Pascal Gourdel
email: pascal.gourdel@univ-paris1.fr]

Nadia Maagli
email: maagli.nadia@gmail.com]

New approach of the hairy ball theorem

Keywords: Hairy ball theorem, fixed point theorems, approximation methods, homotopy, topological degree, connected components

In this paper, we establish an equivalent version of the hairy ball theorem in the form of a fixed point theorem. By using a version of Mas-Colell theorem [6] and by applying homotopy and approximation methods, we obtain our main result.

Introduction

The aim of this paper is to establish and prove the following fixed point theorem equivalent to the hairy ball theorem.

If f : S → S is continuous and satisfying for any x ∈ S, f (x).x ≥ 1 2 , then it possesses a fixed point. The proof of this theorem is based on constructing of an explicit continuous homotopy F between a nontrivial function α and the function f . A component starting from fixed points of α will lead us to a fixed point of f . Yet, using a result of Mas-Colell [START_REF] Mas-Colell | A note on a theorem of F.Browder[END_REF], we prove the result for a twice continuous differentiable function. Our point is that the existence of a smooth path can easily be implented on computers. Then, using approximation technics, we recover all fixed points of f . The paper is organized in the following way. Section 2 contains preliminaries and notations. The equivalent fixed points theorems are set in section 3. The proof of the main result is given in section 4.

Preliminaries and notations

Throughout this paper, we shall use the following notations and definitions. Let S = S n = x ∈ R n+1 : x = 1 be the unit n-sphere. For any 0 < r < 1, we denote by B(S, r) = x ∈ R n+1 : 1r ≤ x ≤ 1 + r . For a subset X ⊂ R n , we denote by X the closure of X, by X c the complement of X, by int(X) the interior of X, and by ∂X = X\int(X) the boundary of X. We denote by x 0 = (0, 0, • • • , 1) and -x 0 = (0, 0, • • • , -1), respectively the north and south pole of S and by 'deg' the classical topological degree.

Definition 1. We denote by H the set of continuous functions F : [0, 1] × S → S such that F (0, x 0 ) = x 0 . For a continuous function F : [0, 1] × S → S, we denote by 1. C F := {(t, x) ∈ [0, 1] × S : F (t, x) = x}.

2. The translation F of F is defined by F (t, x) = F (t, x)x.

For getting our main result, we need the following topological degree properties.

Proposition 1.

Let Ω be a bounded open set of R m , f : Ω -→ R m be a continuous function and y ∈ R m such that y / ∈ f (∂Ω). Then, we have

i. deg(., Ω, y) is constant in g ∈ C(Ω)\ g -f < r where r = d(y, f (∂Ω)). ii. Let Ω 1 be an open set of Ω. If y / ∈ f (Ω\Ω 1 ), then deg(f, Ω, y) = deg(f, Ω 1 , y). iii. Let V be an open and bounded set of [0, 1]×R m , V (t) := {x ∈ R m : (t, x) ∈ V } and F : V → R n , with f t = F (t, .) ∈ C 1 (V (t)). Suppose that
(f t , V (t), p t ) is constant in t ∈ [0, 1].
We need also the following.

Proposition 2. In a compact set, each connected component is the intersection of all open and closed sets that contain it.

Proof. See the appendix.

In the sequel, we will suppose that the integer n is even. The aim of this note is to provide a variant proofs of the hairy ball theorem. First, we recall the hairy ball theorem.

Theorem 1. (Hairy ball theorem) An even dimensional sphere does not admit any continuous field of non-zero tangent vectors.

In other terms, if g : S → S is continuous and for every x ∈ S, we have g(x).x = 0, then there exists x such that g(x) = 0. In what follows, we establish two equivalent versions of the hairy ball theorem presented as fixed point theorems.

Equivalent versions

In the following, we state the first equivalent version to the hairy ball theorem.

Theorem 2. If f : S → S is continuous, then either f or -f possesses a fixed point.

As a first step, we prove that Theorem 2 is equivalent to the hairy ball theorem.

Proof. First, we claim that the hairy ball theorem implies Theorem 2. Indeed, let f : S → S be a continuous function and consider g : S → S given by g(x) = f (x) -(x.f (x))x. By the hairy ball theorem, g has a zero x on S. That is, f (x) = (x.f (x))x. Now, using that f (x) is collinear to x and that both of them belongs to the sphere, we conclude that either f (x) = x or f (x) = -x. Conversely, let g : S → S be a continuous function such that for any x ∈ S, g(x).x = 0. Then for x ∈ S, g(x) + x = 0. So, we consider the function f (x) = x+g(x) x+g(x) . By Theorem 2, there exists x such that f (x) = ǫx, with ǫ ∈ {-1, 1}. This implies that g(x) = 0. Now, we state the main result and we prove that it is equivalent to the hairy ball theorem. Theorem 3. If f : S → S is continuous and satisfying for any x ∈ S, f (x).x ≥ 1 2 , then it possesses a fixed point.

2

Documents de travail du Centre d'Economie de la Sorbonne -2014.51

Proof. Obviously, Theorem 2 implies Theorem 3. In fact, since x = 1 on S, then it is trivial to see that f (x) = -x is impossible. Conversely, we will prove that Theorem 3 implies the hairy ball theorem. Indeed, let g : S → S be a non easy continuous function such that for any x ∈ S, g(x).x = 0. Let M = sup x∈S g(x) . So, put α = √ 3

M and consider the function f α (x) = x+αg(x) x+αg(x) . We have f α (x).x = 1 x+αg(x) > 0 and by simple calculus, we obtain

(f α (x).x) 2 = 1 x + αg(x) 2 = 1 x 2 + α 2 g(x) 2 ≥ 1 1 + α 2 M 2 = 1 4
.

By Theorem 3, f α possesses a fixed point x. Setting f α (x).x = 1 above implies that g(x) = 0, and the result follows.

Remark 1. Let us observe that the choice of the real 1 2 is arbitrary. The main idea of the theorem is that we can allow a radial component if it is not fully opposite to x. Here we state the general version. For any real λ ∈ (-1, 1), we denote by P λ the following proposition If f : S → S is continuous and for any x ∈ S, f (x).x ≥ λ, then f has a fixed point.

It is not difficult to prove that P λ is equivalent to Theorem 2. To sum up, we have Theorem 3 is equivalent to the hairy ball theorem. Thereafter, providing a proof of Theorem 3 will enable us to have a new proof of the hairy ball theorem which differs from the classical proofs [START_REF] Milnor | Analytic proofs of the "Hairy ball theorem" and the Brouwer fixed point theorem)[END_REF].

Proof of Theorem 3

The proof of Theorem 3 will depend on the two following results.

Lemma 1. If f : S → S is continuous such that for any x ∈ S, f (x).x ≥ 1 2 , then there exists F ∈ H such that F (1, .) = f . Theorem 4. There exists a connected component Γ subset of

C F such that Γ ∩ ({0} × S) = ∅ and Γ ∩ ({1} × S) = ∅. Consequently, F (1, .) has a fixed point.
Once we prove Lemma 1 and Theorem 4, then Theorem 3 is deduced immediately.

Proof of Lemma 1

For any (t, x) ∈ [0, 1] × S, consider the function F given by

F (t, x) = tf (x) + (1 -t)α(x) tf (x) + (1 -t)α(x) . (1)
Before introducing the function α, we can easily remark that F satisfies the conclusion of Lemma 1 provided that for any (t, x) ∈ [0, 1] × S, the function α met the three following properties

(P 1 ) tf (x) + (1 -t)α(x) = 0. (P 2 ) α is continuous.
3 Documents de travail du Centre d'Economie de la Sorbonne -2014.51 (P 3 ) "α(x) is positively collinear to x " is equivalent to " x = ±x 0 ". Now, we will construct gradually the function α. First, let us define the function β : S → S by β(x) = y, where

x = (x 1 , • • • , x n+1 ) and y = (y 1 , • • • , y n+1 ) such that ∀i ∈ {1, • • • , n} , y i = x i x 2 1 + • • • + x 2 n = x i 1 -x 2 n+1 ,
and

y n+1 = x n+1 2 -x 2 n+1 .
Second, consider the following rotation R θ of S ⊂ R n+1 = R 2k+1 , north-south axis, whose matrix is given by

R θ =            cos θ -sin θ • • • 0 0 sin θ cos θ . . . . . . . . . cos θ -sin θ . . . 0 sin θ cos θ 0 . . . . . . . . . . . . 0 • • • • • • • • • 1            where 0 < θ < π 2 . Finally, we define the function α : S → S by α(x) = R θ (β(x)).
Next, our objective is to prove that the properties (P 1 ), (P 2 ) and (P 3 ) are satisfied. In order to prove (P 1 ), it suffices to prove that (tf (x) + (1t)α(x)).x > 0. Moreover, since f (x).x > 1 2 , then we just need to prove that α(x).x > 0. This follows from the expression of α. Indeed, we have

α(x) = R θ (β(x)) =                     1 -x 2 n+1 (x 1 cos θ -x 2 sin θ) 1 -x 2 n+1 (x 1 sin θ -x 2 cos θ) 1 -x 2 n+1 (x 3 cos θ -x 4 sin θ) 1 -x 2 n+1 (x 3 sin θ -x 4 cos θ) . . . 1 -x 2 n+1 (x n-1 cos θ -x n sin θ) 1 -x 2 n+1 (x n-1 sin θ -x n cos θ) x n+1 2 -x 2 n+1                    
Therefore, we obtain that

α(x).x = (1-x 2 n+1 ) 3 2 cos θ+x 2 n+1 2 -x 2 n+1 ≥ (1-x 2 n+1 ) 3 2 cos θ+x 2 n+1 ≥ min( 2 3 , cos θ) > 0.
On the other hand, since all the components of the function β are continuous, then (P 2 ) is trivial. This completes the proof of (P 1 ) and (P 2 ). In order to prove (P 3 ), note that if α(x) = λx then, |λ| = 1. Thus, 4

Documents de travail du Centre d'Economie de la Sorbonne -2014.51

we have x n+1 2x 2 n+1 = λx n+1 . Now, since β(x) = λR -θ (x) and in virtue of [START_REF] Prabhu | Fixed points of rotations of n-sphere[END_REF], the only fixed points of R -θ are ±x 0 , then x n+1 = 0. Therefore, we obtain that 2x 2 n+1 = λ = 1. Squaring the last equality implies that x = ±x 0 , as required.

