A New Tree Distance Metric for Structural

Comparison of Sequences

Matthias Gallé

Symbiose Project IRISA/INRIA Rennes-Bretagne Atlantique, France Abstract. In this paper we consider structural comparison of sequences, that is, to compare sequences not by their content but by their structure. We focus on the case where this structure can be defined by a tree and propose a new tree distance metric that capture structural similarity. This metric satisfies nonnegativity, identity, symmetry and the triangle inequality. We give algorithms to compute this metric and validate it by using it as a distance function for a clustering process of slightly modified copies of trees, outperforming an existing measure.

Motivation

Comparison of sequences is a recurrent task in computer science related applications, and had been a striking force in interdisciplinary fields like bioinformatics. Most of the existing comparison methods are based on the content and the sequential composition of these sequences. However, those measures do not apply well for comparing different sequences that may have a similar structure.

In this paper we consider the problem of comparing sequences by their structure rather than by their content. We suppose that the structure of a sequence is represented by an ordered tree whose leaves read from left to right yields the sequence. We do not consider how to obtain this structure, a great challenge itself [START_REF] Jr | Three great challenges for half-century-old computer science[END_REF], but on how to compare it, if it can be expressed in the form of a tree.

Our interest in comparing trees that represent the structure of a sequence is motivated from the use of straight-line context free grammars. This are special grammars that generate only one sequence. Several algorithms exists that, taken a string as input, tries to find a "good" grammar, where "good" is generally defined to be as smallest as possible (the problem of finding a grammar of minimal size that generates a given string is NP-Hard [START_REF] Charikar | The smallest grammar problem[END_REF]). Interest in this algorithms comes from their compression capacity [START_REF] Kieffer | Grammar-based codes: a new class of universal lossless source codes[END_REF][START_REF] Apostolico | Off-line compression by greedy textual substitution[END_REF] and their potential to be used in structure discovery [START_REF] Nevill-Manning | Inferring Sequential Structure[END_REF]. However, results in structure discovery using this tools has been very limited and mostly picked up by eye. We think that this is partly due to that no meaningful measure exists that can be used in an automatic fashion to capture similarity.

Comparing trees is however a recurrent task in fields like natural language processing. One of the main problems in this area is, given a sequence of letters, words or part-of-speech tags, to automatically find the correct syntax tree. This is then compared to a manually annotated, "correct" tree, the gold standard. Despite of criticism [START_REF] Sampson | A proposal for improving the measurement of parse accuracy[END_REF], the Parseval metric [START_REF] Abney | Procedure for quantitatively comparing the syntactic coverage of english grammars[END_REF], or an unlabeled variant of it [9, Section 2.2], seems still to be the standard metric. This measure seems to work for natural-language processing, where the sentence are not very long, and a small change in the brackets suppose a huge change in the meaning. But it is not so clear how well this extends to bigger sequences, and when there are no clues about the underlying structure. The Parseval metric is based on an exact comparison between nodes, and is not flexible to contemplate matches due to similar nodes. We address this issue in our distance metric.

In this paper we present a distance metric for trees that captures their similarity if they are considered to be the structure of a sequence. It is an easy-to-implement metric which is flexible to small differences and a straightforward extension permits to compare trees of different size. This paper is structured as follows: Sect. 2 defines the notation we use, Sect. 3 is the central part of this paper, where we define our metric and prove that it is a proper distance function. In Section 4 we analyze algorithms to compute it and in Section 5 we test its ability to classify tree that were slightly modified.

Preliminaries

We index sequences starting from 1. Trees will be denoted by capital letters, a node is called internal if it has at least one daughter and leaf otherwise. The yield of tree T is the sequence composed by its leaves, read from left to right. Any node n defines a subtree T n where n is the root. Clearly, yield(T n ) is a substring of yield(T). We define the interval of a node n to be the interval corresponding to the positions of yield(T n ) in yield(T ). Finally the bracket set of a tree T is the set of intervals given by the internal nodes (except the root) of T . Brackets will be denoted by lowercase letters. With the exception of the label of the nodes, the parse tree can be fully recovered from this set of intervals, so we will use as equivalent parse trees and bracket set. Example Consider Fig. 1. The bracket set of tree [START_REF] Allen | Subtree transfer operations and their induced metrics on evolutionary trees[END_REF][START_REF] Bille | A survey on tree edit distance and related problems[END_REF], [START_REF] Apostolico | Off-line compression by greedy textual substitution[END_REF][START_REF] Bille | A survey on tree edit distance and related problems[END_REF]} and the one of tree

T 1 is {[1, 5], [1, 2],
T 2 is {[1, 5], [1, 4], [1, 2], [2, 3]}.
The unlabeled version of the Parseval metric used to evaluate syntax tree is the Unlabeled F-Measure (the harmonic mean between the precision and recall). It is equal to the Dice coefficient:

Definition 1 (Dice coefficient). Given sets X, Y . Dice(X, Y ) = 2 * |X ∩ Y | |X| + |Y | In Fig 1, Dice(T 1 , T 2 ) = 0.5

A Similarity Measure for Trees

In this paper we propose a similarity metric between trees that ignores the labels of the nodes. It is inspired by the unlabeled version of the Parseval metric, but our measure turns out to be much more robust to small changes of the brackets. The motivation itself is also different: while the goal of the Parseval metric is to evaluate how close a proposed tree comes to a gold standard, the objective of our measure is to compare different trees over possibly different sequences. It tries to extract all possible similarities, even if the matches of brackets are not perfect.

As a first step we focus on comparison between two brackets, which are represented as intervals. In order to measure how similar they are, we use the well-known Jaccard coefficient that gives a measure of similarity of two sets: Definition 2 (Jaccard Coefficient). Given sets x,y,

J(x, y) = |x ∩ y| |x ∪ y|
Here, we suppose that the brackets are intervals over the integer line, so that each bracket defines a finite set of integers.

This gives a similarity coefficient between 0 and 1, 0 being completely different (empty intersection) and 1 total equality (x = y). The corresponding distance measure is d(x, y) = 1 -J(x, y). The first three are trivially true. For the triangle inequality see [START_REF] Lipkus | A proof of the triangle inequality for the Tanimoto distance[END_REF]. In order to be able to compare the brackets of two trees (set of brackets) X and Y , we suppose an assignment function f : X → Y ∪ {∅}. Moreover, we require this function to be injective, except possibly for ∅. This means, every bracket y from Y has at most one x from X such that f (x) = y. Let R(f ) denote the range of function f . Note that R(f ) may be only a subset of Y . The role of the empty set in the image is to permit to assign brackets from X which otherwise would not be assigned. If f (x) = ∅ we refer to this as a null-assignment. Note that d(x, ∅) = 1, the maximal value for d.

We compare two set of brackets as follows:

Definition 3. D f (X, Y ) = x∈X d (x, f (x)) + |Y \ R(f )|
This gives a penalty of a maximal distance for every bracket of Y to which no bracket of X was assigned. This is the symmetric part of assigning ∅ to a bracket of X.

In order to find the shortest distance between two trees, we are interested in finding the "best" possible assignment function. This is the function f * = arg min f D f (X, Y ). Then, we define:

Lemma 2 (Distance measure). D(X, Y ) = D f * (X, Y )

Symmetry

In order to prove symmetry, we define f -1 the inverse of a function in a non-standard way. Let f be as defined before. Then:

Definition 4 (Inverse function). f -1 : Y → X ∪ {∅} f -1 (y) = x if y ∈ R(f ) and f (x) = y ∅ if y ∈ R(f )
So, each bracket from the image that was not assigned by f , receives a nullassignment of f -1 .

Lemma 3. D f (X, Y ) = D f -1 (Y, X) Proof. D f -1 (Y, X) = y∈Y d y, f -1 (y) + X \ R f -1 = {Definition 4} x∈X∩R(f -1 ) d(f (x), x) + y∈Y \R(f ) d(y, ∅) + |X \ R(f -1 )| = {d(y, ∅) = 1} x∈X∩R(f -1 ) d(f (x), x) + |Y \ R(f )| + x∈X\R(f -1 ) d(x, ∅) = {symmetry of d} x∈X d (x, f (x)) + |Y \ R(f )| ⊓ ⊔
Now, if f does not assign brackets y from Y , the y is null-assigned by f -1 . So, if f * minimizes D f (X, Y ), then (f * ) -1 minimizes D g (Y, X). We have as corollary:

Corollary 1 (Symmetry). D(X, Y ) = D(Y, X)

Triangle Inequality

For the proof of the triangle inequality, we will proceed similarly as before, and redefine the composition of function.

Definition 5 (Composition). If f : X → Y ∪ {∅} and g : Y → Z ∪ {∅}, then g • f denotes the function: g • f : X → Z ∪ {∅} (g • f )(x) = g(f (x)) if f (x) = ∅ ∅ otherwise
See Figure 2 for illustration. The intuition behind is that, if while going from X to Z over Y an x gets a null-assignment, then the final assignment is also null.

We will make use of the two following lemmas: Proof.

Lemma 4. |Z \ R(g • f )| ≤ |Z \ R(g)| + |Y \ R(f )| Proof. By Definition 5, |Z \ R(g • f )| ≤ |Z \ R(g)| ⊓ ⊔
x∈X d (x, (g • f )(x)) = x∈X,f (x) =∅ d(x, (g • f )(x)) + x∈X,f (x)=∅ d(x, ∅) ≤ {triangle inequality of d} = x∈X,f (x) =∅ (d (x, f (x)) + d (f (x), (g • f )(x))) + x∈X,f (x)=∅ d(x, f (x)) = {Definition 5} = x∈X,f (x) =∅ d(x, f (x)) + x∈X,f (x)=∅ d(x, f (x)) + x∈X,f (x) =∅ d(f (x), g(f (x))) x∈X d(x, f (x)) + y∈R(f )\{∅} d(y, g(y)) ≤ x∈X d(x, f (x)) + y∈R(f )\{∅} d(y, g(y)) + y∈Y \R(f ) d(y, g(y)) = x∈X d(x, f (x)) + y∈Y d(y, f (y)) ⊓ ⊔ Corollary 2 (Triangle Inequality). D(X, Y ) + D(Y, Z) ≥ D(X, Z)
Proof. Suppose f, g, h are, respectively, the functions that minimize D e (X, Y ), D e (Y, Z) and D e (X, Z).

