Christophe Chesneau

Adaptive wavelet estimator for a function and its derivatives in an indirect convolution model

Keywords: Deconvolution, Function estimation, Rate of convergence, Wavelets, Hard thresholding. 2000 Mathematics Subject Classification 62G07, 62G20

We consider an indirect convolution model where m blurred and noise-perturbed functions f 1 , . . . , f m are randomly observed. For a fixed ω ∈ {1, . . . , m}, we want to estimate f ω and its derivatives. An adaptive nonlinear wavelet estimator using a singular value decomposition is developed. Taking the mean integrated squared error over Besov balls, we prove that it attains a fast rate of convergence.

Motivations

We observe n stochastic processes Y 1 (t), . . . , Y n (t), t ∈ [0, 1], such that, for any v ∈ {1, . . . , n}, dY v (t) = (f Xv ⋆ g)(t)dt + dW v (t),

where

(f Xv ⋆ g)(t) = 1 0 f Xv (t -u)g(u)du,
W 1 (t), . . . , W n (t) are n unobserved independent standard Brownian motions, g : [0, 1] → R is a known 1-periodic function, X 1 , . . . , X n are n unobserved independent discrete random variables such that, for any v ∈ {1, . . . , n}, the distribution of X v is known, the set of possible values of X v is

X v (Ω) = {1, . . . , m}, m ∈ N * ,
Laboratoire de Mathématiques Nicolas Oresme, Université de Caen Basse-Normandie, Campus II, Science 3, 14032 Caen, France. E-mail: chesneau@math.unicaen.fr and, for any d ∈ {1, . . . , m}, f d : [0, 1] → R is an unknown 1-periodic function.

We suppose that X 1 , . . . , X n , W 1 (t), . . . , W n (t) are independent. For a fixed ω ∈ {1, . . . , m}, we want to estimate f ω and its q-th derivative f (q) ω from Y 1 (t), . . . , Y n (t). An application of this estimation problem is the following: m blurred and noise-perturbed signals f 1 , . . . , f m are randomly observed, and only f ω is of interest. The blurring process is achieved through the convolution operator T (h) = (h ⋆ g)(t) = 1 0 h(t -u)g(u)du. Remark that, when X 1 , . . . , X n are constant random variables with X 1 = . . . = X n = 1, (1) gives

d Y (t) = (f 1 ⋆ g)(t)dt + 1 √ n d W (t), (2)
where

Y (t) = 1 n n v=1 Y v (t), W (t) = 1 √ n n v=1 W v (t).
Since W (t) is a standard Brownian motion, (2) becomes the standard convolution model. In this case, the estimation of f 1 has received a lot of attention. See e.g. [START_REF] Cavalier | Sharp adaptation for inverse problems with random noise[END_REF], [START_REF] Johnstone | Wavelet deconvolution in a periodic setting[END_REF], [START_REF] Donoho | Translation invariant deconvolution in a periodic setting[END_REF], [START_REF] Kerkyacharian | Adaptive boxcar deconvolution on full Lebesgue measure sets[END_REF] and [START_REF] Chesneau | Wavelet estimation via block thresholding: A minimax study under the L p risk[END_REF].

However, to the best of our knowledge, the estimation of f ω (and f (q) ω ) from the general model (1) has not been addressed earlier. Note that the estimation of f (q) ω is of interest to detect possible bumps, concavity or convexity properties of f ω . In the literature on nonparametric functional estimation, the problem of estimating the derivatives have been investigated by several authors starting with [START_REF] Bhattacharya | Estimation of a probability density function and its derivatives[END_REF]. For references using wavelet methods, see e.g. Prakasa [START_REF] Rao | Nonparametric estimation of the derivatives of a density by the method of wavelets[END_REF], [START_REF] Chaubey | Wavelet based estimation of the derivatives of a density for m-dependent random variables[END_REF], [START_REF] Chaubey | Wavelet based estimation of the derivatives of a density with associated variables[END_REF][START_REF] Chaubey | Wavelet based estimation of the derivatives of a density for a negatively associated process[END_REF] and [START_REF] Chesneau | Wavelet estimation of the derivatives of an unknown function from a convolution model[END_REF].

Considering the ordinary smooth case on g (see (4)), we estimate f (q) ω (including f (0) ω = f ω ) by two wavelet estimators: a linear nonadaptive and a nonlinear adaptive based on the hard thresholding rule. Our adaptive estimator uses a singular value decomposition (SVD), some algebraic tools on the distributions of X 1 , . . . , X n and a new version of the "observations thresholding" introduced by [START_REF] Delyon | On minimax wavelet estimators[END_REF]. To evaluate their performances, we adopt the mean integrated squared error (MISE) over Besov balls. Under mild assumptions on the distributions of X 1 , . . . , X n , we prove that our hard thresholding estimator attains a sharp rate of convergence, close to the one attained by our linear wavelet estimator.

The paper is organized as follows. Assumptions on (1) and some notations are introduced in Section 2. Section 3 briefly describes the wavelet basis and the Besov balls. The estimators are presented in Section 4. The results are set in Section 5. Technical proofs are given in Section 6.

Assumptions and notations

Assumptions on f 1 , . . . , f m and g.

-We assume that f 1 , . . . , f m and g belong to L 2 per ([0, 1]) defined by

L 2 per ([0, 1]) = h; h is 1-periodic and 1 0 |h(x)| 2 dx 1/2 < ∞ .
-We suppose that there exists a known constant C * > 0 such that sup d∈{1,...,m}

sup x∈[0,1] |f (q) d (x)| ≤ C * . (3)
Assumptions on the smoothness of g. Any function h ∈ L 2 per ([0, 1]) can be represented by its Fourier series

h(t) = ℓ∈Z F ℓ (h)e 2iπℓt , t ∈ [0, 1],
where the equality is intended in mean-square convergence sense, and F ℓ (h) denotes the Fourier coefficient given by

F ℓ (h) = 1 0 h(x)e -2iπℓx dx, ℓ ∈ Z,
whenever this integral exists. The notation • will be used for the complex conjugate.

We consider the ordinary smooth case on g: there exist three constants, c g > 0, C g > 0 and δ > 1, such that, for any ℓ ∈ Z, F ℓ (g) satisfies

c g (1 + ℓ 2 ) δ/2 ≤ |F ℓ (g)| ≤ C g (1 + ℓ 2 ) δ/2 . ( 4
)
This assumption controls the decay of the Fourier coefficients of g, and thus the smoothness of g. It is a standard hypothesis usually adopted in the field of nonparametric estimation for deconvolution problems. See e.g. [START_REF] Pensky | Adaptive wavelet estimator for nonparametric density deconvolution[END_REF], [START_REF] Fan | Wavelet deconvolution[END_REF] and [START_REF] Johnstone | Wavelet deconvolution in a periodic setting[END_REF]. Assumptions on X 1 , . . . , X n . Recall that X 1 , . . . , X n are unobserved and, for any v ∈ {1, . . . , n}, we let

w d (v) = P(X v = d), d ∈ {1, . . . , m}.
We suppose that the matrix

Γ n = 1 n n v=1 w k (v)w ℓ (v) (k,ℓ)∈{1,...,m} 2 satisfies det(Γ n ) > 0.
For the considered ω (the one which refers to the estimation of f (q) ω ) and any v ∈ {1, . . . , n}, we set

a ω (v) = 1 det(Γ n ) m k=1 (-1) k+ω γ n ω,k w k (v), (5)
where γ n ω,k denotes the determinant of the minor (ω, k) of the matrix Γ n . Then (a ω (v)) v∈{1,...,n} satisfy (a ω (1), . . . , a ω (n)) = argmin

(b1,...,bn)∈∩ m d=1 U ω,d 1 n n v=1 b 2 v , (6)
where

U ω,d = (b 1 , . . . , b n ) ∈ R n ; 1 n n v=1 b v w d (v) = δ ω,d
and δ ω,d is the Kronecker delta. Technical details can be found in [START_REF] Maiboroda | Estimators of components of a mixture with varying concentrations[END_REF]. We set

ρ n = 1 n n v=1 a 2 ω (v) (7)
and we suppose that lim n→∞ ρ n /n = ∞.