In the next section, we will interpret geometrically the properties of the function α.

The geometrical properties of the function α

For any x n+1 ∈ [-1, 1], let us denote by P xn+1 , the following set P xn+1 := {x ∈ S such that x n+1 = x n+1 }. By analogy with the unit sphere of R 3 , we call this set a parallel. Remark that except at the poles, it is a sphere of dimension n -1.

• The image by α of a parallel of altitude x n+1 is a parallel of altitude x n+1 2x 2 n+1 and closer to the corresponding pole. • The image by α of a polar cap of altitude x n+1 is a polar cap of altitude x n+1 2x 2 n+1 and closer to the corresponding pole. Therefore, the parallel of altitude -1 (reduced to the south pole), of altitude 0 (reduced to the equator) and of altitude 1 (reduced to the north pole) are the only one that are globally invariants. In addition, since α(x) and x are on the sphere and belongs both either to the north semi-sphere or to the south semi-sphere, then co-linearity means equality. As a consequence of the computation of the image of a parallel, we have the following. Proposition 3. For any 0 < µ < 1 2 , we have

α(B(x 0 , µ) ∩ S) ⊂ B(x 0 , µ 2 ) ∩ S.
Proof. See the appendix.

The following section provides the proof of Theorem 4.

Existence of fixed points

In order to prove Theorem 4, we willneed the following theorem of Mas-Colell [START_REF] Mas-Colell | A note on a theorem of F.Browder[END_REF]. Let X be an open subset of R n . Let A ⊂ X be open and such that A ⊂ X and let F be the set of twice continuously differentiable functions: F : [0, 1] × X → A.

Theorem 5 (Mas-Colell ).

There is an open and dense set F ′ ⊂ F such that for every F ∈ F ′ , any non empty component The proof of this theorem is based on a transversality argument which make his result only generic. In addition, it is important to notice that Mas-Colell established his result on a convex set X. However, a careful reading of the proof shows that this assumption was only used in a subsequent part. In order to prove Theorem 4, we need first to prove the following result.

Γ of C F with Γ ∩ ({0} × X) = ∅ is diffeomorphic to a segment.
Proposition 4. Let X = B(S, 1 2 ), A = B(S, 3 8 ), p ≥ 2 and F given by( 1) of Lemma 1. Then there exists an open dense set

F ′ ⊂ F and a function G p : [0, 1]×X → A such that G p ∈ F ′ and G p (t, x) -F (t, x x ) ≤ 1 p , for any (t, x) ∈ [0, 1] × X.
Moreover, there exists a connected component

W p ⊂ C Gp such that W p ∩ ({0} × X) = ∅ and W p ∩ ({1} × X) = ∅.
Proof. In order to apply Theorem 5, we present here a constructive proof.

Let F be the function given by ( 1) and consider the function

F : [0, 1] × X → S defined by F (t, x) = F (t, x x )
. By Stone Weierstrass approximation method, for any integer p ≥ 2, there exists a C 2 function

F p : [0, 1] × X → A, such that F -F p ∞ ≤ 1 2p . Let F ′ be an open dense set given by Mas-Colell's theorem such that F ′ = F. Since F p ∈ F, then there exists G p ∈ F ′ such that G p : [0, 1] × X → A and G p -F p ∞ ≤ 1 2p . So, we obtain that G p -F ∞ ≤ 1 p . (2)
It remains to show that C Gp = ∅. Indeed, let us prove that for any 0 < r < 1 3 , there exists x p ∈ B(x 0 , r) ⊂ X, such that G p (0, x p ) = x p . By inequality (2), we have G p (0, x) -F (0, x) = G p (0, x)α( x x ) ≤ 1 p . This implies that for each x ∈ B(x 0 , r), we have

G p (0, x) -x 0 ≤ G p (0, x) -α( x x ) + α( x x ) -x 0 ≤ 1 p + α( x x ) -x 0 .
A simple calculus * , shows that for any x ∈ B(x 0 , r), we have α(

x x ) ∈ B(x 0 , r √ 2 √ 1+ √ 1-r 2 )
. Therefore, by Proposition 3, we conclude that G p maps B(x 0 , r) into itself and by Brouwer theorem, there exists x p ∈ B(x 0 , r) such that x p ∈ C Gp . At this step, applying Theorem 5 to G p , we conclude that any component Γ p ∈ C Gp starting from (0, x p ) is diffeomorphic to a segment. Now, it remains to show that there exists a component Γ p such that Γ p ∩ ({1} × X) = ∅. First, it is important to notice that for any component in C Gp , we have either it starts from 0 and belongs to the family denoted by Γ p,i , or it doesn't intercept {0} × * The proof is related in the appendix 6 Documents de travail du Centre d'Economie de la Sorbonne -2014.51 X and in this case belongs to the second family denoted by Γ ′ p,i . Thus, we have

CC Gp = {Γ p,i } i∈I1 ∪ Γ ′ p,i i∈I2
, where CC Gp is the set of all components of C Gp . Now, let G p be the translation of G p . In the sequel, we abuse notation by writting C Gp = C Gp := (t, x) ∈ [0, 1] × X such that G p (t, x) = 0 . We argue by contradicition. Suppose that there is no component Γ p,i intersecting ({1} × X), then there exists t i < 1 such that Γ p,i ⊂ [0, t i ) × X. By Proposition 2, we have Γ p,i = ∩ j∈J U i,j , where for any j ∈ J, U i,j is open and closed in

C Gp . Let D = C C Gp \([0, t i ) × X), then (∩ j∈J U i,j ) ∩ D = ∅.
Since D is compact and the family U i,j has the finite intersection property, we have that (∩ j∈J1⊂J U i,j ) ∩ D = ∅, for J 1 finite. Moreover, note that U = ∩ j∈J1 U i,j is open and closed set containing Γ p,i , then there exists

j 0 ∈ J 1 such that U = U i,j0 included in [0, t i ) × X.
For the second type of components Γ ′ p,i of C Gp , by a similar argument, there exists

j ′ 0 ∈ J 2 finite such that U = U ′ i,j ′ 0 open and closed included in (r i , 1] × X. The family (U i,j0 ) i∈I1 , (U ′ i,j ′ 0 ) i∈I2
V 1 and V 2 in [0, 1] × Ω such that E ⊂ V 1 , and E c ⊂ V 2 .
The rest of the proof is ruled out in four steps collected in the following lemma.

Lemma 2. We have 1. deg(G p (0, .), Ω, 0) = deg(F (0, .), Ω, 0).

deg(G

p (0, .), Ω, 0) = -2. 3. deg(G p (0, .), Ω, 0) = deg(G p (0, .), V 1 (0), 0). 4. deg(G p (t, .), V 1 (t), 0) is constant in t ∈ [0, 1].
Assuming this Lemma through, we claim that there exists a component

W p ⊂ C Gp such that W p ∩ ({0} × X) = ∅ and W p ∩ ({1} × X) = ∅. Indeed, by Lemma 2 (4), we have deg(G p (t, .), V 1 (t), 0) = deg(G p (0, .), V 1 (0), 0) = -2.
Yet, by construction for t close enough to 1, we have V 1 (t) = ∅. Therefore, deg(G p (t, .), V 1 (t), 0) = 0, which establish a contradiction, as required. The proof of Lemma 2 is based on the degree properties and will take the rest of this section.

Proof. of Lemma 2 1. Let f and g given by the following: for any x ∈ Ω, f (x) = F (0, x) = F (0, x)x = α( x x )x and g(x) = G p (0, x). We claim that 0 / ∈ f (∂Ω). Indeed, if there exists x ∈ ∂Ω such that f (x) = 0, then x is a fixed point of α. That is, x = ±x 0 / ∈ ∂Ω, which yields a contradiction. On the other hand, for any x ∈ Ω, we have

G p (0, x) -α( x x ) ≤ 1 p . That is, g(x) -f (x) ≤ 1 p ≤ r, where r = d(0, f (∂Ω)) > 0. By Proposition 1 (i), we have deg(G p (0, .), Ω, 0) = deg(F (0, .), Ω, 0). 2. It suffices to prove that deg(F (0, .), Ω, 0) = -2. We recall that deg(F (0, .), Ω, 0) = x∈F -1 (0,.)({0}) sgn(det DF (0, x)) = x∈{x0,-x0}
sgn(det DF (0, x)).