D(X, Z)

= D h (X, Z) ≤ {h minimizes D e (X, Z), so in particular D h (X, Z) ≤ D g•f (X, Z)} x∈X d(x, (g • f )(x)) + |Z \ R(g • f )| ≤ {Lemma 5 and Lemma 4} x∈X d(x, f (x)) + y∈Y d(y, f (y)) + |Z \ R(g)| + |Y \ R(f )| = D(X, Y ) + D(Y, Z) ⊓ ⊔

Computation

In this section we consider an algorithm to compute the measure D and analyze its computational complexity.

The Jaccard coefficient can be computed in constant time because we restrict the sets to be intervals. Given the assignment function f

, D f (X, Y ) is computable in O(n),
where n = max(|X|, |Y |). Note that we only consider trees where every node has at least two daughters, so the size of the set of brackets (|X|) is linear in the number of leaves (the length of the yield).

The computation of f * can be mapped to optimize an assignment problem and can be solved, for example, by the Hungarian algorithm [START_REF] Kuhn | The Hungarian method for the assignment problem[END_REF], which is in O(|X| 3 ). So, D(X, Y ) can be computed in O(n 3 ), where n = max(|X|, |Y |).

Experimentations

In order to test our distance measure and compare it to existing metrics we want to see if it is capable to distinguish groups of similar trees. For this, we start with a small set of radical different trees. We then modify copies of these original trees and use a clustering algorithm based on the distance metric to regroup them. The nature of the change is always the same, but is parametrized by a random distribution.

Starting from k sets of brackets, we copy each of them m times with some modifications. We use k = 3: a left-branching tree ({(i, n) : 1 ≤ i < n -1}), a right-branching tree ({(0, i) : 1 < i < n}) and a centered tree ({(i, ni) : 1 < i < ⌊n/2⌋}). In our set-up, n = 30 and we generate m = 32 modified copies of each one, resulting at the end in 99 bracket sets (we keep the 3 original trees). The modifications are obtained by changing each bracket with probability p. A change consists in a shift of the bracket to the left or to the right (choosing randomly). Shifting a bracket [a, b] by ℓ consist in replacing it by [a + ℓ, b + ℓ]. The value ℓ is given by a geometric distribution with parameter q (plus one, to ensure that the bracket is changed). After each change, all overlapping brackets are shorten to avoid overlap. In order to not give preference to any bracket, the order in which they are considered is determined randomly. Throughout our experiments we use q = 0.5 and different values for p. For the clustering algorithm, we compute the square distance matrix and use a k-medoid algorithm (k = 3), taking the cluster with lowest total sum of distances to the center after 20 runs (each run starts with a random selection of centers).

Note that the Dice dissimilarity metric (1 -D(X, Y ) does not satisfy the triangle inequality and is inadequate to be used to compute a distance matrix. It is however closely related to the Tanimoto distance (Dice(X, Y ) = 2 * J(X,Y ) 1+J(X,Y ) ), so we compared our distance metric to the Tanimoto distance, but this time applied to set of brackets. In Fig. 3, we plot p (the probability that a bracket is changed) against the number of hits of the final cluster. Each point is the average over 250 runs, each run consisting in a copy-modify-cluster-count step. As it can be appreciated, as the probability of modification increases, the Tanimoto distance becomes less accurate to discriminate the correct clusters. This reveals the binary nature of a match in the Tanimoto distance: or a bracket matches or not. Our distance is more flexible: this reveals to be counterproductive if there are only few changes, but if p increases, it reveals to be very well suited to cluster the right groups. Both measures results in the same number of hits for p = 0.5, but from there on the number of hits using the Tanimoto distance decreases considerable. Our D distance continues to improve, getting more or less stable at 92 correct hits.

Conclusions

We presented a new distance metric to compare trees and prove that it is a proper metric. The aim of this metric is to compare sequences based on their tree structure. The advantage over previous approaches is its flexibility to compare trees that intuitively are highly similar, but where existing similarity metrics fail.

Our experiments show that this metric permits to distinguish groups of similar parse tree. Starting with a group of radical different trees, we modified each bracket slightly. When the probability of changing a bracket is greater then 0.5, our metric outperforms considerable a classical distance metric.

It would be interesting to analyze how this measures behave with respect to a treeedit distance. Differently from the Parseval measure and ours, they are not based on the similarity between the yields of the nodes, but on the number of operations that are necessary to go from one tree to another one. The standard edit-distance [START_REF] Bille | A survey on tree edit distance and related problems[END_REF][START_REF] Touzet | A linear tree edit distance algorithm for similar ordered trees[END_REF][START_REF] Akutsu | Approximating tree edit distance through string edit distance[END_REF] contemplates operations of insertion, deletion and renaming of nodes. In order to have a meaningful measure, an operation on edges that would permit to re-branch nodes (known as prune and regraft [START_REF] Allen | Subtree transfer operations and their induced metrics on evolutionary trees[END_REF]) should also be considered.

Finally, we think that this measure is suitable to be used together with algorithms that output straight-line grammars to discover similarity that is not captured by comparing only the content of them.

Fig. 1 .

 1 Fig. 1. Two different trees with the same yield.

Lemma 1 .

 1 d is a proper metric, that is, it satisfies: non-negativity (d(x, y) ≥ 0), identity (d(x, x) = 0), symmetry (d(x, y) = d(y, x)) and triangle inequality (d(x, y) + d(y, z) ≥ d(x, z))

Fig. 2 .

 2 Fig. 2. In this case (g • f )(x2) = z1, (g • f )(x3) = z2 and the rest are null-assignments.

Fig. 3 .

 3 Fig. 3. Number of true positives of the final cluster against probability of change of a bracket.

Acknowledgements I am grateful to A. Apostolico and M. Comin for fruitful discussions.


Fabien Lauer

Gérard Bloch

René Vidal

A continuous optimization framework for hybrid system identification

Keywords: hybrid system, identification, robustness to outliers, large-scale

We propose a new framework for hybrid system identification, which relies on continuous optimization. This framework is based on the minimization of a cost function that can be chosen as either the minimum or the product of loss functions. The former is inspired by traditional estimation methods, while the latter is inspired by recent algebraic and support vector regression approaches to hybrid system identification. In both cases, the identification problem is recast as a continuous optimization program involving only the real parameters of the model as variables, thus avoiding the use of discrete optimization. This program can be solved efficiently by using standard optimization methods even for very large data sets. In addition, the proposed framework easily incorporates robustness to different kinds of outliers through the choice of the loss function.

Introduction

Consider a class of discrete-time ARX hybrid systems of the form

y i = f λi (x i ) + v i , (1)
where x i = [y i-1 . . . y i-na , u i-n k . . . u i-n k -nc+1 ] T is the continuous state (or regression vector) of dimension p containing the lagged n c inputs, u i-k , and n a outputs, y i-k , λ i ∈ {1, . . . , n} is the discrete state (or mode) determining which one of the n subsystems, {f j } n j=1 , is active at time step i, and v i is an additive noise term. Two classes of hybrid models can be distinguished on the basis of the evolution of the discrete state λ i . In particular, Switched ARX (SARX) models assume that the system switches arbitrarily, while PieceWise ARX (PWARX) models consider a dependency between the discrete state and the regression vector. The latter are usually defined by piecewise affine maps of the type f (x) = f j (x), if x ∈ S j , j = 1, . . . , n, where {f j } are affine functions ⋆ A preliminary version of this paper was presented at the 15th IFAC symposium on system identification, Saint-Malo, France, July 6-8, 2009. ⋆⋆ This work was partially supported by ANR project ArHyCo, Programme "Systèmes Embarqués et Grandes Infrastructures" -ARPEGE, ANR-2008 SEGI 004 01-30011459, and by the grant NSF CNS 0931805. and {S j } are polyhedral domains defining a partition of the regression space R p . This paper concentrates on the problem of finding a hybrid model f = {f j } n j=1 of the form (1) from inputoutput data {(x i , y i )} N i=1 . We assume that the number of models n and their orders are known and focus on the identification of SARX models. However, the proposed estimators are able to deal with PWARX models without any modification. They provide an estimate of the parameters of an SARX model. Then, determining the partition of the regression space amounts to a pattern recognition problem [START_REF] Duda | Pattern Classification[END_REF], where the labeling of the points is given by the estimated discrete state.

Related work. One of the main challenges in hybrid system identification is that one needs to simultaneously classify the samples into their respective modes and estimate the model parameters for each mode. A general and straightforward approach to this problem is to optimize over both the model parameters and a set of binary variables, that control the assignment of the samples to the modes. This, however, amounts to a nontrivial and non-continuous optimization problem.

Two main classes of methods have been proposed to solve directly this problem. The first class includes the clustering-based approach, using either k-means [START_REF] Ferrari-Trecate | A clustering technique for the identification of piecewise affine systems[END_REF] or Expectation Maximization (EM) [START_REF] Nakada | Identification of piecewise affine systems based on statistical clustering technique[END_REF], and the Bayesian approach [START_REF] Juloski | A Bayesian approach to identification of hybrid systems[END_REF]. These methods alternate between solving the classification problem for fixed model parameters and solving the estimation problem for a fixed classification. Hence they are prone to lead to local minima and are sensitive to initialization. On the other hand, methods in the second class optimize over both continuous and discrete variables simultaneously. This involves solving a combinatorial optimization problem, which can be prohibitively time consuming. The mixed integer programming (MIP) approach [START_REF] Roll | Identification of piecewise affine systems via mixed-integer programming[END_REF] and the bounded-error approach [START_REF] Bemporad | A bounded-error approach to piecewise affine system identification[END_REF] fall into this category.

Beside these methods, the algebraic approach [START_REF] Vidal | An algebraic geometric approach to the identification of a class of linear hybrid systems[END_REF][START_REF] Ma | Identification of deterministic switched ARX systems via identification of algebraic varieties[END_REF][START_REF] Vidal | Recursive identification of switched ARX systems[END_REF] circumvents the aforementioned issues thanks to a continuous approximation of the general optimization problem. This approximation results in a closed form solution to the identification of SARX systems. However, the algebraic method is rather sensitive to noise compared to the other approaches. Inspired by the algebraic approach, the Support Vector Regression (SVR) approach [START_REF] Lauer | A new hybrid system identification algorithm with automatic tuning[END_REF][START_REF] Lauer | Switched and piecewise nonlinear hybrid system identification[END_REF] provides a convenient way of dealing with noisy data and small sample sizes by incorporating regularization into the optimization framework. However, it optimizes over a number of variables that grows with the number of data points, thus it is limited to small data sets. We conclude this short review of the related methods by noting that, other than the bounded-error approach [START_REF] Bemporad | A bounded-error approach to piecewise affine system identification[END_REF], none of them explicitly deals with outliers in the data.

