Antoine Deza
email: deza@mcmaster.ca

Jean-Baptiste Hiriart-Urruty
email: jean-baptiste.hiriart-urruty@math.univ-toulouse.fr

Lionel Pournin
email: lionel.pournin@univ-paris13.fr

AND Jean-Baptiste Hiriart-Urruty

J.-B Hiriart-Urruty

Polytopal balls arising in optimization

come

Introduction

A family of norms on R d resembling the usual Euclidean norm, yet polytopal in the sense that the balls for these norms are polytopes, were introduced in [START_REF] Watson | Linear best approximation using a class of polyhedral norms[END_REF] as a tool to solve linear approximation problems.

These norms, defined by (1.1)

x (k) = inf{ u 1 + k v ∞ : u + v = x},
where x is a vector from R d and k is a parameter that belongs to the interval [1, d], were later considered in the context of robust optimization [START_REF] Bertsimas | Robust linear optimization under general norms[END_REF], a method to deal with linear optimization under uncertain constraints. As shown in [START_REF] Bertsimas | Robust linear optimization under general norms[END_REF], defining the uncertainty constraints using these norms, referred to as D-norms in this case, allows for an efficient way to solve robust optimization problems. It is further observed in [START_REF] Gotoh | Two pairs of families of polyhedral norms versus p-norms: proximity and applications in optimization[END_REF][START_REF] Pavlikov | CVaR norm and applications in optimization[END_REF] that these norms are naturally connected with the conditional value at risk, a popular metric used in quantitative finance: just as the conditional value at risk, these norms, called the CVaR norms in this other context, put the emphasis on the largest coordinates of a vector from R d . In particular, it is shown in [START_REF] Pavlikov | CVaR norm and applications in optimization[END_REF] that these norms are a solution to an optimization problem regarding the conditional value at risk. The same norms also appear in optimization problems over sets of matrices [START_REF] Wu | On the Moreau-Yosida regularization of the vector k-norm related functions[END_REF], where they are called vector k-norms, and in sparse optimization [START_REF] Gaudioso | Feature selection in SVM via polyhedral k-norm[END_REF][START_REF] Gotoh | DC formulations and algorithms for sparse optimization problems[END_REF]. In the latter case, one is faced with the sparsity constraint on the solutions to a problem: the desired solutions-vectors from R d -are required to have a prescribed number of non-zero coordinates. This happens for instance in data science, in machine learning [START_REF] Pokutta | Deep neural network training with Frank-Wolfe[END_REF], in mathematical imaging, or in statistics among other fields. The number of non-zero coordinates of a vector x of R d is often denoted by x 0 in the optimization literature. Formally, if x 1 to x d denote the coordinates of x,

x 0 = |{i : x i = 0}| .

Date: (date1), and in revised form (date2). Despite what the notation suggests, this quantity does not define a norm since it is not absolutely homogeneous. In fact, the map x → x 0 is not convex, or even continuous and as a result, it is often replaced by x → x 1 in order to make sparse optimization problems computationally tractable [START_REF] Juditsky | Statistical inference via convex optimization[END_REF]. Another approach is to replace x 0 by the difference of two of the norms introduced above. Indeed, it is observed in [START_REF] Gaudioso | Feature selection in SVM via polyhedral k-norm[END_REF][START_REF] Gotoh | DC formulations and algorithms for sparse optimization problems[END_REF] that the sparsity constraint x 0 ≤ k is equivalent to the equality

x (k) -x (l) = 0
for any l such that k < l ≤ d. Hence, the norms defined by (1.1) allow for a computationally effective way to estimate sparsity.

As we mentioned above, the balls for these norms are polytopes. The purpose of this article is to study the combinatorics of these polytopes and their duals, by which we mean their f -vector, as well as some of their geometric properties such as their volume and the volume of their boundary.

Throughout the article, we denote by γ d the d-dimensional hypercube [-1, 1] d and by β d the cross-polytope whose vertices are the center of the facets of γ d , following the notation used by Coxeter [START_REF] Scott | Regular polytopes[END_REF]. Note that the former is the unit ball for the ∞-norm and the latter the unit ball for the 1-norm.

We consider the family of polytopes

(1.2) ρ d,k = conv β d ∪ 1 k γ d
when k ranges from 1 to d. It is shown in [START_REF] Gaudioso | Feature selection in SVM via polyhedral k-norm[END_REF] that ρ d,k is the unit ball for the norm defined by (1.1). Note that these polytopes interpolate between the hypercube and its dual cross-polytope (for another example of polytopes with that property, in a loose sense, see [START_REF] Lee | Min-up/min-down polytopes[END_REF]). In particular, ρ Recall that the norm defined by (1.1) can also be viewed as the support function of the polytope ρ d,k polar to ρ d,k [START_REF] Hiriart-Urruty | Fundamentals of convex analysis[END_REF]. Since γ d is the polar of β d , the polytopes ρ d,k provide another way to continuously deform β d into γ d , and give rise to alternate (dual) norms. These polytopes have been considered in [START_REF] Black | One-dimensional projections and monotone paths on cross-polytopes and other regular polytopes[END_REF] in the context of monotone path computation within certain polytopal graphs. It follows from (1.2) that

(1.3) ρ d,k = (kβ d ) ∩ γ d .
By duality, the f -vector of ρ d,k is obtained by reversing that of ρ d,k . However, the volume of ρ d,k requires a separate computation, which we will also provide here. While we are mainly interested in the polytopes ρ d,k and ρ d,k when k is an integer, most of our results hold for any k within the interval [1, d]. The combinatorics of ρ d,k and ρ d,k is studied in Section 2. The volume of ρ d,k and of its boundary are computed in Section 3. The same two volumes, but for ρ d,k instead of ρ d,k are computed in Section 4.

The combinatorics of ρ d,k and ρ d,k

As we mentioned earlier, the number of the (d -i -1)-dimensional faces of ρ d,k is equal to the number of the i-dimensional faces of ρ d,k . Therefore, we only need to compute the number of the faces of one of them. In order to do that, we will give a close look at the continuous deformation of γ d into β d via the polytopes given by (1.2). It will be convenient to consider the dilate kρ d,k instead of ρ d,k itself. Recall, in particular that a polytope has the same combinatorics as any of its dilates by a non-zero coefficient.

