Sophie Dabo-Niang
email: sophie.dabo@univ-lille3.fr

Mustapha Rachdi
email: mustapha.rachdi@upmf-grenoble.fr

Anne-Françoise Yao
email: anne-francoise.yao@univmed.fr

Kernel regression estimation for spatial functional random variables

Keywords: Regression estimation, Random fields, Functional variables, Infinite dimensional space, Small balls probabilities

Given a spatial random process (X i , Y i ) ∈ E × R, i ∈ Z N , we investigate a nonparametric estimate of the conditional expectation of the real random variable Y i given the functional random field X i valued in a semi-metric space E. The weak and strong consistencies of the estimate are shown and almost sure rates of convergence are given. Special attention is paid to apply the regression estimate introduced to spatial prediction problems.

Introduction

Nowadays, the progress of informatics tools permits the recuperation of increasingly bulky data. These large data sets are available essentially by real time monitoring and computers can manage such databases. The object of statistical study can then be curves (consecutive discrete recordings are aggregated and viewed as sampled values of a random curve) not numbers or vectors. Functional data analysis (see Bosq [START_REF] Bosq | Linear Processes in Function Spaces: Theory and applications[END_REF], Ferraty and Vieu [START_REF] Ferraty | Nonparametric Functional Data Analysis[END_REF], Ramsay and Silverman [START_REF] Ramsay | Applied functional data analysis; Methods and case studies[END_REF]) can help to analyze such high-dimensional data sets. The statistical problems involved in modeling functional random variables offer increasing interests in recent literature. For prediction problems, functional data analysis techniques outdo the other approaches because they take advantage of past information. See for example Bosq [START_REF] Bosq | Linear Processes in Function Spaces: Theory and applications[END_REF], Besse et al. [START_REF] Besse | Autoregressive forecasting of some functional climatic variations[END_REF], Cardot and Johannes [START_REF] Cardot | Thresholding projection estimators in functional linear models[END_REF], Fernandez de Castro et al. [17], Ferraty and Vieu [START_REF] Ferraty | Nonparametric Functional Data Analysis[END_REF], Dabo-Niang and Ferraty [START_REF] Dabo-Niang | Functional and Operatorial Statistics[END_REF], Dabo-Niang and Rhomari [START_REF] Dabo-Niang | Kernel regression estimate in a banach space[END_REF], Ferré and Yao [START_REF] Ferré | Smoothed functional inverse regression[END_REF], Rachdi and Vieu [START_REF] Rachdi | Nonparametric regression for functional data: automatic smoothing parametrer selection[END_REF] for some regression and forecasts results obtained by several functional models (linear, nonlinear, neural networks and semi-parametric models) for non spatial variables. It is proved in some of these papers that the best predictions were generally obtained by the functional methods (either autoregressive or functional kernel).

To the best of our knowledge, although potential applications of regression estimation (or prediction) to functional spatial data are without number, only the papers of Dabo-Niang and Yao [START_REF] Dabo-Niang | Kernel regression estimation for continuous spatial processes[END_REF], Delicado et al. [START_REF] Delicado | Point-wise kriging for spatail prediction of functional data[END_REF] and Nerini et al. [START_REF] Nerini | A cokriging method for spatial functional[END_REF], have paid attention to study regression or prediction for functional random fields.

The last two works deal with spatial kriging (linear predictor) methods for spatial functional data. Dabo-Niang and Yao [START_REF] Dabo-Niang | Kernel regression estimation for continuous spatial processes[END_REF] introduced a nonparametric predictor for functional random fields, for which the behavior has not been investigated. We want to go further and extend functional data nonparametric analysis techniques to the spatial domain. We suggest a nonparametric regression estimation approach which is to aggregate over space.That is, we are mainly concerned with kernel regression methods for functional random fields (spatial curves). Note that Lakasaci and Fouzia [START_REF] Lakasaci | Nonparametric estimation of conditional quantiles for functional and spatial dependent variables[END_REF] considered the case of conditional quantile estimation where the regressor take values in a semi-metric space and showed the strong and weak consistency of the conditional quantile.

Spatial regression estimation is an interesting and crucial problem in statistical inference for a number of applications (as in a variety of fields, including soil science, geology, oceanography, econometrics, epidemiology, environmental science, forestry,...), where the influence of a vector of covariates on some response variable is to be studied in a context of spatial dependence. The literature on parametric spatial modeling is relatively abundant and we refer to Chilès and P. [START_REF] Chilès | Geostatistics. Modeling spatial uncertainty[END_REF], Guyon [START_REF] Guyon | Random Fields on a Network -Modeling[END_REF], Anselin and Florax [START_REF] Anselin | New Directions in Spatial Econometrics[END_REF], Cressie [START_REF] Cressie | Statistics for Spatial Data[END_REF] or Ripley [START_REF] Ripley | Spatial Statistics[END_REF] for a list of references. However, the nonparametric treatment of functional or multivariate spatial data is limited. Actually, if some references on spatial nonparametric regression estimation in the multidimensional setting (key refererences are: Lu and Chen [START_REF] Lu | Spatial nonparametric regression estimation: Non-isotropic case[END_REF][START_REF] Lu | Spatial kernel regression estimation: Weak concistency[END_REF], Hallin et al. [START_REF] Hallin | Local linear spatial regression[END_REF], Hallin and Yu [START_REF] Hallin | Local linear spatial quantile regression[END_REF], Carbon et al. [START_REF] Carbon | Kernel regression estimation for random fields[END_REF], Dabo-Niang and Yao [START_REF] Dabo-Niang | Kernel regression estimation for continuous spatial processes[END_REF], Li and Tran [START_REF] Li | Nonparametric estimation of conditional expectation[END_REF],...) already exist, no work have been devoted to nonparametric regression for functionnal spatial data. The aim of this paper is to study the behavior of the functional spatial counterpart of the Nadaraya-Watson Nadaraya [START_REF] Nadaraya | On estimating regression[END_REF], Watson [START_REF] Watson | Smooth regression analysis[END_REF]'s estimator.

The paper is organized as follows. In Section 2, we provide the notations, assumptions and introduce a kernel estimate of the conditional mean. Section 3 is devoted to convergence in probability and strong convergence of the kernel estimate. To check the performance of our estimator, simulations results will be given in Section 4. Conclusion is given on Section 5 and Proofs and technical lemmas are given in the last section.

General setting

Let Z N (N ≥ 1), be the integer lattice points in the N -dimensional Euclidean space. A point in bold i = (i 1 , ..., i N ) ∈ Z N will be referred as a site. In fact, spatial data can be seen as realizations of a measurable strictly stationary spatial process Z i , defined on a probability space (Ω, A, P). In this paper we deal with process of the feature:

Z i = (X i , Y i ), i ∈ Z N such that: the Z ′ i s have the same distribution as a variable Z = (X, Y ),
where Y is a real-valued and integrable variable and X valued in a separable semi-metric space (E, d(., .)) (of eventually infinite dimension).

In the following, we will denote by µ the probability distribution of X and ∀ i, j by ν i,j the joint probability distribution of (X i , X j ), . will denote any norm over Z N , C an arbitrary constant and B(x, ρ) the opened ball of center x and radius ρ. We will write n → +∞ if min k=1,...,N n k → +∞ and we set n = n 1 × ... × n N and N N , will denotes subspace of Z N of vectors with non-negative components.

For a seek of simplicity, we will suppose that the variable Y is bounded. Instead of this condition one can assume (as in Tran [START_REF] Tran | Kernel density estimation on random fields[END_REF], Dabo-Niang and Yao [START_REF] Dabo-Niang | Kernel regression estimation for continuous spatial processes[END_REF] and Ferraty and Vieu [START_REF] Ferraty | Nonparametric Functional Data Analysis[END_REF]) that one of the following assumptions holds:

• For all i = j, E [Y i Y j |(X i , X j )] ≤ C for some constant C > 0. • For all m ≥ 2, E [Y m |X = x] ≤ g m (x) < ∞,
where g m is a continuous function at

x ∈ E.

The proofs given here remain valid in one of these above cases.

The spatial kernel regression estimator for functional data

We aim to estimate the regression function r

(x) = E (Y |X = x) of Y given X.
To do this, we propose the kernel estimate of the function r based on observations of the process (Z i ) in some region I n and without the lost of generality, we suppose that I n is a rectangular region

I n := i ∈ N N : 1 ≤ i k ≤ n k , k = 1, ..., N : r n (x) =    i∈In Y i W i,x if i∈In W i,x = 0; 1 b n i∈In Y i else.
where

W i,x = K (d(X i , x)h -1 n ) j∈In K (d(X j , x)h -1 n )
.

This estimate can also be written as follows:

r n (x) =    ϕ n (x)/f n (x) if i∈In W i,x = 0; 1 b n i∈In Y i else. with ϕ n (x) = 1 nE (K (d(X, x)h -1 n )) i∈In Y i K d(X i , x)h -1 n , x ∈ E, f n (x) = 1 nE (K (d(X, x)h -1 n )) i∈In K d(X i , x)h -1 n , x ∈ E,
where

lim n→+∞ h n = 0 (+) , the kernel K is a function from R + to R + .
That is an extension of the well known of Nadaraya [START_REF] Nadaraya | On estimating regression[END_REF], Watson [START_REF] Watson | Smooth regression analysis[END_REF]'s estimate introduced for i.i.d observations.

Since we are dealing with spatial data, as any spatial model, ours must take into account the dependance between observations. Let us now consider some measures of dependence.

Spatial dependence measures

As it often occurs in spatial dependent data analysis, one needs to defined the type of dependence. We consider here the following two dependence measures:

Local dependence condition

We assume also that for all i, j ∈ N N the joint probability distribution ν i,j of X i and

X j satisfies ∃ǫ 1 ∈ (0, 1], ν i,j (B(x, h n ) × B(x, h n )) = (p x hn ) 1+ǫ 1 , (2.1)
where p x hn = P (X ∈ B(x, h n )) = µ(B(x, h n )). We recall that ν i,j is the joint distribution of (X i , X j ).

Remark 1. Note that this condition leads to ν i,j (B(x,

h n ) × B(x, h n )) -p x hn 2 = (p x hn ) 2 (p x hn ) ǫ 1 -1 -1 ≥ 0 so ν i,j (B(x, h n ) × B(x, h n )) -p x hn 2 = (p x hn ) 2 (p x hn ) ǫ 1 -1 -1 ≤ (p x hn ) 1+ǫ 1 ≤ 1.
Then, it can be link with the classical local dependence condition met in the literature when X and (X i , X j ) admitted respectively the densities f and f i,j :

|f i,j (x, y) -f (x) f (y)| ≤ C,
for some constant C > 0 and for all x, y ∈ E and i, j ∈ N N , i = j.

Actually, this condition can be replace by the more general version (which is not necessary here):

∃ǫ 1 ∈ (0, 1], P ((X i , X j ) ∈ B(x, h n ) × B(y, h n )) = (p x hn p y hn ) 1+ǫ 1 2 ;
then, ν i,j (B(x, h n ) × B(y, h n ))p x hn p y hn = (p x hn p y hn ) (p x hn p y hn )

ǫ 1 -1 2 -1 ≤ (p x hn p y hn ) 1+ǫ 1 2 ≤ 1.
Such local dependency condition is necessary to reach the same rate of convergence as in the i.i.d. case.