The sequence (a ω (v)) v∈{1,...,n} is usually used in the field of nonparametric statistics for mixture density estimation problems. See e.g. [START_REF] Maiboroda | Estimators of components of a mixture with varying concentrations[END_REF][START_REF] Pokhyl'ko | Wavelet estimators of a density constructed from observations of a mixture[END_REF] and Prakasa [START_REF] Rao | Wavelet linear estimation for derivatives of a density from observations of mixtures with varying mixing proportions[END_REF].

3 Wavelets and Besov balls Periodized Meyer wavelets. We consider an orthonormal wavelet basis generated by dilations and translations of a "father" Meyer-type wavelet φ and a "mother" Meyer-type wavelet ψ. The main features of such wavelets are: 1. the Fourier transforms of φ and ψ have compact supports with

       supp (F(φ)) ⊂ - 4π 3 , 4π 3 , supp (F(ψ)) ⊂ - 8π 3 , - 2π 3 ∪ 2π 3 , 8π 3 ,
2. the functions φ and ψ are C ∞ .

For the purpose of this paper, we use the periodized Meyer wavelet bases on the unit interval. For any x ∈ [0, 1], any integer j and any k ∈ {0, . . . , 2 j -1}, let

φ j,k (x) = 2 j/2 φ(2 j x -k), ψ j,k (x) = 2 j/2 ψ(2 j x -k)
be the elements of the wavelet basis, and

φ per j,k (x) = l∈Z φ j,k (x -l), ψ per j,k (x) = l∈Z ψ j,k (x -l),
be their periodized versions. There exists an integer j * such that, for any integer j c ≥ j * , the collection B = {φ per jc,k , k ∈ {0, . . . , 2 jc -1}; ψ per j,k , j ∈ N -{0, . . . , j c -1}, k ∈ {0, . . . , 2 j -1}} forms an orthonormal basis of L 2 per ([0, 1]). In what follows, the superscript "per" will be dropped to lighten the notation. Let j c be an integer such that j c ≥ j * . A function h ∈ L 2 per ([0, 1]) can be expanded into a wavelet series as

h(x) = 2 jc -1 k=0 α jc,k φ jc,k (x) + ∞ j=jc 2 j -1 k=0 β j,k ψ j,k (x), x ∈ [0, 1],
where

α j,k = 1 0 h(x)φ j,k (x)dx, β j,k = 1 0 h(x)ψ j,k (x)dx. (8)
See (Meyer 1992, Vol. 1 Chapter III.11) for a detailed account on periodized orthonormal wavelet bases. Besov balls. Let M > 0, s > 0, p ≥ 1 and r ≥ 1. A function h belongs to B s p,r (M ) if and only if there exists a constant M * > 0 such that the wavelet coefficients (8) satisfy

2 j * (1/2-1/p)   2 j * -1 k=0 |α j * ,k | p   1/p +    ∞ j=j *   2 j(s+1/2-1/p)   2 j -1 k=0 |β j,k | p   1/p    r    1/r ≤ M * .
For a particular choice of parameters s, p and r, these sets contain the Hölder and Sobolev balls. See [START_REF] Meyer | Wavelets and Operators[END_REF].

Estimators

Estimators for wavelet coefficients. The first step to estimate f (q) ω consists in expanding f (q) ω on B and estimating its unknown wavelet coefficients. For any integer j ≥ j * and any k ∈ {0, . . . , 2 j -1},

-we estimate α j,k = 1 0 f (q) ω (x)φ j,k (x)dx by α j,k = 1 n n v=1 a ω (v) ℓ∈Cj (2iπℓ) q F ℓ (φ j,k ) F ℓ (g) 1 0 e -2iπℓt dY v (t), (9)
where C j = supp (F(φ j,0 )) = supp (F (φ j,k )) and a ω (v) is defined by ( 5), -we estimate

β j,k = 1 0 f (q) ω (x)ψ j,k (x)dx by β j,k = 1 n n v=1 Q v 1 {|Qv|≤ηj } , (10)
where

Q v = a ω (v) ℓ∈Dj (2iπℓ) q F ℓ (ψ j,k ) F ℓ (g) 1 0 e -2iπℓt dY v (t), (11)
D j = supp (F(ψ j,0 )) = supp (F (ψ j,k
)), for any random event A, 1 A is the indicator function on A, a ω (v) is defined by ( 5),

η j = θ2 (δ+q)j nρ n ln(n/ρ n ) ,
ρ n is defined by ( 7) and

θ = 3C 2 * + 2 δ 2(2π) 2q c 2 g 8π 3 2(δ+q) . (12)
(C * , c g and δ are those in (3) and ( 4)). Statistical properties of α j,k and β j,k are investigated in Propositions 1, 3, 4 and 5.

Remarks.

-The constructions of α j,k and β j,k are based on a SVD technique (see Proposition 1). -The idea of the thresholding in ( 10) is to operate a selection on the observations: when,

for v ∈ {1, . . . , n}, Q v is too large, Y v (t) is neglected.
From a technical point of view, this allows us to estimate β j,k in an optimal way under mild assumptions on (a ω (v)) v∈{1,...,n} and, a fortiori, on the distributions of X 1 , . . . , X n . Such a thresholding method has been introduced by [START_REF] Delyon | On minimax wavelet estimators[END_REF] for regression wavelet estimation. Note that, in the simplest case where X 1 , . . . , X n are constant random variables, such a selection is not necessary. We consider two wavelet estimators for f (q) ω : a linear estimator and a hard thresholding estimator.

Linear estimator. Assuming that f (q) ω ∈ B s p,r (M ) with p ≥ 2, we define the linear estimator f

(q) L by f (q) L (x) = 2 j 0 -1 k=0 α j0,k φ j0,k (x), ( 13
)
where α j,k is defined by ( 9), j 0 is the integer satisfying

1 2 n ρ n 1/(2s+2δ+2q+1) < 2 j0 ≤ n ρ n 1/(2s+2δ+2q+1)
, ρ n is defined by ( 7) and δ is the one in (4). Note that f (q)

L is not adaptive since it depends on s.

Hard thresholding estimator. We define the hard thresholding estimator f

(q) H by f (q) H (x) = 2 j * -1 k=0 α j * ,k φ j * ,k (x) + j1 j=j * 2 j -1 k=0 β j,k 1 {| β j,k |≥κλj } ψ j,k (x), ( 14
)
where α j * ,k is defined by ( 9), β j,k by ( 10), j 1 is the integer satisfying

1 2 n ρ n 1/(2δ+2q+1) < 2 j1 ≤ n ρ n 1/(2δ+2q+1)
, κ ≥ 8/3 + 2 + 2 16/9 + 4, λ j is the threshold

λ j = θ2 (δ+q)j ρ n ln(n/ρ n ) n , ( 15
)
ρ n is defined by ( 7), θ by ( 12) and δ is the one in (4).

Note that f (q)

H is adaptive. The feature of the hard thresholding estimator is to only estimate the "large" unknown wavelet coefficients of f (q) ω which contain the main characteristics of f (q) ω . Hard thresholding estimators for other deconvolution problems than (1) can be found in [START_REF] Fan | Wavelet deconvolution[END_REF], [START_REF] Johnstone | Wavelet deconvolution in a periodic setting[END_REF], [START_REF] Willer | Deconvolution in white noise with a random blurring effect[END_REF] and [START_REF] Cavalier | Wavelet deconvolution with noisy eigenvalues[END_REF].

Results

Upper bounds for f (q)

L and f (q) H are given in Theorems 1 and 2 below. Further details on our statistical approach can be found in [START_REF] Tsybakov | Introduction à l'estimation nonparamétrique[END_REF].

Theorem 1 Consider (1) under the assumptions of Section 2. Suppose that f (q) ω ∈ B s p,r (M ) with s > 0, p ≥ 2 and r ≥ 1. Let f (q) L be (13). Then there exists a constant C > 0 such that

E 1 0 f (q) L (x) -f (q) ω (x) 2 dx ≤ C ρ n n 2s/(2s+2δ+2q+1)
.

The proof of Theorem 1 uses a moment inequality on (9) and a suitable decomposition of the MISE. Note that f (q)

L is constructed to minimize the MISE as much as possible. For this reason, our benchmark will be the rate of convergence (ρ n /n) 2s/(2s+2δ+2q+1) .