First observe that kρ d,k is obtained by pulling the centers of each facet of the hypercube γ d away from the hypercube along the axes of coordinates, until they are at a distance k -1 from the hypercube, and by taking the convex hull of these pulled points together with the vertices of γ d . In particular, when k is greater than 1 but close enough to 1, kρ d,k is obtained from the hypercube γ d by glueing pyramids over each of its facets. It immediately follows that, except for its facets, all the proper faces of γ d , are still faces of kρ d,k . Moreover, all the facets of kρ d,k are pyramids over a (d -2)-face of γ d . More precisely, if γ d-2 is a (d -2)-dimensional face of γ d , the two facets of kρ d,k incident to it are the two pyramids over γ d-2 whose apices are the points pulled from the center of the facets of γ d incident to γ d-2 . This describes the boundary complex of kρ d,k whenever 1 < k < 2. when k = 2 the two facets of kρ d,k incident to γ d-2 merge into a single facet, a bipyramid over γ d-2 whose two apices are the points pulled from the center of the facets of γ d incident to γ d-2 . By our description, all the facets of 2ρ d,2 are built this way. In particular, they are pairwise isometric. Now let us describe how the boundary complex of kρ d,k get modified when 2 < k < 3. In this case, the two apices of the bipyramid over γ d-2 are further pulled away from the hypercube, which splits that bipyramid into the convex hulls of the line segment that joins these two apices with each of the facets of γ d-2 . Again, all the facets of kρ d,k are isometric to such a convex hull. Now recall that the facets of γ d-2 are (d -3)-dimensional faces of γ d . In particular these faces are (d -3)-dimensional hypercubes. Let γ d-3 be one of these hypercubes. Observe that γ d-3 is contained in exactly three facets of kρ d,k because it is incident to exactly three (d -2)-dimensional faces of γ d . Upon reaching k = 3, these three facets merge into a single facet of 3ρ d,3 , obtained as the convex hull of γ d-3 and the equilateral triangle whose vertices are the points pulled from the centers of the three facets of γ d incident to γ d-3 . As above, all the facets of 3ρ d,3 are obtained this way. That process repeats when k belongs to an interval between two consecutive integers. In particular, when k is an integer, each of the (d -k)-dimensional faces of the hypercube gives rise to a facet of kρ d,k and all the facets of kρ d,k are obtained this way. More precisely, we obtain the following. It is noteworthy that the regular simplex mentioned in the statement of Theorem 2.1 is a face of the cross-polytope kβ d . Remark: Let us assume that k is an integer. According to Theorem 2.1, the number of facets of ρ d,k is equal to the number of (d -k)-dimensional faces of a d-dimensional hypercube. As a consequence, (2.1)

f d-1 (ρ d,k ) = 2 k d k .
By polarity, it further follows from Theorem 2.1 that the vertices of ρ d,k are exactly the 2 k d k points from {0, 1, -1} d with k non-zero coordinates. When k is not an integer, the facets of ρ d,k are obtained, in combinatorial terms, by splitting each of the facets of the polytope k ρ d, k into as many facets as a (d -k )-dimensional hypercube has. Based on Theorems 2.1 and 2.3, we now compute the f -vector of ρ d,k . From there on, we denote by f i (P ) the number of i-dimensional faces of a polytope P . According to our description, ρ d,k has 2 d +2d vertices when 1 < k < d. Recall that ρ d,1 is a d-dimensional hypercube and ρ d,d a d-dimensional cross-polytope, whose number of vertices are 2 d and 2d, respectively.

As mentioned above, in the case when k is an integer,

f d-1 (ρ d,k
) is the number of (d -k)-dimensional faces of a d-dimensional hypercube. By Theorem 2.3, when k is not an integer the number of facets of ρ d,k is the product of the number of (d-k -1)-dimensional faces of a d-dimensional hypercube with the number of facets of a k -dimensional simplex, that is

f d-1 (ρ d,k ) = 2 k+1 d k + 1 ( k + 1).
In order to complete the f -vector of ρ d,k when k is an integer, let us remark that the faces of ρ d,k are of three types: they can be faces of the hypercube γ d /k, faces of the cross-polytope β d , or neither. We first compute the number of the i-dimensional faces of ρ d,k of the latter type.

Lemma 2.4. If k is an integer, and i satisfies 1 ≤ i ≤ d -2, then the number of the i-dimensional faces of ρ d,k that are neither a face of the hypercube

γ d /k, nor a face of the cross-polytope β d is d k u j=l 2 d-j d-k j k i-j d-i d-k-j , where l = max{0, i -k + 1} and u = min{i -1, d -k -1}.
(i -j -1)-dimensional face α i-j-1 of α k-1 . Since γ d-k is a (d -k)-dimensional cube, f j (γ d-k ) = 2 d-k-j d -k j . Since α k-1 is a (k -1)-dimensional simplex, f i-j-1 (α k-1 ) = k i -j . Therefore, conv(γ d-k ∪ α k-1 ) admits exactly 2 d-k-j d -k j k i -j
faces of such as conv(γ j ∪α i-j-1 ). Now observe that if we would multiply this quantity by the number of facets of ρ d,k , conv(γ j ∪α i-j-1 ) would be counted as many times as the number of facets of ρ d,k it is incident to. Let us compute this number. The facets of ρ d,k incident incident to conv(γ j ∪ α i-j-1 ) are obtained by choosing one of the (d-k)-dimensional faces F of the hypercube γ d /k incident to γ j and contained in all the facets of γ d /k the vertices of α i-j-1 are pulled from, and by then taking the convex hull of its union with the (k -1)-dimensional simplex whose vertices are the points pulled from the centers of the k facets of γ d incident to F . Hence, there are d -i d -k -j possible choices for F , and conv(γ j ∪ α i-j-1 ) is incident to that number of facets of ρ d,k . According to these observations, there are As a consequence of these observations, we obtain the following.

d k 2 d-j d-k j k i-j d-i d-k-j faces of ρ d,
2 i+1 d i + 1 faces of dimension i when 0 ≤ i < k -1

and these polytopes do not have any common face of dimension i when

k -1 ≤ i < d.
The following is an immediate consequence of Lemmas 2.4 and 2.5.