Paper contribution. We propose a continuous optimization framework for hybrid system identification that benefits from the advantages of the algebraic and SVR approaches, while also tackling their respective weaknesses. By continuous optimization we refer to the optimization of a continuous cost function over a nondiscontinuous domain, which excludes for instance integer programs. In particular, two reformulations of the mixed integer program at the core of hybrid system identification are considered. The first one is based on a non-differentiable cost function involving min operations. The second one offers a differentiable approximation using products of error terms, as in the algebraic method. These reformulations give rise to the following contributions.

• The proposed framework can include any suitable loss function. Thus robustness to outliers and leverage points is easily incorporated through the choice of a robust loss function as defined in the analysis. In addition, the derivation of the method in the maximum likelihood framework allows the loss function to be chosen with respect to the noise density model as in standard estimation theory.

β ij l(y i -f j (x i )) (2) s.t. β ij ∈ {0, 1}, i = 1, . . . , N, j = 1, . . . , n, n j=1 β ij = 1, i = 1, . . . , N,
where β ij is a binary variable and l(y if j (x i )) is a suitable loss function, e.g., the squared loss, l(y if j (x i )) = (y if j (x i )) 2 . The discrete variables β ij encode the assignment of point i to submodel j, while continuous variables encode the parameters of the submodels f j .

One way to solve this mixed program is to use alternating minimization: i) given the submodels {f j }, compute the assignment of points to submodels according to β ij = 1, if j = arg min k=1,...,n l(y i -f k (x i )), and 0 otherwise; and ii) given the assignments β ij , compute one submodel for each group of points. This approach is effective when the submodels are linear, i.e., f j (x i ) = θ T j x i , and a convex loss function is used, because the estimation of each submodel is a linear system identification problem. However, this approach is sensitive to initialization. Note that problem (2) can also be solved directly by using mixed integer programming techniques, as proposed in [START_REF] Roll | Identification of piecewise affine systems via mixed-integer programming[END_REF] for hinging hyperplane models. These latter optimization techniques can guarantee to find the global minimum, but, due to their high complexity, they can only be used in practice for small data sets.

Continuous optimization approach

This section presents the proposed estimators for hybrid systems. In particular, two closely related estimators are derived in the maximum likelihood ( §3.1) and the error minimization ( §3.2) frameworks, respectively. In order to remain efficient on large data sets, these estimators are devised so as to lead to continuous optimization programs with a small number of variables, which does not depend on the number of data. The section ends with the description of a smooth approximation to the proposed estimators and an analysis of the robustness to outliers.

Maximum likelihood framework

Let the random variables (x, y) be described by a joint probability density function (pdf) p(y, x). Assume that the conditional pdf p(y|x) takes the functional form p(y|x, f ), dependent on the model f . Then, the maximum likelihood approach consists in finding the model f that most likely generated the data. For N i.i.d. samples, this is equivalent to maximizing the log-likelihood, N i=1 ln p(y i |x i , f ), with respect to f . In the context of hybrid systems, the Maximum Likelihood (ML) estimator assigns each data sample (x i , y i ) to the most likely submodel f j , i.e., the one with maximal likelihood of the sample w.r.t. f j . This leads to λi = arg max j=1,...,n p(y i |x i , f j ).

(

This allows the likelihood of a sample with respect to f to be written as p(y

i |x i , f ) = p(y i |x i , f λi ) ∝ max j=1,...,n p(y i |x i , f j ).
The log-likelihood is thus given by

N i=1 ln p(y i |x i , f ) ∝ N i=1 ln max j=1,...,n p(y i |x i , f j ), (4)
which is equivalently maximized by solving

maximize {fj } N i=1 max j=1,...,n ln p(y i |x i , f j ). (5)
Finally, the ML estimator is given as the solution to

minimize {fj } J M L = N i=1 min j=1,...,n -ln p(y i |x i , f j ) . ( 6
)
Note that the minimum of a finite set of continuous functions of some variables is a continuous function of these variables (discontinuities only occur in the derivatives). Therefore, if the submodels {f j } are given by continuous functions of their parameters and if the likelihood function p(y i |x i , f j ) is continuous in f j , then the minimum over j of the negative log-likelihood functions ln p(y i |x i , f j ) in ( 6) is a continuous function of the parameters to be estimated. As a consequence, the cost function in ( 6) is a continuous function of the variables parametrizing the submodels {f j } and ( 6) is a continuous optimization problem.

Minimum-of-errors estimator

In the following, we derive the Minimum-of-Errors (ME) estimator in the framework of loss function minimization and show its relationship to the ML estimator.

The formulation in (2) relies on the assumption that sample x i must be assigned to the submodel that best estimates the target output y i . In order to minimize the overall cost, we thus have to set the estimated mode as λi = arg min j=1,...,n

l(y i -f j (x i )), i = 1, . . . , N. (7)
Explicitly including this result in (2) leads to the ME estimator obtained by solving minimize {fj }

J M E = N i=1 min j=1,...,n l(y i -f j (x i )) . ( 8
)
The relationship between ML and ME estimators appears when choosing the loss function l in the ME estimator (8) according to l(y i -

f j (x i )) = -ln p(y i |x i , f j ).
This provides the choice of the loss function in the case of a known noise distribution.

Using a similar continuity argument for the cost in [START_REF] Lauer | Nonlinear hybrid system identification with kernel models[END_REF] as for that in (6), we see that, for all loss functions l(e) that are continuous in their argument, ( 8) is a continuous optimization problem with respect to the parameters of the submodels {f j }. Thus instead of solving [START_REF] Duda | Pattern Classification[END_REF], which is a non-continuous problem due to the discrete variables {β ij }, we can equivalently solve the continuous problem [START_REF] Lauer | Nonlinear hybrid system identification with kernel models[END_REF], which only involves real variables. After solving for the submodels {f j }, the mode estimates are simply recovered by using (7) (or λi = arg min j=1,...,n |y i -f j (x i )|, if the loss function l cannot yield the decision).

Product-of-errors estimator

For a smooth loss function l, the Product-of-Errors (PE) estimator is obtained by solving the smooth optimization program minimize {fj }

J P E = N i=1 n j=1 l(y i -f j (x i )). ( 9
)
The cost function of the PE estimator in (9) can be seen as a smooth approximation to the ME cost function in [START_REF] Lauer | Nonlinear hybrid system identification with kernel models[END_REF]. In particular, for noiseless data, they share the same global minimum J M E = J P E = 0.

Remark 1: Note that for the particular case of linear submodels f j and squared loss function l(e) = e 2 , the cost function of the PE estimator in (9) coincides with the cost minimized by the algebraic approach [START_REF] Ma | Identification of deterministic switched ARX systems via identification of algebraic varieties[END_REF]. However, the PE estimator aims at directly minimizing this cost with respect to the model parameters, while the algebraic approach minimizes this cost with respect to the tensor product of the parameters. This allows the algebraic algorithm to obtain the solution more efficiently by solving a linear system, but introduces errors in the case of noisy data, because the tensor product of the parameters is an overparametrization of the space of parameters.

Remark 2:

The PE estimator given by ( 9) is also equivalent to the (unregularized) SVR-based approach described in [START_REF] Lauer | A new hybrid system identification algorithm with automatic tuning[END_REF] when using the ε-insensitive loss function defined as l(e) = max(0, |e|ε), for a threshold ε. In this case, the proposed method allows to reduce the number of optimization variables from n(p + N ) to np.

Robustness to outliers and leverage points

Two types of outlying observations, i.e., points which deviate significantly from the rest of the data, are commonly distinguished. Arbitrary large values |y i | in the response variable are simply called outliers, while arbitrary large values in the regression vector x i are called leverage points. It is worth noting that, for the estimation of ARX model parameters, due to the presence of lagged outputs in the regression vector, the same abnormal value can be both an outlier and a leverage point.

For an estimator to be robust to outliers, the effect of a single point on the estimation must be bounded. For instance, in classical regression problems, the influence function of the squared loss, i.e., its derivative with respect to y i , is unbounded. On the other hand, the absolute loss, l(e) = |e|, has an influence function bounded by 1 and thus leads to more robust estimators.

In this subsection, the different cases are investigated for the ME and PE estimators in the context of hybrid systems. l ′ (e) = dl(e)/de denotes the derivative of the loss function l with respect to its scalar argument e.

Minimum-of-errors estimator. For a given x i with an arbitrary large |y i |, we can consider the estimated mode as fixed (either given by λ i = arg max j f j (x i ) or λ i = arg min j f j (x i ), depending on the sign of y i ). Then we can write the influence of this point as

∂J M E ∂y i = ∂l(y i -f λi (x i )) ∂y i = l ′ (y i -f λi (x i )). ( 10
)
This implies that the influence of an outlier is bounded if l has a bounded derivative l ′ (e), as is the case with l(e) = |e|. On the other hand, the influence of leverage points

∂J M E ∂x i = -l ′ (y i -f λi (x i )) df λi (x i ) dx i , (11)
with λ i = arg min j=1,...,n l(y if j (x i )), is unbounded even with bounded derivative l ′ (e).

Product-of-errors estimator. The sensitivity of the cost function in [START_REF] Lauer | A product-of-errors framework for linear hybrid system identification[END_REF] with respect to outliers is

∂J P E ∂y i = n j=1 l ′ (y i -f j (x i )) k∈{1,...,n}\j l(y i -f k (x i )), ( 12
)
and its sensitivity with respect to leverage points is

∂J P E ∂x i = - n j=1 l ′ (y i -f j (x i )) df j (x i ) dx i k∈{1,...,n}\j l(y i -f k (x i )).
(13) In these cases, the terms l(y if k (x i )) are unbounded, hence both sensitivity functions are unbounded.

To summarize, the only way to guarantee that the influence of outliers and leverage points on both the ME and PE estimators is bounded and obtain robust estimators is to use a loss function l leading to l ′ (e) = 0, for large values of the error |e|.