Next, we compute the differential of F (0, .) at the points x 0 and

-x 0 . Define h : R n+1 \ {0} → R n+1 by h(x) = β( x x ), then we have for x = 0, DF (0, .)(x) = R θ (Dh(x)) -I n+1 . Let us denote by x = (x ′ , x n+1 ), we have h(x ′ ) = x ′ x ′ x 2 ,
and

(h(x)) n+1 = x n+1 2 x ′ 2 + x 2 n+1 x 2 .
The function h is differentiable at x 0 and Dh(x 0 ) = 0. Indeed,

we have h(x) -h(x 0 ) = ( x ′ x ′ x 2 , xn+1(x 2 n+1 +2 x ′ 2 ) 1 2 x 2 -1). It suffices to prove that h(x) -h(x 0 ) = o( x -x 0 ). Since we have x -x 0 2 = x ′ 2 + (x n+1 -1)
2 , then we have only to prove that every component belongs to o( x ′ ). It is clear that

x ′ x ′ x 2 = o( x ′
). On the other hand, for the second component of h(x)h(x 0 ), we have

x n+1 (x 2 n+1 + 2 x ′ 2 ) 1 2 x 2 -1 = x 2 n+1 (1 + 2 x ′ 2 x 2 n+1 ) 1 2 -x 2 x 2 = x 2 n+1 (1 + x ′ 2 x 2 n+1 + o( x ′ 2 x 2 n+1 )) -x ′ 2 -x 2 n+1 x 2 = o( x ′ 2 )x 2 n+1 x 2 .
Hence, h is differentiable at x 0 and Dh(x 0 ) = 0. A similar calculus leads to Dh(-x 0 ) = 0. Finally, we can conclude that DF (0, .)x 0 = DF (0, .)(-x 0 ) = -I n+1 . Thus, we have deg(F (0, .), X, 0) = (-1) n+1 + (-1) n+1 = -2, since n is even, and the result follows.

8 Documents de travail du Centre d'Economie de la Sorbonne -2014.51

3. We recall that V 1 (t) = {x ∈ Ω : (t, x) ∈ V 1 } and let Ω 1 = V 1 (0).
We have 0 / ∈ G p (Ω\V 1 (0)). Indeed, suppose that there exists

x ∈ Ω\V 1 (0) such that G p (0, x) = 0. That is, x / ∈ V 1 (0) and (0, x) ∈ C Gp ⊂ V 1 ∪ V 2 .
This implies that (0, x) ∈ V 2 . Though, by construction, V 2 doesn't contain (0, x) which set a contradiction. By Proposition 1 (ii), we get deg(G p (0, .), Ω, 0) = deg(G p (0, .), V 1 (0), 0).

Let

V (t) = V 1 (t), f t = G p (t, .
) and p t = 0. We have 0 / ∈ f t (∂V 1 (t)). Suppose that there exists x ∈ ∂V 1 (t) such that G p (t, x) = 0, then by definition (t, x)

∈ ∂V 1 = V 1 \V 1 and (t, x) ∈ C Gp = E ∪ E c . Thus, we have two cases. If (t, x) ∈ E, then (t, x) ∈ V 1 , an impossibility since (t, x) ∈ V c
1 . Otherwise, we have (t, x) ∈ E c , then by construction of V 1 and V 2 , we have E c ∩ ∂V 1 = ∅, contradiction. Hence, by By Proposition 1 (iii), we conclude that deg

(G p (t, .), V 1 (t), 0) is constant in t ∈ [0, 1].

Transition smooth-continuous version

To sum up, the result holds for the functions twice continuously differentiable. In this section, we will extend our result to continuous functions and recover the fixed points of F . Now, we come to the heart of the proof of Theorem 4.

Proof. From the above result, there exists x p such that (0, x p ) ∈ W p ∩ ({0} × X) ⊂ C Gp . Since the sequence x p is bounded, then we may assume that it converges to some x. Thus, we have lim p→+∞ G p (0, x p ) = F (0, x) = α( x x )x = 0. As explained before, this yields to x = ±x 0 . Without loss of generality, we can suppose that the sequence (x p ) p≥0 converges to x 0 . We denote by W tr k , the translated component of W k , given by W tr k = W k + (x 0x k ). Thus, in the spirit of the lim sup, we put Z = ∩ p>1 ∪ k≥p W tr k = ∩ p>1 Z p . We prove that Z is connex. Since the components W tr k are connex, for all k, and contains a common point (0, x 0 ), then ∪ k≥p W tr k is connex. Therefore, the set Z p = ∪ k≥p W tr k is the closure of a compact connex set. Hence, Z is connex. Now, we claim that Z ⊂ C F . Let (t, x) ∈ Z, then for any p, we have

(t, x) ∈ Z p . That is B((t, x), 1 p )∩(∪ k≥p W tr k ) = ∅. By definition, we ob- tain that ∃ k(p) ≥ p, such that (t p , z p ) ∈ B((t, x), 1 p ) ∩ W tr k (p) . That is, (t p , z p ) → (t, x) and (t p , z p ) ∈ W tr k(p) . Let z ′ p = z p -(x 0 -x k(p) ), then (t p , z ′ p ) ∈ W k(p) ⊂ C G k(p) . Equivalently, we have G k(p) (t p , z ′ p ) = z ′ p . Since x k(p) converges to x 0 , then we have that (t p , z ′ p ) → (t, x)
. By continuity of F , we obtain that d( F (t, x), F (t p , z ′ p )) < ǫ 4 . We have already established that for any p large enough,

F (t p , z ′ p ) -G k(p) (t p , z ′ p ) ∞ ≤ 1 p < ǫ 4 . This implies that, d( F (t, x), G k(p) (t p , z ′ p )) = d( F (t, x), z ′ p ) < ǫ 2 . Hence, we conclude that d( F (t, x), x) ≤ d( F (t, x), z ′ p ) + d(z ′ p , x) ≤ ǫ.
′ k ) ∈ W k . Let x ′′ tr k = x ′ k +(x 0 -x k ). We have x ′′ tr k ∈ W tr k ⊂ ∪ j≥p W tr j ⊂ Z p . Since Z p is compact, then lim k→+∞ x ′′ tr k = y belongs to Z p . On the other hand, we have G k (1, x ′ k ) = x ′ k .
Passing to the limit leads to F (1, y) = y ∈ S. Therefore, we have (1, y) ∈ Z p ∩ ({1} × S). Consequently, for any finite set J, we have that ∩ p∈J Z p ∩ ({1} × S) = ∅. Thus, applying the finite intersection property, we obtain that Z ∩ ({1} × S) = ∅ proving the main result.

i. Proof of Proposition 2

Let C be a connected component of the compact set K and (K i (C)) i∈I be the family of all open and closed sets of K that contains C. We denote by K(C) = ∩ i∈I K i (C). We have, K(C) is closed in the compact set K, then compact. We prove that C = K(C). We have C ⊂ K i (C), for any i ∈ I. Indeed, since

K i (C) c is also open and closed, then if C ∩ K i (C) c = ∅,
C ⊂ F 1 ⊂ U 1 . Let U = U 1 ∪ U 2 , then K(C) ∩ U c = ∅. That is, ∩ i∈I K i (C) ∩ U c = ∅.
Using the finite intersection property, we obtain that there exists I 1 finite such that ∩ i∈I1⊂I K i (C) ∩ U c = ∅. Therefore, there exists i 0 ∈ I 1 , such that K i0 ⊂ U . However, K i0 ∩ U 1 is open and closed in K containing C but not K(C), which establish a contradiction.

ii. Proof of Proposition 3 Let x ∈ B(x 0 , µ) ∩ S, then xx 0 2 = 2(1x n+1 ) ≤ µ 2 .

On the other hand, we have α(x)x 0 2 = 2(1-x n+1 2x 2 n+1 ) = 4 , for any 0 < µ < 1 2 .

iii. Let 0 < r < 1 3 , then for any x ∈ B(x 0 , r), we have α(

x x ) ∈ B(x 0 , r √ 2 √ 1+ √ 1-r 2 ).
Proof. Let x ∈ B(x 0 , r), then xx 0 2 = x 2 -2x n+1 +1 ≤ r 2 . Let t = x 2 1 + • • • + x 2 n and s = x n+1 , then t + s 2 -2s + 1 = (1s) 2 + t ≤ r 2 . On the other hand, we have

5

  Documents de travail du Centre d'Economie de la Sorbonne -2014.51

  forms an open covering of the compact set C Gp . Therefore, we can extract a finite sub-covering {U 1 , • • • , U ′ n } such that the first k components are included in [0, t[×X and the remaining (nk) components are included in (r, 1] × X, where t = max(t i ) 1≤i≤k and r = min(r i ) k+1≤i≤n . The set E = ∪ i≤k U i is open and closed in the compact set C Gp . So E is compact and contained in [0, t)×X. Moreover, E c = C C Gp \E is also compact, open and contained in (r, 1] × X. We have C Gp ⊂ Im( G p ) ⊂ A ⊂ Ω = B(S, 7 16 ) ⊂ X. Using the separation criteria in the Hausdorff metric space [0, 1] × Ω, there exists two disjoint open sets

7

  Documents de travail du Centre d'Economie de la Sorbonne -2014.51

  Tending ǫ towards zero, we obtain that F (t, x) = x. In addition, we have both C F and C F are included in S. Since C F ⊂ Im( F ) = Im(F ) ⊂ 9 Documents de travail du Centre d'Economie de la Sorbonne -2014.51 S and F and F coincide on S, then C F = C F , as required. It remains to show that Z ∩ ({1} × S) = ∅. Using our previous result, we know that for any k ≥ p, there exists (1, x

2 1+xn+1= 2 ( 1 -

 221 x n+1 ) 2 (1 + x n+1 ) ≤ 4(1x n+1 ) 2 ≤ µ 4 ≤ µ 2

x x -x 0 2 = 2 ( 1 1 -r 2 2 . 2 √√ 1 -r 2 )

 221122212 -xn+1 x ) = 2(1 -s √ t+s 2 ). Put f (t) = 2(1 -s √ t+s 2 ) for any t ∈ (0, r 2 -(1s) 2 ), then f is increasing and f (t) ≤ f (r 2 -(1s) 2 ) = 2(1 -s √ r 2 -1+2s ) = g(s). Nown the function g is defined on (1r, 1) and g reaches its maximum at (1r 2 ), theng(s) ≤ g(1r 2 ) = 2(1 -√ Hence, we obtain that x xx 0 ≤ √ 2r √ 1+ √ 1-r 2 < 1 2 .By Proposition 3, we have 11 Documents de travail du Centre d'Economie de la Sorbonne -2014.51 α( x x ) ∈ B(x 0 , r √ 1+ , as required.

  this will contradicts that C is connex. Therefore, C ⊂ K(C). Conversely, it suffices to prove that K(C) is connex. We argue by contradiction. Suppose thatK(C) = F 1 ∪ F 2 ,where F 1 and F 2 are nonempty, open, closed and disjoints sets. Using the separation criteria, we obtain that there exists U 1 , U 2 two disjoints open sets of K such that F 1 ⊂ U 1 and F 2 ⊂ U 2 . Since C ⊂ K(C), then suppose that

Documents de travail du Centre d'Economie de la Sorbonne -2014.51

Documents de travail du Centre d'Economie de la Sorbonne -2014.51


Promoting universal access to health services in post-conflict situations: what role can large scale cash transfer programmes play for better outcomes?