Mixing conditions

Another complementary dependency condition concerned the mixing condition which measures the dependency by means of α-mixing. We assume that Z i , i ∈ Z N satisfies the following mixing condition: there exists a function ϕ (t) ↓ 0 as t → ∞, such that for E, E

′ subsets of Z N with finite cardinals,

α B (E) , B E ′ = sup B∈B(E), C∈B(E ′ ) |P (B ∩ C) -P (B) P (C)| ≤ χ Card (E) , Card E ′ ψ dist E, E ′ , (2.2)
where B (E)(resp. B E ′ ) denotes the Borel σ-field generated by [START_REF] Carbon | Kernel density estimation for random fields[END_REF][START_REF] Carbon | Kernel density estimation for random fields: The l 1 theory[END_REF] and Biau and Cadre [START_REF] Biau | Nonparametric spatial prediction[END_REF]. We refer to Doukhan [START_REF] Doukhan | Mixing -Properties and Examples[END_REF] and Rio [START_REF] Rio | Théorie Asymptotic des Processus Aléatoires Faiblement Dépendants[END_REF] for discussion on mixing and examples. Concerning the function ψ(.), we will only study the case where ψ(i) tends to zero at a polynomial rate, ie. ψ(i) ≤ Ci -θ , for some θ > 0.

(Z i , i ∈ E) (resp. Z i , i ∈ E ′ ), Card(E) (resp. Card E ′ ) the cardinality of E (resp. E ′ ), dist E,
(2.5)

In the following, we denote by:

θ 1 = -θ + N N (1 + 2 β) -θ , θ 2 = -θ + 2N 2N (β + 1) -θ , θ 3 = -θ + N N (1 + 2β + 2 β) -θ . θ * 1 = 2N -θ 4N -θ , θ * 2 = -θ + N N (2 β + 3) -θ , θ * 3 = -θ + 2N -θ + 2N (β + 2) , θ * 4 = -θ + N N (3 + 2β + 2 β) -θ .
Remark 2. The results obtained below can be extend to the case where ψ(i) tends to zero at an exponential rate: i.e ψ(i) = C exp(-si) for some s > 0.

• Each of the two dependence measures have the following specificity: if the first one control the local dependence, the second one control the dependence of sites which are far from each other.

• Clearly, one has, for a fixed h n (note that the same argument can be easily generalized to the case where one deals with two different bandwidths) :

α( i -j ) ≥ g i,j ∞ with g i,j (x, y) = ν i,j (B(x, h n ) × B(y, h n )) -p x hn p y hn .

Assumptions on the kernel

We assume that the kernel K : R → R + is of integral 1 and is such that:

HK 1 : there exist two constants 0 < C 1 < C 2 < ∞: C 1 I [0,1] ≤ K ≤ C 2 I [0,1] .
where

I [0,1] is the indicator function in [0, 1]. or
The support of K is [0, 1], the derivative K ′ of K exists and satisfies

-∞ < C 1 ≤ K ′ ≤ C 2 < 0 and -∃C > 0, ∃ε 0 > 0, ∀ε < ε 0 , ˆε 0 µ(B(x, z)) dz > Cεµ(B(x, ε)).
In some cases, we will assume that:

HK 2 : K is a Lipschitz function.

Main Results

This section is devoted to the study of the consistency of the regression function: first, locally at a given point x of E and secondly, uniformly in the set C.

Local convergence of the regression function

We study here the consistency of the regression function r at a given x ∈ E. In this intention, we will use the assumption:

HF 1 -The regression function r is continuous at x ∈ E,
and the following preliminary result (proved in the Appendix): 2.2) and HK 1 or HK 2 hold, then:

Lemma 3. Let G n = f n or ϕ n . If assumptions (2.1), (
lim n→∞ ( np x hn )var(G n (x)) < ∞, x ∈ E as soon as the condition (2.3) or (2.4) is satisfied with ∞ i=1 i N -1 (ϕ(i)) a < ∞, for some 0 < a < 1/2.
As consequences of Lemma 3 we have the two following theorems. Theorem 4. Under the assumptions HF 1 , HK 1 , (2.1), (2.2), (2.5), np x hn / log n → ∞ and if the mixing satisfies:

• conditions (2.3) and θ > 2N or • conditions (2.4), θ > N (1 + 2 β) and n(p x hn / log n) θ 1 → ∞ then, |r n (x) -r(x)| converges in probability to 0. (3.1)
The next results give the strong convergence of r n under additional conditions. We set g (2.5), np x hn / log n → ∞ and if the mixing verifies:

(n) = N i=1 (log n i )(log log n i ) 1+ǫ , then we have n∈N N 1/ ( n g(n)) < ∞. Theorem 5. Under the assumptions HF 1 , HK 1 , (2.1), (2.2),
• the conditions (2.3), θ > 4N and n p x hn log b n θ * 1 g(n) 2N 4N -θ 4N -θ 2N → ∞ or • the conditions (2.4), θ > N (3 + 2 β) and n p x hn log b n θ * 2 g(n) 2N N (2 e β+3)-θ N (2 e β+3)-θ 2N → ∞ then, |r n (x) -r(x)| converges almost surely to 0. (3.2)

Uniform convergence of the estimator over a set.

We consider a set C such that C ⊂ C n where C n = dn k=1 B(t k , ρ n ) (note that such set C n can always be built), d n > 0 is some integer, t k ∈ E, k = 1, ..., d n , and ρ n > 0. We assume that C is such that:

H 1 -sup x∈C p x hn = Γ(h n ) > 0 exists H 2 -d n = n β and ρ n ≤ (h n ) κ (log n/( nΓ(h n ))) 1/2 , β > 0κ > 1.
HF 2 -The regression function r is uniformly continuous on C. Under the dependence conditions (2.1), (2.2) and (2.5), we get the following weak and strong uniform consistency results:

Lemma 6. Let G n = f n or ϕ n . Under the condition of Lemma 3 and if H 1 is satisfied, we get lim n→∞ ( nΓ(h n ))var(G n (x)) < ∞, x ∈ C.
The two following theorems are consequences of Lemma 6.

Theorem 7. Under conditions HK 1 , HF 2 , and if

nΓ(h n )/ log n → ∞, H 1 , H 2 and
• the conditions (2.3), θ > 2N (β + 1) and n(Γ(h n )/ log n) θ 2 → ∞ or • the conditions (2.4), θ > N (1 + 2β + 2 β) and n(Γ(h n )/ log n) θ 3 → ∞ hold,
• the conditions (2.3), θ > 2N (β + 2) and n(Γ(h n )/ log n) θ * 3 (g(n)) 2N -θ+2N (β+2) → ∞ or • the conditions (2.4), θ > N (3+2β+2 β) and n(Γ(h n )/ log n) θ * 4 (g(n)) 2N -θ+N (2β+2 e β+3)
→ ∞ hold, we have

sup x∈C |r n (x) -r(x)| converges almost surely to 0. (3.4)
The rate of convergence of r n is given in the next theorems under the following additional conditions on the model.

HF 3 -We assume in the following theorem that r is a Lipschitz function.

Theorem 9. Under the conditions of Theorem 7 except that HK 1 and HF 2 are respectively replaced by HK 2 and HF 3 , we have

sup x∈C |r n (x) -r(x)| = O h n + log n Γ(h n ) n in probability. (3.5)
The strong rate of convergence of r n follows in the two cases of mixing.

Theorem 10. Under the conditions of Theorem 8 except that HK 1 and HF 2 are respectively replaced by HK 2 and HF 3 , we have:

sup x∈C |r n (x) -r(x)| = O h n + log n Γ(h n ) n a.s. (3.6)

Simulations

As it was raised ealier in this paper, if our estimator looks like its i.i.d.'s counterpart, there is a slight difference in practice. Indeed, as mentioned in Dabo-Niang et al. [START_REF] Dabo-Niang | Spatial kernel mode estimation for functional random, with application to bioturbation problem[END_REF], one must take into account the spatial dependency which is supposed here to be measured by mixing condition. In this Section, we are interested both by a way of applying our estimator illustred by some simulation tudies.

The mixing condition in practice.

We are dealing with a non parametric spatial dependence measure. We recall that for any couple of sites (i, j)

α (||i -j||) ≤ χ(1, 1) ψ (||i -j||) .
For the sake of simplicity, let us consider the strong mixing case (which corresponds to the case where χ ≡ 1). Then, we have assumed that ψ (||i -j||) tend to zero at a polynomial rate (or respectively at exponential rate). That is, we can say that ψ (||i -j||) ≤ C||i -j|| -θ (respectively ψ (||i -j||) ≤ exp(-θ||i -j||)) for some θ > 0. Thus, this assumption, might be taken into account when using the regression estimator given in Section 2. This argument leads us to say that actually, we are dealing with the following regression estimator.

The spatial regression estimator in practice.

For all x j (which could be observed a site j),

r n (x j ) =    i∈In Y i W i,x j if i∈In W i,x j = 0; 1 b n i∈In Y i I V j (i) else.
where

I V j is the indicator function of the set V j = ϕ (||i -j||) > C ||i -j|| -θ and W i,x j = K (d(X i , x j )h -1 n ) I V j (i) m∈In K (d(X m , x j )h -1 n ) I V j (m)
.

So we have:

r n (x j ) =      P i∈V j Y i K(d(X i ,x j )h -1 n ) P i∈V j K(d(X i ,x j )h -1 n ) if i∈In W i,x j = 0; 1 b n i∈In Y i I V j (i) else.
Note that V j is the set of Card(V j ) nearest neighbors of j.

Remark 11.

1. The choice of the bandwidth (even in finite or infinite dimensional setting) is a crucial question in non-parametric estimation. We propose to chose the optimal bandwidth by using cross-validation procedure. 2. Another interesting problem is the estimation of the sets V j 's. This problem is the subject of another work in progress. Nevertheless, for simplicity, we will consider that V j 's is a set of an arbitrary k n number of nearest neighbors (by meaning of the euclidean distance).

Let k n be an integer. Then, the regression function estimation at point x j is obtained by using the following algorithm:

Algorithm for spatial regression estimation based on nearest neighbors.

1. Compute the optimal bandwith, h kn,opt , by using cross-validation procedure.

2. Take the k n nearest neighboors of each site.

Compute respectively the of k

n 's reals K d(X j , X i ) h kn,opt and Y i K d(X j , X i ) h kn,opt , i ∈ V j : i∈V j K d(X j , X i ) h kn,opt , i∈In Y i W i,x j . 4. Compute r n (x j ).
This algorithm is illustrated in the following simulation studies. In the following, we will denote by Piid, the procedure of estimation of Ferraty and Vieu [START_REF] Ferraty | Nonparametric Functional Data Analysis[END_REF] and Dabo-Niang and Rhomari [START_REF] Dabo-Niang | Kernel regression estimate in a banach space[END_REF] and by Psdep, our procedure of estimation. Note that Psdep and Piid, coincide as soon as k n = n.

Simulations studies

In order to illustrate our results, we have done some simulations based on observations (X i,j , Y i,j ), 0 ≤ i, j ≤ 25 such that ∀ i, j, :

X i,j (t) = A i,j * (t -0.5) 2 + B i,j and
Y i,j = 4A 2 i,j + ε i,j , (4.1)
where A = (A i, j ), B = (B i, j ) and ε = (ε i,j ) are random variables which will be specified later on. Note that here we have r(X) = 4.X ′′ (where f ′′ denotes the second derivatives of a funtion f )). We are first (on Section 1.2.1.) interesting with the estimation of Model (4.1) based on i.i.d. observations Z i = (X i , Y i ) (the sequences A, B and ε are then i.i.d. random variables); after that we deal on Section 1.2.2, with Model (4.1) generated with the spatial dependence structure.

In each case, we have done 30 simulations of Model (4.1) and compared the quality of estimation of Piid and Psdep. The quality of estimation is meaning by coefficient of determination R 2 . The results are presented in Table 1 where each figure includes on one hand, 30 points (on red) representing the 30 values of the R 2 obtained by Piid. And on the other hand, the 30 curves defined by ((k n , R 2 (k n )) discretized in points k n = 8 + 5.ℓ, ℓ = 1, 2, ..., 29 obtained by Psdep.