Theorem 2.6. If k is an integer greater than 1 and less than d then, for any integer i satisfying

1 ≤ i < d -1, (i) f i (ρ d,k ) = f when i ≥ max{d -k, k -1}, (ii) f i (ρ d,k ) = 2 i+1 d i + 1 + f when d -k ≤ i < k -1, (iii) f i (ρ d,k ) = 2 d-i d i + f when k -1 ≤ i < d -k, (iv) f i (ρ d,k ) = 2 i+1 d i + 1 + 2 d-i d i + f when i < min{d -k, k -1},
where, in the right-hand side of these equalities,

f = d k u j=l 2 d-j d-k j k i-j d-i d-k-j , with l = max{0, i -k + 1} and u = min{i -1, d -k -1}.
Remark: A conjecture by Kalai [START_REF] Kalai | The number of faces of centrally-symmetric polytopes[END_REF] states that a d-dimensional centrallysymmetric polytope always has at least 3 d non-empty faces. The polytopes ρ d,k satisfy this conjecture. Indeed, recall that, when k is an integer such that 1 < k < d, ρ d,k has 2d + 2 d vertices and 2 k d k facets. Ignoring the f terms in the expression of f i (ρ d,k ) provided by Theorem 2.6, one obtains that the number of non-empty faces of ρ d,k is at least

1 + 2d + 2 d + 2 k d k + k-1 i=2 2 i d i + d-k-1 i=1 2 d-i d i ,
a sum that can be rearranged into the binomial expansion of (1 + 2) d .

The geometry of ρ d,k

Let us recall that ρ d,k is introduced in [START_REF] Gaudioso | Feature selection in SVM via polyhedral k-norm[END_REF] as an intersection of half-spaces of R d . We recover this description as a consequence of Theorem 2.1.

Corollary 3.1. If k is an integer, then ρ d,k is the set of the points x in R d such that the absolute value of any k coordinates of x sum to at most 1.

In the remainder of the section, we compute the volume of ρ d,k and that of its boundary. Let us remark that our description of kρ d,k naturally provides a polyhedral subdivision of this polytope into convex hulls of unions of hypercubes and simplices. Consider a (d -l)-dimensional face γ d-l of the hypercube γ d where l < k, and the regular (l -1)-dimensional simplex α l-1 whose vertices are the points pulled from the facets of γ d incident to γ d-l in our description of ρ d,k . As l < k, conv(γ d-l ∪ α l-1 ) is a d-dimensional polytope. When l = 0, we will take as a convention that γ d-l is the whole hypercube γ d and α l-1 is the empty set. The family of these polytopes when l ranges from 0 to k form a subdivision of ρ d,k . This subdivision turns out to be regular. In other words, it can be recovered by projecting to R d the lower faces of a (d + 1)-dimensional polytope [START_REF] Jesús | Triangulations: structures for algorithms and applications[END_REF]. In this particular case, an example of such a polytope can be obtained by identifying R d as the subspace of R d+1 spanned by the first d coordinates, by leaving the vertices of γ d within R d , by lifting the vertices of kβ d in the hyperplane of R d+1 wherein the last coordinate is equal to 1, and by taking the convex hull of all the resulting points.

According to this discussion, the volume of ρ d,k can be obtained from that of conv(γ d-l ∪ α l-1 ). This polytope can be alternatively built by starting from γ d-l , constructing the pyramid over γ d-l whose apex is a vertex of α l-1 , then taking the pyramid over that pyramid whose apex is another vertex of α l-1 , and so on until all the vertices of α l-1 have been used.

As a consequence, in order to obtain the volume of this polytope, we first compute the distance of a vertex x of α l-1 to the affine hull of the union of γ d-l and of i vertices of α l-1 other than x.

Lemma 3.2. The distance between a vertex x of α l-1 and the affine space spanned by γ d-l and by i vertices of α l-1 other than x is

k k 2 -2(i + 1)k + (i + 1)l k 2 -2ik + il .
Proof. Consider a set V of i vertices of α l-1 . We assume without loss of generality that γ d-l is the (d -l)-dimensional face of the hypercube γ d wherein the first l coordinates are equal to 1 and that the vertices of α l-1 contained in V are the ones whose positive coordinate is among the first i coordinates. Let us also translate γ d-l and α l-1 by subtracting the center of γ d-l , which can be done without loss of generality as well.

In this setting, the affine space spanned by γ d-l ∪ V contains the origin. After the translation, the first i coordinates of the points contained in V are given by the columns of the following matrix, their last d -l coordinates are equal to 0, and their l -i intermediate coordinates are equal to -1:

        k -1 -1 • • • -1 -1 -1 k -1 . . . . . . . . . -1 -1 . . . -1 -1 . . . . . . . . . k -1 -1 -1 -1 • • • -1 k -1        
.

By symmetry, the orthogonal projection of x on the affine space spanned by γ d-l ∪ V is a multiple by a coefficient λ of the sum of the points contained in V. Now observe that, by symmetry, the equation

x -λ v∈V v • y = 0,
where y is a point from V does not depend on how y is chosen within V.

Solving that equation for λ yields

λ = -2k + l k 2 -2ik + il .
Finally, we obtain

x -λ v∈V v = k k 2 -2(i + 1)k + (i + 1)l k 2 -2ik + il , as desired.
Observe that the volume of γ d-l is 2 d-l . Together with Lemma 3.2, the expression of the volume of a pyramid in terms of the volume of its base and the distance of its apex to it, provides the following.

Lemma 3.3. The volume of conv(γ d-l ∪ α l-1 ) is 2 d-l k l-1 (k -l) (d -l)! d! .
We obtain the volume of ρ d,k from Lemma 3.3.

Theorem 3.4. The volume of ρ d,k is 2 d k k -d k ! .
Proof. By the above remarks on the decomposition of ρ d,k as a polyhedral complex, and since a hypercube has 2 l d d-l faces of dimension d-l, it follows from Lemma 3.3 that the volume of kρ d,k is

2 d k l=0 k l-1 (k -l) l! .
Dividing this quantity by k d , we recover the volume of ρ d,k . In addition,

k l=0 k l-1 (k -l) l! = k k k ! ,
and we obtain the desired result.

Remark: It is noteworthy that the volume of ρ d,2 does not depend on d: by Theorem 3.4, this volume is equal to 2. Further note that ρ d,2 is a nonzonotopal parallelotope (see for instance [START_REF] Dolbilin | Parallelohedra: a retrospective and new results[END_REF] and references therein), and that ρ 4,2 is the 24-cell, a 4-dimensional regular polytope. In particular, ρ 4,2 is self-dual (in the sense that its face lattice and the face lattice of its polar are isomorphic). It turns out that, when d is greater than 2, the polytope ρ d,k is only self-dual in this special case.