Making well-informed decisions about how best to achieve MDGs depends on the ability of public policy makers in accessing the best available evidence about what is known to work and what could be potential benefits, and ways to integrate solutions into complex and often under-resourced health systems. Conditional cash transfer programmes have been largely explored as a policy for improving the education and health outcomes of poor children in developing countries as well as a tool for long-term poverty alleviation; but needs to be appropriate to the context and both fiscally and politically affordable. In DRC, the crisis and conflicts of the past decades severely affected the health status of the population and degraded the health system. Consequently, efforts in reducing infant and under-five morbidity and mortality are seriously hampered by widespread poverty and economic deregulation.

The aim of this paper is to question the feasibility and affordability of cash transfer compared to 2 alternatives: an outreach health and nutrition programme with a behavioral change communication component and the elimination of basic health care user fees. The results show that children health outcomes may instead be driven by the equal distribution of quality services through outreach health and nutrition programmes (20% to real GDP) than a large scale health-oriented conditional cash transfer (25.9% to real GDP). A number of issues is outlined with regard to the country's socio-economic and political context : (i) health-oriented conditional cash transfer cannot operate in DRC due to supply-side constraints and lack of health supply strategy, logistics and engineering ; (ii) targeting is somewhat time consuming and irrelevant in such context with a headcount averaging 77% ; (iii) outcomes of a geographic focused cash transfer programme could be expected on improvement of nutritional status, but less on the rise of health demand and would be difficult to scale-up.

Introduction

The challenges relating to the realization of child rights in the DRC are enormous. According to the World Bank, over 70% of the population lives below the international poverty line of $1 per capita per day [START_REF] Berlinski | The effect of pre-primary education on primary school performance[END_REF] . A recent study revealed that poverty is overwhelmingly affecting children more than adults and reports that nearly 77% of the children in the country live below the poverty line [START_REF]Tableau des Opérations Financières et Économiques[END_REF] . The national poverty line has been set back to $ 0.72 per person per day and the proportion of children living in poverty is still very high (56.6%). Despite the national economic growth estimated at 12% in 2008, the average income per capita remains as low as US$ 102. [START_REF]Final Report on the DRC Demographic and Health Survey (DHS) 2007. Ministry of Plan in collaboration with Ministry of Health, Macro International[END_REF] Children are disproportionately represented among the income-poor and many suffer from severe deprivation. Also, the resources devoted to the social sector remain inadequate and do not fit in a long-term perspective despite the fact that the country adopted a Poverty Reduction Strategy Paper (PRSP, 2006) together with a four years Medium-Term Expenditure Framework (2007 -2010). The DRC ranks 167 out of 177 on the Human Development Index (HDI). The problematic and highly vulnerable position of children is further reflected in their lack of access to health services, quality education, and their exposure to physical and psychological violence. They are paying a heavy tribute as they are the most vulnerable to widespread poverty, deterioration in the quality of life, high unemployment, rising food prices, and grave inequity in access to basic social services.

With a low State budget allocation to health care and the low demand from the population due to extreme poverty (both monetary and non monetary), the country continues to face major difficulties as the levels of morbidity and mortality remain high. It is not surprising that reducing infant and under-five (U5) mortality is seriously hampered under this condition. In the past six years, the under-five mortality rate (U5MR) declined, but by not more than 4% [START_REF]Final Report on the DRC Demographic and Health Survey (DHS) 2007. Ministry of Plan in collaboration with Ministry of Health, Macro International[END_REF] . Even though the recently completed Demographic Health Survey (DHS, 2007) confirms a new U5MR of 148 out of 1,000 live births, the reduction of the mortality rate will not be enough for DRC to be on course for achieving the MDGs by 2015.

These morbidity and mortality rates are exacerbated by the appearance of new challenges facing the health sector: (i) strong proliferation of HIV / AIDS whose prevalence is estimated at 5.5% in 2007 (DHS, 2007) and affects primarily the portion of the population living in poor conditions, (ii) resurgence of tuberculosis, which usually accompanies HIV/AIDS, (iii) persistence of malaria, which currently accounts for 40 to 50% of consultations and 28% of hospitalizations, (iv) low coverage of obstetric care, marked by increased maternal and neonatal mortalities. These situations are likely to increase the risk of mortality, especially among children and women with low nutritional status. On the other hand, the population's inaccessibility to basic health care services affects both social and economic activities by reducing their performance and ability to work. This leads to an economic loss of the potential labor force, both in terms of gross volume and productivity and, consequently, economic growth as it has been clearly demonstrated by Sen (1995) in his human capital theory.

In the recent years, a number of strategies have been explored in order to enhance human capital accumulation and impact household welfare in view of breaking intergenerational transmission of poverty. An illustration of this is the approach of social risk management advocated by the World Bank (social contracts) including direct cash transfers and approaches seeking complementarities between public (social security) and community systems (palliative systems), health insurance and health mutuals. This paper attempts to analyze the potential impact of a child-focused cash transfer programme with a health-oriented conditionality in reducing physical vulnerability (with a focus on child malnutrition) among the under-five to meet 2 objectives: (i) one that directly offsets the income effect of the shock through transfer of an amount of cash per child under-five in each household and ; (ii) one that seeks to mitigate the consequences of vulnerabilities on human development outcomes such as nutrition status of children. Our analysis therefore aims to answer 2 essential questions: (i) Is a universal or targeted cash transfer to reduce child morbidity and mortality fiscally and administratively affordable for the DRC given its actual macroeconomic context? (ii) Is a child-focused health-oriented universal cash transfer program the most

Problem identification: Child and Infant Morbidity and Mortality

Graph 1: Underweight distribution by province (DHS, 2007) The crisis and conflicts of the past decades severely affected the health status of the population of DRC, especially in such fragile environment conducive to health system degradation. As a result, the overall coverage of the basic health services along with the utilization of these services were highly affected.

The main health sector-related MDG indicators remain very poor although there have been some improvements since the end of major conflicts, but the Country is still struggling to reach the pre-1990 levels. Under-five mortality during the period 2002-2006 is estimated from the 2007 Demographic and Health Survey at 148 per 1,000 live births (figure 3), among the highest in the African region.

Data from DHS survey indicate some improvement over the past few years. The estimate for the period 1998-2001 is 165 per 1,000 live births, against 172 in 1993-97. However, most of the improvement is concentrated among ages 1-4 years, likely reflecting better care since the end of the civil war, while infant mortality, thought more sensitive to health care services, hardly changed. At the same time, chronic malnutrition among under-five children estimated at 31.1 % in 2007 (figure 2) may have increased. Consistent with this very poor nutritional situation, 48% of children aged 6-59 months had moderate or severe anemia in 2007, while the prevalence was 18% among women aged 15-49 years.

Using underweight as a proxy for socio-economic conditions, figure 2 illustrates how the prevalence of malnutrition among children were dramatically deteriorated during the period of war as well as the increase in child and infant mortality (figure 3). The projections towards the 2015 MDGs horizon suggest that the country would not be able to achieve health-related MDGs and could only reach the pre-conflict figures [START_REF] Gertler | The medium term impact of Oportunidades on child growth and development in rural areas of Mexico[END_REF] , far from the expected benchmarks ("halve, between 1990 and 2015, the proportion of people living extreme poverty and hunger"). Even with such assumptions, the indicators hide disparities between poorest (1 st quintile: Q1) and better-offs (5 th quintile: Q5), and between rural and urban areas. These also lead to large differences in health outcomes between different socio-economic groups and areas. While 52% of children under-five in rural areas were chronically-malnourished (stunted) in 2007, prevalence was 36% in urban areas. Reflective of the very poor nutritional situation, 50% of children aged 6-59 months in rural areas had moderate or severe anemia in 2007, compared to 45% in urban areas.

Malaria is also highly endemic and contributes a great deal to child morbidity and mortality. The disease is widespread and reported to account for over half of outpatient visits, one-third of hospital admissions, and 25-30% of child mortality (UNICEF, 2008). Population-representative survey data have similarly shown that fever is associated with 40% of child deaths (IRC, 2004, cited by UNICEF, 2008). [START_REF] Gertler | The medium term impact of Oportunidades on child growth and development in rural areas of Mexico[END_REF] These figures represent the levels at which the country was before the outbreak of hostilities. Further efforts are therefore needed to reach the levels obtained during the commitments to the MDGs (1990) , 2006). Also, it is a known fact that the course of infectious diseases such as malaria and HIV/AIDS is much more detrimental for undernourished than for other categories. Therefore, programmes that promote child development through preventing or improving child nutrition and health (in an overall morbidity reduction perspective) reduce the effects of social, environmental, and infectious risks of morbidity and mortality especially among the under-five. This leads, in a medium term, to consistent cognitive and socio-emotional outcomes and, in a long term to educational performance, employment improvement, and disruption of intergenerational cycle of poverty.

