Issam-Ali Moindjié
email: issam-ali.moindjie@inria.fr

Cristian Preda

Sophie Dabo-Niang

Fusion regression methods with repeated functional data

Keywords: linear models, regression, classification, variable fusion, fused lasso, multivariate functional data, repeated functional data, group lasso

Linear regression and classification methods with repeated functional data are considered. For each statistical unit in the sample, a real-valued parameter is observed over time under different conditions. Two regression methods based on fusion penalties are presented. The first one is a generalization of the variable fusion methodology based on the 1-nearest neighbor. The second one, called group fusion lasso, assumes some grouping structure of conditions and allows for homogeneity among the regression coefficient functions within groups. A finite sample numerical simulation and an application on EEG data are presented.

Introduction

Let X be a functional random variable valued in some Hilbert space of real-valued functions defined on the time interval [0, T ], T > 0. Without loss of generality, we assume that this space is the set of squared integrable functions L 2 ([0, T ]) [START_REF] Ramsey | Functional Data Analysis[END_REF]. The setting we consider in this paper assumes that X is observed under p different conditions {C 1 , . . . , C p }, p ≥ 1. For instance, these conditions can represent times or/and locations (regions) in some metrical space (S, d), typically (R s , ∥ • ∥ 2 ), for some natural integer s ≥ 1. Thus, proximity or grouping structures of conditions can be considered through the distance d(•, •). As an example, EEG data [START_REF] Ruiz | The great multivariate time series classification bake off: a review and experimental evaluation of recent algorithmic advances[END_REF] measure the brain activity through the electric field intensity over a time interval of T = 500ms and at different regions of the brain using p = 28 electrodes/sensors evenly distributed (Figure 1). Let denote with X (j) the observation of X under the condition C j , j = 1, . . . , p, and with X, the random vector X = (X (1) , . . . , X (p) ) ⊤ .

The realizations of X (1) , . . . , X (p) are known as repeated functional data. Principal components analysis (PCA) were developed to deal with the dependency between the components X (j) . In [START_REF] Chen | Modeling repeated functional observations[END_REF] the authors use a double PCA exploiting the metric structure of the space of conditions {C 1 , . . . , C p } belong. In [START_REF] Jacques | Model-based clustering for multivariate functional data[END_REF] that structure is ignored and X is viewed as a p-dimensional functional random vector of which principal components are used for clustering and visualization.

In this paper we are interested in the estimation of the linear regression model with scalar or binary response Y and predictor X taking into account the topology of the measurement conditions through some neighborhood or group belonging relationship (defined eventually by the distance d). Thus, as a difference of the classical framework of linear regression with multivariate functional data, in our approach, we use the structure of the space the conditions of measurement of components X (j) belong, for the model estimation. To the best of our knowledge, there are no proposed methods in the multivariate functional data framework that explicitly take into account the information carried by the spatial feature of data. The existing contributions mainly consider X as a p-dimensional functional vector and methodologies were designed to take into account the dependence between the dimensions of X (see e.g [START_REF] Yi | Classification of eeg signals: an interpretable approach using functional data analysis[END_REF], [START_REF] Górecki | Classification problems based on regression models for multi-dimensional functional data[END_REF], [START_REF] Beyaztas | A robust functional partial least squares for scalaron-multiple-function regression[END_REF], [START_REF] Moindjié | Classification of multivariate functional data on different domains with partial least squares approaches[END_REF]).

We tackle the problem from the interpretation point of view in the sense that components X (j) 's spatially close should provide similar information in the regression model. Our motivation application is on electroencephalography recordings (EEG) classification. Each subject is writing a text and the electric field intensity X is measured simultaneously at p = 28 spatial positions (sensors) of the scalp during T = 500ms. Two groups of subjects are considered: the right-handed (Y = 0) and left-handed (Y = 1) writers. The question is to know and interpret in what measure the ability of a person to be left or right-handed is associated with some different activity of the brain. In statistical terms, to understand the relationship between Y and X.

The standard functional linear regression model assumes that there exists β (0) ∈ R and the regression coefficient function β = (β (1) , . . . , β (p) ) ⊤ ∈ H = {L 2 ([0, T ])} p such that E(Y |X) ≈ β (0) + p j=1 ⟨X (j) , β (j) ⟩ L 2

(1)

where ⟨X (j) , β (j) ⟩ L 2 = T 0 X (j) (t)β (j) (t)dt, for j = 1, . . . , p.

If {(X i , Y i )} i=1:n is an i.i.d. sample of size n, n ≥ 1, drawn from the same distribution as (X, Y ) and {(x i , y i )} i=1:n is an observation of that sample, the estimation of the model ( 1) is based on the minimization of the mean of the squared errors (MSE), that is,

( β(0) , β) = arg min (ψ (0) ,ψ)∈R×H 1 n n i=1 y i -ψ (0) + p j=1 ⟨x (j) i , ψ (j) ⟩ L 2 2 . ( 2
)
Because of the non-invertibility of the covariance operator, the direct estimation of the coefficient β under the minimization of the MSE criterion is an ill-posed problem [START_REF] Cardot | Functional linear model[END_REF]. The principal component regression (PCR) and the partial least squares (PLS) have been successful alternatives in this case (see e.g [START_REF] Escabias | Modeling environmental data by functional principal component logistic regression[END_REF], [START_REF] Aguilera | Using principal components for estimating logistic regression with high-dimensional multicollinear data[END_REF], [START_REF] Preda | Régression pls sur un processus stochastique[END_REF], [START_REF] Moindjié | Classification of multivariate functional data on different domains with partial least squares approaches[END_REF]). However, in these approaches, the estimated coefficient functions are sometimes difficult to interpret: why do two components X (j) and X (j ′ ) that are associated to close conditions C j and C j ′ , i.e. the distance d(C j , C j ′ ) is small, have very different associated coefficient functions β (j) and β (j ′ ) ? This situation occurs especially when p is large. In [START_REF] Godwin | Group lasso for functional logistic regression[END_REF], the authors propose to add the constraint P = p j=1 ||ψ (j) || L 2 in the regression model. In this case, P is a generalization to functional multivariate variables of the penalty group lasso (GL), originally introduced in the multivariate data case [START_REF] Meier | The group lasso for logistic regression[END_REF], [START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF]).

This penalty leads to achieving a trade-off between a minimum number of contributing components X (j) and model fit. Our hypothesis is that closeness between components X (j) , in the sense of the distance d between the corresponding conditions C j , can help for a better interpretation of β. For this purpose, the fusion penalty was introduced in the finite multivariate setting in [START_REF] Land | Variable fusion: A new adaptive signal regression method[END_REF].

Let v be a surjective function, v : {C 1 , . . . , C p } → {1, . . . , K}, K ≤ p, and define the fusion penalty (in the functional framework) as

P(β) = K k=1 j∈I k β (j) -βI k 2 L 2 ,
where for each k = 1 :

K, I k = {j : v(C j ) = k} and βI k (t) = 1 |I k | j∈I k β (j) (t) for t ∈ [0, T ] and |I k | denotes the cardinal of I k for k = 1, . . . , p.
Then, the proximity between conditions C 1 , . . . , C p can be integrated through the function v and the distance d : v -1 (k) represents all the conditions closest to C k . Obviously, this penalty favors close dimensions of X to have similar corresponding dimensions of the regression function (the β (j) 's functions).

To our knowledge, this penalty has not been explored in the case of regression with repeated functional variables (nor multivariate functional variables). In the classical multivariate setting, the models that have this penalty are known as the variable fusion model (FU) [START_REF] Land | Variable fusion: A new adaptive signal regression method[END_REF] and, when a lasso penalty is added, as the fused lasso method (FL) [START_REF] Tibshirani | Sparsity and smoothness via the fused lasso[END_REF]. More recently, the group fusion method (GFL) introduced in [START_REF] Bleakley | The group fused lasso for multiple change-point detection[END_REF] extended this penalty from unique conditions to groups of conditions. However, in these cases, the function v has been oriented as a way to integrate consecutive conditions (or dimensions), i.e. v is defined as v(C j ) = j + 1, for 1 ≤ j ≤ p -1 and v(C p ) = p. Even if this case can be well-suited for the setting S ⊂ R (s = 1), it limits the number of applications as for multivariate locations, i.e. s ≥ 2. In this case, the function v can be defined through the distance d and can be used naturally to extend the variable fusion method [START_REF] Land | Variable fusion: A new adaptive signal regression method[END_REF] to such a spatial structure of components of X.