Model (4.1) with i.i.d. observations

In this Section, Model (4.1) is simulated with i.i.d. observations. Namely, the sequences (A i,j ), (B i, j ) and (ε i,i ) are 25×25 i.i.d. random variables such that ∀i, j, A i,j ∼ N (0, 1) , B i,j ∼ N (0, .1) and ε i,j ∼ N (0, 2). We have done 30 simulations of this model, the results are presented in Table 1,Figure A.

These results shows that (as it is expected), procedure Piid leads to better estimation of Model (4.1) than Psdep. Futhermore, the quality of the estimation obtained by Psdep is improved as k n increases and tends to the quality of estimation of Piid. That is explained by the fact that as k n increases, one tends to the situation where Psdep and Piid coincide (k n = n).

With spatial dependency

This time, Model (4.1) is simulated with spatial dependence structure. Thereafter, we denote by GRF (m, σ 2 , s) a stationary Gaussian random field with mean m and covariance function defined by G = GRF (0, 5, 3) and

C(h) = σ 2 exp(-( h s ) 2 ), h ∈ R
D i = 1 n j exp -i-j a (D (i,j) = 1 25×25 1≤m,t≤25 exp -(i,j)-(m,t) a
). The function D is here to ensure and control the spatial mixing condition (even if using the Gaussian Random Fields also brings some spatial dependency). Indeed, our model can be seen verifying a mixing condition with ψ (h) → 0 at exponential rate. Then, the greater is a, the weaker is the spatial dependency. Futhermore, if a → ∞ , D i → 1. Now, let us respectively consider cases a = 50, 20, 5. The case a = 50 corresponds to the one we discuss just before since D i ≃ 1. The results are presented on Table 1-Figure B where, whatever the values of k n , one has a good quality of estimation both with Psdep and Piid and the values are almost equal. The fact that quality of estimation by Piid is as good (despite the existence of dependence) is explained by the high valued of a and the number of independent observations is then not negligeable. Actually, this later case corresponds to A ≃ sin( G 2 + .5) and Model (4.1) is based both on spatial dependent observations and nearly i.i.d. observations. In fact, since (in these conditions) Model , observations of sites i and j with ij < 15 are spatial dependent and nearly independent from ij ≥ 15. So, our observations are a mixture of i.i.d. and dependent observations. Thus, to move away from independence, it suffises to lower the value of a. That is done in the context of Table1, figures C and D respectively with a = 20 and a = 5. One can see that the quality of estimation of Piid deteriorate as a decreases and is very bad with a = 5.

Other interesting results are the evolution of quality estimation of Psdep in Table 1, figures B, C and D which are different from Figure 4.1.A. In fact, as one can see on Figure 4.2 there is an optimal k n around which the quality of estimation is better and quality is increasingly bad when away from this values and tends to that of Piid. These results are not visible in the previous figures the discretization is to coarse.

Conclusion and discussion

In this paper, we have developped a new method in non-parametric spatial modelling (for functional random fields). Then, when the observations are high-dimensional spatial data (as are curves), this method appears as a good alternative to existing ones.

More precisely, we have studied theoretically the asymptotic behavior of our method and, illustrate its pratical use through some finite size simulations. All this makes the proposal very attractive.

In addition, this work offers very interesting perspectives of investigation. In fact, as mentioned above, we have two main problems with our procedure of estimation: the choice of both the bandwith and k n . To solve this problem, we have chosen an optimal bandwith (using cross validation) for each fixed k n as shows Figure 5.1 (and of course tends to the optimal bandwidth of Piid as k n tends to n). Then, a question raised : " Does the results fundamentally change when choosing simultaneously k n and the optimal bandwith?" So, an outlook of this work is the statement of theoretical properties with respect to the choice of these two parameters using cross-validation method in functional random fields modelling (as it is in the i.i.d. setting by for exampleRachdi and Vieu [START_REF] Rachdi | Nonparametric regression for functional data: automatic smoothing parametrer selection[END_REF] for bandwidth selection).

Finally, this work is a step towards functional random fields models taking into account both the functional and spatial dependency feature of the data. The results obtained here are encouraging to pursue investigations in this topic. Namelly, in a work in progress, we aim to apply this method to apply spatial prediction and real data problem.

Appendix

This section is devoted to prove the consistency result stated in the previous sections. For that, we recall three lemmas which can be find to Carbon et al. [START_REF] Carbon | Kernel regression estimation for random fields[END_REF] which will be used in the following. As previously, along this section we will denote by C a positive generic constant.

Lemma 12. Suppose E 1 , ..., E r be sets containing m sites each with dist(E i , E j ) ≥ γ for all i = j where 1 ≤ i ≤ r and 1 ≤ j ≤ r. Suppose Z 1 , ..., Z r is a sequence of real-valued r.v.'s measurable with respect to B(E 1 ), ..., B(E r ) respectively, and

Z i takes values in [a, b].
Then there exists a sequence of independent r.v.'s Z * 1 , ..., Z * r independent of Z 1 , ..., Z r such that Z * i has the same distribution as Z i and satisfies

r i=1 E|Z i -Z * i | ≤ 2r(b -a)χ((r -1)m, m)ψ(γ) (6.1)
Lemma 13. (i) Suppose that (2.2) holds. Denote by L r (F) the class of F-measurable r.v.'s X sat- isfying X r = (E|X| r ) 1/r < ∞. Suppose X ∈ L r (B(E)) and X ∈ L r (B(E ′ )). Assume also that 1 ≤ r, s, t < ∞ and r -1 + s -1 + t -1 = 1. Then |EXY -EXEY | ≤ C X r Y s {χ(Card(E), Card(E ′ ))ψ(dist(E, E ′ ))} 1/t . (6.2)
(ii) For r.v.'s bounded with probability 1, the right-hand side of (6.2) can be replaced by

Cχ(Card(E), Card(E ′ ))ψ(dist(E, E ′ )). Lemma 14. If (2.5) holds for θ > 2N , then ∞ i=1 i N -1 (ψ(i)) a < ∞ (6.3)
for some 0 < a < 1/2.

Proofs of the results for the local convergence case.

Proof. of Lemma 3: Let

Z i, x = K i ∆ x K d (X i , x) h -1 n -E K i ∆ x K d (X i , x) h -1 n , S n = i∈In Z i, x ,
where

∆ x = E (K (d (X, x) h -1 n )), K i = 1 (for the case of f n ) or K i = Y i (for the case of ϕ n ). We have var(G n (x)) = var(S n / n), then var(G n (x)) ≤ n -2 i∈In E(Z 2 i,x ) + i∈In j∈In, i k =j k for some k |E(Z i,x Z j,x )| .
Let us first consider the case where

K i = 1 and set J 1,x,n = ( np x hn ) n -2 i∈In E(Z 2 i,x
) . Then, we have the inequality

J 1,x,n ≤ p x hn (∆ x ) 2 E(K 2 d (X, x) h -1 n )
which leads by assumption HK 1 to:

J 1,x,n ≤ C p x hn ∆ x E (K (d (X, x) h -1 n )) ∆ x = C p x hn ∆ x .
Furthermore, it is easy to see that 0 < C ≤ p x hn /∆ x ≤ C ′ (see for example Lemma 4.3 or 4.4 of Ferraty and Vieu [START_REF] Ferraty | Nonparametric Functional Data Analysis[END_REF]) where C and C ′ are two real constants, so:

lim n→∞ J 1,x,n < ∞.
To continue the proof, we set:

J 2,x,n = np x hn )   n -2   0<dist(i,j)≤mn E(Z i,x Z j,x ) + dist(i,j)>mn E(Z i,x Z j,x )     = J 1 2,x,n +J 2 2,x,n .
where m n is a positive real (depending on n). Now, since

E(Z i,x Z j,x ) ≤ C (∆ x ) 2 |P ((X i , X j ) ∈ B(x, h n ) × B(x, h n ))| , by assumptions HK 1 , where 0 < ǫ 1 < 1 one get: E(Z i,x Z j,x ) ≤ C (∆ x ) 2 (p x hn ) 1+ǫ 1 ≤ C(p x hn ) ǫ 1 -1
and by the way the inequality:

J 1 2,x,n ≤ C( np x hn ) nm N n n -2 (p x hn ) ǫ 1 -1 = Cm N n p x hn ǫ 1 .
Setting

m n = p x hn -(1-γ)ǫ 1 /ν , where ν = -N -ǫ 1 + (1 -γ)ǫ 1 N a -1 with γ and ǫ 1 some small positive numbers such that a -1 ǫ 1 -(N + ǫ 1 )(N (1 -γ)) -1 > 1 (this is possible since 0 < a < 1/2), we get: J 1 2,x,n ≤ C(p x hn ) 1-N (1-γ)/ν , thus lim n→∞ J 1 2,x,n = 0 since ν > N (1 -γ). Let us turn to J 2 2,x,n and let γ ′ = 1-(1-γ)ǫ 1 , δ = 2(1-γ ′ )/γ ′ .
Notice that γ ′ = 2/(2+δ) and 1γ ′ = δ/(2 + δ). We apply Lemma 2.1 of Tran [START_REF] Tran | Kernel density estimation on random fields[END_REF]with r = s = 2 + δ; h = (2 + δ)/δ and get the inequality:

|E (Z i,x Z j,x )| ≤ C 1 (∆ x ) 2+δ E 2+δ K d (X k , x) h -1 n γ ′ (ψ( i -j )) 1-γ ′ ;
which leads to:

J 2 2,x,n ≤ ( n -1 p x hn ) i-j >mn C 1 (∆ x ) 2+δ E 2+δ K d (X k , x) h -1 n γ ′ (ψ( i -j )) 1-γ ′ . ≤ C( n -1 p x hn )(p x hn ) -γ ′ (1+δ) i-j >mn (ψ( i -j )) 1-γ ′ ≤ C( n -1 p x hn )(p x hn ) -γ ′ (1+δ) n i >mn (ψ( i )) 1-γ ′ ≤ C(p x hn ) -1+γ ′ i >mn (ϕ( i )) 1-γ ′ . It comes from assumption ∞ i=1 i N -1 (ψ(i)) a < ∞ that ψ(i) = o(i -N/a
) and since ϕ is a decreasing function we have ϕ(t) = o(t -N/a ) as t → ∞ and

i ν (ψ( i )) 1-γ ′ = i ν o( i -N (1-γ ′ )/a ) = o( i -N -ǫ 1 ), because ν = -N -ǫ 1 + (1 -γ)ǫ 1 N a -1 > 0. Then, i∈In i ν (ψ( i )) 1-γ ′ < ∞. Furthermore (p x hn ) 1-γ ′ m -ν n = 1 so: lim sup J 2 2,x,n ≤ C lim sup (p x hn ) -1+γ ′ i >mn (ψ( i )) 1-γ ′ ≤ C lim sup(p x hn ) -1+γ ′ m -ν n i >mn i ν (ψ( i )) 1-γ ′ ≤ C lim sup i >mn i ν (ψ( i )) 1-γ ′ .
This last term tends to zero as m n tends to infinity. For the case K i = Y i the proof is the same as the previous one since Y is bounded. This yields the proof.

Proof. of Theorem 4.

We consider the decomposition:

r n (x)-r(x) = 1 f n (x) {ϕ n (x) -E(ϕ n (x)) -(r(x) -E(ϕ n (x)))}- r(x) f n (x) {f n (x) -1} , x ∈ E.
(6.4) Let:

I 1 (x) = ϕ n (x) -E(ϕ n (x)), I 2 (x) = r(x) -E(ϕ n (x)).
and

I 3 (x) = f n (x) -1.
Then, we have for a given k

∈ N N I 2 (x) = r(x) -E Y ∆ x K d (X, x) h -1 n = r(x) -E E Y k ∆ x K d (X k , x) h -1 n |X k = r(x) -E r(X k ) K (d (X k , x) h -1 n ) ∆ x = E (r(x) -r(X k )) K (d (X k , x) h -1 n ) ∆ x .
Since the support of the function K is [0, 1], we have:

r(x) -r(X k ) ≤ sup u∈B(x,hn) |r(x) -r(u)|.
So, I 2 (x) ≤ sup u∈B(x,hn) |r(x)r(u)| converges to zero by the continuity assumption of r at x.