Problem analysis

Structural causes of child morbidity and mortality

Though the deterioration of the key health indicators in the DRC comes from a conjectural situation, it is also fair to acknowledge that several structural factors stand to be the direct causes of child morbidity and mortality: (i) the country's poor health policies, (ii) a multitude of supply constraints, (iii) the recent food crisis and, (iv) the loss by several households of their livelihood.

After a long period of structural adjustment marked by "belt-tightening" measures combined with poor governance and disappearance of the Welfare State, the Government spending on social sectors failed to keep up with population growth and to adequately respond to increasing health needs and demands. Care has become less and less affordable for a growing number of people, especially for the poorest. Concurrently, the ideas of "costsharing" and "community financing" (Bamako Initiative, 1987) adopted by the country and operationalised through formal and informal direct payments imposed on the users of public health care services (= user fees) have been detrimental to a large proportion of the population, especially for the poorest. However, to a large extent, these schemes have failed to meet expectations with respect to additional resource generation, improvement of quality of services and the predicted increase of health service utilization. On the contrary, a dramatic and sustained drop in the use of services was observed, given that non-creditworthy remained de facto excluded from the system.

At the opposite of the strong health need and demand, the health services supply remain poor, especially in rural areas due to numerous disruptions, lower-than-projected production growth, and small inventories operating illegally. This situation has worsened considerably, leading the informal service providers to play a significant role on household-level utilization of health services without any guarantee on their quality. The 2001 MICS found that more than 25% of children with fever received care from drug sellers.

The estimated annual requirements to develop and support health services shown in the medium term expenditure framework are in the range of US$ 430 million in the period 2008-2010. This is around US$ 6.50 per capita annually, a level of public spending on health care that would be consistent with a country with an annual GDP per capita of around US$ 200, but still in an unrealistic range.

Regarding social protection scheme, DRC like most sub-Saharan African countries has no explicit policy with regard to national solidarity and redistribution of wealth. The vast majority of the population, estimated at 95% (ILO, 2001), has no social security coverage. The option taken by policymakers was to restrict the coverage to public sector employees under the labor code, registered and able to pay regular contributions. The informal sector, rural people, self employed and unemployed were excluded from the public system, which makes the need to strengthen social safety nets in assisting poor families to cope with declining income without sacrificing the expenses related to school and health care more than necessary.

On another level, the social affairs sector has experienced a paltry budget, with a record of US $ 33 million in 2008 (see figure 5), equivalent to US $ 0.6 per capita. On this line of public spending, nearly half (48%) is devoted to the central administration costs. Thus, the amount budgeted is not necessarily spent and, therefore, is likely to overestimate the social protection expenditure as % of GDP.

A fact that deserves to be stressed is that conflicts have repeatedly led to spontaneous changes due to multiple displacements (DRC abounds more than half of IDPs in the sub-region), the loss of speed, vulnerability in a context of general insecurity. Households are therefore struggling daily to ensure their livelihood mainly based on small-scale agricultural activities and livestock. The continuing fragility of the ecosystem due to inadequate rainfall, the increase in food prices due to the dual financial and food crisis are all factors that explain the negative effects on food sovereignty with visible consequences on child nutrition and health.

As a result, the living conditions of the poorest households are deteriorating, with an increase in the level of household indebtedness and a reduction in the quality and frequency of meals, thus increasing food insecurity and malnutrition. For example, the price of imported rice in DRC in July 2008 was more than double its level a year earlier. Similarly, cooking oil and fresh vegetables have increased by over 16 percent. This is greatly limiting accessibility to nutritious food for rural and urban households (especially those headed by women or children). These crises, if left unattended, have the potential to undo the progress of the last 10 years and put the Country right back at pre-1990 levels. Malnutrition is known to have long-term cognitive and productivity effects on the children involved and may generate cumulative economic costs, up to 3% of GDP annually. This outweighs any short-term fiscal savings from direct nutrition action to prevent malnutrition and promote adequate growth in young children.

During the years 1990-2005, the proportion of people living below the poverty line rose from 80% in 2001 to 70.8% in 2005 (Table 1; ref. PRSP, 2006); the influence of insufficient resources on access to health care has impacted on the lower level of child mortality. Given these factors, changes in the mortality of children under 5 years can be explained by differences in access to care due to the situation of parents who are not remedied by programs addressing poverty and disparities. Furthermore, the probability for a child to die before his 5th birthday decrease as the educational level of mothers increases. Thus, infant mortality increased from 135 ‰ in children whose mothers have not studied at 67 ‰ in those whose mothers have reached the level of higher education.

The largest part of GDP comes from agriculture (especially cash crops) and industry 31.3% and 29.7% in 1990, and 39.8% and 22.3% in 2005 respectively. The share of services remains close to 31%. This means in a long run that improving the situation of children does not automatically come from a change in the structure of the economy or increase in real income, but rather on fair social service offer.

Policies options to overcome child morbidity and mortality

To identify the optimal option in terms of cost-benefit and cost-effectiveness on the reduction of child morbidity and mortality, we considered 3 different approaches/alternatives which are in fact non exclusive : (i) a healthoriented large scale cash transfer programme with a monthly growth monitoring for children under two (and a quarterly visit for those from 3 to 5) at a therapeutic feeding center, nutrition education for mothers, and the provision of vitamin A supplements and deworming; (ii) an outreach health and nutrition programme with a behavioral change communication component given the deterioration of the health system and the per capita allocation to health care (2% of the total government expenditure); (iii) the mobilization of resources from the government to support the elimination of basic health care user fees.

The first approach relies on a rights-based approach to emphasize not only the consequences of the program but also the process through which these are achieved with multiple objectives, namely: the alleviation of current poverty through targeted cash transfers, and the generation of a sustained decrease in poverty by conditioning these transfers on the accumulation of human capital (i.e. nutrition and health status) by households. Two other approaches which have gained prominence over the last decade are the basic needs (Streeten et al., 1981) and capabilities approach (Sen, 1985; Sen, 1987). Both of these approaches distinguish between income as a "mean" or as an "end", and highlight the commonly observed lack of strong correlation between income and other outcomes that enter into one's concept of development.

Methodology and data

The objective is to determine whether potential health-oriented cash transfer program, targeted or universal, is economical and cost effective, taking into account the direct and administrative costs to implement such a programme. The costs of alternatives have been calculated based on public expenditure rather than budget allocations. As the cost-benefit analysis requires calculating the opportunity cost of spending an equivalent amount on other programs, the question is whether the marginal benefit of an additional dollar spent more effectively on a program of cash transfer compared to the marginal benefit of an extra dollar raised to the elimination of user fees in the health sector. We have chosen to make calculations based on quantitative techniques to measure the cost of a potential cash transfer programme and determine whether it is economical and cost-effective in order to reduce poverty and accumulate human capital.

The data used in our calculations are those from the general population Census projections (1984), i.e. well before the conflicts started. High levels of mortality for the period 1990-2000, population displacement and various migrations make theses values unreliable. However, in the absence of recent and relevant data, the figures are in fact considered as being official. From a simulation based on informal sector and poverty survey (1-2-3), we estimated a US $ 10 per child per month for the least interesting scenario and US $ 15 per child per month for the better one, the direct costs of the transfer. Administrative costs have been difficult to determine because such programme in DRC should be multi-sectoral involving several administrations (as no administration has the formal institutional competency to deliver such a social programme alone). To simplify our calculations, an administrative cost of 12% of the delivering costs of the program would be reasonable, even if it seems high compared to other experiments in which these costs were approximately 5%. Because of poor governance and administrative malfunction, leakages will also be high (estimated at about 30%).

Results and discussion

Design and structure of transfers: size, conditionality and coverage

Social transfers constitute subsidies in cash or kind to households or individuals for the purpose of supplementing their income or consumption. In this context, they are primarily an instrument for poverty and vulnerability reduction of the under-fives. Also, transfer sizes related to health and nutrition conditions are generally lump sum. There have been a number of different approaches to the design of the transfer. In most cases, the amount of the health/nutrition transfer represents the difference between the consumption of an average extreme poor household and the food poverty line, with some variations (i.e., the per capita poverty gap for children under 6 only). The goal of this type of transfer is toon average -move households living in indigence to a minimum consumption level. In addition to a basic needs and political issue, minimum consumption is seen as a pre-requisite to human capital investment. Another approach is to base the transfer amount on the opportunity costs of accessing health care, which has resulted in a lower average monthly transfer. For the purposes of the present paper, it is important to note that the health and nutrition grant lump sum structure favors smaller families.

The determination of the size of the transfer is consistent with the country context marked by a significant increase in food prices and experiences in neighboring countries where the size of the transfer was averaging US$ 12 per child per month, although some countries have transferred more than US$ 25 per child per month as is the case in Lesotho and South Africa. Taking into account the government previsions in the Medium Term Expenditure Framework for the year 2010 (which takes into account the level of growth and the inflation rate at its current level), the real GDP was estimated at US$ 12.2 billion. We opted for a transfer to cover the food basket for a child per month in urban areas (US$ 10) and with non foods (US$ 15). Declination to urban / rural was not deemed necessary in our analysis because the most relevant data were lacking. It is clear that a transfer of US$ 15 per child per month would have more impact with positive externalities in terms of access to basic health care and basic education.

In order to receive the month-to-month conditional cash transfer, beneficiary households must comply with a set of conditions related to the use of preventive health services and attendance to a monthly growth monitoring for children under two (and a quarterly visit for those from 3 to 5) at a therapeutic feeding center, nutrition education for mothers, and the provision of vitamin A supplements and deworming. This package of services will enable the country not only to cope with the challenging issues of child malnutrition, but also to contribute to the African strategy for child survival and development (SASDE), a strategy that the country is preparing to implement.

