUHESEQ (Box-Cox Directed AUtoregressive HEteroskedastic Single EQuation) regression[END_REF] to jointly estimate parameters of Eq. (2-B)-(2-C), our analysis of maintenance cost by segment m t tests a number of specifications of the R matrix and refers to further tests with two distributed processes  reported in the source paper.

Basic data and results of the reference maintenance cost model

Table 1 presents the main data of the uncensored sample of 700 observations, representing about 70% of total network segments, a number reduced to 580 segments in the cost estimation model. (503) 1 Out of a total network of about 30 000 km. 2 Available over the period 2000-2010 for a subset of 608 observations. 3 Available over the period 1995-2007 for all 700 observations. Table 2 presents the main variants of the maintenance cost reference model9 estimated from the censored (cruise phase) data. We make four principal comments: i) specification of variables and flexible BCT forms. Column 1 presents a starting formulation with a fixed multiplicative Log-Log form as close in contents as possible to that of the UK Office of Rail Regulation (ORR) function, described in [START_REF] Ongaro | CATRIN (Cost Allocation of TRansport INfrastructure cost), D8 -Rail Cost Allocation for Europe -Annex 2 -Railway Infrastructure Wear and Tear for Freight and Passenger Trains in Sweden[END_REF]. Under this form specification, all signs are as expected except for the time proxy variable (agerail) negative sign which progressively corrects itself with the form and content enrichments of the next three columns; but this form-sensitive variable of expected positive sign is never really significant and never obtains a strong elasticity.

Concerning fit, massive Log Likelihood gains of 111 points are obtained in Column 2 by estimating only 4 BCT, all in fact very different from the logarithmic starting values assumed in Column 1: 0,25 for the dependent variable m, 0,53 for the three technical state variables e, 0,63 for total tonnage (the sum of cumulative and current values (W+w)) and 2,11 for V, a measure of maximum operation speed constructed from assumed maximum speeds by category of train 10 c weighted by their shares of total traffic [cumulative (W c ), measured at the beginning of each year, and current (w c )] measured in tons. ii) the role of control variables and the infrastructure manager's speed of adjustment. In the third column, the addition of the Service Target and Trajectory Correction control variables in accordance with (1-B) yields the expected positive and negative signs for these variables -with acceptable t-statistics-and a significant gain of 10 points in Log Likelihood for a difference of only 2 degrees of freedom.

But the linear 11 forms of these added terms used are the lagged values of target 12 track condition service levels   , both selected primarily because they are more statistically significant 13 than contemporaneous values 10 We assumed that trains are driven at the lowest of 90% of maximum allowed speed v on the segment and of the maximum speed allowed by train type, assumed to be 200 km/h for main intercity (GL) and regional (TER) trains, 100 km/h for freight (FRT) trains and 50 km/h for locomotives (HLP). 11 BCT cannot be used because these variables are not strictly positive. 12 For the derivation and estimation of target track condition Sc, see source paper. 13 The correlation between current and lagged Target Service variables is -0,21 and that between Trajectory Correction variables is 0,81; one could envisage using both current and lagged Trajectory correction variables if Maintenance cost data were available for more than the single year 2007.

  1 (( ))

. Reasonably, lagged values imply a response that, for various reasons including the discrete yearly nature of the formulation and of data, is not immediate.

Other elasticities are also reasonable and stable except for that of segment length, unfortunately much too high and significant until the introductions of spatial correlation in Column 4. In particular, the total ton traffic variable (W+w) is highly significant with BCT values between 0,39 and 0,63 resembling the 0,49 power for the axle weight variable in the referenced UK ORR formula for track damages.