Theorem 2 Consider (1) under the assumptions of Section 2. Let f

(q) H be (14). Suppose that f (q) ω ∈ B s p,r (M ) with r ≥ 1, {p ≥ 2 and s > 0} or {p ∈ [1, 2
) and s > (2δ + 2q + 1)/p}. Then there exists a constant C > 0 such that

E 1 0 f (q) H (x) -f (q) ω (x) 2 dx ≤ C ρ n ln(n/ρ n ) n 2s/(2s+2δ+2q+1)
.

The proof of Theorem 2 is based on several probability results (moment inequalities, concentration inequality,. . . ) and a suitable decomposition of the MISE.

Theorem 2 shows that, besides being adaptive, f

H attains the same rate of convergence than the one of f (q) L up to a logarithmic term. Naturally, in the simplest case where X 1 , . . . , X n are constant random variables with X 1 = . . . = X n = 1 (so ρ n = 1 and f

(q) ω = f (q)
1 ), the rate of convergence attained by f (q)

H becomes the standard one for (2) i.e. (ln n/n) 2s/(2s+2δ+2q+1) . See [START_REF] Johnstone | Wavelet deconvolution in a periodic setting[END_REF] and [START_REF] Chesneau | Wavelet estimation of the derivatives of an unknown function from a convolution model[END_REF].

Conclusion and perspectives. Considering (1), we have developed a new adaptive estimator f (q)

H for f (q) ω . It is based on a SVD, wavelets and the hard thresholding rule. It attains a sharp rate of convergence for a wide class of functions. Possible perspectives of this work are -to investigate the case where the distributions of X 1 , . . . , X n are unknown, -to potentially improve the estimation of f (q) ω by considering other kinds of thresholding rules as the block thresholding one (BlockJS, . . . ). See e.g. [START_REF] Cai | Adaptive Wavelet Estimation: A Block Thresholding And Oracle Inequality Approach[END_REF][START_REF] Cai | On adaptive wavelet estimation of a derivative and other related linear inverse problems[END_REF], [START_REF] Pensky | Functional deconvolution in a periodic setting: Uniform Case[END_REF] and [START_REF] Petsa | Minimax convergence rates under the L p -risk in the functional deconvolution model[END_REF].

All these aspects need further investigations that we leave for a future work.

Proofs

In this section, C represents a positive constant which may differ from one term to another.

Auxiliary results

Proposition 1 For any integer j ≥ j * and any k ∈ {0, . . . , 2 j -1}, let α j,k and β j,k be the wavelet coefficients (8) of f (q) ω . Then -α j,k defined by ( 9) is an unbiased estimator of α j,k , -for (Q v ) v∈{1,...,n} defined by (11), we have

E 1 n n v=1 Q v = β j,k .
Proof of Proposition 1. We have

1 0 e -2iπℓt dY v (t) = 1 0 e -2iπℓt (f Xv ⋆ g)(t)dt + 1 0 e -2iπℓt dW v (t)
with

1 0 e -2iπℓt (f Xv ⋆ g)(t)dt = F ℓ (f Xv ⋆ g) = F ℓ (f Xv )F ℓ (g). ( 16
)
Since E

1 0 e -2iπℓt dW v (t) = 0, we have

E 1 0 e -2iπℓt dY v (t) = E (F ℓ (f Xv )) F ℓ (g) + E 1 0 e -2iπℓt dW v (t) = m d=1 w d (v)F ℓ (f d )F ℓ (g). (17)
Note that, since f ω is 1-periodic, for any u ∈ {0, . . . , q}, f

(u) ω is 1-periodic and f (u) ω (0) = f (u)
ω (1). By q integrations by parts, for any ℓ ∈ Z, we obtain

(2iπℓ) q F ℓ (f ω ) = F ℓ f (q) ω . ( 18
)
It follows from ( 17), ( 6), ( 18) and the Parseval-Plancherel theorem that

E ( α j,k ) = 1 n n v=1 a ω (v) ℓ∈Cj (2iπℓ) q F ℓ (φ j,k ) F ℓ (g) E 1 0 e -2iπℓt dY v (t) = 1 n n v=1 a ω (v) ℓ∈Cj (2iπℓ) q F ℓ (φ j,k ) F ℓ (g) m d=1 w d (v)F ℓ (f d )F ℓ (g) = m d=1   ℓ∈Cj (2iπℓ) q F ℓ (φ j,k )F ℓ (f d )   1 n n v=1 a ω (v)w d (v) = ℓ∈Cj F ℓ (φ j,k )(2iπℓ) q F ℓ (f ω ) = ℓ∈Cj F ℓ (φ j,k )F ℓ f (q) ω = 1 0 φ j,k (x)f (q) ω (x)dx = α j,k .
Similarly, taking ψ instead of φ, we prove that

E 1 n n v=1 Q v = β j,k
. This completes the proof of Proposition 1.

Proposition 2 For any integer j ≥ j * and any k ∈ {0, . . . , 2 j -1}, set

R j,k = ℓ∈Cj (2iπℓ) q F ℓ (φ j,k ) F ℓ (g) 1 0 e -2iπℓt dY v (t).
Then

E(|R j,k | 2 ) ≤ θ 2 2 2(δ+q)j ,
where θ is defined by ( 12).

Proof of Proposition 2. We have

E(|R j,k | 2 ) = E(E(|R j,k | 2 |X v )) = E(V(R j,k |X v )) + E(|E(R j,k |X v )| 2 ). ( 19
) Let us bound E(V(R j,k |X v )) and E(|E(R j,k |X v )| 2 ) in turn. Upper bound for E(V(R j,k |X v )). Since (e -2iπℓt ) ℓ∈Z is an orthonormal basis of L 2 per ([0, 1]) and X 1 , . . . , X n , W 1 (t), . . . , W n (t) are independent, 1 0 e -2iπℓt dY v (t) ℓ∈Cj conditionally to X v are also independent. Therefore V (R j,k |X v ) = ℓ∈Cj (2πℓ) 2q |F ℓ (φ j,k )| 2 |F ℓ (g)| 2 V 1 0 e -2iπℓt dY v (t)|X v . ( 20
)
Using the elementary inequality (x + y) 2 ≤ 2(x 2 + y 2 ), (x, y) ∈ R 2 , the independence of X 1 , . . . , X n , W 1 (t), . . . , W n (t), ( 16) and the fact that (e -2iπℓt ) ℓ∈Z is an orthonormal basis of L 2 per ([0, 1]), we obtain

V 1 0 e -2iπℓt dY v (t)|X v ≤ E 1 0 e -2iπℓt dY v (t) 2 |X v ≤ 2 1 0 e -2iπℓt (f Xv ⋆ g)(t)dt 2 + E 1 0 e -2iπℓt dW v (t) 2 = 2 |F ℓ (f Xv )| 2 |F ℓ (g)| 2 + 1 . Hence E V 1 0 e -2iπℓt dY v (t)|X v ≤ 2 E |F ℓ (f Xv )| 2 |F ℓ (g)| 2 + 1 . (21)
It follows from ( 20) and ( 21) that

E(V(R j,k |X v )) ≤ 2 (A + B) , (22)
where

A = ℓ∈Cj (2πℓ) 2q |F ℓ (φ j,k ) | 2 E |F ℓ (f Xv )| 2 and B = ℓ∈Cj (2πℓ) 2q |F ℓ (φ j,k ) | 2 |F ℓ (g)| 2 .
Let us now bound A and B in turn.

Upper bound for A. It follows from ( 18) and ( 3) that

(2πℓ) 2q E |F ℓ (f Xv )| 2 = E |F ℓ f (q) Xv | 2 ≤ C 2 * . The Parseval-Plancherel theorem gives ℓ∈Cj |F ℓ (φ j,k ) | 2 = 1 0 |φ j,k (x)| 2 dx = 1. So A ≤ C 2 * ℓ∈Cj |F ℓ (φ j,k ) | 2 = C 2 * . ( 23
)
Upper bound for B. The assumption (4) and the elementary inequality |x

+ y| δ ≤ 2 δ-1 (|x| δ + |y| δ ), (x, y) ∈ R 2 , imply that sup ℓ∈Cj (2πℓ) 2q |F ℓ (g)| 2 ≤ (2π) 2q c 2 g sup ℓ∈Cj ℓ 2q (1 + ℓ 2 ) δ ≤ 2 δ-1 (2π) 2q c 2 g sup ℓ∈Cj ℓ 2q (1 + ℓ 2δ ) ≤ 2 δ-1 2(2π) 2q c 2 g 8π 3 2(δ+q)
2 2(δ+q)j .