Let us turn our attention to computing the volume of the boundary of ρ d,k . In the remainder of the section, we assume that k is an integer. Since all of the facets of ρ d,k are isometric and we know their number, we only need to compute the volume of one of these facets in order to establish the volume of the boundary of ρ d,k . Consider the same γ d-l and α l-1 as above but, this time, assuming that k and l coincide. In this case, conv(γ d-l ∪ α l-1 ) is a facet of kρ d,k . Observe that the polytopes conv(γ d-l ∪ S) where S ranges over the facets of α l-1 collectively define a polyhedral subdivision of conv(γ d-l ∪ α l-1 ). The volume of these polytopes can be obtained from Lemma 3.2. As a consequence, we obtain the volume of conv(γ d-l ∪ α l-1 ). Lemma 3.6. If k = l, then the volume of conv(γ d-l ∪ α l-1 ) is

2 d-k k k-1/2 (d -k)! d! .
Since the number of facets of ρ d,k is 2 k d k and these facets are all isometric, the volume of the boundary of ρ d,k is obtained as an immediate consequence of Lemma 3.6. As above, the volume of a facet of kρ d,k should be divided by k d-1 in order to get the volume of a facet of ρ d,k . Theorem 3.7. If k is an integer, then the boundary of ρ d,k has volume

2 d k! k k-d+1/2 .

The geometry of ρ d,k

By symmetry, the volume of ρ d,k is 2 d times the volume of its intersection with the hypercube [0, 1] d . That intersection is precisely made up of the points within [0, 1] d whose sum of coordinates is at most k. It turns out that an explicit formula is known for the volume of the intersection of [0, 1] d with a half-space bounded by an arbitrary affine hyperplane.

(-1) σ(v) (c -a•v) d d!π(a)
, where the sum is over the vertices v of [0, 1] d contained in H -, σ(x) is the sum of the coordinates of a point x in R d and π(x) is their product.

Using this formula, we derive the volume of

ρ d,k ∩ [0, 1] d . Proposition 4.2. The volume of ρ d,k ∩ [0, 1] d is k i=0 (-1) i (k -i) d i!(d -i)! .
Proof. The desired expression is obtained from Theorem 4.1, where c is replaced by k and a by the vector whose coordinates are all equal to 1. In this case the terms of the sum in the numerator of (4.1) only depend on the sum of the coordinates of the associated vertex v of [0, 1] d . Rearranging these terms by first summing over the d i vertices of [0, 1] d whose coordinates sum to i and then letting i range from 0 to k provides the desired result.

Remark: Note that, when k is an integer, ρ d,k ∩ [0, 1] d can be naturally decomposed into hypersimplices. More precisely, consider an integer l such that 0 ≤ l < k. The portion of [0, 1] d made up of the points whose sum of coordinates is between l and l + 1 is an hypersimplex. It is known [START_REF] Laplace | OEuvres complètes[END_REF][START_REF] Stanley | Eulerian partitions of a unit hypercube[END_REF] that the volume of hypersimplices is obtained by dividing Eulerian numbers [START_REF] Euler | Institutiones calculi differentialis[END_REF] by d!. Therefore, when k is an integer, the volume of ρ d,k ∩ [0, 1] d can also be expressed in terms of a sum of the Eulerian numbers.

The volume of ρ d,k is obtained as a consequence of Proposition 4.2.

4 d k k -d k ! k i=0 (-1) i (k -i) d i!(d -i)!
when k is an integer. We have computed this quantity up to d = 100 for all integers k such that 1 ≤ k ≤ d and found that, in these cases, the Mahler volume of ρ d,k is at least 4 d /d! as Mahler's conjecture states [START_REF] Iriyeh | Symmetric Mahler's conjecture for the volume product in the 3-dimensional case[END_REF][START_REF] Mahler | Ein Minimalproblem für konvexe Polygone[END_REF][START_REF] Nazarov | A remark on the Mahler conjecture: local minimality of the unit cube[END_REF]. Finally, let us compute the volume of the boundary of ρ d,k . Unlike ρ d,k , the facets of ρ d,k are not pairwise isometric. The facets of ρ d,k contained in a facet of γ d are isometric to ρ d-1,k-1 , and their volume is given by Theorem 4.4. All the other facets of ρ d,k are isometric to the intersection δ d-1,k of the hypercube [0, 1] d with the hyperplane made up of the points whose sum of coordinates is equal to k. It is noteworthy that, when k is an integer, δ d-1,k is an hypersimplex and, as we mention above, its volume can be computed from the Eulerian numbers. In fact, a formula for the volume of the intersection of an hypercube with an arbitrary affine hyperplane is established in [START_REF] Frank | Hyperplane sections of the n-dimensional cube[END_REF] based on [START_REF] Ball | Cube slicing in R n[END_REF].

Theorem 4.6 (Theorem 2 from [START_REF] Frank | Hyperplane sections of the n-dimensional cube[END_REF]). If a is a vector from R d whose every coordinate is non-zero, c a real number, and H the half space of R d made up of the points x satisfying a•x = c, then the volume of [-1, 1] d ∩ H is Proof. Recall that (4.2) is a sum over the vertices of [-1, 1] d . By a straightforward change of variables, that sum can be transformed into a sum over {0, 1} d that provides the volume of the intersection of the hypercube [0, 1] d with a hyperplane. Just as in the proof of Proposition 4.2, our special case is such that the terms in that sum only depend on the sum of the coordinates of the point from {0, 1} d they correspond to. These terms can therefore be rearranged as we did in the proof of Proposition 4.2, by first summing over the points whose coordinates sum to i and then, by letting i range from 0 to d. The resulting expression for the volume of δ d,k is

√ d 2(d -1)!   k i=0 (-1) i d i (k -i) d-1 - d i= k +1 (-1) i d i (k -i) d-1   .
However, it is well-known (see for instance [START_REF] Sebastián | An algebraic identity leading to Wilson's theorem[END_REF]) that

d i=0 (-1) i d i (k -i) d-1 = 0.
As a consequence,

Figure 1 .

 1 Figure 1. The polytope ρ 3,k , when k is equal to 1, 3/2, 2, 5/2, and 3 (from left to right).

, 1

 1 coincides with the hypercube γ d because β d is a subset of that hypercube. Similarly, ρ d,d is equal to the cross-polytope β d as this cross-polytope admits the dilated hypercube γ d /d as a subset. More precisely, the vertices of γ d /d are exactly the center of the facets of β d , just as the vertices of β d are the centers of the facets of γ d . As we shall see, this observation can be generalized, allowing to determine the whole face lattice of ρ d,k .

Figure 2 .

 2 Figure 2. The polytope ρ 3,k , when k is equal to 1, 3/2, 2, 5/2, and 3 (from left to right).

Theorem 2 . 1 .