We now focus on the convergence of I 1 (x). Note that the proof of I 3 (x) is derive from the one of I 1 (x) by setting Y i = 1.

Let :

Q n (x) = ϕ n (x) -E (ϕ n (x)) = i∈In Z i, n, x -E (Z i, n ) = i∈In Z ′ i, n, x , x ∈ E,
where

Z i, n, x = Y i ∆ i n , ∆ i = K (d (X i , x) h -1 n ) EK (d (X, x) h -1 n ) 20
and p ≥ 1 and integer. Without loss of generality assume that for some integers t 1 , ..., t N , n i = 2pt i , i = 1, ..., N.

(6.5)

The following spatial blocking idea here is that of Tran [START_REF] Tran | Kernel density estimation on random fields[END_REF]. If it is not the case that n i = 2t i p, i = 1, ..., N for some integers t 1 , ..., t N , then a term say T (n, x, 2 N + 1), containing all the Z i, n, x 's at the ends not included in the blocks above can be added, see also Biau and Cadre [START_REF] Biau | Nonparametric spatial prediction[END_REF]. The random variables Z ′ i, n, x are now into blocks of different sizes. Let

U (1, n, x, j) = (2j k +1)p i k =2j k p+1, 1≤k≤N Z ′ i, n, x , U (2, n, x, j) = (2j k +1)p i k =2j k p+1, 1≤k≤N -1 2(j N +1)p i N =(2j N +1)p+1 Z ′ i, n, x , U (3, n, x, j) = (2j k +1)p i k =2j k p+1, 1≤k≤N -2 2(j N -1 +1)p i N -1 =(2j N -1 +1)p+1 (2j N +1)p i N =2j N p+1 Z ′ i, n, x , U (4, n, x, j) = (2j k +1)p i k =2j k p+1, 1≤k≤N -2 2(j N -1 +1)p i N -1 =(2j N -1 +1)p+1 2(j N +1)p i N =(2j N +1)p+1 Z ′ i, n, x ,
and so on. Note that

U (2 N -1 , n, x, j) = 2(j k +1)p i k =(2j k +1)p+1, 1≤k≤N -1 (2j N +1)p i N =2j N p+1 Z ′ i, n, x .
Finally,

U (2 N , n, x, j) = 2(j k +1)p i k =(2j k +1)p+1, 1≤k≤N Z ′ i, n, x .
Setting T = {0, ..., t 1 -1} × ... × {0, ..., t N -1}, we define for each integer l = 1, ..., 2 N , T (n, x, l) = j∈T U (l, n, x, j).

Then we obtain the following decomposition for each l = 1, ..., N . Without loss of generality we will show (6.6) for l = 1.

Q n (x) = ϕ n (x) -Eϕ n (x) =
Let us prove that given an arbitrary large positive constant c, there exists a positive constant C such that for any η > 0

P |T (n, x, 1)| > η log n np x hn ≤ C( n -c + β 1b n )
where

β 1b n = χ( n, p N )ϕ(p)ǫ -1 n . Let T (n, x, 1) = j∈T U (1, n, x, j).
be the sum of

t = t 1 × ... × t N (6.7)
of the U (1, n, x, j)'s. Note that U (1, n, x, j) is measurable with respect to the σ-field generated by X i with i belonging to the set of sites

I i, j = {i : 2j k p + 1 ≤ i k ≤ (2j k + 1)p, k = 1, ..., N }
These sets of sites are separated by a distance greater than p. Enumerate the random variables's U (1, n, x, j) and the corresponding σ-field with which they are measurable in an arbitrary manner and refer to them respectively as V 1 , ..., V b t and B 1 , ..., B b t . By Lemma 6.1 of Carbon et al (1997), we approximate V 1 , ..., V b t by V * 1 , ..., V * b t . We have

|V i | = |U (1, n, x, j)| < Cp N n -1 . (6.8) Let ǫ n = η log b n b np x hn
where η > 0 is a constant to be chosen later.

Since we have T

(n, x, 1) = b t i=1 V i , then P [|T (n, x, 1)| > ǫ n ] ≤ P   b t i=1 V * i > ǫ n /2   + P   b t i=1 |V i -V * i | > ǫ n /2   . (6.9)
By Markov's inequality and using (6.8), (6.1) and recall that the sets of sites with respect to which V i 's are measurable are separated by a distance greater than p, we get

P   b t i=1 |V i -V * i | > ǫ n   ≤ C tp N n -1 χ( n, p N )ψ(p)ǫ -1 n ∼ β 1b n . ( 6
lim n→∞ np x hn (R n(x) + U n(x) ) < C (6.13)
where C is a positive constant and

U n(x) = i∈In E(Z i,n,x ) 2 R n(x) = i∈In l∈In, i k =l k for some k |Cov(Z i,n,x , Z l,n,x )|.
By (6.3) and (6.13), we have

λ 2 n b t i=1 E(V * i ) 2 ≤ C np x hn (U n(x) + R n(x) ) log n < C log n.
Using (6.8), we get |λ n V * i | < 1/2 for large n. We deduce from Berstein's inequality that

P   b t i=1 V * i > ǫ n   ≤ 2 exp(-λ n ǫ n + λ 2 n b t i=1 E(V * i ) 2 ) ≤ 2 exp((-η + C) log n) ≤ n -c
(6.14) for sufficiently large n. We deduce from (6.9), (6.10) and (6.14) that

P [|T (n, x, 1)| > ǫ n ] ≤ C( n -c + β 1b n )
To complete the proof, we will show that β 1b n → 0. Recall that

β 1b n = χ( n, p N )ψ(p)ǫ -1 n . (6.15) Furthermore, condition (2.3) allows to show that θ > 2N is equivalent to (β 1b n ) -1 → ∞.
Actually, we have

β 1b n ≤ Cχ( n, p N )p -θ ǫ -1 n ≤ C np x hn log n (1/2)-(θ/2N ) ( np x hn / log n) 1 2 = C np x hn log n -θ+2N 2N
Analogously to the condition (2.3), we have under (2.4):

β 1b n ≤ Cψ( n, p N )p -θ ǫ -1 n ≤ C n e β np x hn log n (1/2)-(θ/2N ) = C n(p x hn / log n) θ 1 -θ+N (2 e β+1) 2N . Then, condition n(p x hn / log n) θ 1 → ∞ leads to β 1b n → 0.
Proof. of Theorem 5 Under (2.3), we get (see the proof of Theorem 4):

β 1b n ≤ C np x hn log n -θ+2N 2N , then β 1b n ng(n) ≤ n p x hn log n θ * 1 g(n) 2N 4N -θ 4N -θ 2N
.

Analogously to (2.3), we have under (2.4): → ∞ implies that β 1b n ng(n) → 0, this last yields that n∈N N β 1b n < ∞ and the proof is complete by Borel-Cantelli's Lemma.

Proofs of results of uniform convergence case.

Proof. of Lemma 6: It is similar to that of Lemma 3, it suffices to replace p x hn by Γ(h n ).

To show theorems 7-10, we consider the fact that the set C is covered by d n balls B k = B(x k , ρ n ) of radius ρ n and center at x k and we define:

S 1n = max

Proof. of Theorem 7

We have:

P [sup x∈C |T (n, x, 1)| > ǫ n ] ≤ Cd n ( n -c + β 1b n ),
where p x hn is replaced by Γ(h n ) in the expressions of ǫ n and β 1b n of the previous proof. To complete the proof, we will show that d n n -c → 0 and d n β 1b n → 0. We have

d n n -c ≤ C n β-c
which goes to zero if c > β. We get the following inequality, under (2.3): .

d n β 1b n ≤ C n β nΓ(h n ) log n -θ+2N 2N = n(Γ(h n )/ log n) θ 2 -θ+2N (
(6.17)

This yields the proof.

Proof. of Theorem 8

We have under (2.3):

d n β 1b n ≤ C n β nΓ(h n ) log n -θ+2N 2N
, Then,

d n β 1b n ng(n) ≤ C n Γ(h n ) log n θ * 3 g(n) 2N N (2β+4)-θ N (2β+4)-θ 2N
.

In the case where (2.4) is satisfied, we get: The proof is therefore complete.

d n β 1b n ng(n) ≤ C n Γ(h n ) log n θ *

2

 2 and s > 0. Then, we have then simulated Model (4.1) with A = D * sin( G 2 + .5), B = GRF (2.5, 5, 3), ε = GRF (0, .1, spatial dependency,with a=20 D. With spatial dependency,with a=5

Figure 4 . 1 :

 41 Figure 4.1: Covariance function with σ 2 = 5 and s = 5

  (4.1) is based on Gaussian random fields with covariance function C and scale s ≤ 5 (see

Figure 4 . 2 :

 42 Figure 4.2: kn versus R 2 with a = 20

Figure 4 . 1 )

 41 Figure 4.1), observations of sites i and j with ij < 15 are spatial dependent and nearly independent from ij ≥ 15. So, our observations are a mixture of i.i.d. and dependent observations. Thus, to move away from independence, it suffises to lower the value of a. That is done in the context of Table1, figures C and D respectively with a = 20 and a = 5. One can see that the quality of estimation of Piid deteriorate as a decreases and is very bad with a = 5.

25 A

 25 realization of the random field A A realiszation of the ramdom field B

Figure 4 .

 4 Figure 4.3: Simulation

Figure 5 . 1 :

 51 Figure 5.1: k n versus h kn,opt with a = 20: the value of hopt for Piid is 0.113

2 N l=1 T

 2l=1 (n, x, l).To prove that Q n (x) = O log b n b np x hnin probability or a.s., it is sufficient to show that T (n, x, l)

  n (x)ϕ n (x k )|, S 2n = max 1≤k≤dn sup x∈B k |Eϕ n (x k ) -Eϕ n (x)|, S 3n = max 1≤k≤dn |ϕ n (x k ) -Eϕ n (x k )|.Then,sup x∈C |ϕ n (x) -Eϕ n (x)| ≤ S 1n + S 2n + S 3n .It easy to see that underH 2 and HK 2 , S 1n and S 2n are equal to o log b n Γ(hn)b n in probability (or almost surely). It remains to show that S 3n = O log b n Γ(hn)b n in probability (resp. almost surely) which is equivalent to show that max 1≤j≤dn |T (n, x j , 1)| = O log b n Γ(hn)b n probability (resp. almost surely).

16 )

 16 Analogously to (2.3), we have under (2.4):d n β 1b n ≤ C n β n e β nΓ(h n ) log n (1/2)-(θ/2N ) = n(Γ(h n )/ log n) θ 3 -θ+N (2β+2 e β+1) 2N

4 g.

 4 Then, condition n(Γ(hn )/ log n) θ * 3 (g(n)) 2N -θ+2N (β+2) → ∞ or n(Γ(h n )/ log n) θ * 4 (g(n)) 2N -θ+N (2β+2 e β+3)→ ∞ is equivalent to ng(n)β 1b n → 0 which implies n∈N N β 1b n < ∞, then the theorem follows by Borel-Cantelli Lemma.Proof. of Theorem 9According to Theorem 7, we have:sup x∈C |ϕ n (x) -E(ϕ n (x))| = O log n Γ(h n ) n , in probability.andsup x∈C |f n (x) -1| = O log n Γ(h n ) n , in probability.We get also by HF 3 (see the proof of Theorem 4):sup x∈C |r(x) -E(ϕ n (x))| ≤ sup x∈C sup u∈B(x,hn) |r(x)r(u)| = O (h n ) .Proof. of Theorem 10According to Theorem 8 and HF 3 , we have:sup x∈C |ϕ n (x) -E(ϕ n (x))| = sup x∈C |f n (x) -1| = O log n Γ(h n ) n , a.s and sup x∈C |r(x) -E(ϕ n (x))| ≤ sup x∈C sup u∈B(x,hn) |r(x)r(u)| = O (h n ) .