Using again the Parseval-Plancherel theorem, we obtain

B ≤ 2 δ-1 2(2π) 2q c 2 g 8π 3 2(δ+q) 2 2(δ+q)j ℓ∈Cj |F ℓ (φ j,k ) | 2 = 2 δ-1 2(2π) 2q c 2 g 8π 3 2(δ+q)
2 2(δ+q)j .

(24)

Putting ( 22), ( 23) and ( 24) together, we have

E(V(R j,k |X v )) ≤ 2 C 2 * + 2 δ-1 2(2π) 2q c 2 g 8π 3 2(δ+q) 2 2(δ+q)j . ( 25
) Upper bound for E(|E(R j,k |X v )| 2 ).
We have E 1 0 e -2iπℓt dW v (t) = 0 and, by ( 16),

E 1 0 e -2iπℓt (f Xv ⋆ g)(t)dt|X v = F ℓ (f Xv )F ℓ (g).
Therefore, using (18), we obtain

E(R j,k |X v ) = ℓ∈Cj (2iπℓ) q F ℓ (φ j,k ) F ℓ (g) E 1 0 e -2iπℓt dY v (t)|X v = ℓ∈Cj (2iπℓ) q F ℓ (φ j,k ) F ℓ (g) F ℓ (f Xv ) F ℓ (g) = ℓ∈Cj F ℓ (φ j,k )F ℓ f (q)
Xv .

The Parseval-Plancherel theorem gives

E(R j,k |X v ) = 1 0 φ j,k (x)f (q) Xv (x)dx.
It follows from ( 3) and the Cauchy-Schwarz inequality that

E(|E(R j,k |X v )| 2 ) ≤ E 1 0 |φ j,k (x)||f (q) Xv (x)|dx 2 ≤ C 2 * 1 0 |φ j,k (x)| 2 dx = C 2 * . (26)
Putting ( 19), ( 25) and ( 26) together, we obtain

E(|R j,k | 2 ) ≤ 2 C 2 * + 2 δ-1 2(2π) 2q c 2 g 8π 3 2(δ+q) 2 2(δ+q)j + C 2 * ≤ θ 2 2 2(δ+q)j .
The proof of Proposition 2 is complete.

Proposition 3 For any integer j ≥ j * and any k ∈ {0, . . . , 2 j -1}, let α j,k be the wavelet coefficient (8) of f (q)

ω and α j,k be (9). Then there exists a constant

C > 0 such that E | α j,k -α j,k | 2 ≤ C2 2(δ+q)j ρ n n .
Proof of Proposition 3. By Proposition 1, we have E ( α j,k ) = α j,k . Therefore, using the independence of Y 1 (t), . . . , Y n (t), we obtain

E | α j,k -α j,k | 2 = V ( α j,k ) = 1 n 2 n v=1 a 2 ω (v)V   ℓ∈Cj (2iπℓ) q F ℓ (φ j,k ) F ℓ (g) 1 0 e -2iπℓt dY v (t)   . ( 27
)
Proposition 2 implies that

V   ℓ∈Cj (2iπℓ) q F ℓ (φ j,k ) F ℓ (g) 1 0 e -2iπℓt dY v (t)   ≤ E    ℓ∈Cj (2iπℓ) q F ℓ (φ j,k ) F ℓ (g) 1 0 e -2iπℓt dY v (t) 2    ≤ θ 2 2 2(δ+q)j . ( 28
)
It follows from ( 27) and ( 28) that

E | α j,k -α j,k | 2 ≤ θ 2 2 2(δ+q)j 1 n 2 n v=1 a 2 ω (v) = C2 2(δ+q)j ρ n n .
The proof of Proposition 3 is complete.

Proposition 4 For any integer j ≥ j * and any k ∈ {0, . . . , 2 j -1}, let β j,k be the wavelet coefficient (8) of f (q) ω and β j,k be (10). Then there exists a constant C > 0 such that

E β j,k -β j,k 4 ≤ C2 4(δ+q)j (ρ n ln(n/ρ n )) 2 n 2 .
Proof of Proposition 4. Thanks to Proposition 1, we have

β j,k = E 1 n n v=1 Q v = 1 n n v=1 E(Q v 1 {|Qv|≤ηj } ) + 1 n n v=1 E(Q v 1 {|Qv|>ηj } ). ( 29
)
By the elementary inequality (x + y) 4 ≤ 8(x 4 + y 4 ), (x, y) ∈ R 2 , we have

E β j,k -β j,k 4 = E   1 n n v=1 Q v 1 {|Qv|≤ηj } -E Q v 1 {|Qv|≤ηj } - 1 n n v=1 E(Q v 1 {|Qv|>ηj } ) 4   ≤ 8(A + B), (30)
where

A = E   1 n n v=1 Q v 1 {|Qv|≤ηj } -E(Q v 1 {|Qv|≤ηj } ) 4   and B = 1 n n v=1 E(Q v 1 {|Qv|>ηj } ) 4 .
Let us bound A and B in turn.

Upper bound for A. We need the Rosenthal inequality presented in lemma below (see [START_REF] Rosenthal | On the subspaces of L p (p ≥ 2) spanned by sequences of independent random variables[END_REF]).

Lemma 1 (Rosenthal's inequality) Let p ≥ 2, n ∈ N * and (U v ) v∈{1,...,n} be n zero mean independent random variables such that, for any v ∈ {1, . . . , n}, E(|U v | p ) < ∞. Then there exists a constant C > 0 such that

E n v=1 U v p ≤ C max   n v=1 E (|U v | p ) , n v=1 E U 2 v p/2   .
Set, for any v ∈ {1, . . . , n},

U v = Q v 1 {|Qv|≤ηj } -E Q v 1 {|Qv|≤ηj } .
Then, for any v ∈ {1, . . . , n}, we have E(U v ) = 0 and, using Proposition 2 with ψ instead of φ, for any b ∈ {2, 4},

E |U v | b ≤ 2 b E |Q v | b 1 {|Qv|≤ηj } ≤ 2 b η b-2 j E |Q v | 2 ≤ 2 b θ 2 η b-2 j 2 2(δ+q)j a 2 ω (v). It follows from the Rosenthal inequality and ρ n < n/e that A = 1 n 4 E   n v=1 U v 4   ≤ C 1 n 4 max   n v=1 E |U v | 4 , n v=1 E |U v | 2 2   ≤ C 1 n 4 max   2 4 θ 2 η 2 j 2 2(δ+q)j n v=1 a 2 ω (v), 2 2 θ 2 2 2(δ+q)j n v=1 a 2 ω (v) 2   ≤ C 1 n 4 max 2 4(δ+q)j n 2 ρ 2 n ln(n/ρ n ) , 2 4(δ+q)j n 2 ρ 2 n = C2 4(δ+q)j ρ 2 n n 2 . (31)
Upper bound for B. Using the inequality:

1 {|Qv|>ηj } ≤ (1/η j )|Q v |,

and again

Proposition 2 with ψ instead of φ, we obtain

1 n n v=1 E |Q v |1 {|Qv|>ηj } ≤ 1 η j 1 n n v=1 E |Q v | 2 ≤ 1 θ2 (δ+q)j ln(n/ρ n ) nρ n θ 2 2 2(δ+q)j ρ n = θ2 (δ+q)j ρ n ln(n/ρ n ) n . (32)
Hence

B ≤ C2 4(δ+q)j (ρ n ln(n/ρ n )) 2 n 2 . ( 33
)
It follows from ( 30), ( 31), (33) and

ρ n < n/e that E β j,k -β j,k 4 ≤ C 2 4(δ+q)j ρ 2 n n 2 + 2 4(δ+q)j (ρ n ln(n/ρ n )) 2 n 2 ≤ C2 4(δ+q)j (ρ n ln(n/ρ n )) 2 n 2 .
This completes the proof of Proposition 4.