The nutrition outcome is expected to result from the combination of the increased income from the transfers, the growth monitoring and the nutrition knowledge provided to caregivers. Since therapeutic feeding centers are not located in all health zones in the country, a pilot experience in areas where such centers could be used is needed, but the scaling-up of such programme will remain a major challenge. The behavior change communication strategy focuses on infant and young child feeding practices as recommended by WHO and UNICEF, disease preventive measures, home-based care, recognition of danger signs, and care seeking for sick children and incorporates not only mothers but also other influential caregivers such as grandmothers. In addition, the management of acute malnutrition which includes the use therapeutic foods has to be strengthened in all the intervention areas.

Universal or targeted programme?

The issue of children in situations of vulnerability is a complex type in an environment such as the DRC. It is the consequence of a heterogeneous group of children whose common point is to live in an environment unfavorable to their socio-emotional development: street children, abused children, children separated from their parents, children in conflict with the law, exploited children, refugee children, disabled children, etc. It should be noted that these subgroups are not exclusive. It is possible for a child to be both disabled and on the street or on the street and abused, etc. This further complicates any targeting. It is therefore difficult to project a reduction of child poverty and vulnerability by acting exclusively on family structures as it is the case in Latin American countries. Interventions of this nature can only touch a group of children (living in households). A good class of children said Orphans and Vulnerable Children -OVC living outside a structured household will not be affected by the cash transfer programme even though they culminate worst poverty and vulnerability.

Tackling child poverty reduction and vulnerability will be effective if it develops a balanced mix of universal (at the household level) and targeted services (specific groups), including outreach (social treatment on streets, career sites exploiting children, legal protection, etc..), and preventive measures (to combat the risk of social exclusion)

for children from disadvantaged situations in many ways. Can we really talk about targeting in a context where 77% of the population lives below the poverty line? Even relying on the existing social systems, targeting in this context aim to strengthen the social divide and increase administrative costs and leakages. The cash transfer can not be selective in a post-conflict environment like DRC because it will not help riding the targeted groups. Cash transfer directed to U5 caregivers is already a criterion of discrimination in the allocation of subsidies. Thus, going for a universal programme is worthwhile as it will normally reach a theoretical maximum of 3 children per household, without limiting the number of children that can benefit. Also, it should be made easier for migrants and IDPs to enroll in the programme and draw benefits in their new areas of residence. Clearly, that requires the establishment of an integrated management information system for civil registration, but that is an investment that would be well worth making to improve the efficiency and security of the whole civil registration system, as well as to make it more flexible and convenient for citizens who move residence.

Costs and potential benefits

To inform decisions about those options, it is important to look at some cost comparisons, financial affordability, and political feasibility. The number of children estimated to qualify for the program is 12.6 million and reinforces the idea that this type of program is difficult to implement in a post-conflict context with widespread poverty. Giving a monthly direct cash transfer of US$ 15 to each child under age 5, would cost US$ 1.75 billion per year, while refining the transfer to US$ 10, would cost 1.52 billion per year. The total cost of the program is likely to be in the range of US$ 2.1 billion to US$ 3.2 billion based on delivery costs which includes the cost of the transfers, the administrative costs and leakages. The evaluation of costs should also take into account the costs of the program in the broader context of social protection if the implementation is universal. Simulations based on both scenarios and calculation of costs as indicated in Table 2 reported that the programme is not affordable for the government because that would mean 17.7% to 25.9% of its real GDP in 2010 and the entire budget of the Ministry of Social Affairs (investments + current expenditures), while it is known that for Social affairs, current expenditure in absolute terms do not exceed 52%.

Discussions

Given the challenges, it is not surprising that children's health outcomes may instead be driven by the equal distribution of quality services through outreach health and nutrition programmes (20% to real GDP) than a conditional cash transfer (25.9% to real GDP) assuming that all alternatives lead to the same benefits in term of life quality gained. This raises the following question: is a large-scale cash transfer programme financially and administratively affordable and feasible in DRC given its actual socio-economic and political situation? While both supply and demand side factors are likely to be important, without up front analysis, it is difficult to determine which policy or what combination of policies is most cost-effective for DRC government investment without a descriptive and bivariate analysis on expected outcomes.

The estimated annual requirements to develop and support health services as planned in the MTEF are in the range of US$ 430 million in the period 2008-2010, around US$ 6.50 per capita annually, a level of public spending on health care that would be consistent with countries with an annual GDP per capita of around US$ 200. The achievement of health-related MDGs therefore suggests an exponential increase in resource levels and requires annual public spending of US$ 16 to US$ 22 per capita, rising to US$ 30 to US$ 40. By this standard, current estimated total domestic and international public spending of around US$ 5 to US$ 6 in DRC implies a minimum annual gap of US$ 10 per capita, or US$ 600 million additional required resources.

The deterioration of the macroeconomic balances and the dollarization of the economy are seriously jeopardizing the ability of the government to use the traditional monetary instruments, translating any domestic borrowing immediately to inflation and exchange rate depreciation. Precariously low official reserves (equivalent of only 4 weeks of non aid-related imports at end-2007) also meant that the authorities had little margin of maneuver to ensure high and shared growth and create fiscal space for carrying out poverty-fighting programs. Potential availability of funding is therefore not enough to ensure the successful introduction of cash transfers as a national programme. This also requires a favorable political and social climate, with proponents who will advocate for the necessary share of the budget. Beside, at the institutional level, there are many ingredients to achieve these objectives: improved fiduciary control over revenue collection and public expenditures, decentralization, reforms to create professional and accountable civil service, transparent natural resource management, to mention a few.

The successful experiences of cash transfers as a policy option in Latin America, Asia or in some Eastern African countries have taken into account the ramifications and issues at the policy making level. This one advocates for a political solution with key players such as the Ministry of Finance and the Ministry of Planning, where the financial implications of the program should be sufficiently analyzed before opting for the programme or its alternatives. The establishment of such mechanism also requires a rather impressive logistics and engineering to render the system transparent and fair, this is what is still lacking in the ability of government to efficiently implement at a reasonable scale.

A number of health policies have focused on access to care for the poor with a central question: "how to increase demand of care for the poor?" There is an existing UNICEF, World Bank, WHO, initiative on maternal, new born and young child health, and DRC was selected to be one of the pilot countries at the beginning of 2006. The remaining question is how to identify the core minimum package which could be offered free of charge for under 5 children and pregnant women within that initiative? The partnership is also trying to produce a health financing strategy with an emphasis on improving the levels and management of public financing. It seeks to improve national government and donor financing rather than user fees (50.43% to real GDP). More work on estimating the cost of implementing the DRC health sector development strategy is needed. The health sector development strategy adopted in 2006 intends to phase support geographically, aiming at extending progressively the geographic coverage of a comprehensive package of services. Thus, the financial requirements to implement this strategy will depend on how geographic expansion targeting is defined. First, the package of health services to be supported requires costing. Second, realistic projections of expansion of the coverage of this package are needed. Further work on the MTEF will need to reflect such analysis at the same time as remaining realistic in terms of both the likely available resource envelope and absorption capacity.

User fees and drug costs raise significant barriers to access to care for the poor, and one of the aims of domestic and international public financing is to substitute for the current level of out-of-pocket payments in order to improve access for the poor. The 2007 target is already inoperative, as the health allocation in the 2007 budget represents 5.7% of the total (excluding debt payments). Given the budget execution record described previously, even this level of spending is unlikely to be achieved. If these targets were met (and assuming that total government expenditures are around 20% of GDP, which is consistent with recent years), then government health spending would be US$ 150 million in the first year, US$ 275 million in the second, and US$ 325 million in the third.

Conclusion

Although the cash transfer and elimination of user fees are being debated, available resources would not allow public provision of basic health services free-of-charge to the entire population, even targeting the U5. Evidence of the severe impact on access to health services by poor populations in DRC during the conflict led to calls for elimination of user fees (MSF, 2001) and cash transfer conducive to human capital accumulation. Currently, the issue has been raised in the context of development programs. In any case, the paucity of available resources makes this debate somewhat moot: currently, it is neither possible to provide the basic package of health services free-of-charge to the entire population, nor is it to provide cash to households in such a post conflicts situation where the Country is struggling to stabilize its economy. Reducing financial barriers to utilization and the burden on households will require some combination of exemption policies and the provision of free or highly-subsidized selected high-impact services. Such strategies are, to some extent, already in place in DRC and they require strengthening in combination with the necessary external subsidies. This leads to the following conclusive remarks: (i) conditional cash transfer cannot operate in areas like DRC with supply-side constraints without a comprehensive health supply strategy, logistics and engineering to lower leakages administrative costs; (ii) In a context of generalized poverty (headcount = 77%), targeting is time consuming and irrelevant; (iii) The outcomes of a geographic focused cash transfer programme could be expected on improvement of nutritional status, but less on the rise for health demand and would be difficult to scale-up.

  and cost-effective way to meet universal access to basic health care especially in this post-conflict situation?