  E

				′
	the Euclidean distance between E and E	′ and χ : N 2 → R + is a nondecreasing symmet-
	ric positive function in each variable. Throughout the paper, it will be assumed that χ
	satisfies either
	χ (n, m) ≤ C min (n, m) , ∀n, m ∈ N	(2.3)
	or
	χ (n, m) ≤ C (n + m + 1) β , e	∀n, m ∈ N	(2.4)
	for some β ≥ 1 and some C > 0. If χ ≡ 1, then Z i is called strongly mixing. Many
	stochastic processes, among them various useful time series models satisfy strong mixing
	properties, which are relatively easy to check . Conditions (2.3)-(2.4) are weaker than
	strong mixing condition and have been used for finite dimensional variables in (for exam-
	ple) Tran [33], Carbon et al.

Table 1 :

 1 Values of the coefficient of determination R 2

  It is clear that, λ n ǫ n = η log n.

								.10)
	Let
		λ n = np x hn log n	1/2 ,			(6.11)
	p =	np x hn 4λ n	1/N	∼	np x hn log n	1/2N	.	(6.12)
	If (2.5) holds for θ > 2N , we have (by Lemma 3)


Giuseppe Munda
email: giuseppe.munda@uab.es

Michaela Saisana
email: michaela.saisana@jrc.it

Methodological Considerations on Regional Sustainability Assessment based on Multicriteria and Sensitivity Analysis

Keywords: regional sustainability, composite indicators Regional Sustainability, Composite Indicators, Multi-Criteria Evaluation, Sensitivity Analysis JEL Classification Numbers: A12, C43, Q01, R11

This paper proposes the use of a non-compensatory multicriteria approach combined with sensitivity analysis for constructing composite indicators of sustainability. An illustrative example on Spanish and selected Mediterranean regions is used. The sensitivity analysis shows that excluding an indicator from a 29-indicator dataset (which represents in principle a small structural change) has a much lower impact on the regional ranking if that is based on a non-compensatory multicriteria approach than on the classical linear aggregation, for example the weighted arithmetic average. An alternative approach that employs endogenous weighting (region-specific weights) and is based on data envelopment analysis is discussed.

Introduction

The world population increase and the rapid development of economic activity are the main causes of the environmental tensions that exist in all socio-economic systems.

Problems such as the greenhouse effect and climatic change, the depletion of the ozone layer, acid rain, the loss of biodiversity, and the pollution and depletion of renewable and nonrenewable natural resources are clear symptoms of a possible environmental unsustainability (e.g. [START_REF] Allen | Supply-side sustainability[END_REF][START_REF] Barbier | The conditions for achieving environmentally sustainable growth[END_REF][START_REF] Yahe | The economic geography of the impacts of climate change[END_REF].

Awareness of the real and potential conflicts between economic growth and the environment has paved the way for the concept of sustainable development, which is highly attractive in part because unlike Daly's idea of "zero growth" [START_REF] Daly | Steady-State economics[END_REF], it does not pit economic growth against environmental conservation, rather it defends the idea of harmonisation between or simultaneous realisation of economic growth and environmental concerns. The objective of "green accounting" is to furnish information on the sustainability of the economy, but there is no established doctrine on how the different, and at times even contradictory, variables and indicators are to be combined so that they are made immediately useful for policy making in the same way that GDP or other macroeconomic statistics are.

How can the destruction of resources be accounted for if they are not inventoried or if there are no property rights? How can monetary values be included along with intangible impacts (due to trade or international externalities such as exports of carbon dioxide) beyond borders? Setting aside monetary values, how can physical indicators be included? Different physical indicators can show contradictory tendencies, and as a result "aggregating" them (in order to classify the situation) becomes difficult due to the subjective nature of the problem.

Our starting point is the broadly accepted notion that sustainability is a multidimensional concept. In the majority of research proposals or projects aimed at developing sustainability indicators, this multidimensional character is present, although it is not always included in the same way nor with the same interpretation (e.g., Faucheux and O'Connor, 1998;[START_REF] Gibbs | Prospects for Environmental Economic Geography: Linking Ecological Modernization and Regulationist Approaches[END_REF][START_REF] Musu | National accounts and the environment[END_REF][START_REF] Pearce | Measuring sustainable development: progress on indicators[END_REF].

It is precisely the existence of multiple dimensions, along with that of their multiple interrelationships, that explains the difficult task of analysing sustainability. Taken as a whole, there is no generally accepted way of framing the analysis within which a study of sustainability should be performed. However, the common feature of all the proposals reviewed is the three dimensional structure of sustainability encompassing social, economic/institutional and environmental aspects. A second important issue, still controversial in the literature, is how to reduce or synthesise the number of relevant indicators. Moreover, the existence of different levels and scales at which a hierarchical system can be analyzed implies the unavoidable existence of non-equivalent descriptions of it [START_REF] Giampietro | Using hierarchy theory to explore the concept of sustainable development[END_REF].

By understanding, sustainable development has a global dimension. However, the existence of mutual interaction between local and global processes is also increasingly recognised. In particular, regions and cities are open systems impacting on all other areas and on the earth as a whole. In this paper, we adopt the geographical scale of a regional level; thus its peculiarities have to be considered (see e.g. [START_REF] Munda | Social multi-criteria evaluation for urban sustainability policies[END_REF][START_REF] Nijkamp | Decision Support Models for Regional Sustainable Development[END_REF]Giaotzi, 1993, Wallis et al., 2007).. We propose a methodological framework based on a non-linear/non-compensatory multicriteria approach [START_REF] Munda | Multi-Criteria Decision Analysis and Sustainable Development[END_REF][START_REF] Munda | Non-compensatory/non-linear composite indicators for ranking countries: a defensible setting[END_REF] and combined with sensitivity analysis [START_REF] Saisana | Uncertainty and sensitivity techniques as tools for the analysis and validation of composite indicators[END_REF][START_REF] Saltelli | Sensitivity analysis, Probability and Statistics series[END_REF][START_REF] Munda | Social multi-criteria evaluation (SMCE)": methodological foundations and operational consequences[END_REF]2008). The approach we present here can be used as a policy guide to:

• identify priority sustainability issues;

• determine where current policies are producing good results or where they are insufficient;

• provide a baseline for cross-region performance comparisons;

• identify leaders and laggards on an issue-by-issue basis; and

• identify best practices and successful policy models.

Since the importance of combining sensitivity analysis with the non-compensatory approach can be best gauged by means of numerical examples, we start by explaining in Section 2 how dimensions and indicators have been chosen in a hypothetical framework of sustainability developed for Spanish and selected Greek and Italian regions. Section 3 describes methodological issues of the nonlinear/noncompensatory multicriteria ranking system and sketches upon an alternative approach that employs data envelopment analysis. Section 4 presents the regional rankings obtained by means of the multicriteria approach. The impact of excluding an indicator at-a-time from the framework under the non-compensatory multicriteria approach is assessed and confronted with that of the classical linear (and compensatory) aggregation. The ranking produced by the multicriteria approach is finally compared to the ranking produced by Data Envelopment Analysis. Section 5 summarises the main conclusions of the study. We start from the assumption that on-the-ground conditions are the ultimate gauge of sustainability performance, thus measurable outcomes that can be linked to policy targets and, in principle, tracked over time are needed.

An illustrative example

In our study, we will use a hypothetical example to illustrate that even few statistical approaches that are versatile and entail different methodological assumptions can gauge the technical quality of a ranking system. The illustrative framework is built around the three main dimensions of sustainability -environment, society, economy-and has been based on 29 indicators from the regional database REGIO of Eurostat and the Spanish National Statistical Office. The choice of the indicators was based on: (a) relevance to the issue of sustainability, (b) reliable measurement data, (c) few or no missing values (2004, or nearest year) for the Spanish regions and selected Greek and Italian regions that have similar climatic and economic conditions with the Spanish regions. Table 1 presents the list of the indicators considered in the illustrative example. The full data set for the Spanish regions, together with the relevant data sources can be found in the Annex.

Table 1: Dimensions, Indicators and Optimal performance

We discuss next a few directional issues for some of the indicators, so as to make explicit the assumptions made in this analysis. Agricultural land use has less impact on the environment than urbanized area, thereafter the higher the indicator value the better the region's performance from a sustainability point of view. In fact, the building industry in Spain has been the most important source of negative environmental impact in recent years. Municipal waste collected is used as an indicator of overall waste production and should thus be minimised. Aging population is expressed as percentage on the total population, thus it is considered as a minimization problem since high percentages of aging population may be considered a socio-economic source of un-sustainability in most European countries (as a result of the low birth rates and not of a longer life of people, which of course is a positive feature). Population density has to be minimized in the current context since it is considered as a quality of life indicator. As an ecological indicator, on the contrary, it should be maximized. [START_REF] Munda | Multicriteria evaluation in a fuzzy environment[END_REF][START_REF] Munda | Social multi-criteria evaluation for a sustainable economy[END_REF].

F

With this analogy in mind, [START_REF] Munda | Non-compensatory/non-linear composite indicators for ranking countries: a defensible setting[END_REF] present an aggregation convention for (nonlinear and noncompensatory) composite indicators able to rank different countries (or regions, cities and so on). We will use this approach in the present paper. The discrete multicriteria problem is based on the information included in the impact matrix of N regions (or alternatives in general) × M indicators (or criteria in general). The type of information needed to solve the so-called multi-criterion problem is:

• Intensity of preference (when quantitative indicator scores are present).

• Number of indicators in favour of a given region.

• Weight attached to each indicator.

• Relationship of each region with all the other regions.

Combinations of these pieces of information generate different aggregation conventions, i.e.

manipulation rules of the available information to arrive at a preference structure. The aggregation of several indicators implies taking a position on the fundamental issue of compensability [START_REF] Bouyssou | Some remarks on the notion of compensation in MCDM[END_REF][START_REF] Bouyssou | Noncompensatory and generalized noncompensatory preference structures[END_REF] ). Summing up, an additive aggregation implies full compensability among the indicators, whilst a multiplicative aggregation entails partial compensability.

In this paper, we employ a fully noncompensatory aggregation rule [START_REF] Munda | Non-compensatory/non-linear composite indicators for ranking countries: a defensible setting[END_REF] that is not assuming either additive or multiplicative aggregation and can be synthesised as follows. Given a set of individual indicators

{ } M m g G m ,..., 2 , 1 , = = and a finite set { } N n a A n ,..., 2 , 1 , = =
of regions, let's assume that the indicator value of each region n a with respect to an indicator m g is based on an ordinal, interval or ratio scale of measurement. For simplicity of exposition, we assume that a higher indicator value is preferred to a lower one, that is:

   = ⇔ > ⇔ ) ( ) ( ) ( ) ( a g a g a I a a g a g a P a k m j m k j k m j m k j (2)
where P and I indicate a preference or an indifference relation respectively, both fulfilling the transitive property. Let's also assume further the existence of a set of indicator weights

{ } ∑ = = = = M m m m w M m w W 1 1 with ,..., 2 , 1 ,
that are derived as importance coefficients. The mathematical problem to be dealt with is how to use the available information to rank in a complete pre-order (i.e. without any incomparability relation) all the regions from best to worst. The mathematical aggregation convention proposed can be divided into two main steps:

1. Pair-wise comparison of regions according to the whole set of indicators used. An N N × matrix, E, called outranking matrix [START_REF] Arrow | Social choice and multicriterion decision making[END_REF]) can be built, in which any generic element, k j e jk ≠ , is the result of the pair-wise comparison, according to all the M indicators, between regions j and k . Such a global pair-wise comparison is obtained by means of equation ( 2) and the value of the jk e is calculated by:

1 1 ( ) ( ) 2 M jk m m jk jk m P I e w w =   = +     ∑ (3)
Property ( 4), although obvious, is very important since it allows us to consider the outranking matrix E as a voting matrix i.e., a matrix where instead of using indicators, alternatives are compared by means of voters' preferences (with the principle one agent one vote). This analogy between a multi-criterion problem and a social choice one, as noted by [START_REF] Arrow | Social choice and multicriterion decision making[END_REF], is very useful for tackling the step of ranking the N regions in a consistent axiomatic framework.