iii) directed correlation of residuals: within region effects. The addition of first order directed autocorrelation in Column 4 yields huge further gains (31 points of Log Likelihood for the single additional parameter 1  =0,59) under the assumption that error terms of segments are correlated if they belong to the same SNCF administrative region. As indicated in Table 3, the number of track segments present in the sample varies across the 23 administrative regions of 2007, with numerical implications for the structure 14 of 1 R . iv) other forms of directed autocorrelation. A systematic attempt (with a slightly larger data set of 673 observations) to define a second type of directed autocorrelation 2 R based on regions sharing a border with that of the given segment, a number which varies from 2 to 8 as indicated in Table 3, and to define yet another type 3 R under the assumption that the error term of a segment could be correlated with those of other segments sharing one or more train lines, yielded no significant correlation in models specified as those found in Columns 3 and 4 of Table 2. The attempt to further estimate matching proximity 15 parameters 2  and 3  in addition to 2  and 3  , in accordance with Eq. (2-E), also failed to provide statistically significant gains in fit beyond those of models 3 or 4, whether the additional AR-C-D processes were considered separately or jointly with 1  .

In this latter 3 R case, the fact that segment residuals are uncorrelated with use by shared train lines either close (i.e. 3  estimates do not differ from 0) or far (i.e. 3  estimates do not differ from 1) suggests that the underlying representation of the track damage function in Eq. (1-B) is adequate.

Everything considered, the hypothesized role of within-region SNCF administrative practices appears considerable in explaining reported total maintenance cost by segment; and unaccounted for influences of neighboring regions have a negligible effect. We now wish to probe this stunning improvement in fit obtained with 1 R in Column 4 and try to make some sense of it.

14 Consider Region 10, with the smallest number of segments, and the Region 43 with the highest: in matrix 1 R , the 5 rows corresponding to the former have 4 values of 1,00 and the 51 rows corresponding to the latter have 50 such values. In the row-normalized 1 R , the former values become 1/4 and the latter 1/50. 15 A parameter 1  would not be meaningful: the matrix 1 R partitioned by region is symmetric and block diagonal. It is not clear what its powers would mean.

possibly because the dryer and warmer climate reduces maintenance costs. Moreover, one notes that these 5 regions with negative dummy signs are geographically dispersed, in contrast with 3 of the regions with positive t-values around 0,60 (54,Lyon;56,57,Chambéry), which are contiguous: this timid similarity may perhaps be due to their significant snowfall, absent from the specification, which increases maintenance costs 18 .

Missing climate variables?

The first clue obtained from this addition of dummy variables is therefore climatic, but it is a weak clue. Firstly because of the small explanatory power of the additional 18 dummy variables and secondly because the autocorrelation is only slightly weakened: the estimated value of 1  decreases from 0,59 to 0,44 but further removing this autocorrelation parameter (in run 138, not shown) massively lowers the Log Likelihood by 10 points: regional dummies then, of little interest in terms of fit, are not a substitute for within-region correlationstill in need of an interpretation. Displacing the mean error does not affect correlation among errors.

The second reason for doubting the importance of climatic variables is that, in a model that did not contain a Cumulative Traffic variable W, Rail Concept (2011, p. 46, Test 4) found only weak impacts of geographic dummy variables defined by climatic zone19 : oceanic, semi-oceanic, continental, mountainous, Mediterranean. The question cannot be solved without trying climatic variables within the new model formulation containing W, but the above dummy variable tests make the perspective unpromising.

Differences of autocorrelation coefficients across administrative regions.

To test the assumption, maintained up to this point, that the 23 within-region correlation coefficients are all equal, we grouped 21 administrative regions into 4 larger geographical units (East, North, South-West and South-East) detailed in Table 4, each containing a sufficient numbers of segments for estimation purposes and including its Paris station head region. These 4 geographical units, jointly accounting for all administrative regions except two (30, Paris St-Lazare; 33, Rouen) with too few observations together to yield robust estimates of own autocorrelation coefficients, are then used to define 4 pairs of complementary sub-sets of 1 R , i.e. of sub-matrices D R and D R , all documented in Table 4.