Proposition 5 For any integer j ≥ j * and any k ∈ {0, . . . , 2 j -1}, let β j,k be the wavelet coefficient (8) of f (q) ω , β j,k be (10) and λ j be (15). Then, for any κ ≥ 8/3 + 2 + 2 16/9 + 4,

P | β j,k -β j,k | ≥ κλ j /2 ≤ 2 ρ n n 2 .
Proof of Proposition 5. Using (29), we have

| β j,k -β j,k | = 1 n n v=1 Q v 1 {|Qv|≤ηj } -E Q v 1 {|Qv|≤ηj } - 1 n n v=1 E Q v 1 {|Qv|>ηj } ≤ 1 n n v=1 Q v 1 {|Qv|≤ηj } -E Q v 1 {|Qv|≤ηj } + 1 n n v=1 E |Q v |1 {|Qv|>ηj } .
By (32) we obtain

1 n n v=1 E |Q v |1 {|Qv|>ηj } ≤ θ2 (δ+q)j ρ n ln(n/ρ n ) n = λ j . Hence P | β j,k -β j,k | ≥ κλ j /2 ≤ P 1 n n v=1 Q v 1 {|Qv|≤ηj } -E Q v 1 {|Qv|≤ηj } ≥ (κ/2 -1)λ j . ( 34
)
Let us now present the Bernstein inequality (see [START_REF] Petrov | Limit Theorems of Probability Theory: Sequences of Independent Random Variables[END_REF]).

Lemma 2 (Bernstein's inequality) Let n ∈ N * and (U v ) v∈{1,...,n} be n zero mean independent random variables such that there exists a constant M > 0 satisfying, for any v ∈ {1, . . . , n}, |U v | ≤ M < ∞. Then, for any λ > 0, we have

P n v=1 U v ≥ λ ≤ 2 exp - λ 2 2 n v=1 E (U 2 v ) + λM 3 .
Set, for any v ∈ {1, . . . , n},

U v = Q v 1 {|Qv|≤ηj } -E Q v 1 {|Qv|≤ηj } .
Then, for any v ∈ {1, . . . , n}, we have

E(U v ) = 0, |U v | ≤ |Q v | 1 {|Qv|≤ηj } + E |Q v |1 {|Qv|≤ηj } ≤ 2η j
and, using again Proposition 2 with ψ instead of φ,

n v=1 E |U v | 2 = n v=1 V Q v 1 {|Qv|≤ηj } ≤ n v=1 E |Q v | 2 ≤ θ 2 2 2(δ+q)j n v=1 a 2 ω (v) ≤ θ 2 2 2(δ+q)j nρ n .
It follows from the Bernstein inequality that

P n v=1 U v ≥ n(κ/2 -1)λ n ≤ 2 exp   - n 2 (κ/2 -1) 2 λ 2 j 2 θ 2 2 2(δ+q)j nρ n + 2n(κ/2-1)λj ηj 3   . (35)
Remark that

λ j η j = θ2 (δ+q)j ρ n ln(n/ρ n ) n θ2 (δ+q)j nρ n ln(n/ρ n ) = θ 2 2 2(δ+q)j ρ n and λ 2 j = θ 2 2 2(δ+q)j ρ n ln(n/ρ n ) n .
Combining ( 34) and ( 35), for any κ ≥ 8/3 + 2 + 2 16/9 + 4, we obtain

P | β j,k -β j,k | ≥ κλ j /2 ≤ 2 exp   - (κ/2 -1) 2 ln(n/ρ n ) 2 1 + 2(κ/2-1) 3   = 2 n ρ n - (κ/2-1) 2 2 ( 1+ 2(κ/2-1) 3 ) ≤ 2 ρ n n 2 .
This completes the proof of Proposition 5.

Proofs of the main results

Proof of Theorem 1. We expand the function f (q)

ω on B as

f (q) ω (x) = 2 j 0 -1 k=0 α j0,k φ j0,k (x) + ∞ j=j0 2 j -1 k=0 β j,k ψ j,k (x),
where

α j0,k = 1 0 f (q) ω (x)φ j0,k (x)dx, β j,k = 1 0 f (q) ω (x)ψ j,k (x)dx.
We have

f (q) L (x) -f (q) ω (x) = 2 j 0 -1 k=0 ( α j0,k -α j0,k ) φ j0,k (x) - ∞ j=j0 2 j -1 k=0 β j,k ψ j,k (x).
Since B is an orthonormal basis of L 2 per ([0, 1]), we have

E 1 0 f (q) L (x) -f (q) ω (x) 2 dx = A + B,
where

A = 2 j 0 -1 k=0 E | α j0,k -α j0,k | 2 , B = ∞ j=j0 2 j -1 k=0 |β j,k | 2 .
Using Proposition 3, we obtain

A ≤ C2 j0(1+2δ+2q) ρ n n ≤ C ρ n n 2s/(2s+2δ+2q+1)
.

Since p ≥ 2, we have

B s p,r (M ) ⊆ B s 2,∞ (M ). Hence B ≤ C2 -2j0s ≤ C ρ n n 2s/(2s+2δ+2q+1)
.

Therefore

E 1 0 f (q) L (x) -f (q) ω (x) 2 dx ≤ C ρ n n 2s/(2s+2δ+2q+1)
.

The proof of Theorem 1 is complete.

Proof of Theorem 2. We expand the function f (q)

ω on B as

f (q) ω (x) = 2 j * -1 k=0 α j * ,k φ j * ,k (x) + ∞ j=j * 2 j -1 k=0 β j,k ψ j,k (x),
where

α j * ,k = 1 0 f (q) ω (x)φ j * ,k (x)dx, β j,k = 1 0 f (q) ω (x)ψ j,k (x)dx.
We have

f (q) H (x) -f (q) ω (x) = 2 j * -1 k=0 ( α j * ,k -α j * ,k )φ j * ,k (x) + j1 j=j * 2 j -1 k=0 β j,k 1 {| β j,k |≥κλj } -β j,k ψ j,k (x) - ∞ j=j1+1 2 j -1 k=0 β j,k ψ j,k (x).
Since B is an orthonormal basis of L 2 per ([0, 1]), we have

E 1 0 f (q) H (x) -f (q) ω (x) 2 dx = R + S + T, (36)
where

R = 2 j * -1 k=0 E | α j * ,k -α j * ,k | 2 , S = j1 j=j * 2 j -1 k=0 E β j,k 1 {| β j,k |≥κλj } -β j,k 2 and T = ∞ j=j1+1 2 j -1 k=0 |β j,k | 2 .
Let us bound R, T and S in turn.

Using Proposition 3 and the inequalities: ρ n < n/e, ρ n ln(n/ρ n ) < n and 2s/(2s + 2δ + 2q + 1) < 1, we obtain

R ≤ C2 j * (1+2δ+2q) ρ n n ≤ C ρ n n ≤ C ρ n ln(n/ρ n ) n 2s/(2s+2δ+2q+1)
.

For r ≥ 1 and p ≥ 2, we have B s p,r (M ) ⊆ B s 2,∞ (M ). Since ρ n ln(n/ρ n ) < n and 2s/(2s + 2δ + 2q + 1) < 2s/(2δ + 2q + 1), we have

T ≤ C ∞ j=j1+1 2 -2js ≤ C2 -2j1s ≤ C ρ n ln(n/ρ n ) n 2s/(2δ+2q+1) ≤ C ρ n ln(n/ρ n ) n 2s/(2s+2δ+2q+1)
.

For r ≥ 1 and p ∈ [1, 2), we have

B s p,r (M ) ⊆ B s+1/2-1/p 2,∞ (
M ). Since s > (2δ + 2q + 1)/p, we have (s + 1/2 -1/p)/(2δ + 2q + 1) > s/(2s + 2δ + 2q + 1). So, using again

ρ n ln(n/ρ n ) < n, T ≤ C ∞ j=j1+1 2 -2j(s+1/2-1/p) ≤ C2 -2j1(s+1/2-1/p) ≤ C ρ n ln(n/ρ n ) n 2(s+1/2-1/p)/(2δ+2q+1) ≤ C ρ n ln(n/ρ n ) n 2s/(2s+2δ+2q+1)
.