The maximum likelihood principle selects as a final ranking the one with the maximum pair-wise support, which also involves the minimum number of pair-wise inversions. The

∈ = ≠ ! ... , 2 , 1 , (5)
The final ranking ( * r ) is the one which maximises equation ( 6), which is: (6)

An alternative weighting and aggregation scheme based on Data Envelopment

Analysis

In absence of reliable information about the true weights to be attached to the underlying indicators of sustainability, one can endogenously assign region-specific weights that maximize the composite indicator score for a given region using the Data Envelopment Analysis (DEA) method and a distance to best performer scaling [START_REF] Melyn | Towards a synthetic indicator of macroeconomic performance: Unequal weighting when limited information is available[END_REF][START_REF] Cherchye | Legitimately diverse, yet comparable: On synthesising social inclusion performance in the EU[END_REF]. This gives the following linear programming problem for each region n in the dataset: 2004) for a survey). We preferred to attach restrictions on the shares (instead of the weights), because shares (i) do not depend on the measurement unit of the indicators and (ii) directly reveal the contribution of an indicator to the composite indicator score [START_REF] Cherchye | Studies 35 and Robustness Analysis: the case of the Technology Achievement Index[END_REF]Wong and Beasley, 1990). Formally, the m -th share for a region n is given as the product with m L and m U the respective lower and upper bounds for the pie-shares.

Results and Sensitivity Analysis

In this section we discuss the results of the multicriteria ranking system and assess the impact on the benchmarking results of three main assumptions in the development of the regional composite indicator of sustainability: (a) number of indicators included in the framework, (b) aggregation function, (c) weights assigned to the indicators.

Results using the noncompensatory multicriteria approach

In our example, we assume that all indicators within each dimension receive equal weights and that all three dimensions are equally weighted, too. While each region has unique socio-economic and geographic characteristics, environmental policy priorities and development goals, cross-regional comparisons between different countries can nevertheless yield useful insights. To this end, we selected 4 Italian and 4

Greek regions that are similar to the Spanish regions regarding socio-economic development, climate, land area, and population density. Table 3 shows the ranking of the Spanish regions before and after the inclusion of the Italian and Greek regions.

Among the twenty-five regions studied, Lombardy (the Italian region which hosts Milan) performs best in overall sustainability. Madrid and Catalonia follow, with Tuscany (IT) It is interesting to note that once Italian and Greek regions enter in the non-compensatory ranking system the order of the Spanish regions is only slightly affected. Catalonia and Navarra shift between 2 nd and 3 rd rank, and Aragon and Murcia shift between 7 th and 9 th rank. This result can be in part explained by the fact that the non-compensatory algorithm depends on the set of regions compared. As a consequence, this ranking procedure may not always respect the axiom of independence of irrelevant alternatives [START_REF] Arrow | Social choice and individual values, 2d edition[END_REF]. However, the literature suggests that a Condorcet consistent rule, such as the one we used here, has the lowest probability of occurrence of a rank reversal compared to any of the Borda consistent rules [START_REF] Moulin | Axioms of co-operative decision making[END_REF][START_REF] Young | Condorcet's theory of voting[END_REF]. A further explanation of why the relative position of the Spanish regions changes when the Italian and Greek regions are introduced in the analysis is related to the different number of indicators available for the Greek and Italian regions ( 22indicators) compared to the 29 indicators considered for the Spanish regions.

No particular pattern is revealed between the population density in the regions and their level of sustainability. In fact, the most populated region, Attiki (GR), ranks 20 th , whilst the least populated region, Castilla la Mancha, ranks 17 th . This result makes it clear that population density is not determining sustainability. This argument is of course linked to the validity of the theoretical framework and the selection of the indicators, but as discussed in Section 2, the case study is illustrative. However, the argument whether population density undermines sustainability can be in principle studied in a similar way as proposed here, provided that a peer-reviewed theoretical framework has been chosen. Every aggregate measure or ranking system has a subjective nature related, for example, to the selection of indicators, the choice of aggregation rule, or the weights attached to the indicators. Because the quality of a ranking system depends on the soundness of its assumptions, good practice requires assessment of the uncertainties in the development process. By acknowledging a variety of methodological assumptions that are intrinsic to policy research, sensitivity analysis can determine whether the main results of a ranking system change substantially when those assumptions are varied over a reasonable range of possibilities [START_REF] Saisana | Uncertainty and sensitivity techniques as tools for the analysis and validation of composite indicators[END_REF], Saltelli et al. 2008;[START_REF] Saisana | Composite Learning Index: Robustness Issues and Critical Assessment, EUR 23274 EN[END_REF][START_REF] Brand | Comparative Analysis of Alcohol Control Policies in 30 Countries[END_REF]).

The validity of the ranking system developed here is assessed by studying its sensitivity To begin with, we compare the impact on the ranking of excluding a single indicator from the framework and using either a linear/compensatory or a non-linear/noncompensatory multicriteria aggregation rule, while maintaining equal weights for the indicators within each dimension and equal weights for the three main dimensions.

Normalisation is not needed in case of the multi-criteria approach (only ordinal information is used), while a min-max scaling in (0, 1) was undertaken prior to the linear aggregation.

There were thus 30 scenarios analysed for each type of aggregation, one with the entire set of twenty-nine indicators, and 29 sets of twenty-eight indicators each. Table 4 provides statistics for the regions rank range, i.e. the difference between the worst and the best case scenario, in either the linear or the multicriteria approach. An interesting feature revealed by Table 4 is the sensitivity of the linear-based ranking system to the exclusion of a single indicator. Instead, the results based on the non-compensatory multicriteria approach are much more stable. The significant impact of such a small structural change on the regional ranking based on the linear aggregation rule is due to the compensation effects among indicators. In fact, in the linear system only 2 regions are not sensitive to the exclusion of a single indicator (shift ≤ 2 positions), whilst 20 regions shift more than (≥) 5 positions. On the contrary, in the nonlinear/noncompensatory multicriteria system 10 regions are not sensitive, whilst only 5 regions (as opposed to twenty under the linear-based system) shift more than 5 positions in the overall 25-rank classification. To complement these results, Figure 1 and Catalonia. The wide rank range for Veneto and Catalonia is due to several indicators, whilst only two indicators influence the rank of Tuscany and Sicily. To be more specific, Sicily's rank is sensitive to "Infant mortality rate" and "Patent Application", while Tuscany's rank is sensitive to "Employment" and "Foreigners with tertiary education". In the linear ranking system, the regions that present the widest rank range (more than 10 positions) are Attiki (GR), Sicily (IT) and Canary Islands. These results have shown that the nonlinear/noncompensatory ranking system is robust to small changes in the indicators' set (exclusion of one indicator at-a-time in a 29-indicator dataset), which provides a further argument in favour of a regional sustainability ranking system based on a noncompensatory multicriterion approach.

Table 4: Statistics for the rank range (difference between best and worst case scenario)

of the twenty five regions studied.

Linear Aggregation of Indicators

We finally study the impact on the ranking of the weights to be assigned to the indicators. We employ data envelopment analysis (DEA) as discussed in Section 3.2 [START_REF] Charnes | Measuring the efficiency of decision making units[END_REF][START_REF] Charnes | Preface to Topics in Data Envelopment Analysis[END_REF] which uses linear optimisation rules to calculate region-specific weights for the indicators, accepting that there is no (expert) consensus on the appropriate set of weights for the indicators. Moreover, several authors have argued that differential weighting may be desirable in composite indicators, e.g.

because of different environments or political attitudes in different countries or regions (e.g. [START_REF] Veenhoven | Happy Life Expectancy: A comprehensive measure of quality-of life in nations[END_REF] or because the very idea of imposing weights may be inconsistent with the subsidiarity principle [START_REF] Cherchye | Legitimately diverse, yet comparable: On synthesising social inclusion performance in the EU[END_REF]. Basically, such worries are then overcome by rendering the weight selection problem endogenous for each observation. That is, the relative weight assigned to each indicator is endogenously determined in this type of performance evaluation models, so as to reflect the associated relative performance for the region under evaluation [START_REF] Melyn | Towards a synthetic indicator of macroeconomic performance: Unequal weighting when limited information is available[END_REF]. In practice, endogenous weighting is attaching the higher weights to the indicators that show the best performance for a given region; thus if this region is still occupying a low position in the ranking, it is possible to state that this poor performance is reliable.

Figure 4 plots the multicriteria based ranking versus the DEA-based ranking. In our DEA application we require that the relative share of each indicator (i.e. product of indicator value and the respective weight) is between 3% and 20% of the total aggregate score. We added these constraints to avoid allowing regions to achieve a high score simply by assigning zero weight to those indicators for which they have low performance, or by assigning an unreasonably high weight to a single indicator (Cherchye et al., 2007;[START_REF] Brand | Comparative Analysis of Alcohol Control Policies in 30 Countries[END_REF].

The rank order correlation coefficient between the two rankings is slightly lower than before, r = 0.564. The four Italian regions are those that are most affected by the DEA system, and 17 placed in a much lower rank than their multicriteria equivalent. Again, given the resemblance of the data envelopment analysis to a linear aggregation system, the impact on the regions ranks of excluding a single indicator from the dataset when using DEA is pronounced and similar to the one presented in Figure 3. In our opinion, sensitivity analysis helps to gauge the robustness of the results obtained, to increase the transparency of the ranking system, to identify the regions that improve or decline under certain assumptions, and to help the framing of the debate around the use of a conceptual framework.

Correlation with the GDP

Having assessed the sensitivity of the regional sustainability ranking system to methodological assumptions, we finally come to tackle the question whether sustainability must necessarily be sacrificed to achieve economic success. The possible compatibility between economic growth and environmental protection is the core assumption of the socalled environmental Kuznets curve hypothesis (e.g., [START_REF] Andreoni | The simple analytics of the environmental Kuznets curve[END_REF][START_REF] Arrow | Economic growth, carrying capacity, and the environment[END_REF][START_REF] Deacon | Does the environmental Kuznets curve describe how individual countries behave?[END_REF][START_REF] Plassmann | Preferences, technology, and the environment: Understanding the environmental Kuznets curve hypothesis[END_REF]. Figure 5 shows that there is a statistically significant and high correlation between GDP per capita and the regional sustainability rank (r = -0.834). The higher a region's GDP per capita, the more sustainable it is (of course by using our data and thus with all the limitations we already pointed out). Nevertheless, at every income level there is variation in the ranks, which is greatest at the lower income level. For example, Aragon does far better than St. Ellada (GR) at a similar level of income 20,000 EUR/cap. The most economically developed regions with GDP per capita above 20,000 EUR consistently score in the top ten of regional

Conclusions

The study aimed primarily to present a methodological framework for assessing regional sustainability and benchmarking relative performance of the regions considered. In a realm plagued by uncertainty and often dominated by rhetoric rather than systematic analysis, this paper aims at showing how data-driven policymaking at regional level might enable movement towards a more fact-based, empirical, and analytically rigorous approach to sustainability.