In this paper, we introduce two new fusion-like penalties for the functional linear regression estimation under the mean squares error criterion. The first one is an extension of the variable fusion method (FU) where the 1-nearest neighbor graph (1-NN) is used to define v. We show that the estimation of FU in this setting is equivalent to a group lasso method such as those studied in [START_REF] Godwin | Group lasso for functional logistic regression[END_REF]. The second method we introduce takes into account a more general grouping structure of conditions. We call it the group fusion lasso (GFUL) model. It allows also us to test the equality among the dimensions of the coefficient regression function belonging to the same cluster of conditions.

The paper is organized as follows. Section 2 presents the proposed methodologies and their estimation strategies using basis function expansion techniques. A comparison study of the two methods and the group lasso approach is performed using simulated data in Section 3.1. A real data application from the EEG classification task is presented in Section 3.2. The paper ends with a discussion in Section 4.

Two new fusion methods for linear regression with multivariate functional data

Without loss of generality assume that X and Y are zero mean random variables. Moreover, we consider that {(x i , y i )} i=1,...,n is an observation of {(X i , Y i )} i=1,...,n , an i.i.d sample of size n ≥ 1 drawn from the joint distribution as (X, Y ). Under the zero mean assumption of X and Y , the intercept β (0) in (1) vanishes and the mean square criterion (2) becomes:

β = arg min ψ∈H 1 n n i=1 y i - p j=1 ⟨x (j) i , ψ (j) ⟩ L 2 2 .
Remind that for each i = 1, . . . , n, y i ∈ R and x i is a multivariate function defined on [0, T ],

x i (t) = x

(1)

i (t), . . . , x (j) i (t), . . . , x (p) i (t) ⊤ , t ∈ [0, T ],
where each dimension x (j) i is observed under the condition C j , j = 1, . . . , p. Let us now introduce our first model of penalty based on distance among conditions.

Fusion method based on the neighbor relationship among conditions

The basic idea is that if two conditions C j and C j ′ are close in the space S (with respect to distance d), then the contributions brought by the components X (j) and X (j ′ ) in the linear model (1), i.e., β (j) and β (j ′ ) , might be comparable. Allowing for identical coefficients β (j) associated with close conditions, the variable fusion methodology is a candidate to obtain a parsimonious model and to compete with existing linear model approaches [START_REF] Land | Variable fusion: A new adaptive signal regression method[END_REF], [START_REF] Tibshirani | Sparsity and smoothness via the fused lasso[END_REF], [START_REF] Bleakley | The group fused lasso for multiple change-point detection[END_REF]).

When the conditions C j belong to R s with s ≥ 2, the distance d defines a neighbor relationship between conditions and thus it can be used to estimate the regression coefficient functions accordingly. More precisely, following the ideas in [START_REF] Land | Variable fusion: A new adaptive signal regression method[END_REF], the 1-nearest neighbor (1-NN) variable fusion model can be formulated as the following optimization problem:

βλ = arg min β∈H 1 2 n i=1 y i - p j=1 ⟨x (j) i , β (j) ⟩ L 2 2 + λ p j=1 ||β (j) -β (v(C j )) || L 2 (3)
where λ ≥ 0, v : {C 1 , . . . , C p } → {1, . . . , p} denotes the neighbor function

v(C j ) = arg min i∈{1,...,p}\{j} d(C i , C j ), j = 1, . . . , p. (4)
The function v helps to integrate into the estimation process of β the information brought by the conditions (locations, spatial distributions). Notice that if the set of arg min in (4) is not unique, then we choose randomly or experimentally an element of this set. For ease of notation, let denote with ⟨•, •⟩ H the inner product in H defined by :

⟨f, g⟩ H = p i=1 ⟨f (i) , g (i) ⟩ L 2 ,
for all f, g ∈ H.

The penalty function in (3) can then be rewritten as

p j=1 ||β (j) -β (v(C j )) || L 2 = ||Lβ|| L 2 ,1 ,
where L = W -I p×p and W = (w i,j ) 1≤i≤p,1≤j≤p is the adjacency matrix with elements

w i,j = 1 if v(C i ) = j 0 otherwise.
I p×p is the p × p identity matrix and || • || L 2 ,1 is the norm on H defined as :

||f || L 2 ,1 = p i=1 ||f (i) || L 2 , f ∈ H.
For illustrative purposes, consider the toy example shown in Figure 2, with S ⊂ R 2 and p = 8. It represents p = 8 points corresponding to the conditions C j ∈ R 2 , j = 1, . . . , p.

The neighborhood relationship among the conditions is given by the following v function:

v(C 1 ) = 8, v(C 2 ) = 5, v(C 3 ) = 4, v(C 4 ) = 5, v(C 5 ) = 4, v(C 6 ) = 1, v(C 7 ) = 1 and v(C 8 ) = 1.
Remark that the rank of the matrix L is generally lower than p, since symmetric relationships are possible (contrary to consecutive conditions case, see e.g Land and Friedman (1997)). For example, Figure 2 shows that C 1 is the neighbor of C 8 and C 8 is the neighbor of C 1 , the same for the couple (C 5 , C 4 ). Lemma 1. If r is the rank of the matrix L, there exists a r × p full rank matrix L 0 such as

||Lf || L 2 ,1 = ||L 0 f || L 2 ,1 , f ∈ H.
(5)

Thus, L 0 avoids redundancy. Its construction consists of finding the couples of rows corresponding to symmetric relations and then, for each such a couple, replace it with a row representing the double of the replaced ones. The rank of the matrix L coincides with the number of vertices of the undirected version of the 1-NN graph. As an illustration, in our toy example (Figure 2), we have the following matrices

L =             -1 0 0 0 0 0 0 1 0 -1 0 0 1 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 1 -1 0 0 0 1 0 0 0 0 -1 0 0 1 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 -1             and L 0 =         -2 0 0 0 0 0 0 2 0 -1 0 0 1 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 -2 2 0 0 0 1 0 0 0 0 -1 0 0 1 0 0 0 0 0 -1 0         .
Hence, lemma 1 implies that there's an alternative reformulation of (3). Similarly to the variable fusion methodology [START_REF] Land | Variable fusion: A new adaptive signal regression method[END_REF], Proposition 1 shows that (3) can be resolved using a lasso method.

Proposition 1. The solution of (3) is given by

βλ = D -1 ψλ , where ψλ = arg min f ∈H 1 2 n i=1 y i -⟨(D -1 ) ⊤ x i , f ⟩ H 2 + λ r j=1 ||f (j) || L 2 , (6)
D = L 0 T , L 0 is the r × p reduced matrix of L and T is a (p -r) × p matrix which rows form a basis of the null space of L 0 , L 0 T ⊤ = 0 r×(p-r) and 0 r×(p-r) is the r × (p -r) matrix of zeros.
Note that the estimation of the non-penalized part of f in (6), f (r+1) , . . . , f (p) , might lead (putting maximum weights on the non-constrained part of β) to model overfitting. To fix this issue, we propose to modify the penalty term in (6) as:

||Lβ|| L 2 ,1 + √ p -r η ||Tβ|| L 2 ,2 ,
where η is the Frobenius matrix norm of T, and

|| • || L 2 ,2 denotes the Frobenius norm of H: ||f || L 2 ,2 = p i=1 ||f (i) || 2 L 2 ,
for f ∈ H. Thus, the optimization problem (6) becomes :

ψλ = arg min f ∈H 1 2 n i=1 y i -⟨(D -1 ) ⊤ x i , f ⟩ H 2 +λ   r j=1 ||f (j) || L 2 + √ p -r η p j=r+1 ||f (j) || 2 L 2 1/2   .
(7) This methodology is based on only one neighbor. In the next section, we introduce a similar methodology based on more than one neighbor, we called it the group fusion lasso.

The group fusion lasso

Let consider the example represented in Figure 3. In this example, we assume that the conditions are labeled according to K = 3 groups: the yellow group (C 3 , C 4 , C 7 ), the red group (C 1 , C 6 , C 8 ) and the black group(C 2 , C 5 ). For this configuration, more than one neighbor must be considered. Indeed, the following sets ( C 1 ,C 6 ,C 8 ) (C 3 ,C 4 ,C 7 ),(C 2 ,C 5 ) have symmetric neighborhood relations (i.e. C 8 has (C 1 ,C 6 ) as neighbours,C 6 has (C 1 ,C 8 ) as neighbours,etc.). Rather than examining the interactions of the conditions individually, we propose in this section to test the resulting group relations.

where K is a number of groups, K ≤ p.