The results of these tests, shown in Table 5, demonstrate that there in fact exists no correlation among residuals within the East and North groupings: not only does one make significant gains (of 3.6 and 2.2 points, respectively) in Log Likelihood by using two orders of autocorrelation for these groupings, but the estimated small own correlation coefficients are negative 20 but not significantly different from 0. By contrast, this kind of splitting has no impact whatsoever for the South-West and South-East groupings where all correlation coefficients remain statistically indistinguishable from the maintained value of 0,590 found in the reference model, as demonstrated by the trivial gains in Log Likelihood (of less than one point) yielded by these pairs of splits of R 1 . Moreover, the union of East and North matrices yields gains of 6.05 points, i.e. a value close to the sum of the gains indicated for each grouping considered separately, and estimates of autocorrelation coeficients compatible with Minkowsky's triangle inequality condition that 12 1   , implying stationarity. The absence of correlation among residuals of Eastern and Northern groupings of regions considered separately or together, combined with strongly resilient positive correlation within regions of the rest of France, is puzzling. The 4 right-hand side columns of Table 4 indicate that segments in those East and North regions are shorter and contain more switches than the average segment in the sample. But these two variables have already been accounted for in the explanations of the total maintenance cost per km, as made clear in Table 2. Moreover, as implied in Section 2.2 above, it would be unwise to make the elements of R l functions of continuous variables, such as segment length or switches per segment, because the resulting estimates of  then become conditional on units of measurement of the selected continuous variable (Bolduc et al., 1989, p. 369), an unacceptable dependency.

Hidden cultural factors?

The SNCF is a centralized firm with «horizontal» accounting controls applied across all administrative regions: a finding of significant differences between the East and North groupings of regions and the rest of France then calls for an explanation. Could residuals within Eastern and Northern groupings be uncorrelated because of differences in accounting practices imputable to local cultural factors? Consider regional firm and family traditions. 20 Splitting R 1 has no impact on the segment length variable elasticity and on the t-statistic of its coefficient; in fact, neither does it have any impact on any of the other coefficients. Note that estimates for the North grouping satisfy Minkowsky's triangle inequality condition for stability, but that this is not the case for all other groupings. The implications of this fact in spatial models seems to be even more difficult to ascertain than in time series models. In our examples, estimated elasticities do not seem to differ between the stable and unstable cases.

One possibility is that accounting traditions differed in the original regional railway firms merged into the SNCF conglomerate in 1938. Indeed, the East grouping closely matches the network of the original Compagnie de l'Est, including its section later managed by l'Administration des chemins de fer d'Alsace et de Lorraine 21 ; and the North grouping resembles the network of the Compagnie du Nord. The finding concerning the Eastern grouping has nothing to do with the fact that, in Alsace-Lorraine, rail traffic runs on the opposite track from that used in the rest of France! Rather, could it be that, in both East and North cases, firm-specific accounting traditions have long lives? On this point, consulting firms 22 hired in 2011 by Réseau Ferré de France (RFF) and the regional transport authority Syndicat des transports d'Île-de-France (STIF) to study the value of train regularity found that administrative forms used to register the causes of late train arrivals varied across the main large stations of Paris, each using its reporting formats defined before 1936. Our historical hypothesis concerning the similar survival of distinct accounting practices could only be confirmed by a detailed study, well beyond the scope of this paper, that might also explain in passing differences in average management section length.

Another testable possibility, more sociological, postulates that local family structures might influence employee attitudes towards centralized administrative norms and affect the assignment of work invoices to segments. To test this wild card assumption, we use [START_REF] Todd | La nouvelle France. Nouvelle édition mise à jour. 288 p[END_REF] map of prevailing family structures found in Table 6 : it is defined in terms of two categorical dimensions, namely amount of parental authority and inequality in the treatment of siblings. Todd (1990, p. 88).