The application proposed here is based on a nonlinear/noncompensatory multicriteria approach of the Condorcet type applied to an illustrative example of 29 indicators grouped in three dimensions: Environment, Society, and Economy. The sensitivity analysis results show that impact on the ranks of the exclusion of an indicator, which represents a small structural change in a 29-indicator dataset, is significantly lower in the case of the noncompensatory approach than in the case of a linear aggregation or data envelopment analysis. Thus, we can be reasonably confident in the robustness of the multicriteria rankings and the indication they provide about which regions are performing well in response to the challenges of pursuing sustainability objectives. Analysis of the results obtained and underlying data reveal a number of key points:

• Despite some data shortcomings and the conceptual complexity of bringing the range of issues that fall under the sustainability rubric into a single ranking system, this application shows that sustainability performance can be tracked in rigorously and quantitatively.

• The cross-region comparisons provide a useful way to identify leaders, laggards, and best practices on an issue-by-issue and aggregate basis. Every region lags in performance on some issues on which it can learn from the success of peer regions Our analysis has shown that, at least in the Mediterranean regions studied, there might be a simultaneous realisation of economic growth and sustainability concerns encompassing all three dimensions of sustainability (economy, environment, society). Furthermore, no particular pattern is revealed between the population density and the level of sustainability in the 25 Mediterranean regions studied, which may support the argument that population density is not necessarily undermining sustainability. According to the data we used, it seems that: economic growth tends to alleviate sustainability problems once a region's per capita income exceeds 20,000 EUR. But of course this result has to be taken with a lot of prudence.

The results obtained depend heavily on the problem's structuring phase [START_REF] Munda | Social multi-criteria evaluation (SMCE)": methodological foundations and operational consequences[END_REF]). In the application presented here, main delicate issues are:

1. Quality of the information available (in our case the REGIO database of Eurostat was used).

2. Choice of indicators (i.e. which representation of reality we are using considering that a set of indicators is simply a descriptive model of it. In our case the choice of the indicators served for illustration purposes only).

3. Direction of each indicator (i.e. the higher the better or vice versa. This choice is not always obvious).

4. Relative importance of the indicators (in our case region-specific weights estimated by data envelopment analysis were also used), 5. Ranking method used (in our case linear aggregation or fully noncompensatory rules).

This paper has discussed methodological issues on the development of a regional ranking system of sustainability and thus it is not meant to suggest real-world policy lessons. When composite indicators are used for policy consumption, the structuring process should heavily be based on social preferences, thus public participation becomes an essential ingredient (see e.g. [START_REF] Munda | Social multi-criteria evaluation for a sustainable economy[END_REF]. Finally it is important to remember that there is no "optimal level of sustainability", thus precise benchmarks cannot be set. This limits the interpretation of findings for the relative performance of regions. In fact even regions in the bottom part of the ranking might have satisfactory levels of sustainability in absolute terms or vice versa top performing regions might indeed be very far from sustainability in absolute ideal reference values.

F

4

 4 Quantitative measurement is needed to create a context for sound policy-making. Indicators that permit cross-region comparisons provide a further foundation for evaluating results, benchmarking performance and clarifying what might be achieved in particular circumstances. In this paper, we use an illustrative numerical example to demonstrate how few statistical approaches that are versatile and deal with different

  of regions in a complete pre-order.

.

  adaptation of the maximum likelihood ranking procedure to the ranking problem we are dealing with is reasonably simple. The maximum likelihood ranking of regions is the ranking supported by the maximum number of individual indicators for each pair-wise comparison, summed over all pairs of regions considered. More formally, all the ) Call R the set of all ! N possible complete rankings of regions,

  In this basic programming problem, the weights are non-negative and a region's score is between 0 (worst) and 1 (best). The non-negativity restriction on the weights, however, allows for extreme scenarios. If a region has a value in a given indicator that dominates the values of other regions, this region would always obtain a score of 1.0 even if it has very low other indicators. Furthermore, it may lead to a situation where a large number of regions score 1.0, rendering the benchmarking exercise meaningless. Therefore, some additional constraints on the weights are needed, as recommended by several DEA supporters (seeThanassoulis et al. (

  the sum of the shares equals the n CI . The constraints placed on the shares are thus expressed as:

  4 th place. In general, three of the four Italian regions we analysed are among the top eight. Sicily (IT) performs lower than the other three Italian regions, although it has a middle rank (14 th ) in the overall sustainability ranking. All four Greek regions, including Attiki (the most urbanised region of Greece which hosts Athens, recognised as the "business capital" of the country), rank 19 th or lower.

  to three main sources of uncertainty (or decisions): (a) number of indicators in the dataset, (b) aggregation rule, and (c) weights of the indicators.

Figure 2

 2 Figure2present the median, best and worst rank across the 30 scenarios in the multicriteria

Figure 1 :Figure 2 :

 12 Figure 1: Multi-criteria based ranking

Figure 3 :

 3 Figure 3: Non-Compensatory Multi-Criteria Aggregation (MCA) of Indicators v.

Figure 4

 4 Figure 4: Non-Compensatory Multi-Criteria Aggregation (MCA) of Indicators v. Data Envelopment Analysis (DEA) Aggregation of Indicators

  , St. Ellada (GR) is close to this turning point but its sustainability performance is one of the lowest in the dataset.

Figure

  Figure 5: Non-Compensatory Multi-Criteria Aggregation (MCA) of Indicators v. GDP per capita

  or in other Mediterranean countries (e.g., Italy and Greece in our illustrative application).

Figure 2 :

 2 Figure 2: Additive (linear) based ranking * * Black marks correspond to the median of the simulated ranks. Whiskers show best and worst rank across 30 scenarios produced either by considering all twenty-nine indicators, or by excluding one indicator at-a-time.

The Measurement Framework 3.1 A Regional Sustainability Composite Indicator Based on Non-compensatory Multicriteria Evaluation

	indicator (Condorcet, 1785). When various individual indicators are used to compare the
	overall performance of two regions, some indicators are in favour of one region and some of
	the other region. As a consequence a conflict among the indicators exists. How can this
	conflict be treated in the light of a nonlinear/noncompensatory logic? This is the classical
	multicriteria discrete problem
		Although various functional forms for the aggregation of indicators into a composite
	o r indicator (here the term composite indicator is used as synonymous of index) have been developed in the literature (e.g. Journal of Economic and Social Measurement, 2002; Saisana CI for a given country F o and Tarantola, 2002), in the standard practice, a composite indicator n n, can be considered a weighted linear aggregation function applied to a set of m r
	(	m	=	, 1	,..., 2	M	P ) normalised variables: P
	e e r y is usually a scale adjusted variable (e.g. GDP per capita) normalized between ∑ = M m mn m y w 1 (1) where mn = n CI e e r
	R e v i e w w a weight attached to mn zero and one, and m y , with ∑ = = M m m w 1 1 and Munda and Nardo (2005) analyse the formal axioms behind linear aggregation (e.g., the 1 0 ≤ ≤ m w . weighted arithmetic average) and its operational implications and conclude that the use of nonlinear aggregation rules to construct composite indicators is compulsory for reasons of theoretical consistency when weights have the meaning of importance coefficients (i.e. the R e v i e w
	O n l y higher the weight the more important the individual indicator) or when the assumption of preferential independence among indicators does not hold. Moreover, in case of linear O n aggregation, compensability among the underlying indicators is always assumed, which implies complete substitutability among the indicators considered. For example, in a composite indicator of sustainability, economic growth can always substitute any l y
	environmental destruction or within a given dimension, e.g., the environmental dimension,
	clean air can compensate for a loss of potable water (Munda, 2005). From a normative point
	of view, such a complete compensability is often not desirable (see Munda, 2008 for a
	thorough discussion). For all these reasons, here we use a nonlinear/noncompensatory
	Condorcet consistent aggregation rule for computing the regional sustainability composite
							7

6

3.

  Balearic islands. The lowest five ranked regions are Asturias, Andalucia, Castilla la Mancha, Extremadura and Galicia. Mid-ranked performers include the remaining seven regions -Pais Vasco, Murcia, Valencia, Aragon, Cantabria, Castilla y Leon and Canary Islands. The geographic pattern of sustainability, as measured by the proposed framework, confronts the more sustainable north-east Spanish regions, with the less sustainable south and west regions.It is interesting to note that the top performing regions do not necessarily have top performance in all three dimensions. In fact, Catalonia has middle performance in two dimensions (Environment and Society), whilst the Balearic Islands have middle performance in Economy and low performance in Society. On the other hand, the bottom-five performing regions do not necessarily have the lowest performance in all three dimensions. To make an

	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	12

Table

2

presents the overall noncompensatory ranking and the ranking for each of the three sustainability dimensions.

Among the 17 Spanish regions, the top five are Madrid, Navarra, Catalonia, Rioja and the example, Extremadura has a top-five performance in Society, while Andalucia and Castilla la Mancha have middle performance in two dimensions (Environment and Society). For the middle-rank regions, performance is medium in all three sub-dimensions for Aragon and Cantabria. Exceptionally, Murcia, despite its top-five performance in two dimensions (Environment and Society), it is ranked among the mid-performers. The opposite is noticed for three regions -Castilla y Leon, Canary Islands and Asturias.

Table 2 : Rank of the Spanish regions in sustainability and its three dimensions based on the non-compensatory multicriteria approach

 2

Table 3 : Sustainability Ranking of the Spanish and Selected Italian and Greek Regions based on the non-compensatory multicriteria approach 4.2 Sensitivity Analysis Results

 3

	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	14

Table 1 : Dimensions, Indicators and Optimal performance Dimensions Indicators Optimal performance

 1

		Agricultural area		max
		Forest area		max
		Distances driven by trucks		min
		Municipal waste collected		min
	Environment	o r Forest area affected by fires Non-differentiated urban waste Differentiated urban waste		min min max
		Cement (consumption &sales) P Abstraction of total fresh water by public water supply	min min
		Investments in waste water collection and treatment facilities e Population affected by diseases of the respiratory system e r Aging population	max min min
	Society	Infant mortality rate Population density Crude birth rate Population affected by alcoholic abuse R e v i w Population affected by mental and behavioural disorders e Number of hospital beds Number of physicians /doctors	min min max min min max max
		Participation in General Elections Population in prison Employment Gross Domestic Product Households with minimum subsidy or no income	max min max max l O n y min
	Economy	Foreigners with tertiary education Life long learning		max max
		Population employed in Hi-Tech		max
		Patent application to the EPO		max
		Total intramural R&D expenditure (GERD)		max
		21

Table 2 : Rank of the Spanish regions in sustainability and its three dimensions based on the non-compensatory multicriteria approach

 2

		Overall	Environment	Society	Economy
		Rank	Rank	Rank	Rank
	Madrid	1	5	4	1
	Navarra	2	1	1	2
	Catalonia	3	7	7	3
	Rioja	4	3	5	4
	Balearic islands	5	2	14	9
	Pais Vasco	6	6	8	5
	Murcia	7	4	3	11
	Valencia	8	17	6	6
	Aragon	9	10	12	7
	Cantabria	10	9	10	12
	Castilla y Leon	11	15	13	8
	Canary Islands	12	13	15	10
	Asturias	13	14	17	14
	Andalucia	14	11	9	16
	Castilla la Mancha	15	12	11	15
	Extremadura	16	8	2	17
	Galicia	17	16	16	13

Table 3 : Sustainability Ranking of the Spanish and Selected Italian and Greek Regions based on the non-compensatory multicriteria approach

 3

	All Regions studied	Only Spanish Regions
		Rank	Rank
	Lombardy (IT)	1
	Madrid	2	1
	Catalonia *	3	3
	Tuscany (IT)	4
	Navarra *	5	2
	Rioja	6	4
	Balearic Islands	7	5
	Veneto (IT)	8
	Pais Vasco	9	6
	Aragon **	10	9
	Valencia	11	8
	Murcia **	12	7
	Cantabria	13	10
	Sicily (IT)	14
	Castilla y Leon	15	11
	Andalucia	16	14
	Castilla la Mancha	17	15
	Canary Islands	18	12
	Kriti (GR)	19
	Attiki (GR)	20
	Asturias	21	13
	Extremadura	22	16
	Galicia	23	17
	St. Ellada (GR)	24
	Thessalia (GR)	25
	Note the rank reverse in the cases of :
	* Catalonia and Navarra (shift between 2nd and 3rd rank), and
	**Aragon and Murcia (shift between 7th and 9th rank).