We recall the definition of the sets of index

I k = {j ∈ {1, . . . , p}, v(C j ) = k}, k = 1, . . . , K.
Let denote the size of each group by

p k = |I k |, ∀k ∈ 1, . . . , K.
The idea behind the group fusion methodology (GFU) is to introduce criteria that favor having similar coefficients for the components corresponding to conditions belonging to the same group. In the example presented in Figure 3, p = 8, K = 3 and

-v(C 1 ) = v(C 6 ) = v(C 8 ) = 1, the "red" group, -v(C 2 ) = v(C 5 ) = 2, the "black" group -v(C 3 ) = v(C 4 ) = v(C 7 ) = 3 the "yellow" group.
Then, as in the lasso regularization framework, this estimation methodology forces the clusters of conditions to have close coefficient functions and, eventually, some of them be exactly the same:

{β (1) = β (6) = β (8) } and/or {β (2) = β (5) } and/or {β (3) = β (4) = β (7) }.
For this purpose, let modify the criterion (3) by adding a term penalty for each group k of coefficient functions, P k (•), k = 1, . . . , K, as follows:

βλ = arg min β∈H 1 2 n i=1 (y i -⟨x i , β⟩ H ) 2 + λ K k=1 P k (β), (9)
where

P k (β) = √ p k i∈I k ||β (i) -βI k || 2 L 2 and βI k (t) = 1 p k j∈I k β (j) (t), t ∈ [0, T ].
Remark. If for some k ∈ 1, . . . , K, I k = {j}, then there is no penalty on the j-th component (dimension) of the corresponding coefficient function, β (j) .

As in the previous criterion (3), the optimization criterion ( 9) might lead to model overfitting (see Proposition 2): the fusion penalties have no control over all terms in the norm of β. To overcome this difficulty we introduce the group fusion lasso (GFUL) methodology as a modified version of the elastic-net strategy [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF], that is,

βλ,α = arg min β∈H 1 2 n i=1 (y i -⟨x i , β⟩ H ) 2 + λ K k=1 P α,k (β), ( 10
) with P α,k (β) = (1 -α)P k (β) + α|| βI k || L 2 , α ∈ (0, 1).
The purpose of GFUL is related to the group lasso methodology where, given some grouping structure of predictor variables, the objective is to force to zero all the coefficients of variables within some group(s) (for more details see [START_REF] Meier | The group lasso for logistic regression[END_REF], [START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF]). From this perspective, GFUL aims to obtain some group conditions with the same coefficient functions, which is a more general statement.

Remark. The penalty function is composed of two terms: the first one, P k (β) is of fusion type

; P k (β)is zero if only if β (j) = β (k) , ∀j, k ∈ I k ; the second term, || βI k || L 2 is a group- lasso-like penalty.
As for FU methodology (see Proposition 1), we show now that GFUL estimation reduces to a group-lasso one.

In the GFUL methodology, the membership of conditions to groups is a central notion. Let define the indicator matrix M = (m k,j ) 1≤k≤K,1≤j≤p as

m k,j = 1 if j ∈ I k 0 otherwise.
In the toy example (Figure 3), the matrix M is given by

  1 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 1 0   .
In a general case, up to a permutation of columns, M can be written as

M =     ⃗ 1 ⊤ p 1 ⃗ 0 ⊤ p 2 ... 0 ⃗ 0 ⊤ p 1 ⃗ 1 ⊤ p 2 ... 0 ... ... ⃗ 0 ⊤ p 1 ⃗ 0 ⊤ p 2 ... ⃗ 1 ⊤ p K     ,
where ⃗ 1 p k , ⃗ 0 p k are respectively the p k column vector of ones and the p k column vector of zeros.

Let denote by M the standardized version of M, i.e. M = diag(1/p 1 , 1/p 2 , . . . , 1/p K )M. Then, similarly as in Lemma 1, the following result holds:

Lemma 2. Let f ∈ H, α ∈ (0, 1) and p k ≥ 2, for k = 1, . . . , K. Consider 2K synthetic groups { Ĩk } 2K k=1 , defined as Ĩk =      j ∈ {1, . . . , p} 1 + k-1 l=1 (p l -1) ≤ j ≤ k l=1 (p l -1) k = 1, . . . , K {k + p -2K} k = K + 1, . . . , 2K.
Up to a permutation of dimensions, the penalty function of GFUL can be written as

K k=1 P α,k (f ) = 2K k=1 i∈ Ĩk ||(G α f ) (i) || 2 L 2 , f ∈ H (11)
where G α is the p × p non-singular matrix given by:

G α = (1 -α)R α M ,
with R is the block diagonal matrix composed of the following elements

√ p 1 R 1 , . . . , √ p K R K ,
and for k = 1, . . . , K, R k is the upper triangular (p k -1) × p k matrix obtained from the reduced rank QR decomposition of

P k = I p k ×p k - 1 p k 1 p k ×p k ; here 1 p k ×p k denotes the p k × p k matrix of ones.
Using the non-singularity of G α , the following proposition provides a way to estimate GFUL using a simpler model.

Proposition 2. For α ∈ (0, 1), the solution of (10), holds βα,λ = G -1 α ψλ , where

ψλ = arg min f ∈H 1 2 n i=1 (y i -⟨(G -1 α ) ⊤ x i , f ⟩ H ) 2 + λ 2K k=1 i∈ Ĩk ||f (j) || 2 L 2 , (12)
The proof of this proposition follows as a direct consequence of the non-singularity of G α and Lemma 2.

Remark. The case of α = 1 or α = 0 can be resolved using the same technique as in Proposition 1. Indeed, the non-null part of G α is full rank for α ∈ {0, 1}

The direct estimation of β, under least squares regression, is generally an ill-posed inverse problem [START_REF] Cardot | Functional linear model[END_REF], [START_REF] Aguilera | Using principal components for estimating logistic regression with high-dimensional multicollinear data[END_REF]). The basis expansion technique, a wellknown dimension reduction technique as an alternative to solve this problem, is presented in the next section.

Computational aspect: Basis expansion

The basis expansion technique assumes that there exists a set of linearly independent functions {ϕ k } M k=1 , such as, for each i = 1, . . . , n, x i can be written as

x (j) i (t) = M k=1 a (j) i,k ϕ k (t) = (a (j) i ) ⊤ ϕ(t), t ∈ [0, T ] (13)
where a (j)

i,k ∈ R for i = 1, . . . , n, j = 1, . . . , p and Note that for each x i , we have that

• a (j) i = a (j) i,1 . . . a (j) i,M ⊤ , • ϕ = ϕ 1 . . . ϕ M ⊤ is
x i =    x (1) i . . . x (p) i    = Φa i
where Notice that in the expression in (13), we use the same basis ϕ for all dimensions of X. This seems realistic since X (1) , . . . , X (p) measure the same parameter X. However, that is not mandatory, each dimension X (j) can be expressed on its own basis. Similarly, we assume that the coefficient function β can also be expressed as

a i =    a (
β(t) = Φ(t)b, t ∈ [0, T ] where b =    b (1) . . . b (p)    , with b (j) ∈ R M .
Remark that the predictors x i and β admit also the equivalent matrix notations

β(t) = Bϕ(t) and x i (t) = A i ϕ(t) (14)
where A i and B are the following matrices of size p × M ,

B = b (1) . . . b (p) ⊤ and A i = a (1) i a (2) i ... a (p) i ⊤
, for all i = 1, . . . , n.

Proposition 3. The following statements hold

1. ||β|| L 2 ,1 . = p j=1 ||β (j) || L 2 = ||BF 1/2 || 2,1 , where || • || 2,1 is the (2, 1) matrix norm and F 1/2
is the square root matrix of F= {⟨ϕ i , ϕ j ⟩} i,j .

2. Let k be an integer in {0, . . . , p -1} and Z be a matrix of size (p -k) × p. Define β 0 (t) = Zβ(t), for all t ∈ [0, T ]. Then

β 0 (t) = b 0 Φ(t), ∀t ∈ [0, T ],
where b 0 = (Z ⊗ I M ×M )b and ⊗ denotes the Kronecker product.

The first statement says that the norm of β depends on the vector b and the basis {ϕ k } M k=1 via the matrix F.