Partition according to parent-children relationship

We obtain a good match between areas of Authoritarian or Liberal parental attitudes and SNCF administrative regions, but find it impossible to achieve a 4-split partition of the latter also matching differences in the degree of hierarchy among siblings. As defined in any case, this sociological hypothesis concerning subtle influences of family structure on employee attitudes turns out to be useless, as the results of the last column of Table 5 make clear. 21 We tried to split 1 R between Alsace-Loraine (regions 18 and 17) and the rest of France, but the conditioning of the complementary matrix for the rest of France did not allow for a stationary value of its correlation coefficient. 22 Stratec, a Belgium firm, and Significance, a Dutch firm.

Conclusion

Using as reference a recent France-wide model of rail infrastructure maintenance cost where regression model residuals associated to track segments are assumed to be similarly correlated among themselves within the 23 administrative regions of the national firm (SNCF), we attempt to explain the presence of the strong positive and stationary correlation coefficient estimated23 in this manner and to probe the extent to which the assumption of a common correlation coefficient across administrative regions might be refined and interpreted.

We first find that country-wide within-region correlation among residuals is not weakened if regional dummy variables are added to the model in the hope of finding interpretative clues, notably of the presence of climate effects unaccounted for in the model specification. The implicit regionspecific weather effects indirectly so represented turn out to be extremely weak, if present at all, and do not affect the strength of extant within-region correlation or the need to make sense of it.

We then explore differences in correlation coefficient estimates among regions and show that, within the reference maintenance cost model, two large geographic groupings of regions, each comprising in the East or in the North about 15% of total available track segments, in fact have own residuals that are uncorrelated among themselves, in contrat to the more numerous 70% of segment residuals remaining in the rest of regions, which as a group remain robustly and positively correlated, and in a stationary manner (with 01   ). As the two groupings with uncorrelated residuals closely correspond to the networks of regional firms merged into the SNCF conglomerate in 1938, we hypothesize, faute de mieux and in the absence for the moment of refined local climate variables to pursue unpromising missing variable tests, that within-firm accounting traditions might have survived centralized management control of the assignment of track surveillance, maintenance and repair invoices to track sections centrally defined for accounting management purposes.

  1. The context: European rail infrastructure maintenance cost modeling ........................................ 3 2. Background of the model selected for tests of spatial correlation ............................................... 3 2.1. First wave rail track segment models .................................................................................... 3 2.2. A particular second wave track segment model .................................................................... 5 3. Basic data and results of the reference maintenance cost model ................................................. 7 4. Within-region correlation among regression errors ................................................................... 10 5. Conclusion ................................................................................................................................. 14 6. References .................................................................................................................................. 14 List of tables Table 1. Mean values of principal variables used in reference model ................................................. 7 Table

Table 1 . Mean values of principal variables used in reference model Variables (2007 sample size 700) m t Current maintenance expenses

 1

	total cost per km (current Euros)	52 432
	surveillance	14 821
	maintenance	37 611
	e 0 Technical state variables K length of segment (meters)	28 219 1
	length of all tracks (meters)	46 186
	number of switches per segment	25,48
	h 0 Initial standing maximum allowed speed (km/h)	114,69
	t t Current condition average age of rails (years)	30,56
	S t Service quality of the track NL index of longitudinal track rectitude (mm)	1,451 2

q t Traffic per day: trains and gross tons trains 3 weight 3 (per train)

	GL : long distance passenger trains (VFE)	17,99	10 255	(570)
	TGV: high speed trains	9,25	5 612	(601)
	Classic intercity trains (Corail)	8,74	4 643	(531)
	TER : regional passenger trains	24,44	5 559	(228)
	IdF : Île-de-France passenger trains	16,21	6 215	(384)
	Fret : freight trains	13,70	15 484	(1131)
	HLP : locomotives	2,84	275	(97)
	Total for the six categories of trains	75,16	37 788

Table 2 . BCT forms and elasticities of maintenance cost m with respect to 8 regressors