1 DRC Public Expenditure Review, World Bank report 42167 ZR, December 2007 2 Pauvreté des enfants et disparités en RDC, UNICEF RDC, Kinshasa juillet 2008 3 Democratic Republic of Congo, Economic report: Fall 2008, Worldbank. 4 UNICEF MICS 2001, UNICEF SOWC 2008 appropriate

dof U5 Trends in mortality among U5 213 220 185 148 66

	Graph 2 : Trends in underweight among children aged 6-59	Graph 3 : Trends in child and infant mortality (children aged 6-
	months		59 months)
	Trends in underweight
	Orientale 21.4 31.1 2007 % 50 250 35 32.3 19,6 26 15 10 15 20 25 30 1990 2001 2015 100 150 200 U5 Mortality rate	1990	2001	2007	2015
	Equateur 29.2
	Legend : Red = Actual evolution of the indicator ;	Nord Kivu 20
	Sud Kivu 30.8 Green = projection towards the MDGs horizon (2015) and ;
	Maniema 18.1 Orange = Projection expected as per MDGs to the benchmark Kinshasa 14.8
	Sources: DHS 1990, MICS 2001, DHS 2007.
	Bandundu 27.8
	Based on epidemiological evidences, the causal relationship between highest prevalence as stunting, iodine and Kasai Oriental 30.8 iron deficiencies, and inadequate cognitive and social-emotional stimulation was determined (Grantham-
	Bas Congo 25.6
	Katanga 20.2
	Kasai Occidental 30.3

McGregor et al., 2007; Heckman et al., 2006; Berlinski et al.

Table 2 :

 2 Costing of the programme using the 2 different scenarios

	Nb. of	$US	$US
	population	10/month/child	15/month/child

†

As percentage of the delivery cost

Table 3 :

 3 Policies alternatives and costs

	Policies alternatives	Estimated total	% to real GDP
		cost	(2010)
		(billion US$)
	Health-oriented universal cash transfer programme	3.17	25.90%
	Outreach health and nutrition programme*	2.51**	20%
	Elimination of basic health care user fees	6.33	50.43%
	* The distribution of nutritional supplements is now integrated with Vit A and deworming
	** From the Health Sector MTEF, and does not take into account leakages (normally low)


Antoine Laurent
email: laurent@limsi.fr

Nathalie Camelin
email: nathalie.camelin@lium.univ-lemans.fr

Christian Raymond
email: christian.raymond@irisa.fr

Boosting bonsai trees for efficient features combination : application to speaker role identification

In this article, we tackle the problem of speaker role detection from broadcast news shows. In the literature, many proposed solutions are based on the combination of various features coming from acoustic, lexical and semantic information with a machine learning algorithm. Many previous studies mention the use of boosting over decision stumps to combine efficiently these features. In this work, we propose a modification of this state-of-the-art machine learning algorithm changing the weak learner (decision stumps) by small decision trees, denoted bonsai trees. Experiments show that using bonsai trees as weak learners for the boosting algorithm largely improves both system error rate and learning time.

Introduction

In this article, we focus on speaker role detection in broadcast news shows. In the literature, the problem is seen as a multiclass classification problem where each speaker of a show has to be associated with a role label. In this way, some previous studies have tackled the problem using machine learning from mainly lexical features extracted from the transcription [START_REF] Barzilay | The rules behind roles: Identifying speaker role in radio broadcasts[END_REF][START_REF] Damnati | Robust speaker turn role labeling of tv broadcast news shows[END_REF][START_REF] Liu | Initial study on automatic identification of speaker role in broadcast news speech[END_REF], from acoustic / prosodic features [START_REF] Salamin | Automatic role recognition in multiparty recordings: Using social affiliation networks for feature extraction[END_REF][START_REF] Bigot | Speaker role recognition to help spontaneous conversational speech detection[END_REF], or from a combination of lexical and acoustic features [START_REF] Dufour | Combinaison d'approches pour la reconnaissance du rôle des locuteurs[END_REF]. Those studies have highlighted the efficiency of a boosting algorithm over decision stumps to combine the various features. We propose in this work a modification of this algorithm substituting the decision stumps by small decision trees, we call bonsai, in order to improve the combination of these various features. Thus, we propose here a speaker role identification system capitalizing on several features coming from acoustic, lexical or semantic level of description comparable to the system of [START_REF] Dufour | Combinaison d'approches pour la reconnaissance du rôle des locuteurs[END_REF] with the notable exception of our classification algorithm. We show on two speaker role detection databases that our algorithm combines much more efficiently features than the original algorithm while decreasing the training time of the classifier. Speaker role detection accuracy is improved up to 4% absolute while training time could be reduced by a factor of 4.

The paper is organized in the following way: Section 2 presents our speaker role recognition system, especially the set of features involved. Section 2.4 explains the modification of the boosting algorithm we propose and the benefits we expect. Section 3 presents the comparative experiments conducted on the EPAC and REPERE databases.

Speaker Role Identification System

The next three sub-sections describe the feature sets, and the last one presents the characteristics of the machine learning algorithm used.

Automatic spontaneous speech characterization

A method for automatically detecting spontaneous speech in audio documents was proposed by [START_REF] Dufour | Local and global models for spontaneous speech segment detection and characterization[END_REF]. Acoustic (vowel duration, phonemic duration, pitch. . . ) and linguistic (number of repetitions and number of proper names, syntactic pattern size. . . ) features were extracted by an automatic transcription system to evaluate the level of spontaneity of each speech segment.

In [START_REF] Dufour | Investigation of spontaneous speech characterization applied to speaker role recognition[END_REF], an analysis pointed out the relationship between speech type and speaker role. For instance, a presenter tends to prepare his speech, while a guest will often see his speech identified as highly spontaneous. The authors directly applied a spontaneous speech detection system to the particular task of role detection and reached encouraging results with an overall classification precision of 74.4%. We reimplemented this system as the baseline for the role detection task. The specific features used in this baseline system, except the automatic transcription, are called "SPONTA" in the rest of this article.

Social Network Analysis (SNA)

Social Network Analysis (SNA) aims to determine each speaker position in the dialogue. The idea supported by SNA is that a specific role, an actor, interacts with other actors during events. These interactions may help to identify the role assigned to each speaker involved in the network. Some SNA features have been investigated in [START_REF] Garg | Role recognition for meeting participants: an approach based on lexical information and social network analysis[END_REF][START_REF] Wang | Automatic identification of speaker role and agreement/disagreement in broadcast conversation[END_REF]. The goal of this method is to determine the centrality of each speaker [START_REF] Vinciarelli | Speakers role recognition in multiparty audio recordings using social network analysis and duration distribution modeling[END_REF] compared to other speakers in a show. A speaker i is considered as interacting with a speaker j if j occurs just after i in the transcript. Drawing inspiration from [START_REF] Vinciarelli | Speakers role recognition in multiparty audio recordings using social network analysis and duration distribution modeling[END_REF], we propose to compute centrality according to the following equation:

Ci = nb j=1 χDi,j nb j=1 Di,j (1)
With χ = 1 if Di,j = 1, and χ = 0 otherwise, Ci is the centrality of i, nb the number of speakers and Di,j the distance between speakers i and j. This distance is expressed as the number of oriented links to pass through in order to reach node j from node i. In the example of a social network in figure 1, D1,2 = 1 and D1,3 = 2. Some nodes are not connected to any others: this is the case for the node corresponding to speaker 5 in the example. In this situation, the distance between this node and any other node is set to nb + 1. In the remainder, "SNA" denotes the centrality and speaker coverage features.

Ngram patterns

One of the most obvious pieces of information to use to detect the speaker role is the transcription itself. The usual way to extract features from the transcription is to use word Ngram patterns. In the experiments section, the set of features extracted from the transcription is denoted as "TEXT".

Boosting of bonzai trees

The main idea of boosting is to combine many simple moderately inaccurate classifiers into a single, highly accurate one. The base classifiers are trained sequentially. At each iteration of the boosting algorithm, a base classifier is trained with the data, the boosting algorithm gives more weight to samples that have been misclassified by the previous learner, forcing the next learner to focus on them. At the end the final classifier is built by a linear combination of all weak learners.

Many previous studies on speaker role identification [9, 1, 2, 7, 10] showed the efficiency of the AdaBoost.MH [START_REF] Schapire | BoosTexter: A boostingbased system for text categorization[END_REF] algorithm over decision stumps (ie decision trees with 2 leaves) to combine the various features involved in classification based systems.

Although this algorithm has exhibited very good results, the linear combination of decision stumps may have difficulties to capture structures in the training data while boosting full decision trees may be less-efficient due to both the tree instability and data overfitting [START_REF] Quinlan | Bagging, boosting, and C4.5[END_REF].

We choose to experiment the boosting of very small decision trees, named bonsai trees, constrained by their depth. We assess that the maximum depth of the bonsai remains very small to avoid the drawbacks of full decision trees. But we expect that a slightly more complex classifier than a decision stump will be able to capture more robust structures. Figure 2 illustrates the difference between a decision stump and a depth 2 bonsai tree.

Boosting bonsai trees in our situation, where we may want to examine millions of textual features, may benefit both the efficiency and the training time of the algorithm.

While boosting algorithm is iterative, the decision tree induction can be easily sped up with parallel processing. We expect that the use of bonsai trees as weak learners will require less iterations from the boosting algorithm than using decision stumps to reach equivalent performance. Moreover, since each bonsai tree could be induced using several parallel processes, their inductions may be not too much longer than a decision stump.

We expect 3 improvements from this modification:

1. a performance gain : a bonsai tree is a more complex classifier than a decision stump, thus it should be able to capture more complex structures in the data;

2. a training time gain : for the same reason, our boosting algorithm should produce a classifier that performs the same as the original algorithm with much fewer iterations;

3. a fewer number of features used : a bonsai should be able to build implicitly Ngrams from bag of words (unigrams). If we are right, we will not have to generate explicitly Ngrams and evaluate them any more. The consequence would be a drastic reduction of features to be evaluated by the algorithm and therefore again we should observe a training time gain.

In the original algorithm, decision stumps are induced according to the pseudo-loss criterion proposed in [START_REF] Schapire | BoosTexter: A boostingbased system for text categorization[END_REF]. We grow our bonsai by applying the same procedure recursively in each leaf of the tree. This recursion is stopped according to 2 criteria:

1. tree depth: the tree is a bonsai because it is constrained by his depth;

2. pseudo-loss gain: if a node subdivision attempt does not reduce the pseudo-loss, this node become a leaf.