As an example, let consider the following group lasso problem (each dimension represents a group)

βλ = arg min β∈H 1 2 n i=1 (y i -⟨x i , β⟩ H ) 2 + λ p j=1 ||β (j) || L 2 (15) Since ⟨x (j) i , β (j) ⟩ = (a (j) i ) ⊤ Fb (j) and ||β (j) || L 2 = (b (j) ) ⊤ Fb (j)
1 2 , the problem in ( 15) is equivalent to the one of finding the vector bλ

=      b(1) λ b(2) λ . . . b(p) λ      such that bλ = arg min b∈R pM 1 2 n i=1 y i - p j=1 (a (j) i ) ⊤ Fb (j) + λ p j=1 ||F 1/2 b (j) || 2 . ( 16
)
Let denote by γ(j) = F 1/2 b (j) . Then, obtaining γλ as solution of γλ = arg min

γ∈R pM 1 2 n i=1 y i - p j=1 (a (j) i ) ⊤ F 1/2 γ (j) + λ p j=1 ||γ (j) || 2
therefore allows to estimate b (j) as b(j) λ = (F 1/2 ) -1 γ(j) λ , j = 1, . . . , p. The problem (15) is studied in Godwin (2013) using principal component analysis in order to avoid multicollinearity and high-dimension issues [START_REF] Aguilera | Using principal components for estimating logistic regression with high-dimensional multicollinear data[END_REF], [START_REF] Escabias | Modeling environmental data by functional principal component logistic regression[END_REF]).

The second statement in Proposition 3 shows the correspondence (relationship) between expansion coefficients in the basis ϕ after linear transformation of a function in H, in particular for the coefficient function β. In the next section, this relationship helps to estimate the coefficient regression function under the FU methodology by reducing the problem to a group-lasso-like, as in (17).

FU estimation

In order to obtain the solution βλ of the FU criterion (7), we use the second statement in Proposition 3 and Proposition 1. Let γλ be the solution of the minimization problem γλ = arg min

γ∈R pM 1 2 n i=1 y i -a ⊤ i (D ⊗ I M ×M ) -1 F 1/2 γ 2 + λ   r j=1 ||γ (j) || 2 + √ p -r η p j=r+1 ||γ (j) || 2 2   ,
where

F 1/2 =      F 1/2 0 . . . 0 0 F 1/2 . . . 0 . . . . . . 0 0 . . . F 1/2      .
Then, the coefficient function βλ is given by βλ = Φ(D ⊗ I M ×M )(F 1/2 ) -1 γλ .

GFUL estimation

We use a similar procedure as in Section 2.3.1 for the estimation of βλ,α .

Let define the sets G 1 , . . . , G 2K as follows.

G k = j 1 + M k-1 l=1 (p k -1) ≤ j ≤ M k l=1 (p k -1) k = 1, . . . , K, G k = {j |p ′ + M (k -1 -K) + 1 ≤ j ≤ p ′ + M (k -K)} k = K + 1, . . . , 2K,
where p ′ = M (p -K), and K is the number of groups in GFUL. Note that {G k } 2K k=1 correspond to { Ĩk } 2K k=1 (see Lemma 2) under the basis expansion hypothesis, i.e when each β (j) is represented by M expansion coefficients.

For convenient notation, let define the permutation matrix S = (s u,v ∈ {0, 1}) (u,v)∈{1,...,p} 2 , such that

Sβ =     β I 1 β I 2 . . . β I K     ,
where β I k is the vector of components of β corresponding to the set of indexes I k , k = 1, . . . , K.

Then, the group fusion lasso problem reduces to determine γλ,α as solution of the following problem γλ,α = arg min

γ∈R pM 1 2 n i=1 y i -a ⊤ i (G α S ⊗ I M ×M ) -1 F 1/2 γ 2 + λ 2K k=1 ||γ G k || 2 . ( 17
)
Therefore, βλ,α is given by βλ,α = Φ(G α S ⊗ I M ×M )(F 1/2 ) -1 γλ,α .

Remark. The case of binary response can be naturally taken into account in our proposed methodologies. More precisely, as in [START_REF] Meier | The group lasso for logistic regression[END_REF], the MSE criterion is replaced by the likelihood one (multiplied by -1) whereas the penalized terms are the same. In this case, the optimization problem in (3) becomes:

βλ = arg min β∈H - n i=1 (y i ⟨x i , β⟩ H -log(1 + ⟨x i , β⟩ H )) + λ p j=1 ||β (j) -β (v(C j )) || 2 . ( 18
)
3 Numerical experiments

Simulations

We present a simulation study that compares the performance of the proposed methods, FL and GFUL, with competitor lasso methods. Notice that all our models are estimated by using Lukas and Meier (2020) R package. The R code sources of our simulations are available at https://github.com/imoindjie/GFUL-FU.

The simulation setting

The setting of the simulation is as follows. In order to show the efficiency of taking into account the grouping structure of conditions, we consider two scenarios. In the first one, the number of conditions is fixed to p = 12 and we show that all the methods perform equally in terms of MSE criteria. In the second one, we increase the number of conditions to p = 80 and then, we show the efficiency of our methodology with respect to the others. In both scenarios, the number of groups is K = 4 and the number of conditions in each group is Let consider the p conditions C j , j = 1, . . . , p, as points in R 2 and their group structure defined as follows: Figure 4 presents the conditions for p = 12 and p = 80. One can imagine that these conditions correspond to the position of p points in a 10 × 10 squared metal piece where one observes in each point C j , j = 1, . . . , p, the temperature X (j) over the time interval [0, 1].

p 1 = p 2 = . . . = p K = p K . = κ. Next,
Group 1: C j = ζ j + c 1 , j = 1, . . . , κ, Group 2: C j = ζ j + c 2 , j = κ + 1, . . . , 2κ, (a)
(b) The theoretical regression coefficient functions.

The theoretical coefficient regression function β = (β (1) , . . . , β (p) ) is defined as follows:

Groupe 1:

β (j) = 0, j = 1, . . . , κ, Groupe 2: β (j) = √ 2 3 k=1 ∆ k , j = κ + 1, . . . , 2κ, Groupe 3: β (j) = b j 9 k=1 ∆ k , j = 2κ + 1, . . . , 3κ, Groupe 4: β (j) = - √ 2 3 k=1 ∆ k , j = 3κ + 1, . . . , p,
where b j = (-1) j 1+jmodκ κ , the functions ∆ 1 , . . . , ∆ 9 denote the set of functions defined by:

∆ s (t) = (1 -0.2(10t -s) 2 ) + ,
where (.) + is the positive part function. In this setting, only the third group has different coefficient functions.

(c) The predictor and the response variables.

For j = 1, . . . , p, X (j) is generated as

X (j) (t) = 9 s=1 a s ∆ s (t),
where a s ∼ N (0, 1), s = 1, . . . , 9 and t ∈ [0, 1].

The response variable Y is given by

Y = ⟨X, β⟩ H + ϵ,
where ϵ ∼ N (0, σ 2 ϵ ). The values of σ 2 ϵ are set such that the noise to signal ratio,

σ 2 ϵ var(Y )
, is of about 10%.

(d) The two simulation settings Two scenarios are presented according to the size of groups, κ:

(S1) κ = 3, σ ϵ = 1.6 (S2) κ = 20, σ ϵ = 3.6
The theoretical coefficient regression functions β (j) , j = 1, . . . , p for the two scenario are presented in Figure 5. The function predictor X is observed on 100 equidistant sampling time points in the interval [0, 1]. For all dimensions of X, X (j) j = 1, . . . , p, we use as an approximation their expansion into a cubic B-splines basis of size M = 20. To assess model performances, a random training sample of 80% of the data is considered and the remaining 20% is used for prediction. This experiment is repeated I = 100 times.

(e) The competitor methods

The variable fusion methodology (FU) is employed using the 1-NN relationship among conditions whereas the grouping structure is used for the group fusion lasso (GFUL). In order to evaluate their performances, FU and GFUL are compared with two group lasso methods [START_REF] Godwin | Group lasso for functional logistic regression[END_REF]. The first one, denoted by GL1 ("Group Lasso 1"), uses each dimension X (j) of X as a group, as in the classical lasso setting. The second one, denoted by GL2 (Group Lasso 2), uses the same group definitions as in GFUL (see equation ( 8) ).

Group 1

β (1) =β (2) =β (3) Group 2 β (4) =β (5) =β (6) Group 3 β (7) ̸ =β (8) ̸ =β (9)
Group 4 β (10) =β (11) =β (12) S1 β (j) 's β (1) =β (2) =...=β (20) β ( 21) =β (22) =...=β (40) β (41) ̸ =β (42) ̸ =...̸ =β (60) β (61) =β (62) =...=β (80) S2 β (j) 's In addition to these methods, we propose also the regression model HG (Homogeneous Groups) resuming all the conditions within a group I k by their mean function,

m (k) = 1 p k j∈I k X (j) ,
and then fit a multivariate functional linear model

Y = K k=1 T 0 m (k) (t)γ (k) (t)dt + ϵ,
using principal component regression methodology [START_REF] Aguilera | Using principal components for estimating logistic regression with high-dimensional multicollinear data[END_REF]). The idea behind this method is to obtain the same coefficient regression function for all conditions within a group, and that for all the groups: ∀k = 1, . . . , K,

β (j) = (1/p k )γ (k) , ∀j ∈ I k .
The difference with GFUL is that the latter one allows only for some groups to have identical coefficient functions, whereas HG imposes it for all groups. Except for the model HG which doesn't have a penalty term, the hyperparameters (α, λ) in ( 10)) are tuned by 10-fold cross-validation: λ is chosen from the set λ ∈ 0.96 i λ max , i = 0, 1, . . . , 148 ∪ {0} and α ∈ {0.1, 0.2, . . . , 1}, with λ max is determined as in Lukas and Meier (2020).