Such loss functions are considered in the framework of redescending M-estimators, where l ′ (e) is known as the Ψ function and decreases smoothly towards zero when |e| increases. As an example, the Hampel's loss function, defined as

l(e) = δ 2 /π (1 -cos(πe/δ)) , if |e| ≤ δ, 2δ 2 /π, otherwise, (14)
which satisfies that l ′ (e) = 0 for |e| > δ, will be considered in §4.1. Further experiments with other loss functions and the PE estimator can be found in [START_REF] Lauer | A product-of-errors framework for linear hybrid system identification[END_REF].

Optimization of linear hybrid models

All the results developed in the previous sections apply similarly to linear and nonlinear submodels f j . In the following, we focus on linear hybrid models, in which the submodels f j (x) are given in the linear form

f j (x) = θ T j x, j = 1, . . . , n, (15)
where the parameter vectors θ j to be estimated are of dimension p = n a + n c . Note that affine submodels can be equivalently considered by appending a 1 to the regression vector and considering p = n a + n c + 1.

For all continuous loss functions l, the cost functions in ( 8) and ( 9) are both continuous functions of the parameters θ j . Moreover, the number of variables involved in these problems is small and fixed to the number of model parameters, n × p, for any number of data N . These two combined features allow a solver for continuous problems to find a satisfying solution in reasonable time, despite the NP-hard nature of the problem. Note however that the cost functions in ( 8) and ( 9) require to compute a sum over N terms, hence the linear complexity of the algorithm with respect to N .

In the following, we propose to solve ( 8) and ( 9) with the Multilevel Coordinate Search (MCS) algorithm 1 [START_REF] Huyer | Global optimization by multilevel coordinate search[END_REF], that is guaranteed to converge if the objective is continuous in the neighborhood of the global minimizer. This optimizer uses only function values (when required, derivatives are estimated from these) and alternates between global and local search. The local search, done via sequential quadratic programming, speeds up the convergence once the global part has found a point in the bassin of attraction of the global minimizer.

Examples

We now present some illustrative examples and start with the identification of a switched linear system (Sect. 4.1), including a comparison with the algebraic procedure and the study of the robustness to outliers of the estimators. Large-scale experiments (Sect. 4.2) are then presented to analyze the complexity of the method with respect to the number of data and the number of parameters in the model.

Assuming no prior knowledge on the parameters, box constraints that limit the search space in the MCS algorithm are set for all variables θ jk to the quite large interval -100 ≤ θ jk ≤ 100. Beside this, the MCS parameters are all of time-limiting nature (maximum number of iterations or function evaluations), for which the default values led to satisfying results. For all the problems, N samples are generated by

y i = θ T λi x i + v i , i = 1, . . . , N, (16)
where the θ j ∈ R p are the true parameters to be recovered and v i ∼ N (0, σ 2 v ) is a Gaussian noise. The methods are compared on the basis of the Normalized Mean Squared Error on the parameters, NMSE =

n j=1 θ j -θj 2 2 / θ j 2 2
, where the θj are the estimated parameters. In the Tables, all numbers of the form A±B correspond to averages (A) and standard deviations (B) over 100 trials. The Tables also show the number of failures (# fail) of the algorithms, i.e., the number of trials for which the parameter estimates are irrelevant. It is 1 http://www.mat.univie.ac.at/˜neum/software/mcs/.

worth noting that the NMSE is computed without taking these trials into account. All experiments are performed on a standard desktop computer with Matlab.

Switched linear system identification

Consider the example taken from [START_REF] Vidal | Recursive identification of switched ARX systems[END_REF]. The aim is to recover, from N = 1000 samples, the parameters θ 1 = [0.9, 1] T and θ 2 = [1, -1] T of a dynamical system, arbitrarily switching between n = 2 modes, with continuous state x i = [y i-1 , u i-1 ] T and input u i ∼ N (0, 1). The standard deviation of the generated trajectories is σ y ≈ 2. Comparison with the algebraic approach. As described in [START_REF] Vidal | An algebraic geometric approach to the identification of a class of linear hybrid systems[END_REF], the algebraic method is originally implemented as a linear problem solved w.r.t. the tensor product of the parameters. It can also be implemented in the proposed PE framework [START_REF] Lauer | A product-of-errors framework for linear hybrid system identification[END_REF], with squared loss, as the direct minimization over the model parameters (see Remark 1). These two algorithms are compared in Table 1 for different noise levels σ v . The ME estimator (8) with squared loss is also included in the comparison. The results highlight the gain in solving the problem directly for the model parameters θ j versus the optimization over the product of the parameters. However, the gain in NMSE obtained by solving directly for the θ j comes at the cost of solving a nonlinear optimization program instead of a linear problem. This leads to a computing time about 30 times larger with the PE estimator (0.3 sec.) than with the algebraic method (0.01 sec.). Note however, that for N = 1000 points these times remain below one second, whereas directly solving (2) is already prohibitively time consuming.

In practice, the algebraic method estimates the normals b j to the hyperplanes on which the data lie as [x i , -y i ]b j = 0, and assigns a coefficient to the output y i . Thus, the original model parameters are recovered by dividing all the coefficients in b j by the last component of b j , which leads to failure when this number is close Robustness to outliers. We now illustrate the robustness to outliers of the ME (8) and PE ( 9) estimators obtained by using the Hampel's loss function ( 14) instead of the squared loss. The data are corrupted with additive Gaussian noise (σ v = 0.2) and an increasing percentage of outliers by forcing y i , at random time steps i, to take uniformly distributed random values in the interval [-10, 10]. Note that this also introduces outliers in the regressors x i , which are built from lagged outputs. According to the previous results shown in Table 1, the algebraic method cannot handle these highly corrupted data. Table 2 shows the benefit of using the robust versions of the proposed estimators over their standard counterpart. The resulting NMSE remain in the order of 10 -4 to 10 -3 with up to 50% of outliers for both the Robust ME and the Robust PE estimators, while being comparable to the NMSE obtained with the non-robust squared loss in the outlier-free case. However, the number of failures of the Robust ME estimator slightly increases with the number of outliers. Again, the Robust PE estimator is not susceptible to these failures even in the presence of outliers, except for the extreme case of 50 % of outliers. In these experiments, the parameter of the Hampel's loss function was set to δ = 2. The estimation of the optimal value for δ is left to future work.

Large-scale experiments

Large data sets. The performance of the method on large data sets is evaluated on a set of 100 randomly generated problems with n = 2 and p = 3. The true parameters {θ j } are randomly drawn from a uniform distribution in the interval [-2, 2] p , while very loose constraints, θ j ∈ [-100, 100] p , are applied for the estimation. The N data are generated by ( 16) with uniformly distributed random regression vectors x i ∈ [-1, 1] p , a random switching sequence {λ i } and additive Gaussian noise (σ v = 0.2). Due to the noise level in the data, the algebraic method is not included in the comparison, which focuses on the ME (8) and PE (9) estimators with squared loss. Table 3 shows the resulting average NMSE and computing times for an increasing number of data N . These times show that the complexity of the proposed algorithms scales linearly with the number of data. As a result, the method can be applied to very large data sets, which cannot be handled by previous approaches such as the ones described in [START_REF] Bemporad | A bounded-error approach to piecewise affine system identification[END_REF][START_REF] Ferrari-Trecate | A clustering technique for the identification of piecewise affine systems[END_REF][START_REF] Lauer | A new hybrid system identification algorithm with automatic tuning[END_REF][START_REF] Roll | Identification of piecewise affine systems via mixed-integer programming[END_REF]. Note that the programs, including the MCS optimization algorithm, are fully implemented in non-compiled Matlab code and that the variability over the 100 runs comes from the random sampling of the true parameters generating the data, not from the optimizer. However, in 5 runs out of 400 (over all experiments), the optimizer led to local minima for the ME estimator, which failed to yield a correct model. Note that such failures do not occur when using the PE estimator. Overall, the ME estimator leads to models with less error than the PE estimator, but requires twice as much time to compute and can occasionally lead to bad local minima.

Larger model structures. The method has been shown to be very effective on large data sets. However, the computing time heavily relies on the number of model parameters n × p, against which the method is now tested. As before, a set of 100 random experiments is performed for N = 10 000 samples and varying numbers of modes n and parameters per mode p. To be able to compare the computing times of the proposed PE estimator with those of the algebraic method, experiments without noise (σ v = 0), for which both methods perfectly estimate the parameters, are considered. The results appear in Figure 1 for the PE estimator (9) with squared loss and the algebraic method. These curves show that the PE estimator is clearly slower than the algebraic method for small model structures. However, the computing time of the algebraic method increases quickly with the number of parameters and exceeds the one of the PE estimator for n = 6 and p = 8. In addition, similar experiments with noisy data lead to similar time curves for the PE estimator. Thus the cost in computing time of the proposed method is still reasonable considering the gain in accuracy obtained by the PE estimator over the algebraic method in the noisy case (as shown by Table 1).

Conclusion

Two classes of estimators for hybrid systems have been proposed: the ME estimator, based on the minimum of the submodel errors, and the PE estimator, based on the product of the submodel errors. The ME estimator benefits from a straightforward maximum likelihood interpretation and is an exact reformulation of the mixed integer problem (2) into a continuous optimization problem. The PE estimator has been devised as a smooth approximation to the ME estimator, which appears as a convenient substitute in terms of avoiding local minima while maintaining a similar level of accuracy. In terms of efficiency, the proposed estimators have been tailored to tackle large-scale problems and have been shown to yield fast and accurate results in experiments with numerous data. The paper also focused on providing an algorithm which remains robust to high levels of noise compared to previous approaches such as [START_REF] Vidal | An algebraic geometric approach to the identification of a class of linear hybrid systems[END_REF]. In addition, the analysis shows that robustness to outliers either in the output y i or the regression vector x i can be obtained by a proper choice of the loss function. Future work will aim at finding better approximations to the ME estimator than the PE estimator and extend the approach to the estimation of unknown nonlinearities in the context of hybrid systems, as initiated in [START_REF] Lauer | Nonlinear hybrid system identification with kernel models[END_REF].

Fig. 1 .

 1 Fig. 1. Average computing time in seconds for the PE estimator (left) and the algebraic method (right) over 100 trials versus the number of parameters per mode p for different number of modes n.