 21 If k is an integer, then the facets of kρ d,k are exactly the convex hulls of the union of a (d -k)-dimensional face γ d-k of γ d with the (k -1)-dimensional regular simplex whose vertices are the points pulled from the centers of the k facets of γ d incident to γ d-k .

Theorem 2 . 3 .

 23 If k is not an integer, then the facets of kρ d,k are exactly the convex hulls of a (d -k -1)-dimensional face γ d-k -1 of γ d with one of the ( k -1)-dimensional simplices whose vertices are any k of the points pulled from the centers of the k + 1 facets of γ d incident to γ d-k -1 .

  Proof. Assume that k is an integer. According to Theorem 2.1 any facet of ρ d,k is the convex hull of the union of a (d -k)-dimensional face γ d-k of the hypercube γ d /k and the (k -1)-dimensional regular simplex α k-1 whose vertices are the points pulled from the centers of the k facets of γ d incident to γ d-k . By construction the affine hulls of γ d-k and α k-1 are orthogonal subspaces of R d . Therefore, the proper faces of conv(γ d-k ∪ α k-1 ) that are not a face of γ d-k or a face of α k-1 are exactly the convex hulls of the union of a proper face of γ d-k and a proper face of α k-1 . Moreover the dimension this convex hull is greater by one than the sum of the dimension of the faces of γ d-k and α k-1 it is constructed from. Let us consider a face that arises this way from a j-dimensional face γ j of γ d-k and a

  k obtained as the convex hull of the union of a j-dimensional face of γ d /k with a (i -j -1)-dimensional face of β d . Such faces of γ d /k and β d exist if and only if l ≤ j ≤ u with l = max{0, i -k + 1} and u = min{i -1, d -k -1}, which completes the proof. Now recall that, if k > 1, then by our description of ρ d,k , the hypercube γ d /k shares all of its faces of dimension less than d -k with ρ d,k and no other. Similarly, if k < d, then the cross polytope β d shares all of its faces of dimension less than k -1 with ρ d,k , and no other face.

Lemma 2 . 5 .

 25 If k > 1, then ρ d,k and γ d /k share 2 d-i d i faces of dimension i when 0 ≤ i < d -k and they do not share any face of dimension i when d -k ≤ i < d. If k < d, then ρ d,k and β d share

Theorem 4 . 1 (

 41 [START_REF] Barrow | Spline notation applied to a volume problem[END_REF]). If a is a vector from R d whose every coordinate is nonzero, c a real number, and H -the half space of R d made up of the points x satisfying a•x ≤ c, then the volume of [0, 1] d ∩ H -is (4.1)

Theorem 4 . 4 .(- 1 )

 441 The volume of ρ d,k is 2 d k i=0 i (k -i) d i!(d -i)! . Remark: Since ρ d,k and ρ d,k are polar to one another, their Mahler volume is the product of their volumes. Hence, by Theorems 3.4 and 4.4, the Mahler volume of these polytopes is

6 . 4 . 7 .

 647 v + c) d-1 s(a•v + c)π(v) 2(d -1)!π(a), where the sum is over the verticesv of [-1, 1] d , s(x)is the sign of a number x, and π(x) is the product of the coordinates of a point x in R d .We derive the volume of δ d-1,k for any k within[1, d] using Theorem 4.Proposition The volume of δ d-1,k is

7 . 4 . 8 .

 748 i d i (k -i) d-1 = -k i=0 (-1) i d i (k -i) d-1 ,and the desired result follows.Recall that the facets of ρ d,k are either isometric to ρ d-1,k-1 or to δ d-1,k . As ρ d,k has 2d facets isometric to ρ d-1,k-1 and 2 d facets isometric to δ d-1,k , we obtain the following from Theorem 4.4 and Proposition 4.Theorem The volume of the boundary of ρd,k is i (k -1 -i) d-1 i!(d -i -i (k -i) d-1 i!(d -i)!   .

Acknowledgement. The authors wish to thank Miguel Anjos for initiating this work by bringing them together, Jun-ya Gotoh for pointing out references [3,12,26], and Sebastian Pokutta and Christoph Spiegel for pointing out references [15,25]. Antoine Deza is partially supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant Program (RGPIN-2020-06846) and Lionel Pournin by the ANR project SoS (Structures on Surfaces), grant number ANR-17-CE40-0033.


M Durand

A Picot

J Régnier

C Turpin

O Crassous

M Scohy

R Stephan

O Abassie

C Andrieux

Automated Analysis of EIS curves for PEM Fuel Cells using Dynamic Time Warping

Keywords: Dynamic Time Warping, PEM Fuel Cell, Automated Analysis, Aeronautical, Electrochemical Impedance spectroscopy, Hydrogen, Start-up and shut-down P1s

whether they are published or not. The documents may come

I. INTRODUCTION

The aeronautics industry is today driven by the prospect of a more electric aircraft incorporating efficient and environmentally-friendly technologies. In this context, Fuel Cells (FC) offer a number of interesting characteristics and are seen as a substitute for fossil fuel-based solutions currently used [START_REF] Pratt | Proton exchange membrane fuel cells for electrical power generation on-board commercial airplanes[END_REF]. Known for years, hydrogen fuel cell technologies have been recently put again on the front of the stage with GHG emissions reduction targets. The aeronautical sector is particularly interested in greening its fleets as it still represented almost 2% of global carbon dioxide emissions in 2018, according to IEA report [START_REF]Global energy and co2 status report 2018[END_REF].

Recently, research efforts have intensified on a particular fuel cell technology: the high-temperature proton exchange membrane fuel cells (HT-PEM FC) which presents notably the advantages of simplified water management, higher impurities tolerance and more usable heat [START_REF] Araya | A comprehensive review of pbi-based high temperature pem fuel cells[END_REF]. So that this technology can be embedded into aircraft, progress has to be made on both cost and durability. Part of the work involves understanding the origins and limiting the damage caused by the start and stop processes. In this perspective, start-up and shut-down (SU/SD) cycling aging tests have been carried out on a five-cell HT-PEM stack. The development of automated test benches allows repeating predefined SU/SD protocols in a reproducible manner. The evolution of the stack performances can be monitored before, during and after start and stop cycles by combining analysis of polarization curves [START_REF] Wang | A parametric study of pem fuel cell performances[END_REF], cyclic voltammetries [START_REF] Génevé | Voltammetric methods for hydrogen crossover diagnosis in a pemfc stack[END_REF] and electrochemical impedance spectroscopies (EIS) [START_REF] Tang | Recent progress in the use of electrochemical impedance spectroscopy for the measurement, monitoring, diagnosis and optimization of proton exchange membrane fuel cell performance[END_REF]. Particular attention is given to the study of EIS as this technique is feasible in operation and provides a lot of information on FC state of health and degradation behaviors.