Hence, for r ≥ 1, {p ≥ 2 and s > 0} or {p ∈ [1, 2) and s > (2δ + 2q + 1)/p}, we have

T ≤ C ρ n ln(n/ρ n ) n 2s/(2s+2δ+2q+1) . (38)
The term S can be decomposed as

S = S 1 + S 2 + S 3 + S 4 , (39)
where

S 1 = j1 j=j * 2 j -1 k=0 E β j,k -β j,k 2 1 {| β j,k |≥κλj } 1 {|β j,k |<κλj /2} , S 2 = j1 j=j * 2 j -1 k=0 E β j,k -β j,k 2 1 {| β j,k |≥κλj } 1 {|β j,k |≥κλj /2} , S 3 = j1 j=j * 2 j -1 k=0 E |β j,k | 2 1 {| β j,k |<κλj } 1 {|β j,k |≥2κλj } and S 4 = j1 j=j * 2 j -1 k=0 E |β j,k | 2 1 {| β j,k |<κλj } 1 {|β j,k |<2κλj } .
Let us analyze each term S 1 , S 2 , S 3 and S 4 in turn.

Upper bounds for S 1 and S 3 . We have

| β j,k | < κλ j , |β j,k | ≥ 2κλ j ⊆ | β j,k -β j,k | > κλ j /2 , | β j,k | ≥ κλ j , |β j,k | < κλ j /2 ⊆ | β j,k -β j,k | > κλ j /2 and | β j,k | < κλ j , |β j,k | ≥ 2κλ j ⊆ |β j,k | ≤ 2| β j,k -β j,k | . So max(S 1 , S 3 ) ≤ C j1 j=j * 2 j -1 k=0 E β j,k -β j,k 2 1 {| β j,k -β j,k |>κλj /2} .
It follows from the Cauchy-Schwarz inequality and Propositions 4 and 5 that

E β j,k -β j,k 2 1 {| β j,k -β j,k |>κλj /2} ≤ E β j,k -β j,k 4 1/2 P | β j,k -β j,k | > κλ j /2 1/2 ≤ C2 2(δ+q)j ρ 2 n ln(n/ρ n ) n 2 .
Since ρ n ln(n/ρ n ) < n and 2s/(2s + 2δ + 2q + 1) < 1, we have

max(S 1 , S 3 ) ≤ C ρ 2 n ln(n/ρ n ) n 2 j1 j=j * 2 j(1+2δ+2q) ≤ C ρ 2 n ln(n/ρ n ) n 2 2 j1(1+2δ+2q) ≤ C ρ n ln(n/ρ n ) n ≤ C ρ n ln(n/ρ n ) n 2s/(2s+2δ+2q+1) . ( 40
)
Upper bound for S 2 . Using Proposition 4 and the Cauchy-Schwarz inequality, we obtain

E β j,k -β j,k 2 ≤ E β j,k -β j,k 4 1/2 ≤ C2 2(δ+q)j ρ n ln(n/ρ n ) n .
Hence

S 2 ≤ C ρ n ln(n/ρ n ) n j1 j=j * 2 2(δ+q)j 2 j -1 k=0 1 {|β j,k |>κλj /2} .
Let j 2 be the integer defined by

1 2 n ρ n ln(n/ρ n ) 1/(2s+2δ+2q+1) < 2 j2 ≤ n ρ n ln(n/ρ n ) 1/(2s+2δ+2q+1)
.( 41)

We have

S 2 ≤ S 2,1 + S 2,2 , where S 2,1 = C ρ n ln(n/ρ n ) n j2 j=j * 2 2(δ+q)j 2 j -1 k=0 1 {|β j,k |>κλj /2} and S 2,2 = C ρ n ln(n/ρ n ) n j1 j=j2+1 2 2(δ+q)j 2 j -1 k=0 1 {|β j,k |>κλj /2} .
We have

S 2,1 ≤ C ρ n ln(n/ρ n ) n j2 j=j * 2 j(1+2δ+2q) ≤ C ρ n ln(n/ρ n ) n 2 j2(1+2δ+2q)
≤ C ρ n ln(n/ρ n ) n 2s/(2s+2δ+2q+1)

.

For r ≥ 1 and p ≥ 2, since B s p,r (M ) ⊆ B s 2,∞ (M ),

S 2,2 ≤ C ρ n ln(n/ρ n ) n j1 j=j2+1 2 2(δ+q)j 1 λ 2 j 2 j -1 k=0 |β j,k | 2 ≤ C ∞ j=j2+1 2 j -1 k=0 |β j,k | 2 ≤ C ∞ j=j2+1 2 -2js ≤ C2 -2j2s ≤ C ρ n ln(n/ρ n ) n 2s/(2s+2δ+2q+1)
.

For r ≥ 1, p ∈ [1, 2) and s > (2δ + 2q + 1)/p, since B s p,r (M ) ⊆ B s+1/2-1/p 2,∞ (M ) and (2s + 2δ + 2q + 1)(2 -p)/2 + (s + 1/2 -1/p + δ + q -2(δ + q)/p)p = 2s, we have S 2,2 ≤ C ρ n ln(n/ρ n ) n .

So, for r ≥ 1, {p ≥ 2 and s > 0} or {p ∈ [1, 2) and s > (2δ + 2q + 1)/p}, we have

S 2 ≤ C ρ n ln(n/ρ n ) n 2s/(2s+2δ+2q+1) . ( 42
)
Upper bound for S 4 . We have

S 4 ≤ j1 j=j * 2 j -1 k=0 |β j,k | 2 1 {|β j,k |<2κλj } .
Let j 2 be the integer (41). We have .

For r ≥ 1 and p ≥ 2, since B s p,r (M ) ⊆ B s 2,∞ (M ), we have

S 4,2 ≤ ∞ j=j2+1 2 j -1 k=0 |β j,k | 2 ≤ C ∞ j=j2+1 2 -2js ≤ C2 -2j2s ≤ C ρ n ln(n/ρ n ) n 2s/(2s+2δ+2q+1)
.

For r ≥ 1, p ∈ [1, 2) and s > (2δ + 2q + 1)/p, since B s p,r (M ) ⊆ B s+1/2-1/p 2,∞ (M ) and (2s + 2δ + 2q + 1)(2 -p)/2 + (s + 1/2 -1/p + δ + q -2(δ + q)/p)p = 2s, we have .

So, for r ≥ 1, {p ≥ 2 and s > 0} or {p ∈ [1, 2) and s > (2δ + 2q + 1)/p}, we have

S 4 ≤ C ρ n ln(n/ρ n ) n 2s/(2s+2δ+2q+1) . (43)
It follows from ( 39), ( 40), ( 42) and ( 43) that S ≤ C ρ n ln(n/ρ n ) n 2s/(2s+2δ+2q+1)

.

(44)

Combining ( 36), (37), ( 38) and (44), we have, for r ≥ 1, {p ≥ 2 and s > 0} or {p ∈ [1, 2) and s > (2δ + 2q + 1)/p},

E 1 0 f (q) H (x) -f (q) ω (x) 2 dx ≤ C ρ n ln(n/ρ n ) n 2s/(2s+2δ+2q+1)
.

The proof of Theorem 2 is complete.

  (s+1/2-1/p+δ+q-2(δ+q)/p)p ≤ C ρ n ln(n/ρ n ) n 2s/(2s+2δ+2q+1)

Acknowledgments. The author thanks two anonymous referees for their thorough and useful comments. This work is supported by ANR grant NatImages, ANR-08-EMER-009.


H Tollenaar

Struikmans

L Veerbeek
email: l.veerbeek@ikw.nl

L Van Der Geest
email: l.van.der.geest@ikw.nl

M Wouters
email: m.wouters@nki.nl

O Guicherit
email: oguicherit@bronovo.nl

A Does-Den Heijer

J Nortier
email: j.w.r.nortier@lumc.nl

A Marinelli
email: a.marinelli@mchaaglanden.nl

R Tollenaar
email: r.a.e.m.tollenaar@lumc.nl

H Struikmans
email: h.struikmans@mchaaglanden.nl

Enhancing the quality of care for patients with breast cancer: Seven years of experience with a Dutch auditing system

Enhancing the quality of care for patients with breast cancer: Seven years of experience with a Dutch auditing system

study. Nine quality indicators were evaluated over the years. In 2004 and 2005 the hospitals also carried out an intervention project aimed at improvement of the efficiency of both the diagnostic process and the surgical treatment.