Table 1

 1 Average NMSE (×10 -3 ) and number of failures over 100 trials.

	σv	ME	PE	Algebraic
	0.00 NMSE 0.00	0.00	0.00
	# fail	0	0	0
	0.02 NMSE 0.00 ± 0.00 0.00 ± 0.00 0.05 ± 0.28
	# fail	44	0	0
	0.10 NMSE 0.03 ± 0.03 0.08 ± 0.06 2.99 ± 12.32
	# fail	20	0	4
	0.20 NMSE 0.09 ± 0.10 0.27 ± 0.23 4.86 ± 10.38
	# fail	28	0	10
	0.30 NMSE 0.27 ± 0.22 0.77 ± 0.65 10.31 ± 13.46
	# fail	27	0	10

Table 2

 2 Average NMSE (×10 -3 ) over 100 trials for the proposed estimators using either the squared loss or the Hampel's (robust) loss function when applied to data sets with additive Gaussian noise (σv = 0.2) and an increasing percentage of outliers. For the ME estimator, the failures are due to the convergence of the optimizer to bad local minima. However, failures aside, the ME estimator leads to an NMSE that is similar to the one obtained by separate least squares estimations for each mode with knowledge of the true mode λ i . This indicates that good local, if not global, minima are found. Note that in these experiments, the PE estimator does not suffer from the convergence to bad local minima, but leads to slightly less accurate estimates.

	% of outliers		0 %	10 %	20 %	30 %	40 %	50 %
	Robust ME NMSE 0.14 ± 0.12 0.14 ± 0.14	0.27 ± 0.22	0.22 ± 0.21	0.43 ± 0.44	0.71 ± 0.64
		# fail	10	11	15	19	21	27
	ME	NMSE 0.09 ± 0.10 40.3 ± 43.8	118.8 ± 47.7	280.6 ± 72.3 379.6 ± 103.6 496.2 ± 149.3
		# fail	28	15	13	22	11	12
	Robust PE	NMSE 0.15 ± 0.12 0.15 ± 0.11	0.22 ± 0.21	0.26 ± 0.24	0.43 ± 0.39	0.84 ± 0.89
		# fail	0	0	0	0	0	6
	PE	NMSE 0.27 ± 0.23 118.0 ± 0.83.4 251.7 ± 139.4 215.5 ± 96.3 425.2 ± 177.7 501.3 ± 210.0
		# fail	0	0	0	0	4	14
	to zero.

Table 3

 3 Average NMSE and computing time over 100 randomly generated problems with N samples.

			N			NMSE (×10 -6 ) #fail	Time (sec.)
		10 000 ME			1.3 ± 0.9	0	2.7 ± 1.0
				PE			3.0 ± 2.3	0	1.3 ± 0.2
		50 000 ME			0.4 ± 0.5	0	10.0 ± 4.0
				PE			1.1 ± 2.8	0	4.2 ± 0.6
	100 000 ME			0.3 ± 0.2	1	23.4 ± 10.7
				PE			1.0 ± 1.8	0	10.4 ± 1.7
	500 000 ME			0.3 ± 0.8	4 158.5 ± 49.2
				PE			0.7 ± 1.6	0	53.9 ± 6.1
	10	3	4	5	6	7	8


PhD Student, Clotilde Minster
email: minster@inrets.fr

Tackling obesity by urban planning? Recent research and a European case study: some evidence and perspectives

Considerable research has been undertaken in recent years in order to understand the relationship between being overweight or obese and urban design. Almost all researchers focusing on this link work in North America. However this problematic has recently become a research topic in Europe. Thus, one may question the relevance of the current way of thinking and methodology and its transferability from the U.S. to Europe. Recent literature dealing with the issue of the relationship between being overweight or obese and the built environment in urban context is presented in this paper. Theories from this literature have then been compared with evidence from the field, based on a medium-sized town in Germany, a country often described as less suburban. Conclusions reached in North America seem irrelevant to our European case, which shows that U.S. solutions cannot simply be transferred directly. We came up with suggest that one has to understand both morphological and human aspects of cities to gain a deeper understanding of overweight or obesity by a spatial approach on an inner-urban scale.

Introduction

Definition and implications of overweight and obesity

Overweight and obesity are defined as an accumulation of fat due to an imbalance between energy expenditure and caloric intake. This medical designation is typically assigned based on the Quetelet or Body Mass Index (BMI). BMI is obtained by dividing the patient's body mass by the square of the patient's height. The causes of overweight and obesity i are now well-known [START_REF] Basdevant | Médecine de l'obésité[END_REF]. Expressed simplistically, the conditions are related to an imbalance of the 'obesity equation' whose parameters are energy expenditure and calorie intake and include a lack of exercise coupled with the consumption of foods rich in sugar and saturated fat like frozen dinners and fast food (World Health Organization (WHO), 2007b). Overweight and obesity are linked to a number of serious health conditions, including cancer, cardiovascular disease, and type 2 diabetes (World Health Organization, 2007b). Overweight and obesity also have economic consequences. For example, medical expenses for the overweight and obese in the United States exceed $90 million per year [START_REF] Finkelstein | National medical expenditures attributable to overweight and obesity: how much and who's paying?[END_REF]. The Robert Koch Institute has also estimated the direct and indirect costs of obesity for the year 1995 to be between 3.1 and 5.5 % of the total health costs in Germany (Robert Koch Institut, 2003). Experts agree that doing nothing to tackle overweight and obesity is not an economical solution [START_REF] Review | Fair Society, Healthy Lives: A Strategic Review of Health Inequalities in England Post-2010[END_REF].

Overweight and obesity are currently one of the fastest growing public health problems worldwide.

The number of people concerned by overweight or obesity has tripled since the 1980's (World Health Organization, 2007). According to 2005 WHO statistics, there are at least 400 million obese adults worldwide. The WHO forecasts this number will increase to 700 million by 2015. The speed with which this rate has increased is alarming. As a result, the WHO has referred to overweight and obesity as an 'epidemic' since 2004. This semantic shift accompanied a shift in research that started late 1990s, highlights the gravity of the malady and its consequences.

New research directions

Thus, the focus of research has shifted in recent years from trying to explain the triggers of overweight and obesity to understanding the speed with which the overweight and obesity rates are increasing. To this end, investigators have considered new research questions. Researchers are no longer considering overweight and obesity solely as an individual biomedical dysfunction, but rather as a global health problem. Research now includes consideration of food intake, caloric expenditure, and environmental constraints and researchers are trying to identify the importance of each variable (Foresight, 2007). Because of these new research directions, medical doctors are no longer the only academics working in this field. Economists, sociologists, and urban planners are also working to identify and understand, on a variety of scales, the economical or spatial contexts in which the prevalence of overweight or obese citizens is high and these professionals postulate that external factors such as the economic or built or even social environment may have an influence on these health problems.

Since late 1990s, urban planners have been concerned with the relationship between overweight or obesity and urban design. These studies have sparked the interest of a wide audience. For example, the 'Global strategy on diet, physical activity, and health' resolution, adopted at the 57 th World Health Assembly, addresses the issue of the built environment and transport policies as way of increasing physical activity and energy expenditure and combating overweight and obesity.

In this paper, we present an overview of selected literature of recent research on overweight and obesity from a spatial perspective. We draw upon major international studies to examine the relationship between overweight or obesity and the built environment in urban context. We then examine the transfer of solutions from North America to Europe using Germany as a case study.

Recent literature

Many studies have been conducted in developed or developing countries in order to understand the relationship between urban form, means of transportation, and being overweight or obese.

Overweight, obesity and poverty

All studies conducted in developed countries have confirmed a correlation between socio-economic status and health. Overweight and obesity are more prevalent among the poorest individuals. This relationship between low socio-economic status and overweight or obesity has been confirmed by several studies conducted in the fields of medicine, epidemiology, and nutrition [START_REF] Kimm | Race, socioeconomic status, and obesity in 9-to 10-year old girls: The NHLBI growth and health study[END_REF][START_REF] Jeffery | Socioeconomic status, ethnicity and obesity in women[END_REF][START_REF] Crawford | Ethnic Issues in the Epidemiology of Childhood Obesity[END_REF]Office for National Statistics, 2001;[START_REF] Goldstein | Poverty is a predictor of noncommunicable disease among adults in Peruvian cities[END_REF][START_REF] Murasko | Socioeconomic status, height, and obesity in children[END_REF].

The relationship has also been confirmed by research conducted using the spatial perspective, through studies conducted on various scales. Researchers have shown that the overweight and obesity map is largely coterminous with the poverty map in France [START_REF] Salem | Géographie d'une crise française, dossier obésité : l'épidémie[END_REF]. Similar results were obtained in the United States. U.S. academics have demonstrated that area-based measures of socio-economic status accurately predict areas of health and overweight or obesity problems in California [START_REF] Drewnowski | Poverty and childhood overweight in California Assembly districts[END_REF].

While the association has been demonstrated by several studies, the causes of this relationship are controversial. To date, researchers have presented three primary causes for this relationship. A genetic explanation has been expressed by [START_REF] Lobstein | Prevalence of overweight among children in Europe[END_REF]. According to this perspective, fat storage depends on the genetic make-up of a particular individual [START_REF] Lobstein | Prevalence of overweight among children in Europe[END_REF]. The economic explanation is based on the argument that the poor have fewer food options than the rich. Food is one of the primary expenses in the budgets of low income households and such families have few choices available to them. Thus, the poor buy inexpensive food products that are typically high in calories [START_REF] Tanumihard | Poverty, obesity and malnutrition: an international perspective recognizing the paradox[END_REF][START_REF] Darmon | Does social class predict diet quality?[END_REF] and fat. These foods also do little to satiate hunger and thus promote overeating. The third explanation is linked to a sociological perspective. According to this viewpoint, people in middle-and uppersocio-economic groups tend to have more negative opinions regarding overweight and obesity than their counterparts in the lowest socio-economic groups [START_REF] Gortmaker | Social and economic consequences of overweight in adolescence and young adulthood[END_REF]cited in Wardle & Griffith, 2001). A survey (Telefonischer Gesundheitssurvey (2003) cited in [START_REF] Koch | Beiträge zur Gesundheitsberichterstattung des Bundes: Armut, soziale Ungleichheit und Gesundheit[END_REF] conducted in Germany in 2003 confirmed that overweight and obesity are more prevalent among those who hold a secondary school degree than among those who graduated high school: 76.0 % of men and 72.3 % of women with a secondary school education are overweight or obese as opposed to 60.8 % of men and 38.2 % of women with a high school education. These figures are in line with other studies conducted in the U.S. and in Europe, which have identified a negative linear association between obesity and education in both regions (OECD, 2009). Furthermore, education seems independent from other variables such as income: one study has found that cultural standing is more significant than economic standing in the prevalence of overweight and obesity [START_REF] Régnier | Obésité, corpulence et statut social: une comparaison France / Etats-Unis (1970-2000[END_REF].