Electrochemical Impedance Spectroscopy (EIS) is a commonly used measure to study the state of health of the FC. A small amplitude sinusoidal current excitation is imposed to the stack and its voltage response is monitored. Impedance can thus be extracted for different signal frequencies. Compared to other characterization, EIS is not much time consuming, is supposed to have a small impact on degradation and gives a lot of information about running processes.

However, the repetition of aging protocols coupled with automatic characterizations led to a large number of EIS data files raising an issue of post-processing, as aging campaign can last for several thousand hours [START_REF] Tognan | Analyse et modélisation du vieillissement d'un stack de pile à combustible pem h2/o2[END_REF]. A solution could be found thanks to shape analyzing. Information extracted from visual analysis of EIS might be difficult to transcript as one or several indicators. Dynamic Time Warping (DTW) is an algorithm that measures the similarity between 2 sequences. It has been widely used for speech processing [START_REF] Sakoe | Dynamic programming algorithm optimization for spoken word recognition[END_REF], [START_REF] Permanasari | Speech recognition using dynamic time warping (DTW)[END_REF] and more recently for data mining [START_REF] Keogh | Scaling up dynamic time warping for datamining applications[END_REF], [START_REF] Li | Time works well: Dynamic time warping based on time weighting for time series data mining[END_REF]. The present work proposes to use the DWT technique to identify the similarities between different EIS curves and use that as an indicator in order to group them by likeness in order to make the global analysis of the data easier.

The experimental test bench and PEMFC data that were used in this work are described in section II. Then, the proposed method is detailed through an example. The results obtained on the experimental data are presented and discussed in section III. Section IV concludes the paper.

II. MATERIALS AND METHOD A. Experimental setup

The fuel cell used in this work consisted of a stack of five commercial PBI-based membrane-electrode assemblies (MEAs) separated by metal bipolar plates (BP) and thermally regulated around 160∼180˚C. The stack was designed and build by Safran Power Units. This technology is known as High Temperature Proton Exchange Membrane Fuel Cell (HT-PEMFC).

The stack temperature was dynamically regulated with a Presto A40 thermoregulator from Julabo connected to an external oil circuit. The heating and cooling phases took about 20 minutes each. To ensure the repeatability of the measures and make comparison between start or stop protocols possible, an automated control of the test bench in terms of gases flows, temperatures, pressures and currents imposed to the fuel cell was developed. Hydrogen, air and nitrogen flows were controlled and measured with mass flow controllers from Brooks. The current was imposed by two active loads, one EA-EL 9080-400 from Elektro-Automatik for the floor part, and a PBZ60-6.7 from Kikusui for dynamic excitation during EIS.

Fig. 1 displays the experimental bench with the instrumented stack in the middle. The computer that supports the HMI is on the left, in a specific tower besides the bench. This part also house the FC thermoregulator (in its bottom) as well as the electrical power management system. Other specific equipment were also used for characterizations : a Diagnostack, previously developped in collaboration with Helion, controlling the load during polarization curves and EIS plotting, and an Origalys system used for cyclic voltametry.

B. Data campaign

The data used in this work are extracted from SU/SD cycling campaigns. The fuel cell is repeatedly subjected to thermal cycles to reproduce mission stresses. On startup, when a sufficient temperature is reached, current is progressively imposed. A wait for thermal stabilization under constant flows and current conditions is then done. When stability is achieved, measurements are taken to monitor performance trends. Once data are collected, the FC is turned off by gradually reducing the current before flushing the electrodes with nitrogen or air depending on the protocol chosen.

Two different SU/SD protocols were tested: P 1 and P 2. On one hand, P 1 is a laboratory-oriented SU/SD protocol with no specific constraint developed to minimize the impact of SU/SD on the FC. On the other hand, P 2 was designed in collaboration with Safran Power Units in respect with FC industrial constraints.

In order to confirm the observation of degradations by limiting the impact of singularities, each protocol combination was repeated 50 times in a row. Before and after each of these cycling block, the fuel cell was characterized specifically by cyclic voltammetry cell by cell and a polarization curve was plotted. These performance curves were voluntary limited in the current density range explored in order to lessen the impact of charcterization itself on aging. Each time, six points were taken from 0.1 to 0.6 A/cm 2 . It was also decided to only carry out these measurements between cycles of 50 repeated start-stops.

Parallel to this observation, in-operation measurements could be carried out using electrochemical impedance spectroscopy (EIS): a small amplitude sinusoidal excitation is over-imposed on current and the stack voltage response is acquired. For different sinusoidal frequencies it is then possible to plot the real and imaginary parts of the fuel cell impedance. In this study, a constant excitation amplitude of 0.77 A peak-to-peak was used. The spectrum of frequencies explored extendend from 5 kHz to 0.1 Hz. These plots reveal information on fuel cell performances and state of health during start-stop cycling. Among extractable indicators, the high frequency resistance, defined as the first intersection of the impedance plot with the real axis on a Nyquist diagram, characterizes the performances in terms of electrical charge transfer (either positive or negative). Monitoring this parameter thus reveals information on fuel cell performances and state of health during start-stop cycling. Those data can also be used for model fitting in order to improve our understanding of aging phenomena.

Please note that EIS were carried out automatically every 5 minutes as current could be imposed to the stack. This led to variability in operation conditions between characterizations even if the same conditions were achieved at the end of stabilization step. Thus, the data from each protocol can be divided into 2 classes: stabilized P Xs and unstabilized P Xu, where X is the protocol number. Moreover, the data from the break-in phase (BI) were kept in the dataset. A total of 383 EIS curves is obtained. The number of curves in each condition is summed up in table I. Added to the large number of starts and stops studied, a solution had to be developed in order to automatically process the data of the hundreds of EIS carried out. Thus the purpose of this work consists of separating the numerous EIS curves obtained under different operating conditions over aging time.