Table 4 : Statistics for the rank range (difference between best and worst case scenario) of the twenty five regions studied.

 4 Black marks correspond to the median of the simulated ranks. Whiskers show best and worst rank across 30 scenarios produced either by considering all twenty-nine indicators, or by excluding one indicator at-a-time.

		Multi-criteria	Additive (linear)
	Minimum	0 (St. Ellada, Galicia, Thessalia) 2 (Galicia, Navarra)
	Average	3	7
	Maximum	10 (Tuscany)	14 (Canary Islands)
	Standard deviation	2.4	3.0
	Less than (≤) 2 positions	10	2
	More than (≥) 5 positions 5	20

*

  Eurostat, Database: REGIO, http://ec.europa.eu/eurostat/ (B) Instituto de Estadística, Com. de Madrid, http://www.madrid.org/iestadis/fijas/otros/estructu.htm (*) Direction of indicator: 1 indicates that higher values are desirable, -1 indicates the opposite 1 Population aged over 65y (% total population). Eurostat, most recent data: 2003 2 Infant mortality rate. Eurostat, most recent data: 2000 Population affected by mental and behavioural disorders (crude death rate). Eurostat, most recent data: 2002 (GR, IT), 2003 (ES) 8 Population affected by alcoholic abuse, including alcoholic psychosis (crude death rate). Eurostat, most recent data: 2002 (GR, IT), 2003 (ES) 9 Participation in General Elections (%). Instituto de Estadística, most recent data: 2000 10 Population in prison (per 100.000 inhabitants). Instituto de Estadística, most recent data: 2004 General and Regional Statistics -Regions, http://ec.europa.eu/eurostat/ (B) Instituto de Estadística, Com. de Madrid, http://www.madrid.org/iestadis/fijas/otros/estructu.htm (*) Direction of indicator: 1 indicates that higher values are desirable, -1 indicates the opposite 1 Employment (% total active population). Eurostat, most recent data: 2003 2 GDP (€ per capita). Eurostat, most recent data: 2003 3 Households with minimum subsidy or with no income (% total). Instituto de Estadística, most recent data: 1998 4 Foreigners with tertiary education (% total population). Eurostat, most recent data: 2003 5 Life long learning. Eurostat, most recent data: 2004 6 Population employed in Hi-Tech (%). Eurostat, most recent data: 2004 7 Patent application to the EPO (per mio inhabitants). Eurostat, most recent data: 2003 8 Total intramural R&D expenditure -GERD (%GDP). Eurostat, most recent data: 2003 Thanks are due to the fBBVA foundation.

	per capita EVALUATION MATRIX WITH INDICATOR SCORES, DATA SOURCES AND YEAR F o r P e e r R e v i e w O n l y Annex Dimension: Environment Agricultural area 1 Forest area 2 Distances driven by trucks 3 Municipal waste collected 4 Forest area affected by fires 5 Non-differentiated urban waste 6 Differentiated urban waste 7 Cement (consumption &sales) 8 Abstraction of total fresh water by public water supply 9 Investments in waste water collection and treatment facilities 10 Direction of indicator ( * ) 1 1 -1 -1 -1 -1 1 -1 -1 1 Galicia 30.15 59.69 1,412,161 1,568 0.55 191.9 5.6 1,031 12.652 33.185 Asturias 32.62 42.13 888,031 626 1.16 0.5 23.7 822 11.225 NaN Cantabria 38.92 53.17 733,166 225 1.48 117.3 15.2 930 14.013 NaN Pais Vasco 33.51 53.88 2,798,799 1,014 0.11 4.6 40.4 654 26.967 111.502 Navarra 59.64 29.89 682,716 329 0.00 1.2 20.8 1,239 11.779 36.918 Rioja 54.37 27.79 421,038 150 0.01 3.1 17.3 1,362 14.024 36.111 Aragón 50.67 27.73 1,966,061 786 0.01 38.6 17.2 1,013 14.560 8.016 Madrid 43.12 24.28 1,456,527 2,875 0.01 0.5 13.9 700 9.061 42.291 Castilla y Leon 54.42 28.77 6,112,246 1,064 0.25 117.7 10.8 1,231 11.218 4.511 Castilla la Mancha 59.55 25.51 3,111,736 829 0.01 167.3 6.3 1,115 10.406 16.383 Extremadura 53.64 38.38 998,282 531 0.06 31.1 3.9 1,050 12.315 NaN Catalonia 36.86 44.06 8,556,978 3,652 0.05 1.3 23.7 1,003 8.058 1.127 Valencia 35.67 48.05 5,009,795 2,458 0.03 219.7 10.7 1,449 9.259 26.924 Balearic Islands 45.57 32.72 21,879 690 0.02 186.9 16.2 812 12.187 153.311 Andalucia 55.09 29.35 4,578,740 5,031 0.08 21.2 8.0 1,324 9.343 46.209 Murcia 55.54 24.33 679,370 635 0.02 122.1 10.6 1,794 7.523 157.105 F o r P e e r R e v i e w O n l y 8 Consumption and sales of cement (kg per capita). Instituto de Estadística, most recent data: 2005 9 Abstraction of total fresh water (ground+surface) by public supply (mio m³/yr per 100,000 inhabitants). Eurostat, most recent data: 1998 10 Total investments in waste water collection and treatment facilities (public + private sectors) (Mio € per 100,000 inhabitants). Eurostat, most recent data: 1998 11 Diseases include those of the respiratory system (J00-J99) and chronic lower respiratory diseases (J40-J47) (per 100.000 inhabitants). Eurostat, most recent data: varying Dimension: Society Aging population 1 Infant mortality rate 2 Population density 3 Crude birth rate 4 Number of hospital beds 5 Number of physicians /doctors 6 Population affected by mental and behavioural disorders 7 Population affected by alcoholic abuse 8 Participation in General Elections 9 Population in prison 10 Direction of indicator ( * ) -1 -1 -1 1 1 1 -1 -1 1 -1 Galicia 21.13 4.2 91.4 7.6 358.8 262.9 32.4 1.3 69.48 164 Principado de Asturias 21.89 3.4 100.0 6.8 376.2 387.0 48.3 1.3 69.33 122 Cantabria 18.99 2.1 101.9 9.2 388.1 222.6 30.0 0.2 73.32 134 Pais Vasco 18.18 3.1 289.1 9.3 389.5 368.7 33.5 0.2 64.48 63 Comunidad Foral de Navarra 17.86 4.0 54.7 10.9 406.0 548.7 21.6 0.9 67.60 28 La Rioja 19.09 3.8 56.5 10.1 320.9 390.2 24.0 0.4 75.43 142 Aragón 21.15 6.0 25.6 9.3 417.7 485.2 38.8 0.2 72.19 196 Comunidad de Madrid 14.51 4.1 702.5 12.0 335.7 322.5 19.4 0.4 73.33 133 Castilla y León 22.62 4.0 26.1 7.7 421.5 362.2 28.2 0.9 74.37 258 Castilla-la Mancha 19.40 4.0 22.7 10.1 278.5 201.1 30.9 1.1 77.01 115 Extremadura 19.01 4.9 25.6 9.3 366.4 397.1 18.2 1.3 76.67 105 Cataluña 17.12 3.5 204.4 11.5 466.4 317.0 46.7 0.6 64.73 124 Comunidad Valenciana 16.17 3.5 186.7 11.0 272.3 319.8 26.7 0.6 73.38 131 Illes Balears 14.09 4.7 184.1 11.4 399.2 274.0 23.8 NaN 61.91 149 Andalucia 14.63 5.0 85.6 11.7 281.9 329.0 17.6 0.7 69.76 165 Región de Murcia 14.15 6.8 110.4 13.0 318.3 438.2 22.0 0.7 74.48 67 Canarias (ES) 11.91 6.4 247.6 10.2 448.1 282.0 18.1 0.9 61.64 159 F o r P e e r R e v i e w O n 7 Page 32 of 37 Population affected by diseases of the respiratory system 11 -1 185.5 207.4 155.9 135.3 130.7 144.6 160.8 122.1 169.4 166.7 154.1 136.6 148.5 114.0 125.6 134.9 Regional Studies F o r P e e r R e v i e w O n Dimension: Economy Employment 1 Gross Domestic Product 2 Households with minimum subsidy or no income 3 Foreigners with tertiary education 4 Life long learning 5 Population employed in Hi-Tech 6 Patent application to the EPO 7 Total intramural R&D expenditure (GERD) 8 Direction of indicator ( * ) 1 1 -1 1 1 1 1 1 Galicia 37.41 14,619 3.54 0.070 88.9 0.40 2.10 0.69 Principado de Asturias 36.41 15,843 1.24 0.055 22.6 0.37 6.08 0.68 Cantabria 41.45 17,986 3.68 0.075 8.0 0.42 8.12 0.55 Pais Vasco 46.71 23,028 6.08 0.096 80.5 0.44 19.05 1.34 Comunidad Foral de Navarra 52.07 23,481 2.21 0.159 12.6 0.45 35.03 1.03 La Rioja 47.47 20,464 1.16 0.197 3.2 0.45 23.76 0.46 Aragón 46.93 19,841 2.93 0.083 27.0 0.44 16.57 0.70 Comunidad de Madrid 49.21 24,584 2.51 0.193 122.5 0.48 17.17 1.73 Castilla y León 41.25 17,217 2.08 0.141 85.0 0.39 9.78 0.80 Castilla-la Mancha 39.18 14,513 2.57 0.067 45.5 0.39 5.13 0.32 Extremadura 34.00 12,173 3.64 0.050 18.4 0.34 2.91 0.59 Cataluña 48.50 22,415 2.58 0.129 109.4 0.47 35.06 1.11 Comunidad Valenciana 42.40 17,517 3.29 0.142 165.5 0.45 14.15 0.70 Illes Balears 46.06 21,290 1.83 0.137 29.4 0.49 5.69 0.23 Andalucia 34.66 14,135 5.29 0.153 183.4 0.36 6.86 0.61 Región de Murcia 39.32 15,694 3.44 0.068 39.3 0.43 7.11 0.64 Canarias (ES) 39.38 17,371 0.00 0.105 71.0 0.43 4.59 0.51 Sources: (A) Eurostat-F o r P e e r R e v i e w O n F o r P e e r R e v i e w O n F o r P e e r R e v i e w O n F o r P e e r R e v i e w O n Sources: (A) Regional Studies l y l y l y l y l y l y
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3 Population density. Eurostat, most recent data: 2003 4 Crude birth rate. Eurostat, most recent data: 2003 5 Number of hospital beds (per 100.000 inhabitants). Eurostat, most recent data: 2000 (GR), 2002 (ES, IT) 6 Number of physicians/doctors (per 100.000 inhabitants). Eurostat, most recent data: 2001 (GR), 2003 (ES, IT)

Page 31 of 37 http://mc.manuscriptcentral.com/cres Email: regional.studies@fm.ru.nl http://mc.manuscriptcentral.com/cres Email: regional.studies@fm.ru.nl 34 ACKNOWLEDGEMENTS:

Q0 -General < Q -Agricultural and Natural Resource Economics, R11 -Regional Economic Activity: Growth, Development, and Changes < R1 -General Regional Economics < R -Urban, Rural, and Regional Economics