In response to these new ways of interpreting overweight and obesity, researchers have started to consider the relationship between the built environment and being overweight or obese.

Overweight, obesity and the built environment

The starting point for studying the relationship between overweight or obesity and urban form is the assumption that cities influence human behaviour and their inhabitants' way of life. Most articles examine how cities and the build environment interact with the weight equation, defined by energy expenditure and caloric intake. The common theme underlying these studies is that the post-WW II city is responsible for the 'obesity epidemic' because they concentrate all of the triggers of overweight and obesity in one place: supermarkets with frozen dinners, round the clock fast food, and the urban sprawl that leads to a dependence on cars for transportation. Thus, researchers are dealing with three domains: urban sprawl (which is linked to a dependency on cars), the availability of recreational areas (or lack thereof) and the type of foods available [START_REF] Feng | The built environment and obesity: A systematic review of the epidemiologic evidence[END_REF].

Urban sprawl and means of transportation

Nearly all researchers focusing on the link between being overweight or obese and urban sprawl are from North America. Thus, patterns of urban sprawl resulting from land development are one of the main foci of these studies. Urban sprawl is typically characterised by low population density and discontinuous (or decentralised) development [START_REF] Plantinga | A spatial economic analysis of urban land use and obesity[END_REF]. [START_REF] Ewing | Relationship between Urban Sprawl and Physical Activity, Obesity, and Morbidity[END_REF] add the criterion of low-connectivity in defining urban sprawl.

Although some researchers have argued that neighbourhood environmental characteristics influence the means of transportation used by inhabitants, U.S. academics have shown that people living in denser areas with high street connectivity are less likely to be obese than inhabitants of low-density areas (Saelens et al., 2003 a). Many U.S. academics have linked being overweight or obese to urban sprawl [START_REF] Killingsworth | Development and public health: could our development patterns be affecting our personal health[END_REF][START_REF] Ewing | Relationship between Urban Sprawl and Physical Activity, Obesity, and Morbidity[END_REF]. [START_REF] Ewing | Relationship between Urban Sprawl and Physical Activity, Obesity, and Morbidity[END_REF] found that people living in sprawling cities walk less in their leisure time, weigh more, and are more likely to be obese regardless of their age [START_REF] Ewing | Relationship between Urban Sprawl and Physical Activity, Obesity, and Morbidity[END_REF]. Means of transportation is linked to urban forms and researchers in the U.S. pay particular attention to the role played by cars in increasing the overweight and obesity rates [START_REF] Frank | Obesity Relationship with Community Design, Physical Activity, And Time Spent In Cars[END_REF][START_REF] Eid | Fat City: questioning the relationship between urban sprawl and obesity[END_REF].

Urban sprawl is linked to the use of motorised means of transportation because: 'We sit in cars. We don't walk to the store at the corner.

[…] Residents often live on curvy, dead-end streets (often culde-sac) that feed into high-volume highways leading to segregated uses.

[…]' (Pierece, 2001quoted in Sui, 2003). Before WW II, car drivers were often assumed to be overweight, because such transportation was only available to those living in wealth, the upper classes [START_REF] Dupuy | Les territoires de l'automobile[END_REF]. This clichéd image has now largely been dismissed. However, cars as a mean of transportation are still associated with overweight and obesity. Between 1977 and 1985, walking trips decreased from 25 to 10 % to total share of all trips made in the U.S. [START_REF] Sui | Musings on the Fat City: are obesity and urban form linked[END_REF]. During this same period, driving trips increased from 84 to 90% [START_REF] Sui | Musings on the Fat City: are obesity and urban form linked[END_REF]. Similar trends have been observed in the U.K. [START_REF] Tudor-Locke | Active commuting to school: an overlooked source of childrens' physical activity[END_REF] and among children in Germany (Das Deutsche Kinderhilfswerk e.V. & Der Verkehrsclub Deutschland e.V., 2010). The human body is used far less as a power source in modern means of transportation. A study conducted in China using cross-sectional data showed a temporal correlation between the increased overweight and obesity rates, and the increase in the rate of car ownership [START_REF] Bell | The road to obesity or the path to prevention: motorized transportation and obesity in China[END_REF]. After adjusting data with other variables such as age, income, education, work and leisure activity, researchers found that the risk of being overweight or obese was 70 % greater for men and 85 % greater for women living in a household with a car than for those who did not own any motorised vehicle [START_REF] Bell | The road to obesity or the path to prevention: motorized transportation and obesity in China[END_REF]. In addition, [START_REF] Froguel | L'obésité, première maladie non infectieuse de l'histoire[END_REF] found a correlation between the number of cars and the prevalence of overweight and obesity in the U.S. Using data on BMI, time spent travelling by car and walking distance, the influence of cars on overweight and obesity has been quantified by academics who estimate that each hour spent in a car per day increase the risk of becoming obese by 6 % [START_REF] Frank | Obesity Relationship with Community Design, Physical Activity, And Time Spent In Cars[END_REF]. [START_REF] Lopez-Zetina | The link between obesity and the built environment. Evidence from an ecological analysis of obesity and vehicle miles of travel in California[END_REF] has conducted a study focusing on the relationship between overweight, obesity and vehicle travel mileage in California. Authors have found that overweight and obesity are associated with automobile use [START_REF] Lopez-Zetina | The link between obesity and the built environment. Evidence from an ecological analysis of obesity and vehicle miles of travel in California[END_REF]. This supports results published by Saelens et al. (2003 a) and [START_REF] Ewing | Relationship between Urban Sprawl and Physical Activity, Obesity, and Morbidity[END_REF] [START_REF] Lopez-Zetina | The link between obesity and the built environment. Evidence from an ecological analysis of obesity and vehicle miles of travel in California[END_REF].

Equally, the relationship between active transportation and the absence of overweight and obesity has also been confirmed by other studies [START_REF] Gordon-Larsen | Associations among active transportation, physical activity, and weight status in young adults[END_REF][START_REF] Grafova | Neighborhoods and Obesity in Later Life[END_REF]. Some researchers have compared physical activity levels in New York and Geauga Counties [START_REF] Ewing | Relationship between Urban Sprawl and Physical Activity, Obesity, and Morbidity[END_REF]. They argue that high obesity rates are caused by sprawling development because people walk less than those living in a dense area of a city. Inhabitants of New York County, the more densely populated of the two areas, walk about 79 minutes more each month than inhabitants of Geauga County [START_REF] Ewing | Relationship between Urban Sprawl and Physical Activity, Obesity, and Morbidity[END_REF]. [START_REF] Lopez | Neighborhood risk factors for obesity[END_REF] conducted a study using data from the U.S. Behavioral Risk Factor Surveillance System and the U.S. Census. He obtained results similar to those published by Ewing et al: people living in denser areas have lower BMIs. He found that the risk of being overweight or obese decreases with increasing population density [START_REF] Lopez | Neighborhood risk factors for obesity[END_REF]. However, these results differ for different ethnic groups [START_REF] Ewing | Relationship between Urban Sprawl and Physical Activity, Obesity, and Morbidity[END_REF][START_REF] Lopez | Neighborhood risk factors for obesity[END_REF].

In contrast, other academics have suggested that BMIs are higher in areas of urban sprawl due to the personal preferences of obese people who tend to live where they can easily use their cars [START_REF] Plantingra | The association between urban sprawl and obesity: Is it a twoway street[END_REF][START_REF] Eid | Fat City: questioning the relationship between urban sprawl and obesity[END_REF]. According to these researchers, obese people make this decision because they dislike walking [START_REF] Eid | Fat City: questioning the relationship between urban sprawl and obesity[END_REF]. Thus, a link between BMI and urban sprawl has been identified. However, causality has yet to be proven.

Availability of recreational areas

Another domain explored by researchers is the issue of the availability of recreational areas in neighbourhoods. Indeed, urban living arrangements may affect the energy expended by inhabitants.

Researchers have noted that people spend less time outside today than they did 30 years ago: according to [START_REF] Ewing | Relationship between Urban Sprawl and Physical Activity, Obesity, and Morbidity[END_REF], 90 % of North Americans lived in a city and most of them spend 90 % of their lives indoors. Similar results were found in Germany. Similar results were obtained among children and adolescents: more than 50 % of those who live near a public park, green area, or playground practice physical exercise frequently, as opposed to about 40 % of those who do not live near a recreational area (WHO, 2007 a). However, it should be pointed out that similar associations have been found with socio-economic variables (WHO, 2007 a). The LARES conclusions are that individual characteristics such as age or education influence the amount of physical activity undertaken, but the proximity of recreational areas nearby has a positive influence on sport patterns (WHO, 2007 a). However, some studies have suggested that the availability of recreational areas may be a more important determining factor than the socioeconomic status of inhabitants for improving their state of health [START_REF] Giles-Corti | Socioeconomic status differences in recreational physical activity levels and real and perceived access to a supportive physical environment[END_REF][START_REF] Mitchell | Effect of exposure to natural environment on health inequalities: An observational population study[END_REF]cited in Marmot Review, 2010).

Another aspect of a neighbourhood which encourages physical activity -and thus, may have an influence on caloric expenditure -is the safety perception. According to the LARES study, adults seem to be more likely to engage in sports if they perceive their neighbourhood as safe (WHO, 2007 a). Similar trends have been observed in Australia where a study conducted among adolescents showed that the perception of safety influences the types of physical activities adolescents engage in [START_REF] Kerr | Exercise aids, neighbourhood safety, and physical activity in adolescents and parents[END_REF]. Parental perceptions of safety also influence the use of active means of transportation among children [START_REF] Timperio | Perceptions about the local neighbourhood and walking and cycling among children[END_REF].

The availability of recreational areas is also linked to the idea that some neighbourhoods are 'walkable' while others are not. A 'walkable' area is defined as an area that meets at least three criteria: high population density, pedestrian-friendly design, and diversity of land use [START_REF] Cervero | Travel demand and the 3Ds: density, diversity, and design[END_REF]. Another approach adds the criterion of a good public transportation network [START_REF] Frank | Linking objective physical activity data with objective measures of urban form[END_REF]. Some U.S. studies have identified a negative correlation between the risk of being overweight or obese and walkable areas where inhabitants use non-motorised means of transportation. No clear effects of recreational availability on BMI have been found but it should still be highlighted that recreational areas have a positive impact on inhabitants' general state of health [START_REF] Review | Fair Society, Healthy Lives: A Strategic Review of Health Inequalities in England Post-2010[END_REF].