 2

	Elasticity m)	Column	1		2		3		4
	t-statistic=0)*, [t-statistic=1]		m		m		m		m	
	m	Total maintenance cost per km	n.a.		n.a.	0,25	n.a.	0,24	n.a.	0,22
		(dependent variable)	(t=0)		0		(19.38)		(18.18)	(16.83)
			[t=1]			[-59.68]		[-57.33]	[-59.32]
	 0	Intercept		n.a.		n.a.		n.a.		n.a.
			(t=0) (11.60)		(9.50)		(8.62)		(9.98)
	S	Target Δ service : 2006-2005					0.08**		0.10**
		[E(S ct-1 )-E(S ct-2 )] Trajectory correction: (obs.-target) 2005 (t=0)					(4.51) -0.002	1	(5.22) -0.002	1
		[S t-2 -E(S ct-2 )]	(t=0)					(-3.14)		(-3.50)
	e 0	Segment length		-0.121		-0.097		-0.082		-0.038
		Track length Number of switches	(t=0) (-2.91) 0.116 (t=0) (1.05) 0.191	1	(-2.35) 0.178 (2.32) 0.416	0.53 (8.53) [-7.71]	(-2.10) 0.149 (1.93) 0.394	0.58 (8.81) [-6.29]	(-0.98) 0.181 (2.37) 0.440	0.50 (8.39) [-8.48]
			(t=0) (10.01)		(20.21)		(19.53)		(19.94)
	(W+w)	Cumulative+ current total tons	0.279		0.251	0.63	0.273	0.39	0.240	0.40
		(W+w)	(t=0)	(8.21)		(9.35)	(2.93)	(10.39)	(2.63)	(8.75)	(2.37)
			[t=1]				[-1.72]		(-4.14]	[-3.56]
	h 0	Speed V weighted by (W c +w c ) shares (t=0) (0.62) 0.036	0	0.068 (2.18)	2.11 (1.25)	0.016 (1.44)	3.30 (0.47)	0.022 (1.16)	2.81 (0.53)
			[t=1]				[0.69]		[0.33]	[0.34]
	t	Time since last regeneration (agerail) (t=0) (-0.88) -0.095		0.022 (0.26)	0	0.053 (0.63)	0	0.062 (0.80)	0
	R 1	Same region:  1								0.590
			(t=0)							(6.91)
		Log likelihood		-6977.54	-6866.10	-6856.14	-6824.94
		Number of  k estimated		7		7		9		9
		Number of  k estimated		0		0		3		3
		Difference in degrees of freedom	0		4		6		6
		Variant run number		95		94		934	937/139

Table 3 . SNCF regions: neighbors (nb), segments (sg) in the sample and t-statistic of dummies

 3 ** In run 140, t-stat. of 18 dummy variables added to Column 4 (run 139), increasing Log Likelihood from -6824.94 to -6817.76.

	N o	Region*	nb	sg	t-stat.*** N o	Region*	nb	sg	t-stat.***
	10 PARIS EST**	5	5	reference	43 BORDEAUX	4	51	(0.10)
	14 REIMS		7	17	(0.02)	44 LIMOGES	4	17	(0.28)
	17 METZ-NANCY	3	36	(0.11)	46 TOURS		6	19	(0.09)
	18 STRASBOURG	2	31	(-0.47)	47 TOULOUSE	4	33	(-0.03)
	20 PARIS NORD**	7	25	reference	50 PARIS SUD EST**	8	24	reference
	23 LILLE		2	38	(0.34)	53 DIJON		7	30	(0.37)
	24 AMIENS		4	14	(0.10)	54 LYON		5	35	(0.66)
	30 PARIS ST-LAZARE**	5	11	reference	56 CLERMONT-FERRAND	7	27	(0.61)
	33 ROUEN		5	21	(0.23)	57 CHAMBERY	3	31	(0.69)
	34 PARIS RIVE GAUCHE**	5	25	reference	58 MARSEILLE	3	29	(0.39)
	36 RENNES		2	14	(-0.17)	59 MONTPELLIER	4	23	(-3.31)
	37 NANTES		5	24	(-0.16)	NUMBER OF SEGMENTS AND OF DUMMIES	580 sg	18 dummies