(f ) The goodness of fit and homogeneity among conditions coefficient regression functions For each method, the goodness of fit is assessed by the mean squared error (MSE) computed on the test set. Their ability to recover the true equality among coefficient functions β (j) is measured by "sensitivity" (Sens) and "specificity" (Spec) metrics. They are defined as follows. For each pair β (j) , β (k) , j, k = 1, . . . , p, we define

Sens(j, k) = P β(j) = β(k) |β (j) = β (k) ,
and

Spec(j, k) = P β(j) ̸ = β(k) |β (j) ̸ = β (k) .
Thus, Sens(j, k) measures the capability of the method to obtain identical estimated coefficient functions β(j) = β(k) when the theoretical ones verify that equality, β (j) = β (k) . Then, as global measures, let define

Sens = 2 p(p -1) p j=1 k<j Sens(j, k), Spec = 2 p(p -1) p j=1 k<j Spec(j, k).

Results

Scenario 1 Let remind that in this scenario p = 12 and κ = 3. The summary of the obtained metrics in S1 is presented Table 1.

In the first scenario (p = 12), all models give close results except the naive model (HG). Indeed, the HG model gives the highest MSE and the estimation of the coefficient functions is not consistent (see Figure 6). Thus, the naive hypothesis that " dimensions in the same group share the same regression coefficient function" might lead to inconsistent results. Table 1 shows that the variable fusion (FU) and the group fusion lasso methods (GFUL) reach the highest scores of sensibility and specificity. This demonstrates the ability of these methodologies to find true equalities among coefficients as compared to the group lasso methods (GL1 and GL2).

MSE

Sens Spec GL1 6.2(1.59) 0.22(0.13) 1(0.01) GL2 6.07(1.45) 0.29(0.12) 1(0) FU 5.97(1.52) 0.82(0.22) 0.99(0.01) GFUL 5.21(1.81) 0.92(0.22) 1(0) HG 14.85(3.25) 1(0) 0.95(0)

Table 1: Scenario S1: MSE mean and standard error (in parentheses), Sensibility and specificity obtained metrics with I = 100 experiments.

Group 1

β (1) =β (2) =β (3)
Group 2

β (4) =β (5) =β (6)
Group 3

β (7) ̸ =β (8) ̸ =β (9)
Group 4 Scenario 2:

In this scenario, p = 80 and κ = 20. Table 2 presents the results. It shows that GFUL is the best methodology for the MSE metric. The high specificity and the low sensitivity of the competitor methods indicate that they ignore that some groups share the same regression coefficient functions (see Figure 7). This is also reflected in the MSE criteria.

Let observe that all the other methods, including FU, provide quite bad results with respect to GFUL. This can be explained by the fact that these methods are clearly not adapted to consider the grouping structure of conditions.

MSE

Sens Spec GL1 88. 74(23.75) Group 1 20) Group 2 40) Group 3

β (1) =β (2) =...=β (
β (21) =β (22) =...=β (
β (41) ̸ =β (42) ̸ =...̸ =β (60)
Group 4

Application: FingerMovements

In this section, we are interested in a supervised binary classification problem for Finger-Movements1 dataset. These data come from the brain-computer interface domain and are used for binary classification as a benchmark. More precisely, a subject has been asked to type characters using only the index and the pinky fingers of the right (Y = 0) and the left (Y = 1) hands. The challenge is to determine, based on their electroencephalography (EEG) recording (X), the hand that has been used. The EEG signal is recorded by p = 28 sensors located on the scalp during 500 ms. Thus, for any subject, p = 28 curves are available. This dataset has been used in [START_REF] Ruiz | The great multivariate time series classification bake off: a review and experimental evaluation of recent algorithmic advances[END_REF]. The authors showed that the Inception Time (IT) model [START_REF] Ismail Fawaz | Inceptiontime: Finding alexnet for time series classification[END_REF] provides the best predictions among the stateof-the-art models.

In this section we compare the results obtained by our methodologies (FU and GFUL) with the competitors one, i.e. GL1, GL2 and IT.

The FU method is based on the 1-NN graph built with the Euclidean distance between the spatial location of sensors C j ∈ R 3 , j = 1, . . . , 28. For the GFUL method, we used K = 10 clusters of conditions obtained from the k-means clustering algorithm applied to sensor locations. The K = 10 groups correspond to well-defined scalp regions (group 1 = frontal left, group 2 = frontal right, etc). See Figure 8. Two group lasso models are also fitted: GL1 and GL2. Similarly to the simulation study, the GL1 method uses each dimension as a group whereas GL2 uses the same grouping structure as GFUL.

For all dimensions X (j) , j = 1, . . . , 28, a basis of M = 30 B-splines is used to reconstruct the functional form of the predictors. The hyperparameters λ and α are tuned by a 10-fold cross-validation procedure, on the following grids λ ∈ 0.96 i λ max , i = 0, 1, . . . , 148 ∪ {0} and α ∈ {0, 0.1, 0.2, . . . , 1}, where λ max is the minimum value such that the penalty term vanishes (P( βλ,α ) = 0).

Results

FU GFUL

Figure 9: Estimated structures FU: connected points share the same coefficients, GFUL: groups in red share the same coefficients

Table 3 shows that our proposed methodologies perform better than most competitors (GL2 and IT) in terms of accuracy (well-classified rate) estimated on the test sample. Figure 9 shows the grouping structure of the estimated regression coefficient functions obtained with FU and GFUL. Hence, those results provide information about the importance of sensors and their locations (also through the grouping structure) for the prediction of the response. The associated coefficient functions graphs are presented in the appendix A.

Methods Accuracy

FU 64% GFUL 68% IT 56.7% GL1 65% GL2 58%
Table 3: Accuracy obtained on the test dataset

Discussion

In this paper we introduced two new criteria for estimating a linear regression model with the predictor represented by a functional random variable observed under different conditions (eventually spatially distributed). We called that data repeated functional. When some grouping structure of the observation conditions is present, our methods can integrate it in the fitting process through specific penalties: fusion and group fusion-lasso.

The numerical simulation study confirms the efficacy of taking into account such a grouping structure of conditions, as well as the application to Finger movements data. Both proposed methodologies give similar results or outperform the lasso method competitors.

The GFUL method can be seen as a generalization of the fusion method in more than one neighbor. However, GFUL tests group membership at once instead of testing one-on-one interactions. This is a quite strong hypothesis, as it assumes that equality relations (among regression coefficient functions) in a group can be either all true or all false. The use of smaller overlaps between groups could be an alternative model. In this setting, the solution is related to the group lasso with overlap, which is more challenging [START_REF] Yuan | Efficient methods for overlapping group lasso[END_REF]. An extensive study of adapted optimization problems should be done. One can also explore the model group lasso proposed in [START_REF] Jacob | Group lasso with overlap and graph lasso[END_REF]. Yet, it seems that using this approach leads to losing the diffusion between overlapped groups, the penalty is no longer defined on the 2, 1 norm (See [START_REF] Jacob | Group lasso with overlap and graph lasso[END_REF] for details). The integration of sparsity conditions and the study of other types of neighborhood structures in the fusion method can be some future promising developments. 5) β( 6) β( 7) β( 8) β( 9) β( 10) β( 11) β( 12) β(13) = 0 β( 14) β( 15) β( 16) β( 17) β( 18) β( 19) β( 20 5) β( 6) β( 7) β( 8) β( 9) β( 10) β( 11) β( 12) β( 13) β( 14) β( 15) β( 16) β( 17) β( 18) β( 19) β( 20) β( 21) β( 22) β( 23) β( 24) β( 25) β( 26) β( 27) β(28)

Figure 13: GL2 estimated coefficient model

B Proofs

Proof of Lemma 1. Let the neighbor function v: {C 1 , C 2 , . . . , C p } → {1, 2, . . . , p}, as defined in Section 2.1 and V 0 , V 1 defined as:

V 0 = {i ∈ {1, 2, . . . , p}, v 2 (C i ) = i} and V 1 = {i {1, 2, . . . , p}, i > v(C i ), v 2 (C i ) = i}, with v 2 (C i )= v(C v(C i ) ), for i = 1, . . . , p. Observe that i ∈ V 0 is equivalent to v(C i ) ∈ V 0 , and i ∈ V 1 implies that v(C i ) ̸ ∈ V 1 .
In other words, V 0 is the set of indexes corresponding to conditions for which a 2-cycle structure is present in the 1-NN graph. V 1 is a subset of V 0 with cardV 1 = 1 2 card(V 0 ). Then, for all f ∈ H, we have

p j=1 ||(Lf ) (j) || 2 = j∈V 0 ||f (j) -f (v(C j )) || 2 + j̸ ∈V 0 ||f (j) -f (v(C j )) || 2 = j∈V 1 ||2(f (j) -f (v(C j )) )|| 2 + j̸ ∈V 0 ||f (j) -f (v(C j )) || 2 .
Then, there exists a matrix L 0 ∈ R r×p , with r = p -1 2 card(V 0 ), such as

p j=1 ||(Lf ) (j) || 2 = r j=1 ||(L 0 f ) (j) || 2 .
Since v is constructed by the one-nearest neighbor graph-only 2-cycle structures can occur [START_REF] Eppstein | On nearest-neighbor graphs[END_REF]-then L 0 is a full rank matrix.

Proof of Proposition 1. We borrow some reasoning from [START_REF] Tibshirani | The solution path of the generalized lasso[END_REF] (the full-rank matrix case). In their paper, these authors were interested in the case where the penalty is defined using the l1 norm and a linear transformation of the coefficient in the multivariate case. We extend their reasoning to the || • || L 2 ,1 norm and the setting of multivariate functional coefficients. Notice that For the first point (a), direct calculation shows that For point (b), without loss of generality, we assume that

||Df || L 2 ,1 = r j=1 ||(L 0 f ) (j) || L 2 + p j=r+1 ||(Tf ) (j) || L 2 , f ∈ H.
f I k -fI k ⃗ 1 p k = I p k ×p k - 1 p k 1 p k ×p k P k f I k . (21)
M =     ⃗ 1 ⊤ p 1 ⃗ 0 ⊤ p 2 ... 0 ⃗ 0 ⊤ p 1 ⃗ 1 ⊤ p 2 ... 0 ... ... ⃗ 0 ⊤ p 1 ⃗ 0 ⊤ p 2 ... ⃗ 1 ⊤ p K     .
Note that ⃗ 1 p k belongs to the kernel of P k , i.e P k ⃗ 1 p k = ⃗ 0 p k , for all k ∈ {1, . . . , K}. From the definition of R k ,it follows that

P k = Q k R k 0 ⊤ p k
, where Q k is an orthogonal matrix. Then,

P k ⃗ 1 k = 0 p k implies that R k ⃗ 1 k = 0 p k -1 . As R k ⃗ 1 k = 0 p k -1
for all k ∈ {1, . . . , K}, we have

R M⊤ = 0 (p-K)×K .
Finally, as a direct consequence of (a), the matrix G α satisfies the relation (11). Observe that G α is non-singular as a consequence of (b). This concludes the proof.

Proof of Proposition 3.

1. The equation ( 14) implies that for each dimension j, j = 1, . . . , p, we have

β (j) (t) = (b (j) ) ⊤ ϕ(t),
where b (j) ∈ R M t ∈ [0, T ]. Define F= {⟨ϕ i , ϕ j ⟩} i,j and F = (F 1/2 ) ⊤ F 1/2 . Thus, we have

||β (j) || L 2 = ||(F ) ⊤ b (j) || 2 = ||(b (j) ) ⊤ F 1/2 || 2 .
Moreover,

BF 1/2 =     (b (1) ) ⊤ (b (2) ) ⊤ ... (b (p) ) ⊤     F 1/2 =     (b (1) ) ⊤ F 1/2 (b (2) ) ⊤ F 1/2 ... (b (p) ) ⊤ F 1/2     ,

Figure 1 :

 1 Figure 1: FingerMovements data. Each subject is represented by p = 28 EEG recordings (right) corresponding to 28 sensors located on the scalp (left).

Figure 2 :

 2 Figure 2: One nearest neighbor graph: a → b means b is the neighbor of a.

Figure 3 :

 3 Figure 3: Conditions with grouping structure

  the vector of functions. The most common choices of ϕ are Fourier or B-splines functions, depending on the periodicity of X[START_REF] Ramsey | Functional Data Analysis[END_REF].

  . . ϕ M 0 . . . 0 . . . 0 . . . 0 0 . . . 0 ϕ 1 . . . ϕ M . . . 0 . . .

  we present the construction of our simulation study. (a) the conditions and the grouping structure, (b) the theoretical regression coefficient functions, (c) the definition of the predictor and the response variables, (d) the two simulation settings, (e) the competitor methods, (f) the goodness of fit and homogeneity among conditions coefficient regression functions (a) The conditions and the grouping structure.

Figure 4 :,

 4 Figure 4: Conditions when p = 12 (a) and p = 80 (b). The colors are associated with each group of conditions.

Figure 5 :

 5 Figure 5: Theoretical regression coefficient function β for the two scenario

βFigure 6 :

 6 Figure 6: Scenario 1-The estimations of β by the different methods(first simulation).

βFigure 7 :

 7 Figure 7: Scenario 2-The estimations of β by the different methods (first simulation).

  Each curve is summarized by 50 equidistant times points in the interval [0, 500ms]. The Figure 1 (see the Introduction section) presents the curves registered by a sample of 6 sensors (named F1,F2, F3, F4,O1,O2), for a given subject. The dataset is composed of N = 416 subjects and it is split into a training set of n = 316 units and a test set of 100 units.

Figure 8 :

 8 Figure 8: Conditions and groups for the FingerMovements Dataset

Figure

  Figure 10: Fusion variable model

  Figure 12: GL1 estimated coefficients model

  Using the definition of L 0 ,||Df || L 2 ,1 = ||Lf || L 2 ,1 + p j=r+1 ||(Tf ) (j) || L 2 ,1 .Hence, the problem (3) can be written as βλ = arg minf ∈H 1 2 n i=1 (y i -⟨x i , f ⟩ H ) 2 + p j=1 λI(j ≤ r)||(Df ) (j) || L 2 (19) with I(•) is the indicator function. The non-singularity of D implies ψλ ≤ r)||f (j) || L 2 ,(20)with ψλ = D βλ . The equality⟨(D -1 ) ⊤ x i , f ⟩ H = ⟨x , D -1 f ⟩ H concludes the proof.Proof of Lemma 2. To simplify the notation, let denote by f I k the function composed only of the set of dimensions of f ∈ H which belong to I k . The proof of the lemma relies on the following statements:(a) For f ∈ H, ||f Ip k -fI k ⃗ 1 p k || L 2 ,2 = ||R k f I k || L 2 ,2 .for all k ∈ {1, . . . , K}.(b) The matrices M and R are such that R M⊤ = 0 (p-K)×K

  The rank of P k is p k -1. Let R k be the R reduced rank matrix of size (p k -1) × |p k | obtained by the QR decomposition of P k . Since || • || L 2 ,2 is the Frobenius function norm, we have (a), i.e.||P k f I k || L 2 ,2 = ||R k f I k || L 2 ,2 .

  and ||β|| L 2 ,1 = ||BF 1/2 || 2,1 . 2. Notice that a i = vec(A ⊤ i ), b = vec(B ⊤ ) with vec(•) denotes the vectorization operator. It follows that b 0 = vec((ZB) ⊤ ) = vec(B ⊤ Z ⊤ ) = (Z ⊗ I M ×M )vec(B ⊤ ) = (Z ⊗ I M ×M )b,with ⊗ denotes the Kronecker Product.

Table 2 :

 2 Scenario S2: MSE mean and standard error (in parentheses), Sensibility and specificity obtained metrics with I = 100 experiments.

	0.2(0.19) 0.85(0.18)
	GL2 64.29(17.22) 0.08(0.14) 1(0)
	FU 70.58(18.53) 0.07(0.01) 1(0)
	GFUL 31.68(16.33) 0.73(0.4) 1(0)
	HG 69.37(15.61) 1(0)	0.93(0)

https://www.timeseriesclassification.com/description.php?Dataset=FingerMovements


Bad smells in reviewers' reports? Text-mining the MDPI Open Peer Review Corpus

Objective. Malpractice affecting the reviewing process is detrimental to science. We introduce methods to reveal evidence of peer review manipulation, such as template usage, citation manipulations or botched and meaningless reviewer reports. We apply and evaluate these methods on a corpus of reports.