The food environment

Concerns regarding the built environment encompass the issue of the diversity of destinations [START_REF] Cervero | Travel demand and the 3Ds: density, diversity, and design[END_REF] and the types of foods available to inhabitants (and thus their caloric intake). Another important question addressed by investigators in this field is the relationship between the food environment and the prevalence of overweight and obesity. In particular, researchers have focused on the effect of fast food and the types of food readily available at neighbourhood supermarkets on overweight and obesity.

Studies conducted within the U.S. have reported contradictory results. A 2004 study conducted among preschool children in Cincinnati, Ohio examined how distance to playgrounds and fast food restaurants affects BMI. The authors found that distance to fast food restaurants had no effect on BMI [START_REF] Burdette | Neighborhood playgrounds, fast food restaurants, and crime: Relationships to overweight in low-income preschool children[END_REF]. Other researchers have examined the relationship between overweight, obesity and access to different types of food. They found a statistically significant relationship between access to fast food and overweight or obesity, and argue that increased access to supermarkets and decreased access to fast food can reduce inhabitants' weight [START_REF] Chen | Obesity in urban food markets: Evidence from Geo-referenced micro data[END_REF]. Moreover, in a study conducted in the U.S., researchers found that restricting fast food near schools may significantly decrease the prevalence of overweight and obesity among school children. However, implementing similar policies in residential areas does not yield the same results for adults [START_REF] Currie | The effect of fast food restaurants on obesity[END_REF].

Another recent research focus considers access to food rather than distance to fast food restaurants. In one study, researchers examined neighbourhood characteristics and both the number of and the distance to the nearest fast food restaurants or grocery stores. They found that there are fewer restaurant and grocery store options within a three-mile radius of neighbourhoods composed of residents of lower socio-economic status [START_REF] Sharkey | Association between neighborhood need and spatial access to food stores and fast food restaurants in neighborhoods of colonias[END_REF]. Similar results were obtained in a study conducted in Detroit. This study brings to light the fact that the poorest areas have fewer supermarkets [START_REF] Zenk | Neighborhood racial composition, neighborhood poverty, and the spatial accessibility of supermarkets in Metropolitan Detroit[END_REF]. In both case studies, this means that inhabitants have fewer food types and healthy choices in their neighbourhood and have to travel greater distances to access food.

The food environment refers to the question of food security, which is the availability of the types of foods necessary for a healthy diet. However, the link between access to different types of foods and overweight or obesity has not yet been clearly defined. Some relationship has been noted. However, causality cannot yet be clearly assigned.

Conclusions

The results of the studies cited here indicate that urban planning can help combat the 'obesity epidemic', an idea congruent with Oscar Newman's Defensible Space Theory. This theory was developed in the 1970s and posits that urban design can increase inhabitants' security. Indeed, all researchers working in this field accept the idea that external factors affect inhabitants' health. Salutogenesis, a term coined by medical sociologist Aaron Antonovsky , is an important aspect of any city. Antonovsky's goal was to understand why some people stay healthy in an unfavourable environment by exploring the behaviours and other factors that promote good health rather than understanding the causes of disease [START_REF] Lindström | Professor Aaron Antonovsky (1923-1994): the father of the salutogenesis[END_REF]. Antonovsky did not consider good health a state but rather a goal people strive for. This theory was adopted by the WHO in the Ottawa Charter for Health Promotion (1986) [START_REF] Eriksson | A salutogenic interpretation of the Ottawa Charter[END_REF]. Indeed, one area of action identified by the Ottawa Charter is the creation of an environment that encourages healthy lifestyles and has a significant, positive impact on health (WHO, 1986). By supporting the idea that urban planning can help inhabitants stay fit, academics and organizations like the WHO accept that environmental determinants play a role in helping people lead healthy lives (WHO, 2006).

However, the articles considered present variable and, sometimes, contradictory results. This reflects the complicated nature of overweight and obesity but also the variety of indicators and the different types of environmental exposure -which are not always comparable -considered in these different studies. Such indicators include the socio-economic or ethnic groups examined, or the scale of focus of the case studies. The complexity of this question makes it difficult to draw clearcut conclusions regarding the importance of external factors such as those above on the overweight and obesity prevalence. As noticed by [START_REF] Handy | Critical Assessment of the Literature on the Relationship Among Transportation, Land Use, and Physical Activity[END_REF], there are many variables related to the built environment that influence physical activity levels. Researchers have not taken all of these options into consideration, and thus the findings for certain parameters are disproportionately strong. According to [START_REF] Handy | Critical Assessment of the Literature on the Relationship Among Transportation, Land Use, and Physical Activity[END_REF], the association between population density and active means of transport was consistent but the relation between transport systems and means of transport was not so significant. Additionally, most studies are cross-sectional and thus do not define the nature of the relationship (correlation or causality) between the variables, do not observe the change in behaviour after a change in the built environment and do not exclude an interpretation bias [START_REF] Handy | Critical Assessment of the Literature on the Relationship Among Transportation, Land Use, and Physical Activity[END_REF]. [START_REF] Handy | Community Design and Physical Activity: What Do We Know? -and what DON'T we know[END_REF] concluded one paper by writing that the only sure thing is that some changes made in community design will increase the opportunities to practise physical activity but will not necessarily increase the amount of exercise undertaken by the inhabitants. Nevertheless, researchers agree that an 'anti-obesogenic' neighbourhood must in particular have a high population density and a mix land-use in order to promote active means of transportation, and offer many public transportation options. Such neighbourhoods also must have green areas and playgrounds, and provide a number of food choices in order to help inhabitants stay fit and maintain healthy lifestyles.

The 'anti-obesogenic environment' garnered favour first among scientists and then among politicians with the objective of helping inhabitants stay healthy. Overweight and obesity have only recently become a buzz word in the media. Mass media has increased public awareness of this health problem through government campaigns which aim to encourage people to adopt a healthier diet (e.g., the '5 a day' programme in the U.K.) and increase their level of physical activity (e.g., the 'Walking to school' and 'Bike to work' projects in Germany and the 'Active living programs' of the Robert Wood Johnson Foundation in the U.S.A.). Prince Charles, himself a famous public personality, is concerned with overweight and obesity, and his Foundation for the Built Environment is planning a town in Scotland to help inhabitants stay fit [START_REF] Watson | Royal welcome to first 'Trim Town[END_REF]. However, one may question the relevance of our current way of thinking in Europe and how directly we can transfer solutions to this problem from the U.S. to Europe.

A European problem?

Because of its various components, researchers focusing on the link between overweight, obesity and urban design come from many fields. However, it is interesting to note that these researchers come from only a few countries.

Transferring the solutions

The problem of the relationship between urban planning and health status concerns more and more researchers worldwide. However, U.S. studies have promoted this issue and represent an important part of the research conducted. This may be because the 'epidemic' started earlier in North America than it did in Europe. However, different countries have their own unique urban structures, e.g., the size and organization of cities; and distinct cultural behaviours which may bear on the country's overweight and obesity rates. Thus, information gathered in studies from North America may not be readily applicable to the situation in Europe. For example, urban sprawl, eating habits, and mobility preferences vary among the two regions and there is a greater degree of segregation based on socioeconomic status in the U.S. than there is in Europe.

The German paradox

Based on the results published by North American researchers, Germany stands out as a unique case because of its high overweight and obesity rates and its compact city organization.

Overweight and obesity in Germany

'The time when obesity was thought to be a problem on the other side of the Atlantic has gone by', said Mars Di Bartolomeo in March 2005 (cited in [START_REF] Helmert | Die 'Adipositas-Epidemie' in Deutschland -Stellungnahme zur aktuellen Diskussion[END_REF]. This is particularly true for several European countries, including Germany, where according to statistics from the International Association for the Study of Obesity (2010), 45.5 % of men and 29.5 % of women are overweight in Germany. The graphic below (Graph 1), constructed using measured data ii , shows that Germany is one of the European countries most affected by overweight and obesity. , 2009). However, the regional differences in the prevalence of overweight and obesity must be highlighted. For example, the childhood overweight rate varies from 7.2 % in Thuringia to 13.6 % in Mecklenburg-Vorpommern [START_REF] Moss | Prävalenz von Übergewicht und Adipositas bei deutschen Einschulkindern[END_REF]. As shown by Graph 1, German adults are not the most overweight people in Europe but the overweight and obesity rates are high. The overweight rate is even higher in Germany than in the U.S. According to the Robert Koch Institute, the direct and indirect costs of overweight and obesity were between 3.1 and 5.5 % of the country's total health costs in 1995 (Robert Koch Institut, 2003). Experts predict an increase in these costs in the future. As in other countries, the incidence of overweight and obesity is quickly increasing in Germany.

While overweight and obesity are a growing concern in Germany, considerable research has been undertaken in an attempt to understand the epidemic and its causes. However, German studies are almost exclusively dedicated to the pathology of the fact accumulation. Political action focuses on encouraging healthier diets and participation in sports. The built environment is only marginally considered, although German cities typically meet the 'anti-obesogenic' requirements.

Urban culture and mobility

Despite the high rate of overweight and obesity, German cities are typically in line with the recommendations made by U.S. academics for creating a built environment that fights overweight and obesity. In fact, population density is high in Germany (230 inhabitants per square kilometre, compared to 31.15 inhabitants per in the U.S.) and cities are generally described as compact.