* In 2012, a merger of regions 10 and 14 and of regions 20 and 24 reduced the total number to 21. ** The pie-shaped administrative regions located in Île-de-France are treated as first neighbors of one another in matrix R 2 . *

Table 4 . Four geographic partitions of matrix R 1 into pairs of complementary matrices Geographic partitions Average segment length (km) Average number of switches per segment Impact matrix criterion: East North South-West South-East N o Administrative region East Other North Other South-West Other South-East Other

 4

	10 PARIS EST	5			5		5		5	20,38		72,60
	14 REIMS 17 METZ-NANCY	17 36			17 36		17 36		17 36	31,93 29,78	25,59	25,53 32,94	32,36
	18 STRASBOURG	31			31		31		31	18,07		28,94
	20 PARIS NORD		25	25			25		25	21,90		42,20
	23 LILLE		38	38			38		38	18,14	22,31	21,53	30,36
	24 AMIENS		14	14			14		14	34,35		33,21
	30 PARIS ST-LAZARE 33 ROUEN		11 21		11 21		11 21		11 21	7,36 23,81 32,43	1,64 19,06 28,19
	34 PARIS RIVE GAUCHE		25		25	25			25	25,52		33,68
	36 RENNES		14		14	14			14	43,95		30,07
	37 NANTES		24		24	24			24	32,37		23,58
	43 BORDEAUX		51		51	51			51	34,83	34,51	21,67	25,52
	44 LIMOGES		17		17	17			17	39,21		23,47
	46 TOURS		19		19	19			19	33,25		32,79
	47 TOULOUSE		33		33	33			33	36,67		21,64
	50 PARIS SUD EST		24		24		24	24		35,84		31,54
	53 DIJON		30		30		30	30		37,16		30,77
	54 LYON		35		35		35	35		32,65		36,31
	56 CLERMONT-FERRAND		27		27		27	27		28,39	32,10	16,22	29,01
	57 CHAMBERY		31		31		31	31		29,21		26,29
	58 MARSEILLE		29		29		29	29		23,49		30,62
	59 MONTPELLIER		23		23		23	23		39,90		29,61
	NUMBER OF SEGMENTS	89	491	77	503	183	397	199	381	28,22	25,48
	MATRIX NAME RE	RNE	RN	RNN	RRG	RNRG	RSE	RNSE	R1	--	--	--

Table 5 . Testing geographic and sociological decompositions of matrix R 1

 5

Table 6 . Family structure in France and the Liberal/Authoritarian partition of SNCF regions

 6

*Source:

Journal of Economic Literature: C61, D61, D92, H54, R41, R48

The authors thank Cong-Liem Tran for computing assistance, Gérard Mathieu for comments, and are grateful to Société nationale des chemins de fer français (SNCF) for financial support and for allowing use of a database constructed by Pascaline Boyer.

They applied a single common BCT to all individual variables of a Trans-Log specification.

As these models contain no speed variable by train type, results by train type combines weight, speed and other effects of actual weights per axle, un-sprung mass and technical features absorbing the effect of gross axle weights.

Unfortunately, many previous applications of BCT to zero observations, such asCaves et al. (1980a[START_REF] Caves | Network effects and the measurement of returns to scale and density for U.S. railroads[END_REF], do not take these precautions and, as a result, estimate values of BCT dependent on the units of measurement of the (traffic) variables containing the zeroes. This fault makes their BCT form estimates uninteresting and unusable for readers.

As did a cumulative traffic variable used in the ad hoc road example mentioned above.

For details on the use of two BCT on the same variable of a regression, see[START_REF] Gaudry | Turning Box-Cox, including Quadratic Forms in Regression[END_REF] who show that the procedure makes it possible to model asymmetric turning forms as well as classical symmetric quadratic U shapes.

The elasticities found in Table2differ slightly from those found in the final model due to slight changes later made to some values of a particular regressor. This modification has no effect on spatial correlation test results.