Results

At the end of the project all nine indicators showed significant improvement compared to the start of the project. Discussion of treatment strategy in a multidisciplinary breast cancer team took place more often before surgery (83% versus 56%) as well as after surgery (98% versus 70%). The National Guideline for maximum waiting times was met more often for the outpatient clinic (74% versus 61%), time to diagnosis (92% versus 82%), and surgical treatment (52% versus 34%). More sentinel node procedures were performed successfully (92% versus 69%), and for more patients more than 10 lymph nodes were evaluated in case of axillary lymph node dissection (85% versus 58%). More patients had definitive surgical treatment consisting of one surgical intervention (87% versus 75%), and left the hospital within 7 days after hospital admission (98% versus 66%).

Conclusion

The clinical audit contributed to improvement of the quality of breast cancer care in the Midwestern part of the Netherlands between 2002 and 2008.

Key words: breast cancer, audit, indicators, guidelines, registration project, intervention project

Introduction

In the Netherlands, between 1990 and 2002, the number of women with the diagnosis breast cancer increased annually by more than one percent.[1] In 1990, 77 out of 100 000 women were diagnosed with breast cancer, whereas twelve years later the incidence had increased to 91 out of 100 000 women. [START_REF] Héry | Quantification of changes in breast cancer incidence and mortality since 1990 in 35 countries with Caucasian-majority populations[END_REF] Therefore, hospitals in the Netherlands needed to adjust to increasing volumes of breast cancer patients. During the nineties, breast cancer care started to become a greater burden on tight health care budgets. [START_REF] Van Den Berg | Toe)zicht op ziekenhuizen: vijf jaar presteren met indicatoren[END_REF] As a result, insurance companies demanded that hospitals should work more efficiently and at the same time maintain high level quality care. [START_REF] Bos | Zorginkoop onder de loep[END_REF] In 1999 the National Breast Cancer Council in The Netherlands (NABON) defined various quality indicators for the diagnostic process and the primary treatment of breast cancer. [START_REF] Borstkanker | De organisatie van diagnostiek en behandeling van mammapathologie in Nederland[END_REF] These NABON quality indicators were revised in 2008. [START_REF]NABON)-nota[END_REF] In 2001, the professional network of surgical oncologists affiliated with the Comprehensive Cancer Centre West (CCCW) situated in the Midwestern part of the Netherlands wanted to gain insight into the quality of the diagnostic process and the surgical treatment of their breast cancer patients. They started a regional registration project. This information was fed back to the multidisciplinary breast cancer team (MBT) as a clinical audit. We investigated the extent to which the clinical audit or registration project and the intervention project contributed to the improvement of this care. We measured the level of compliance with the National Guideline using nine quality indicators.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

Methods

Patients

Between January 2002 and December 2008, data were collected on patients diagnosed with a malignancy of the breast (including ductal carcinoma in situ (DCIS)) in one of nine hospitals in the CCCW region. Male patients and patients with lobular carcinoma in situ were not included.

Registration

Quality indicators

A quality indicator gives the percentage patients treated according to the National Guideline. This percentage gives an indication of the quality of care. For each quality indicator, NABON defined so-called norm scores. A norm score is the minimum percentage patients who should be treated according to the National Guideline. Table 1 shows the indicators that were measured.

Benchmark

The intervention project was set up according to the method of Don Berwick. [START_REF] Elferink | Disparities in quality of care for colon cancer between hospitals in the Netherlands[END_REF] This method includes the Plan-Do-Study-Act cycle of Deming to initiate continuous quality improvement. This method is designed to enable care improvement within a short period of time. [START_REF] Berwick | A primer on leading the improvement of systems[END_REF] Each year from 2002 until 2006, hospitals received a written report with regional benchmark information on each indicator. Furthermore, in 2004, 2005, and 2006, the care professionals attended training sessions twice a year. During these training sessions, an anonymous benchmark was presented in which the indicator scores for each hospital were compared with the regional mean score and the norm score. The care professionals generally discussed those indicator scores that clearly deviated from the regional mean score or the norm score with experts in the field. The care professionals were stimulated to improve these aspects.

Finally, in 2006, 2007 and 2008, a member of the MBT presented the benchmark information to the Oncology Committee within each hospital. This presentation for direct colleagues stimulated the care professionals to discuss the results more freely and to initiate improvement initiatives.

Analysis

Regional mean scores of various quality indicators were calculated and compared across three different time periods: 2002-2003, 2004-2005, and 2006-2008. For each quality indicator, the difference between the regional mean scores was tested for significance with the Chisquared test for trend (Linear-by-Linear Association). In order to identify the extent to which clinical practice was in compliance with the National Guideline, the regional mean score was compared with the norm score for each indicator as defined in 2008.

M A N U S C R I P T A C C E P T E D

Results

Increased registration

Yearly a total of approximately 1 400 patients was diagnosed with a malignancy of the breast within the CCCW region. The inclusion of patients in the database increased over the years (Figure 1). In 2002 and 2003, data was collected on in total 880 patients. During this period, only four out of nine hospitals in the region registered data on their patients. In 2004 and 2005, data was collected on in total 1 614 patients. During this period all nine hospitals registered their patient data. Between 2006 and 2008, data was collected on in total 4 364 patients.

During this period the CCCW registered data on all patients.

Local initiatives for care improvement

Several local initiatives for care improvement have been introduced. In order to decrease the waiting times, the opening hours of the outpatient clinics have been extended, and better logistics have been applied to appointments for various disciplines, for example through time slots. Furthermore, for the SN procedure several techniques were used: pre-operative lymphoscintigraphy, perioperative radiocolloid combined with blue dye injection, and occasionally SPECT imaging. The higher success rate of the SN procedure was, however, probably the result of better cooperation within the multidisciplinary teams, leading to better team performance. Besides, the performance may have been refined (with for example massage of the injection site, and a raised arm position during lymphoscintigraphy). Finally, discussion of treatment strategies in a MBT before surgery, and an appointment with the breast cancer-nurse before surgery have become standard procedure.

Quality improvement

In 2002, none of the nine quality indicators met the norm score. In 2004, the norm score was met for two quality indicators: the proportion of patients discussed in the MBT after surgery, and the proportion of patients with only one primary surgical intervention (either breastconserving surgery or mastectomy). In 2008, four of the nine quality indicators met the norm score: the proportion of patients discussed in the MBT after surgery, the proportion of patients who first visited the outpatient clinic at the latest on day seven after contact, the proportion of patients discharged at the latest on day seven after hospital admission, and the proportion of patients with only one primary surgical intervention. All nine indicators exhibited significant improvements (p<0.01) over time (Figure 2). after the patient received the histological diagnosis; and the proportion of patients with only one primary surgical intervention (either breast-conserving surgery or mastectomy), among others. [START_REF] Blauwgeers | Normen voor mammazorg. Kwaliteitsinformatiesysteem toetst de zorg voor patiënten met borstkanker[END_REF] That study showed that most quality indicators had improved in 2004, compared to 2002, as was the case for this registration project. Furthermore, a national publication on the intervention project showed that in 2006 the waiting times had decreased, more patients had been discussed with the MBT before surgery, and the burden of surgical treatment had decreased compared to 2005. [

10]

Awareness

Various factors may have contributed to the improved indicator scores in the current study.

The benchmark information that was presented during the training sessions, and during the meetings of the Oncology Committee within each hospital, probably made the care professionals eager to perform better than the year before and better than other hospitals.

MBT's had started local initiatives for care improvement and were monitoring the effectiveness of these initiatives.

The registration project demanded careful registration of the care provided and offered insight into various quality indicators. In cooperation with the CCCW, hospitals have significantly improved documentation for the hospital information systems, which facilitated better evaluation of breast cancer care. [START_REF] Veerbeek | Verbeterde registratie van de oncologische zorg door samenwerking tussen ziekenhuizen en de kankerregistratie in de regio van het Integraal Kankercentrum West[END_REF] It is likely that both the registration process and the benchmark information increased awareness of the MBT's of the quality of breast cancer care.