Inhabitants are encouraged to walk or cycle in those cities thanks to mixed-use development (and thus shortened distances). There are green areas and playgrounds, and an extensive public transportation system. Several elements encourage pedestrians and non-motorised transportation, including paths or marked lanes designated for use by cyclists, cities are pedestrian-friendly, autofree zones increase pedestrian safety, reduced traffic zones and speed limits in residential areas (ca. 30 km or 18 miles per hour) also encourage pedestrian travel. Thus, pedestrians and cyclists feel confident and are more likely to use active means of transportation than their counterparts in the U.S. [START_REF] Pucher | Promoting safe walking and cycling to improve public health: Lessons from The Netherlands and Germany[END_REF][START_REF] Bassett | Walking, cycling, and obesity rates in Europe, North America, and Australia[END_REF] Pucher and Dijkstra ( 2003) have showed that German pedestrians and cyclists have a safe environment in which to move. German pedestrians are three times less likely to be killed than are American pedestrians [START_REF] Pucher | Promoting safe walking and cycling to improve public health: Lessons from The Netherlands and Germany[END_REF]. The urban design and sense of security also mean that the elderly stay more active: Germans aged 75 and older walk for 48.0 % of their trips compared with only 6.0 % of Americans aged 65 and older [START_REF] Pucher | Promoting safe walking and cycling to improve public health: Lessons from The Netherlands and Germany[END_REF]. According to the European Commission's Directorate-General for Energy and Transport (cited in [START_REF] Bassett | Walking, cycling, and obesity rates in Europe, North America, and Australia[END_REF], each European walked about 382 km (237 miles) during 2000, compared to 140 km (87 miles) for their American counterparts.

The Bottrop case

In order to better understand the 'German paradox', we examine a medium-sized city as a case study. We focus on Bottrop, a city located in North Rhine-Westphalia, the westernmost region of Germany. Bottrop has an area of 10 km 2 and is occupied by some 120,000 inhabitants (Stadt Bottrop, 2007). Density rates vary across the districts of Bottrop, from 61 to 7,970 inhabitants per km 2 (Stadt Bottrop, 2007). For comparison, there are 529 inhabitants per km 2 in North Rhine-Westphalia and 230 inhabitants per km 2 in Germany. We chose to focus on Bottrop because this city has one of the highest overweight and obesity rates in the region [START_REF] Liga Nrw | Übergewicht und Adipositas bei Kindern und Jugendlichen[END_REF]. This city is also interesting because there are 'two Bottrops': the denser, historic city to the south; and the more recent neighbourhood of Kirchhellen to the north, which was its own municipality until 1976 reforms to the structure of local governance (Gebietsreform). U.S. literature pays particular attention to the relationship between population density, overweight and obesity. In contrast, we focus on the different urban contexts represented by this city.

Bottrop was examined using 2007 municipal data. The poverty map is coterminous with the overweight and obesity maps at the statistical district iii scale. Fewer children were overweight or obese in the north, which is the richest area of Bottrop. On average, 13.19 % of children living in southern statistical districts of the city were overweight or obese, compared with an average of 4.36 % for those living in northern statistical districts (Stadt Bottrop, 2007). The level of unemployment and the number of citizens receiving social security benefits were also higher in southern statistical districts than in the north: on average 5.10 % of the south inhabitants were claiming unemployment benefits and 11.18 % were receiving social security benefits (Hartz IV) as opposed to 2.25 % and 3.53 % respectively in the north of Bottrop (Stadt Bottrop, 2007). It should be highlighted that the levels of overweight and obesity in children were lower in the north of the city, which is not as 'anti-obesogenically' designed (lower population density, higher travel distances, low degree of mixed-land use) area. Indeed, we found no association between urban sprawl and overweight or obesity. Higher childhood overweight and obesity rates were found in the higher-density areas; these areas were located in the south and they meet the basic anti-obesogenic requirements (e.g., pedestrian zones, speed reduction zones, playgrounds, green areas, cycling facilities, a sophisticated public transportation system). Although researchers in the U.S. pay particular attention to the effect of cars on the increasing overweight and obesity rate [START_REF] Frank | Obesity Relationship with Community Design, Physical Activity, And Time Spent In Cars[END_REF][START_REF] Eid | Fat City: questioning the relationship between urban sprawl and obesity[END_REF], we found no association between cars, overweight and obesity: the number of cars per inhabitant in the south of Bottrop is lower than that in the north (on average 0.48 in the south as opposed to 0.60 in the north).

After this statistical exploration, it is hard to find a relationship between weight, means of transportation, and urban design based on the conclusions reached by U.S. academics. Thus, we have added a more qualitative dimension to this quantitative analysis. Conducting observations in the city, we recorded inhabitants' use of public space. We found, that children and adults living in the south spend more time outside than those living in the north. According to Mead & Guth (1945, cited in INSERM, 2000), eating habits seem affect weight status more than environmental factors. Indeed, if we found no differences in the food environment, we noticed different eating habits.

There is no 'food desert' in Bottrop, those areas where inhabitants cannot access affordable healthy foods due to physical distance or financial reasons (people do not have the chance to go supermarkets because they, for example, have no car, public transport is limited or the healthy food is too expensive). Conducting observations in two stores of the same supermarket chain within Bottrop where the same products were sold, we observed that food buying habits vary based on neighbourhood of residence. We observed that the socio-economic distribution (north-south) affects the number of calories per gram of food and the quality of products purchased. Field observations showed that people living in the north buy considerably more vegetables and fish, and fewer readymade meals. In the south, people buy more carbonated beverages, ready meals and other popular foods from Western Eurasia.

Conclusions regarding the German paradox

Conclusions reached in North America regarding the relationship between mobility, urban design, and weight status seem irrelevant to this less suburban European country. The case study presented above suggests that the effect of urban planning on weight status is less significant. If active means of transportation for daily travel contributes to better health and helps in weight control, overweight and obesity should be less prevalent in Germany. Thus, more is involved here than just the built environment and overweight or obesity have to be considered as the result of a combination of quantitative or qualitative variables (Foresight, 2007). Non-identifiable factors in a cross-sectional study such as the evolution of eating habits or the evolution of the amount of physical activity undertaken (at work, at home, during spare time) may play a role. Given the case of Germany, which is a paradox according to information based on U.S. studies, the differences in the built environment between North America and Europe may influence its role and importance. The built environment may have less of an impact on overweight and obesity in Europe. Transferring solutions to this problem from North America to Europe will require some adaptations. These may include focusing on the morphological aspect of urban design and on the inhabitants' characteristics, thus taking 'city' to encompass both the buildings and the people that inhabit them [START_REF] Sui | Musings on the Fat City: are obesity and urban form linked[END_REF].

Conclusion

Several studies have addressed the issues of weight status, transportation options, and the built environment. Many studies deal only with a limited number of variables in trying to explain the relationship between overweight, obesity and environmental factors. Many others are crosssectional and thus establish associations but no correlation [START_REF] Handy | Critical Assessment of the Literature on the Relationship Among Transportation, Land Use, and Physical Activity[END_REF]. Regardless of the sometimes contradictory conclusions reached in North America, the 'anti-obesogenic' environment has become an issue that is impossible to ignore. This new direction in the fight against obesity appeared first in the U.S. and was then transferred to Europe. However, this solution cannot simply be transferred directly. Rather, country-specific solutions should be proposed.

Most studies have focused on metropolitan areas in a North American urban context [START_REF] Feng | The built environment and obesity: A systematic review of the epidemiologic evidence[END_REF]. The scale of North American cities is not comparable to that of even the largest European city. In addition, there are several important differences between the two continents, including the construction history, processes involved in populating cities, mobility behaviours, and eating habits. Thus, it is inappropriate to directly apply socially oriented (and thus culture dependent) solutions based on studies conducted in North America to the European context. This is well-illustrated by the German case study. Based on studies conducted in the U.S., Germany meets the 'anti-obesogenic' requirements, yet overweight and obesity are prevalent. To understand this paradox, we must not only focus on the morphological aspects of cities but also strive to gain a deeper understanding of how inhabitants interact with each other and their city.

Solutions posited for North America are also inadequate for addressing the European problem because of differences in the built environment and available means of transportation. Future studies should examine social aspects of overweight and obesity at the individual-and communitylevel. Further research is needed in order to consider individual variables so that we might understand the role that each one plays in promoting overweight and obesity, and consider how environment and health interact in both North American and European contexts.

Graph 1 :

 1 Percentage of women and male adults overweight and obese (Data: International Association for the Study of Obesity, 2010) Overweight and obesity do not only concern German adults. 20.4 % of German boys and 20.1 % of German girls aged 5-17 were overweight in 2002 (International Association for the Study of Obesity

Table 1 : Physical activity frequency and availability of recreational areas (Source: WHO

 1 According to the 'Zu Fuss zur Schule' ('Walking to school') project organisers, 30 years ago children spent 4 hours per day outside. Today, they spend only half an hour outside and 9 hours seated indoors each day (Das Deutsche Kinderhilfswerk e.V. & Der Verkehrsclub Deutschland e.V., 2010). The reduced amount of time spent outside may explain why inhabitants do not burn enough calories to stay fit. Thus, academics have started to examine the link between the availability of recreational areas (playgrounds and green areas that facilitate physical activity and so increase caloric expenditure) and the prevalence of overweight and obesity in neighbourhoods. A positive correlation has been identified between physical activity and a lack of overweight and obesity among inhabitants of North America and Europe regardless of age[START_REF] Krahnstoever Davison | Do attributes in the physical environment influence children's physical activity? A review of the literature[END_REF][START_REF] Nielsen | Do green areas affect health? Results from a Danish survey on the use of green areas and health indicators[END_REF][START_REF] Coombes | The relationship of physical activity and overweight to objectively measured green space accessibility and use[END_REF]. The LARES (Large Analysis and Review of European housing and health Status) project(WHO, 2007a), conducted among 8519 residents in 3373 households by the WHO, quantified this relationship and showed that the availability of a public park or green space is cross-sectionally associated with a higher level of physical activity (Table1).

	Frequency of physical	Whole population	Recreational area nearby (within 100 meters)
	activity	(2-80 years old)	Yes	No
	Seldom	41.8 %	38.3 %	46.4 %
	Occasionally	31.9 %	33.2 %	30.2 %
	Frequently	26.3 %	28.5 %	23.4 %
				, 2007
	a, p. 17)

Acknowledgements

The author wants to thank Professor David Banister and Dr. Moshe Givoni for giving her the opportunity to write this paper and the reviewers for their time and their useful comments that have helped to improve the manuscript. I also thank Professor Gabriel Dupuy (Université Paris 1) for having supervised the thesis where the results were originally presented. The thesis was written as part of a research project conducted at the Université Paris 1 looking at the transferability of North American approaches to Europe. A paper written in French by G. Dupuy, C. Minster, B. Ory, and R. Watel (Université Paris 1) entitled 'Environnement urbain et obésité : peut-on transposer en Europe les approches nord-américaines ?' looks closely at the transferability from the U.S. to Europe by bringing together case studies conducted in several European countries. This paper is in process.