Method. We downloaded the "MDPI Open Peer Review Corpus 2" webscraped by Miłkowski et al. (2023, https://doi.org/10.18150/shkp7b). We focused on 'Round 1' reports, retaining those in plain text (excluding those uploaded as attached files). We computed statistics on report length, identified references suggested by reviewers with regular expressions, extracted frequent word sequences, and analysed the pairs of reports showing an inter-textual similarity higher than 90%.

Results. The MDPI corpus consists of reports for 135,653 accepted articles in MDPI journals from 2011 to 2022. We mined this 170 GB dataset and observed several shortcomings. Some reports appear to be truncated due to failed webscraping. The dataset we analysed contains 135,437 articles and their 339,387 associated reports. The average report has 270 words with a median length of 202. Microscopic reviews consist of one word only, such as 'accept', 'none', 'Nil' or 'N.A.' (n = 230). These seem to be reports (mis)presented by the publisher as 'Round 1' albeit resulting from a peer review ran for an undocumented earlier submission ('reject and resubmit' editorial act). Tiny reports of less than 20 words account for 3.5% of the dataset. We also searched for report templates being constantly reused. Report-report similarities show that 40 reports were almost identical. At least 10 articles share two identical reports. Large chunks of text were reused across unrelated 380 reports sharing at least 300 words in chunks of 10+ words. We also spotted potential coercive citations: some reports contain the same wording suggesting authors to add a set of DOIs/PMIDs.

Conclusion.

We found very few evidence of questionable features in the MDPI reports. More research is needed to improve malpractice detection and assess its prevalence in (open) peer review reports. Once implemented into publishers' workflows, early misconduct detection should help to prevent botched reports, template reuse, and coercive citations.

Objective: 50 words

Malpractices in the reviewing process are detrimental to the scientific literature. We present methods intended to find evidence of peer review manipulations, such as: template usage, citation manipulations or too short and meaning-less reviews. We test these methods on an open peer review corpus to assess its quality.

-> 48 mots

The scientific literature suffers from a variety of research integrity issues. Malpractices in scientific publications have been identified. Malpractices in the reviewing process would also be detrimental to the scientific literature.

We present methods intended to analyze peer reviews searching for evidence of manipulations, such as template usage, citation manipulations, or too short review.

-> 54 mots

Build and validate methods to find evidence of peer review manipulations: template usage (similarity), citation manipulations (DOI/PMID), too short review (none, NA, ...).

Method: 50 words

We downloaded the "MDPI Open Peer Review Corpus 2" [START_REF] Miłkowski | 2 Bad smells in Reviewers' Reports: text-mining the MDPI Open Peer Review Corpus[END_REF]. We excluded reviews consisting solely of an attached file, considering only the 'Round 1' of the review process. We compute statistics on review lengths, search for suggested references using regular expressions, compute similarity scores and frequent n-grams.

-> 50 words

We applied the following methods on the plain text of reviews (i.e., excluding reviews consisting solely of an attached file) only for 'Round 1' of the review process, which are usually more extensive reports and thus more interesting to analyze:

• Compute statistics on review lengths;

• Search for suggested references based on regular expressions based on publication identifiers (i.e., PMID and DOI); • Compute similarity scores between pairs of reviews;

• Search frequent n-gram (n=??) and construct graphs of reviews -> 108 mots :-(

We downloaded the "MDPI Open Peer Review Corpus 2" [START_REF] Miłkowski | 2 Bad smells in Reviewers' Reports: text-mining the MDPI Open Peer Review Corpus[END_REF] covering 135,653 accepted articles (170 GB). Selected only "round 1" reports and excluded attached files.

• stat on length • regexp PMID/DOI • presence of tortured phrases • compute similarity between round 1 reports • compute frequent n-gram (n=??) and graph of review

Acknowledgements. GC and CL acknowledge the NanoBubbles project that has received Synergy grant funding from the European Research Council (ERC), within the European Union's Horizon 2020 program, grant agreement no. 951393.

Data

https://doi.org/10.18150/SHKP7B

Method

• computation of pairwise text similarity, visual inspection of highest similarity scores

• [GC] regexp to identify the most frequent DOIs and PMIDs

Results

• 'Regenerate Response' from ChatGPT https://pubpeer.com/publications/E6F750F5DE06F5C90B0455E1AB4563

• Report = None, acknowledged by authors https://pubpeer.com/publications/1151A6FEE0ED92F4A02469D888AB4A

• Empty report https://pubpeer.com/publications/D8D8B9B3AEF07D3BE02593A2542A64 • Identical reviews in different journals (review template) https://www.mdpi.com/2673-7116/2/2/17/review_report Reviewer 2 https://www.mdpi.com/2071-1050/14/7/4068/review_report Reviewer 1 https://www.mdpi.com/2076-3417/11/23/11187/review_report Reviewer 1 https://www.mdpi.com/2071-1050/14/2/1016/review_report Reviewer 2 (Anonymous) https://www.mdpi.com/2076-3417/12/19/9814/review_report Reviewer 2 (Ali A. Shubbar) https://www.mdpi.com/2076-3417/12/17/8508/review_report Reviewer 2 (Ali Shubbar) https://www.mdpi.com/2073-445X/11/2/312/review_report Reviewer 3 https://www.mdpi.com/2071-1050/13/3/1288/review_report Reviewer 2 the following differing only from suggested references: biomedicines10020240 https://www.mdpi.com/2227-9059/10/2/240/review_report biomedicines10061340 https://www.mdpi.com/2227-9059/10/6/1340/review_report cancers13215505 https://www.mdpi.com/2072-6694/13/21/5505/review_report cancers14010072 https://www.mdpi.com/2072-6694/14/1/72/review_report cancers14040865 https://www.mdpi.com/2072-6694/14/4/865/review_report cancers14071680 https://www.mdpi.com/2072-6694/14/7/1680/review_report cancers14071730 https://www.mdpi.com/2072-6694/14/7/1730/review_report cancers14102386 https://www.mdpi.com/2072-6694/14/10/2386/review_report cancers14122838 https://www.mdpi.com/2072-6694/14/12/2838/review_report curroncol29050266 https://www.mdpi.com/1718-7729/29/5/266/review_report curroncol29060346 https://www.mdpi.com/1718-7729/29/6/346/review_report ijms23010223 https://www.mdpi.com/1422-0067/23/1/223/review_report ijms23052704 https://www.mdpi.com/1422-0067/23/5/2704/review_report ijms23052801 https://www.mdpi.com/1422-0067/23/5/2801/review_report ijms23158133 https://www.mdpi.com/1422-0067/23/15/8133/review_report jcm10081638 https://www.mdpi.com/2077-0383/10/8/1638/review_report jcm11051213 https://www.mdpi.com/2077-0383/11/5/1213/review_report jpm12040540 https://www.mdpi.com/2075-4426/12/4/540/review_report • Some DOIs/PMIDs are (too) frequently suggested (citation abuse?) • [Cyril] Frequent ngrams -histogramme des tailles -détection des tortured phrases • Reports des papiers rétractés ? https://www.mdpi.com/2076-3417/9/11/2316 (trouvé https://www.mdpi.com/2076-3417/9/11/2316/review_report) • jointure avec torturedpapers: uniquement un cas https://doi.org/10.3390/jcp1020017 -From other people: • https://twitter.com/clementFFF/status/1706987985764270231 Final decision Reject -Accept : 26 -Accepted: 7 -None : 59 -No comments : 70 -Nb de reveiw < 20 mots 11751

Results: 200 words

The MDPI corpus consists of reviews for 135,653 accepted articles (170 GB) in MDPI journals from 2011 to 2022. The corpus has some inherent shortcomings. It contains incompletely recovered reviews and files attached to reviews are absent. For these reasons 13,670 reviews are excluded in our final corpus consisting of 339,387 reports about 135,437 distinct articles. Some artefacts appear in the form of very small reviews consisting of only one word, such as 'accept', 'none', 'Nil' or 'N.A.'. They often correspond to reports (mis)presented by the publisher as 'Round 1' but are in fact the result of an additional round of an undocumented previous submission. We noticed 11751 such reviews (3.4%) smaller than 20 words. The 122,607 articles that have all reports in plain text have on average 270 words (median 202).