Remarks

For various reasons, careful interpretation of comparisons between the three study periods is warranted. First, the changes in care need to be interpreted as the possible effect of both the intervention project and the registration project, because, between 2004 and 2006, the intervention project was carried out at the same time as the registration project. It is impossible to identify the possible effects of each project separately.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

8 Secondly, at the beginning of the project data were missing, because hospitals collected data at random and sometimes only during certain periods in time. Therefore the results for this period may not be fully representative, although there is no reason to assume that hospitals registered patient data selectively. Blauwgeers et al. noted that despite changes in the number of patients included, the performance for each hospital remained almost the same in this period. [START_REF] Blauwgeers | Normen voor mammazorg. Kwaliteitsinformatiesysteem toetst de zorg voor patiënten met borstkanker[END_REF] Furthermore, the total number of patients within each period is large enough to rule out the possible effects of differences in patient characteristics.

Thirdly, in the 2006-2008 period independent data clerks took over data registration from the care professionals. This introduced the risk of bias, since independent data clerks do not have the background information the caregivers have about the care provided. It is hard to say how this possible bias may have affected the differences in care before and after 2006.

Fourthly, the caregivers knew about the aim of the registration project. It is conceivable that the improvement initiatives would have been less effective if the caregivers had been unaware that their performance was being measured. The simple fact is that people tend to perform better when they know their actions are being evaluated. [START_REF] Franke | The Hawthorne experiments: First statistical interpretation[END_REF] Fifthly, it is likely to assume that surgeons gained experience over the years, which in itself will have lead to better results over time, independent of possible improvement initiatives. Finally, we can not rule out the possibility that changes in outcome were due to autonomous factors, not in any way related to the registration or the intervention project.

Recommendations

We recommend that hospitals keep investing in the registration and monitoring of breast cancer care, in order to detect possible changes in quality of care and to enable a continuous evaluation of improvement initiatives in their care process over time. Preferably, registration should be carried out by an independent organisation, to decrease the risk of interpretation bias. When the results are used to evaluate improvement initiatives, they are most informative when compared with historic results, regional mean scores, or norm scores.

Conclusions

According to our quality indicators, compliance with the National Guideline improved over time.

  The registration project lasted until 2008. During part of the registration period, in 2004, 2005 and 2006, the Dutch Association of Comprehensive Cancer Centres (ACCC) together with the Dutch Institute for Healthcare Improvement (CBO) coordinated a national intervention project that aimed at improvement of the diagnostic process and surgical treatment.

From

  2002 until 2006, the MBT entered aspects of the diagnostic process and the primary surgical treatment of each patient in a local database. In 2006, independent data clerks from the CCCW took over the registration, and data were collected in the database of the Netherlands Cancer Registry (NCR). Patient characteristics, such as date of birth, date of diagnosis and tumour characteristics, were registered. Tumour characteristics, such as tumour type, sub site (International Classification of Diseases for Oncology (ICD-O-3), histology, stage (Tumour Lymph Node-Metastasis (TNM) clinical classification), grade, and treatment, were obtained routinely from the medical records and registered in the NCR. The NCR is based,among others, on notification of all newly diagnosed malignancies in hospitals by the national automated pathological archive (PALGA). The quality of the information is guaranteed through thorough training of the data clerks and computerized consistency checks at regional and national levels.[START_REF] Elferink | Disparities in quality of care for colon cancer between hospitals in the Netherlands[END_REF]

  indicators have improved significantly. In 2008, four of the nine quality indicators met the norm score, whereas in 2002, none of the quality indicators met the norm score. A former publication on breast cancer benchmark information from this region showed comparable results. Blauwgeers et al. measured fourteen quality indicators over the period 2002-2004. In 2004, six of fourteen quality indicators met the benchmark: the proportion of patients who received the histological diagnosis at the latest on day seven after the first visit to the outpatient clinic; the proportion of patients who underwent surgery at the latest on day 21

  However, five out of nine quality indicators have not yet met the norm score, leaving room for improvement in the level of compliance with the National Guideline. Waiting times for diagnosis and treatment have been reduced significantly. Registration and monitoring quality of care have become an integral part of breast cancer care within each hospital in the region. This registration project increased the caregivers' interest in achieving quality. Therefore we conclude that the clinical audit contributed to improvement of the quality of breast cancer care in the Midwestern part of the Netherlands during the years 2002-2008.

Figure 2 :

 2 Figure 2: Number of patients with a 'yes' as proportion of the total number of patients registered in each time period.

Acknowledgements

The authors would like to thank H.M. Oosterkamp, H. Leezer, M. Honshorst, M. Nienes of the Medical Centre Haaglanden, J.W.S. Merkus, W.A. van Deijk, J.E.A. Portielje, J. Fris, J. Venema, P.C.M. Koper, H.P. Sleeboom, A. Teunissen of the Haga Hospital, L.T. Vlasveld, N.I. Weijl, A. van der Wilden, A. Hogervorst-Apperloo of the Bronovo Hospital in The Hague, A. van Schie, J. Pomp, C.C. van der Pol, E . Maartense of the Reinier the Graaf Group in Delft, R.F. Schmitz, B.A.E.J. Mensch, B.C. Tanis of the Groene Hart Hospital in Gouda, B. Jas, A.C. Ogilvie, E. Bekink-Beuken of the Lange Land Hospital in Zoetermeer, P.A. Neijenhuis, .J.P.M. Jonkers, A.M. Zeillemaker of the Rijnland Hospital in Leiderdorp, R. Vree, E. Batman, E.V. Planken of the Diaconessenhuis, G.M.C. Ranke, A.N. Scholten, E.M.M. Krol-Warmerdam of the Leiden University Medical Centre in Leiden for their annual feedback on the data. Also many thanks to G. Bieger and S. Louw for their kind help with the English language and the lay out. The pilot audit was made possible by a financial support of the ZOLEON foundation and the Comprehensive Cancer Centre West.

Funding

The study was funded by the Comprehensive Cancer Centre West with support from a grant from ZOLEON.

Conflicts of interest statement

The submitted material has not been published and is not under consideration for publication elsewhere. We declare there are no conflicts of interest. The manuscript has been seen and approved by all authors.


EBI Rafik Absi
email: r.absi@ebi-edu.com

Hitoshi Tanaka
email: tanaka@tsunami2.civil.tohoku.ac.jp

SPECIFIC EDDY VISCOSITY FOR WAVE BOUNDARY LAYERS: CALIBRATION BY A TWO EQUATION k-ω MODEL

Coastal zones are of high vulnerability to natural hazards/disasters. Waves, currents and tides make coastal zones areas changing due to erosion and deposition of sediments. Thus, an understanding of sediment transport process in coastal zones is of crucial importance for accurate predictions of coast line evolution and sea-bed changes. However, the modeling of coastal sediment transport needs a compromise between two types of models: detailed mathematical models and engineering approaches. This compromise is imposed by on the one hand the accuracy of predictions and on the other hand the usability in practical applications. In coastal engineering practical accurate engineering models which take into account the more important involved physics, are needed.

In the engineering approach, the net (averaged over the wave period) total sediment transport is obtained as the sum of the net bed load and net suspended load transport rates [START_REF] Fredsoe | Mechanics of coastal sediment transport[END_REF]. For suspended load, the net sand transport is defined as the sum of the net current-related and the net wave-related transport components. These suspended transport components require computation of the time-averaged suspended sediment concentration (SSC) profile and its integration in the vertical direction (van Rijn 2007). Computation of SSC needs the sediment diffusivity s  which is related to the eddy viscosity t  by the parameter β (i.e., the inverse of the turbulent Schmidt number).

The aim of our study is to improve the prediction of SSC in wave boundary layers by using simple analytical tools which take into account the more important involved physics, for practical use in coastal engineering.

Our study will show that:

-For practical applications the period-averaged eddy viscosity could be described by a simple analytical formulation (Absi, 2010) -

The shape of the analytical period-averaged eddy viscosity formulation will be validated by BSL k-ω model [START_REF] Suntoyo | Effect of bed roughness on turbulent boundary layer and net sediment transport under asymmetric waves[END_REF] for sinusoidal (figure 1) and asymmetric waves -Period-averaged eddy viscosity profile depends on the wave non-linearity parameter and requires therefore a specific calibration.

ACKNOWLEDGMENTS

The first author is grateful for the financial support provided by Japan Society for the Promotion of Science (JSPS), within the FY2010 JSPS Invitation Fellowship Program for Research in Japan (No. S-10168). Figure 1 -Dimensionless eddy viscosity for sinusoidal wave; Top: Temporal and Spatial Variation obtained by BSL k-ω model; Bottom: Period-averaged dimensionless eddy viscosity.