Unfortunately, national studies produced by the CATRIN consortium[START_REF] Wheat | CATRIN (Cost Allocation of TRansport INfrastructure cost), Deliverable 8 -Rail Cost Allocation for Europe[END_REF] all tended to show that segment lengths are statistically significant, a result which may everywhere indicate an endogeneity of segment lengths.

The gain from -6824.94 in run 139 to -6817.76 in run 140 (not shown in Table2). In previous work on a 1999 database, using a Generalized Box-Cox specification (2-A) applied to (1-A) -with interactions among explanatory variables accounted for[START_REF] Gaudry | Rail track wear-and-tear costs by traffic class in France[END_REF]) -, we added 22 dummy variables, taking region 10 as the arbitrary reference: this improved the Log Likelihood by 24 points. The comparable exercise carried out here yields hardly any gain, which suggests again that second-wave specification (1-B), estimated on censored observations, is more realistic than first-wave specification (1-A).

In Sweden, as much as 80% of track maintenance costs are attributable to snow clearing[START_REF] Andersson | Marginal railway infrastructure cost estimates in the presence of unobserved effects[END_REF].

Rail Concept also tested the impact of tri-dimensional location (latitude, longitude, altitude), as well as that of frost, precipitations and relative humidity but they were not statistically significant.

This estimated coefficient was accepted, but other possible sources of correlation were rejected, notably the presence of neighbouring influences across region boundaries and that of shared train lines influencing infrastructure costs in physical ways unaccounted for in the reference model.

AJD-145 ----------

Within-region correlation among regression errors

The near-demise of segment length as an explanatory variable. Observations on maintenance expenses per segment result from administrative procedures assigning track work invoices to segments. A natural reaction to the existence of within-region correlation among residuals, positive or negative, is then to think that it reflects some systematic arbitrariness in the assignment of these amounts across track segments of any given region, a view partly sustained by noting that taking correlation into account in Column 4 of Table 2 reduces both the size of the coefficient of the segment length variable and its statistical significance to a low enough level expected from the start 16 . But even granting this, the positive sign of the 1  estimate still requires interpretation.

Substitution/complementarity interpretation.

A first candidate interpretation of that positive sign is that segments within the same region behave as substitutes. To see why 0   could result from the treatment of segments as substitutes, consider the following, based on applying Eq. (2-B)-(2-C) at the level of two observations, numbered 2 and 4, for which one writes:

which, after replacement of 2 v in (3-1), of 4 v by 44 yX   , and a rearrangement of terms, leads to:

If 0   , the impact of X 4 on y 2 in this last relationship will be in the opposite direction from that of X 2 , as occurs between substitute goods in problems where X 2 and X 4 denote their prices and the indices pertain not to observations but to quantities demanded, as given by distinct equations. Similarly, 0   will have the opposite effect, increasing the impact of a unit change in X k , as between complements if the X k again stand for prices. But this is only a simple behavioral analogy.

Missing variable interpretation.

An alternate interpretation comes from time-series analysis where positive serial correlation between residuals is often caused by the absence from regressor lists of variables, notably those standing for dynamic effects (Spanos, 1987(Spanos, -1988)): if a particular X k is missing and its values are correlated over time, then residual t v will contain a missing component

that easily generates serial correlation. One way to explore the nature of variables presumably missing here is to add to a reference model regional dummy variables representing locally missing factors and to probe their coefficients (each expressing a difference with the coefficient of the region selected as arbitrary reference) for meaningful identity clues.

On these lines, adding 18 dummy variables to the Column 4 specification (the remaining 5 pieshaped regions centered on downtown Paris then forming the Île-de-France reference region), increases the Log Likelihood by only 7 points 17 ; also, the t-statistics of the 18 new coefficients, shown in Table 3, obtain 13 positive and 5 negative values, the latter including the single case (59, Montpellier) of a statistically significant difference from the Île-de-France region coefficient,