C. Dynamic Time Warping

Dynamic Time Warping (DTW) is well-known technique for measuring similarity between two time-dependent sequences that might vary in speed. It has been first developed for speech recognition [START_REF] Sakoe | Dynamic programming algorithm optimization for spoken word recognition[END_REF] and then successfully used in other domain such as data mining where it was used as a metric to replace the Euclidian distance [START_REF] Keogh | Scaling up dynamic time warping for datamining applications[END_REF]. The interest of such a technique is to take into account the possible shifting and distortion between to signals. Over the years, the weaknesses of the algorithm in terms of complexity and scale adaptability have been answered to make the DTW more reliable [START_REF] Salvador | Toward accurate dynamic time warping in linear time and space[END_REF].

Let X = (x 1 , x 2 , . . . , x N ) and Y = (y 1 , y 2 , . . . , y M ) be 2 sequences of respective size N and M . The purpose of the DTW algorithm is to find the optimal match between the points of X and Y . This is done by computing first the accumulative cost matrix C = [c ij ] of size N × M where each element c ij represents the minimum accumulative error between x i and y j . This is achieved recursively according to eq. ( 1).

c ij = d(x i , y j ) + min(c i-1,j-1 ; c i-1,j ; c i,j-1 ) (1)
In this equation, d(x i , y j ) is the distance between x i and y j . This distance is calculated as expressed in eq. ( 2). Here, the error between the derivative at instant i and j is also taken into account, where ∂x i and ∂y j are the derivative of X and Y at instant i and j respectively.

d(x i , y j ) = |x i -y j | + |∂x i -∂y j | (2)
In order to increase the computation time, only the points in the "Sakoe-Chiba band" are computed [START_REF] Sakoe | Dynamic programming algorithm optimization for spoken word recognition[END_REF]. This means that the warping path is looked up in a window w around the diagonal i = j. Once the cost matrix C has been computed, the optimal warping path p = (p 1 , p 2 , . . . , p L ) with p k = (i, j) ∈ [1..N ] × [1..M ] giving the correspondence between each elements x i and y j of both sequences is built following 3 criteria:

• Boundaries: the first and the last points of the warping path are the first and last points of the sequences. • Monotonicity: the path points p k are set by increasing order of i and j. • Step size: from p k to p k+1 , i and j can not be increased of more than 1. The optimal warping path computation is detailed in algorithm 1.

Fig. 2 shows an example of the obtained matching between 2 EIS curves. Both EIS curves were obtained on the same fuel cell for different current conditions. The blue and red curves were recorded at two different current densities. The dotted lines display the corresponding points.

It seems obvious from Fig. 2 that the obtained matching between the 2 curves is coherent. In particular, it can be noticed that the local minimum and maxima are correctly matched. From this match, it is now possible to compute the distance along Re(Z) and along Im(Z). These distances are respectively noted DistX and DistY in the following. Fig. 3 displays both DistX and DistY for the example shown in Fig. 2. In this figure, the distance is presented depending on the path index but it could also be displayed depending on the EIS frequencies. Here, the blue curve is used as the reference.

Algorithm 1 Optimal Warping Path p = (N, M ) i = N , j = M while (i > 1) and (j > 1) do if i == 1 then j = j -1 else if j == 1 then i = i -1 else m = min(c i-1,j-1 ; c i-1,j ; c i,j-1 ) if m == c i-1,j then i = i -1 else if m == c i,j-1 then j = j -1 else i = i -1, j = j -1 end if end if p = ((i, j); p) end while
The analysis of distance curves displayed in Fig. 3 is consistent with the observations that can be made on Fig. 2. Along Re(Z), one can notice that the matching points are very near till the first maximum where they start being shifted. This shifting stabilizes after the local minimum. This is exactly the evolution that is described by DistX. It is first very close to 0, then increases until it stabilizes. The same observation can be made on Im(Z). The matching points are very close till the first maximum where the red curve is higher than the blue one. The level of the 2 curves are similar again around the local minimum and then the red curve becomes lower than the blue one before reaching the same level again. This evolution is well depicted by DistY which is almost constant around 0, then shows an increase, goes back to 0, and then describes a decrease before going back to 0 again. This example demonstrates that the DTW can be a powerful tool to describes the visual comparison that can be made between 2 curves.

III. RESULTS AND DISCUSSIONS A. Evaluation process

The DTW method is applied on the EIS extracted from the experimental data presented in II-B. These EIS curves are depicted in Fig. 4. It can be observed in Fig. 4 that it seems to be 5 groups of EIS curves. To differentiate them, these groups have been plotted with 5 different colors: blue, red, black, magenta and green. From the FC perspective, these 5 groups are corresponding to different experimental conditions:

• Red (1): Break-in period of the FC (BI).

• Blue (2): SU/SD protocol 1, EIS are performed before the temperature is stabilized (P1u). • Black (3): SU/SD protocol 1, EIS are performed after the temperature is stabilized (P1s). • Green (4): SU/SD protocol 2, EIS are performed before the temperature is stabilized (P2u).

• Magenta ( 5): SU/SD protocol 2, EIS are performed after the temperature is stabilized (P2s). It should be noted that if three temporal phases can be distinguished depending on wether the characterization is carried out during break-in, first or the second cycle of SU/SD, there is no predetermined order in the sequence of EIS within the same SU/SD cycle. From a temporal point of view, the datasets colored in black and blue in Fig. 4 (respectively green and red) are completely mixed.

The proposed method is evaluated on its ability to classify the 5 groups that were visually identified. To do so, the average absolute distances mean(|DistX|) and mean(|DistY |) are computed from the results of the DTW algorithm on each curve. Then, an unsupervised classification algorithm is performed on mean(|DistX|) and mean(|DistY |). The classification algorithm used here is a classic k-means clustering with 5 classes [START_REF] Likas | The global k-means clustering algorithm[END_REF]. This algorithm was chosen for its simplicity. This choice will not be discussed furthermore in the paper.

B. Classification results

The results of the classification is displayed on Fig. 5. It can be seen in this figure that 5 groups are identified. The colors used in this figure are the same as in Fig. 4. From Fig. 5, one can notice that the classes identified seem to match the visually identified groups. The first EIS (the one the more on the left in 4) is taken as the reference. The red crosses are very close to 0 and then shift to the right with almost no change on mean(|DistY |). It is the same behavior for the red EIS observed in 4. Then, another class is identified as the blue crosses, characterized by a shifting on the mean(|DistX|) axis and a tiny increase on mean(|DistY |), which is what is observed on the blue EIS curves: they are shifted on the Re(Z) axis with a slight increase on amplitude. On the right of the blue crosses class comes the green squares class with still a shifting on the right and a small increase on mean(|DistY |), which is once again similar to one can observe on the green EIS curves. Concerning the black EIS curves, their position is similar to the blue ones on the Re(Z) axis but their amplitude is highly decreased. On Fig. 5, the black stars class have above the blue crosses one, showing a similar distance on Re(Z) but more distance on -Im(Z). The fact that an increase is observed on mean(|DistY |) is due to the absolute operator that is used to compute the distance. However, it indicates an important change in amplitude which is the case here. The same observations can be done on the magenta triangles class compared to the green squares one.