Our implementation is available online [START_REF] Raymond | bonzaiboost[END_REF].

Experiments

Experimental protocol

Our role detection system is evaluated on two corpora : EPAC and REPERE, briefly described in the next sub-sections. In order to evaluate our performance, we compare with two previously published systems : the first described in [START_REF] Dufour | Investigation of spontaneous speech characterization applied to speaker role recognition[END_REF] for the EPAC corpus and the second described in [START_REF] Bigot | Speaker role recognition on tv broadcast documents[END_REF] for the REPERE corpus.

All results are presented in terms of precision, recall and Role Error Rate (RER). The RER corresponds to the ratio between the number of misclassified examples and the total number of examples to be classified. Evaluations are applied on the whole corpus through a 20 cross-fold evaluation for EPAC and a 200 cross-fold evaluation for REPERE.

In all the experiments, 1g or 2g refers to the size of the Ngrams extracted (note that 2g includes 1g) while the d{1..4} refers to the depth of the bonsai used as weak learner in the boosting algorithm. The term ALL denotes that all features are used (ie. TEXT, SPONTA and SNA), otherwise the type of features used is explicitly written.

Brodcast news from EPAC Corpus

Corpus

The EPAC corpus was built during the EPAC project funded by the French Research Agency (ANR) from 2007 to 2010 [START_REF] Estève | The EPAC corpus: manual and automatic annotations of conversational speech in French broadcast news[END_REF]. It is composed of 100 hours of conversational speech manually transcribed. Broadcast news from 3 French radio stations (France Info, France Culture and RFI) were recorded between 2003 and 2004 . In addition to orthographic transcription, a lot of metadata have been manually annotated, including speaker role, function and job (when available). Each speaker has a main role among the following seven:

• Auditor: the radio listener who calls during a program;

• Columnist: the broadcaster or writer who reports and analyzes events in the news during a broadcast show;

• Correspondent: the journalist who reports on particular subjects from a remote location;

• Guest: person who comes to talk about his latest news;

• Interviewee: person who answers questions raised by the presenter;

• Anchor: person who has multiple functions within a show-can host a talk, may take calls from listeners, or has the responsibility to give news, or weather information. . ;

• Voice-over: pre-recorded voice commonly used to introduce programs.

The orthographic transcription that we used for the training and the evaluation comes from an automatic speech recognition system [START_REF] Deléglise | Improvements to the LIUM French ASR system based on CMU Sphinx: what helps to significantly reduce the word error rate?[END_REF] which yields a WER of 17.3% on the EPAC test corpus.

Results

We compare our system to a baseline system trained with the same configuration as used in [START_REF] Dufour | Investigation of spontaneous speech characterization applied to speaker role recognition[END_REF] in which we include the SNA feature as they do in [START_REF] Dufour | Combinaison d'approches pour la reconnaissance du rôle des locuteurs[END_REF]. This system corresponds in our notation to ALL-2g-d1. Experimental results point out interesting conclusions.

Firstly, Table 1 compares results from our proposed system, tuned with a depth of 3, and the baseline system, which is depth 1, trained with different feature combinations. It shows how each feature type (TEXT,SNA,SPONTA) contributes to the performance of the final system. We can observe that our approach outperforms significantly the baseline for all configurations. The results show that a decision stump is too weak to capture some relevant information from data. Figure 3 tends to confirm: indeed, while learning curves for boosting over depth 2/3/4 bonsai converge around the same plateau, boosting over decision stumps gets stuck.

Secondly, Table 3 shows that a depth 2 tree as base learner for boosting gives notably better results than a decision stump. Thirdly, Table 2 details the results for all roles obtained with our depth 3 bonsai system using 1g and 2g as text features. We compare theses configuration to the system proposed in [START_REF] Dufour | Combinaison d'approches pour la reconnaissance du rôle des locuteurs[END_REF] that actually correspond to the notation ALL-2g-d1 in the table. We can observe that a consistent improvment is observed through all different roles. Moreover, these results show that using a bonsai on 1g gets better performance than decision stumps on 2g. Using bonsai on 2g does not bring improvement over bonsai on 1g. This is an interesting result, since it tends to show that it is not necessary to generate explicitly 2g or more when using bonsai trees: Ngrams are captured by the tree. The REPERE corpus was built during the DEFI-REPERE french evaluation campaign from 2010 to 2014 [START_REF] Kahn | The repere challenge: Finding people in a multimodal context[END_REF]. Data from various TV shows were recorded from French TV channels : BFM and LCP. Twenty four hours have been recorded : about 14 hours of broadcast news, about 6.5 hours of debates and 3.5 hours of parliamentary sessions of the French National Assembly.

ALL-2g-d1

The corpus has been manually segmented according to the speaker and manually enriched with annotations relative to five speaker roles:

• R1 : the anchor;

• R2 : journalists, equivalent to the EPAC columnist;

• R3 : reporters, equivalent to the EPAC correspondent and voice-over roles;

• R4 : guests;

• R5 contains all remaining speakers with another role, like EPAC interviewee or auditors, or politicians . . .

The [START_REF] Deléglise | Improvements to the LIUM French ASR system based on CMU Sphinx: what helps to significantly reduce the word error rate?[END_REF] system adapted for REPERE yields a WER of 15.18% on the REPERE test corpus (phase 1).

Results

Similarly to the experiments done on the EPAC corpus we propose in Table 4 different results obtained for different depth of bonsai. We compare our results with the SVM based approach published in [START_REF] Bigot | Speaker role recognition on tv broadcast documents[END_REF], denoted Bigot. They propose 2 different systems based on low level features, one using the show information (denoted Bigot+show) and the other not. It should be noted that the set of features used in their system and ours is not the same. Boosting system over decision stumps exhibits better performance than the SVM approach proposed in [START_REF] Bigot | Speaker role recognition on tv broadcast documents[END_REF].

We point out that once again boosting bonsai outperforms significantly boosting decision stumps with observed RER of 85.1%, 88.1% and 88.6% for respectively bonsai depth 1, 2 and 3. Furthermore, as previously shown on the EPAC corpus, a consistant improvment is observed through all the different roles. We can notice that the ALL-1g-d3 system that does not take into account the show information already outperforms the (Bigot et al.)+show. Taking into account the name of the show as a new feature of our system (ALL-1g-d3+show) allows an extra-benefit of 2.4%.

Finally, to be fair, we also evaluate our system without using the transcription as in [START_REF] Bigot | Speaker role recognition on tv broadcast documents[END_REF]. The system SPONTA+SNA-d3 obtained a RER of 90.2%, which is 7 points better than the reference system.

Conclusion

We have investigated and presented in this paper a modification of the boosting algorithm widely used in the literature for speaker role identification. We propose to replace decision stumps used as weak learner for the boosting algorithm by small decision trees, we called bonsais. A speaker role identification system based on this algorithm is evaluated and compared to state-of-the-art systems published in [START_REF] Dufour | Combinaison d'approches pour la reconnaissance du rôle des locuteurs[END_REF] and [START_REF] Bigot | Speaker role recognition on tv broadcast documents[END_REF] on two different corpora.

Experiment results tend to confirm our 3 expectations about the use of bonsai as weak learner for the boosting algorithm: it allows significant improvement in terms of accuracy while bonsai needs only to be very small to get these improvements. It permits a strong reduction of the number of iterations of the boosting algorithm. It is also able to produce similar results using 1g features and 2g features, because it is able to implicitly build them: this result is quite interesting because it seems that we don't need to generate explicitly complex Ngram features, saving a lot of training time. Actually, taking in consideration the previous facts, we computed that building a boosting based system with depth 3 bonsais is approximately 4 times faster than building a boosting based system on decision stumps.

Figure 1 :

 1 Figure 1: Example of a social network

Figure 2 :

 2 Figure 2: Examples of bonsai trees learned on the EPAC corpus. Each node represents the test selected during the bonsai induction. Each leaf indicates the population P and the majority label along with his frequency.

Figure 3 :

 3 Figure 3: Results in term of 100-RER for learning curves of various depths trees wrt the number of boosting iterations.

Table 1 :

 1 RER of the systems trained with various feature combinations and various depth : d3 corresponds to our system trained with a bonsai of depth 3 and d1 corresponds to the baseline system trained with classical decision stumps.Increasing depth to 3 outperforms again, while depth 4 yields similar performance. Thus these results show that using trees only slightly deeper than the usual decision stumps brings notable improvements to system performance.

			d3	d1
	TEXT -2g		68.9 64.5
	+ SPONTA 75.8 71.6
	+ SNA	77.7 73.9
	+ SNA		75.0 70.9
	SPONTA		67.9 65.2
	+ SNA		74.1 70.9
	SNA		60.0 54.6
	Ngram/tree depth	d1	d2	d3	d4
	1g	72.9 76.2 77.7 78.1
	2g	73.9 77.8 77.7 78.4

Table 3 :

 3 RER of the system trained with every features on EPAC.

Table 2 :

 2 Comparative results, in precision and recall, of our depth 3 bonsai system with[START_REF] Dufour | Combinaison d'approches pour la reconnaissance du rôle des locuteurs[END_REF], denoted ALL-2g-d1 for various roles of EPAC. The first column indicates the number of each role and the last line shows the RER for the three systems.

	ALL-1g-d3	ALL-2g-d3

Table 4 :

 4 Comparative results of our best system, that uses a 3 depth bonsai, with the reference[START_REF] Bigot | Speaker role recognition on tv broadcast documents[END_REF] system on the REPERE corpus. The term show indicates the show-dependent classifier.

	3.3. Brodcast news from REPERE Corpus
	3.3.1. Corpus