The joined analysis of the EIS curves and the classification on the DTW distances demonstrates that DTW is a very interesting tools to transcript the observations one can visually make.

Table II presents the confusion matrix of the classification. A performance of 92.2% correct classifications is reached. The classification rates are very good except for Class BI where only 42.4% are correctly classified. Moreover, misclassified EIS curves are the one on the boundary to another class (BI to P1u, P1u to P2u, and P1s to P2s). As an example, Fig. 6 displays the EIS curves of Class BI (red) and the beginning of Class P1u (blue). It can be noticed from Fig. 6 that the first EIS curves in Class P1u are mixed up with some EIS curves of Class BI. As a consequence it is difficult to separate them visually which explains why the proposed method does not achieve to separate them and class them correctly.

C. Experimental data analysis

EIS is a interesting measure on FC because it gives information on the FC performance [START_REF] Cooper | Electrical test methods for on-line fuel cell ohmic resistance measurement[END_REF]. It is achieved in particular through the tracking of the ohmic resistance R HF that can be extracted when the EIS crosses the Re(Z) axis at high frequency, i.e. on the left of the curve. The lower this resistance, the better performance of the FC [START_REF] Yuan | Ac impedance diagnosis of a 500w pem fuel cell stack: Part i: Stack impedance[END_REF]. The R HF is extracted from the different EIS curves and displayed in Fig. 7 according the classification results presented in section III-B, using the same colors as previously. For confidentiality reasons, normalized values are presented here. First, one can remark that the end of the break-in phase (Class 1 -first 33 recordings) have been misclassified, as mentioned in the previous section. From a global point of view, R HF increases of about 50% along the 2 SU/SD phases (from recording 34 till the end), which represents about 120 h. This fact indicates that the conditions imposed to the FC had a strong impact on the performance in terms of electrical charge transfers.

More specifically, the analysis of R HF during the 2 SU/SD phases shows that R HF is more stable when the temperature is stabilized (black and magenta curves) as R HF values fluctuates more on the blue and the green curves. This underlines the major influence of the temperature on the R HF .

It is interesting to notice that the method was able to classify the EIS curves when the temperature was stable which are more interesting to exploit. Studying the trend of the black and the magenta curves shows indeed that the performance decrease is not the same for the 2 SU/SD strategies. The slope of each curve (black and magenta) was fitted as a first order polynomial. The computed values are displayed in table III. Tab. III confirms that the slope is lower in the case of P2 (magenta) than for P1, which would indicate that P2 is less degrading than P1. This result is quite surprising considering that the P1 protocol is supposed to be more careful with the FC. This result should indeed be studied more deeply to determine if this difference is really due to the change of SU/SD strategy and is linked neither to the general impact of cycling nor specific to a component aging of the FC prototype used. The strong degradation of the R HF is indeed also an indicator of bipolar plates corrosion, which could explain the trend observed.

IV. CONCLUSIONS

With the consideration of a significant development of hydrogen technologies and an increase in research work, the development of relevant analysis tools is more useful than ever. As part of the aging monitoring of HT-PEM FC subjected to repeated start-stop cycles, numerous EIS data were generated. The method without a-priori presented in this article is an efficient way to classify Nyquist plots based on shape evolution recognitions.

A method based on Dynamic Time Warping was presented for the automated analysis of EIS curves. This method allows getting a measure of similitude between 2 curves. This method was evaluated on data from a high-temperature PEM Fuel Cell under 2 different conditions of Start & Stop cycling. The database represents 120 h of working.

The proposed method has proven to be efficient to give numerical indications that confirm the visual observations made on the different curves. The mean distance on Re(Z) and -Im(Z) resulting from the proposed method were used for unsupervised classification of the EIS curves using a classic k-means classifier. The proposed method reaches 92% correct classifications on 383 EIS curves and the misclassified curves are mainly the ones that were not possible to visually distinguish.

From this classification, R HF was extracted from the different EIS curves. The DTW-based classification allowed separating the temperature stabilized and unstabilized EIS curves, which enlightened the impact of temperature on the quality of the R HF extraction. Moreover, the R HF analysis seems to fully benefit from this information.

However, work remains to be done in order to validate the method with new EIS data from the end of the presented aging campaign and from other campaigns on different fuel cell and with other operating conditions. To another extent, the method could be used for on-line diagnosis as EIS can be done during fuel cell operation and the automated analysis can detect degradation signatures.

Figure 1 .

 1 Figure 1. Aging test-bench for start-up & shut-down cycling.

Figure 2 .

 2 Figure 2. Example of DTW matching for 2 EIS curves.

Figure 3 .

 3 Figure 3. Distances DistX and DistY for the example given in Fig. 2.

Figure 4 .

 4 Figure 4. EIS curves from the experimental data campaign.

Figure 5 .

 5 Figure 5. Classification results on the average distances of the DTW results on the EIS curves.

Figure 6 .

 6 Figure 6. EIS curves of Class BI (red) and the beginning of Class P1u (blue).

Figure 7 .

 7 Figure 7. Normalized R HF extracted from the EIS curves according their class (1: BI, 2: P1u, 3: P1s, 4: P2u, 5: P2s).

Table I NUMBER

 I OF EIS CURVES IN EACH CONDITION.

	BI	P1u	P1s	P2u	P2s
	33	67	108	64	111

Table II CONFUSION

 II MATRIX OF THE EIS CURVES CLASSIFICATION.

					Target
			BI	P1u	P1s	P2u	P2s
		BI	14	0	0	0	0
	Output	P1u P1s P2u	19 0 0	59 0 8	0 105 0	0 0 64	0 0 0
		P2s	0	0	3	0	111
	Rate		42.4%	88.1%	97.2%	100%	100%

Table III SLOPE

 III OF THE R HF EVOLUTION IN STABILIZED TEMPERATURE CONDITIONS.

ACKNOWLEDGMENT

This work is part of PIPAA project, included in HyPort meta-project, in partnership with Safran Power Units and financed by BPI France. The authors are very grateful to all the actors at the Hydrogen platform in Toulouse and at Safran Power Units who contributed to get these results.