Nevertheless, the corpus provides multiple information. Although anecdotal with regards to the number of reviews, our study reveals questionable practices: at least 40 reviews entirely (or quasi-) identical, a minimum of 10 articles having two identical reviews, large chunks of text reused across unrelated reviews (380 reviews -1.1%-sharing at least 300 words in chuncks of at least 20 words) and finally we are able to identify potential citation abuses.

->197

, keeping entirely or partially the same reviewers, but without author responses to previous comments We also found Identical reviews for the same paper or for different papers, which question the review process.

Pb corpus (https://pubpeer.com/publications/25353AAFD4FC52E2BEC8C7AD08B259):

• web scraping issues reported to corpus makers (V1->V2)

• incomplete reviews (renamed attached files can't be matched to the reviews)

• artefact: small reviews (like "accept", "none"...) from assessment of resubmission that are miss presented by the publisher as "round1", Same review copy/paste several times (19x,...)

Features:

• 339,387 reports (+13,670 reviews éliminées: "see attached files"/pb scrapping) for 135,653 distinct articles • 12,830 distinct articles with *fullreviews_False • number of papers that have all reports in plain text: 122,607/135+ • stats: average of 225 (Med 165) words per review • 11751 smaller than 20 words ("None", "Regards", "Best", "N.A.", "Nil"...) Anecdotal evidence of questionable practices:

• Report Template or chunk of text are reused from one review to an other • "Regenerate Response" ?? Not for now • Citation abuse : Frequent PMIDs/DOIs, DOI/PMID suggestions for the journal where the paper is submitted • Identical reviews for the same paper (probably an error in the editorial process?) or for different papers (manipulation of the review process)

Conclusion: 50 words

Thanks to our methods, we can identify questionable practices by reviewers and publishers. We found very few evident malpractices in reviewers' reports. Due to frequent false positive automatizing completely malpractice detection appears difficult. Quantifying precisely the phenomenon requires methods improvements.

-> 40

Thanks to our methods, we are able to identify questionable practices of reviewers and publishers. Very few evident malpractices in reviewers' reports appear. Due to false positive it is difficult to automatize completely the detection and to quantity precisely such phenomenon. Nevertheless there is a need for furthered methods.

->49

Very few bad smells and few questionable practices through some anecdotal evidence. thanks to our methods we are able to identify questionable practices. Due to false positive it is difficult to automatize completely the detection and to quantity precisely the phenomenon. Nevertheless it shows that ... better tools are needed ... and publisher could already address some problems ->58


Wendeline Swart

Guillaume Cabanac

Tortured conference series thriving in Computer Science

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Objective. The 'Problematic Paper Screener' (PPS, WCRI'22, https://doi.org/10.48550/arXiv.2210.04895) flagged 12k+ questionable articles featuring tortured phrases, such as 'glucose bigotry' instead of 'glucose intolerance.' It daily screens the literature for 'fingerprints' from a list of 4k tortured phrases known to reflect nonsensical paraphrasing with synonyms. We identified a concentration of 'tortured articles' in IEEE conferences and reported our concerns in November 2022 (https://retractionwatch.com/?p=127299). This WCRI submission unveils 'tortured conference series': questionable articles that keep being accepted in successive conference editions.

Method. We analysed data from the 'Tortured detector' of the PPS (https://bit.ly/PPS-tortured). The considered corpus includes articles from conference proceedings flagged with 5+ tortured phrases (n = 3848). We grouped the records by publisher and by conference. For each conference, we calculated the yearly (i.e., per edition) number of tortured articles included in the conference proceedings.

Results. The 3848 tortured articles were published over 2008-2023 in the conference proceedings of 11 publishers. Those publishing 10+ tortured articles are: IEEE (n = 3227; 84%), IOP Publishing (n = 314; 8%), AIP Publishing (n = 233; 6%), ACM (n = 45; 1%), and Atlantis Press (n = 10; <1%). IEEE (Institute of Electrical and Electronics Engineers) and ACM (Association for Computing Machinery) are the two flagship professional associations of computer scientists: they award the two major prizes in Computer Science (CS) and publish most of CS research. The 45 tortured ACM articles appeared in 35 conference editions, with a top concentration of 4 articles in AICTC'16. In contrast, way higher concentrations affected IEEE: up to 17% (n = 44) tortured articles appeared in the ICERECT'22 proceedings. Even more troubling: we flagged 172 conference series with 2+ tortured articles published over 2+ editions. The worst cases in volume and longevity are 1) ICACCS with 82 tortured articles over 6 years since 2017 and 2) ICCES with 75 tortured articles over 8 years since 2016.

Conclusion.

IEEE has published the highest number of conference proceedings featuring tortured articles. We found evidence of affected conference series and advise IEEE to audit their peer review process to prevent further contamination of the literature.

Acknowledgements. GC acknowledges the NanoBubbles project that has received Synergy grant funding from the European Research Council (ERC), within the European Union's Horizon 2020 program, grant agreement no. 951393.


L Dentz

G Chau

G Hallais

F Hamouda

J P Connolly

A Jaffré

L Vincent

D Bouchier

D Mencaraglia

C Renard

DEVELOPMENT OF MONOLITHICALLY INTEGRATED (Al)GaAs ON SILICON FOR TANDEM SOLAR CELLS

Keywords: Epitaxy, III-V Semiconductors, Silicon, Multijunction Solar Cell

We present here an innovative solution to heterogeneously integrated III-V materials on silicon based on the growth of micrometric GaAs crystals on silicon without any structural defects nor stress, using Epitaxal Lateral Overgrowth on Tunnel Oxide from nanoseeds (ELTOn). This epitaxial method has the advantage of being compatible with cost effective processes, while using innovative growth mechanisms with very low III-V material requirement (a micron or so deposited and no III-V or Ge substrate required) helping keep costs competitive and with low use of critical raw materials. The main focus of this work is to address the issue of the doping determination of micrometric size GaAs crystals which is a pre-requisite for developing design and technology of multijunction solar cells based on an array of GaAs crystals free of structural defects. In a previous work, local electrical characterizations by µ-PL have revealed a rather high non intentional p-type doping. Several growth solutions (effect of V/III ratio and use of alternative precursor) will be presented to lower these high non intentional p-type doping.

INTRODUCTION

The monolithic integration on silicon of GaAs, and more generally of III-V semiconductors, would be a very promising route for the production of high efficiencies multijunctions devices in the manner of those developed for spatial applications on germanium, but at a much lower cost suitable for terrestrial PV applications. The lattice parameter mismatch leading to threading dislocations is however the major problem that many approaches have tried to resolve but to date no one has succeeded in achieving defect-free and cost-effective direct integration of the simplest GaAs binary III-V compound, on silicon.

In previous work, we have demonstrated the perfect integration on silicon of micrometric size GaAs crystals without any structural defects nor stress, using Epitaxal Lateral Overgrowth on Tunnel Oxide from nanoseeds (ELTOn). This technic also leads to a very good electrical connection not only between the GaAs microcrystals and the Si substrate, but also across the GaAs/thin SiO2/Si sandwich (by tunneling assisted transport) surrounding the nanoholes [START_REF] Renard | High current density GaAs/Si rectifying heterojunction by defect free Epitaxial Lateral overgrowth on Tunnel Oxide from nano-seed[END_REF]. The principle of the ELTOn technic is illustrated in Figure 1.

We aim now to extend this strategy to AlxGa1-xAs compounds, more specifically up to an Al content x=18%, which is the composition required to achieve current matching and an optimum conversion efficiency for an AlGaAs/Si tandem photovoltaic solar cell of 32% (derived from our realistic software model called SOL). The assessment of our SOL software has been fully proven by accurate modelling of the published experimental data of record multijunctions solar cells [START_REF] Connolly | Designing III-V multijunction solar cells on silicon[END_REF]. Nevertheless, before investigating AlGaAs alloy we will focus on the control of the residual doping of the simply binary GaAs compound.

The residual doping control in a single GaAs microcrystal is a pre-requisite to developing the future design and technology of multijunction solar cells based on this approach. Local electrical I/V measurements of the GaAs/Si heterojunction performed with CP-AFM showed a high rectification ratio but a high p-type non intentional doping level was suspected and related to high nonintentional carbon incorporation induced by trimethylgallium (TMGa) used as Ga precursor. The precise doping level determination in a single µ-crystal by local I/V or C/V measurements is difficult because in the first case the estimation is too indirect and in the later one, the capacitance values are too small to be easily measured. To obtain a better estimation of the doping level, we have developed a contactless method based on the measurement of the evolution of the bandgap versus temperature [3]. The bandgap value is extracted from the microphotoluminescence spectra of a single GaAs crystal. The bandgap narrowing due to p-type doping is taken into account and the doping density can then be estimated from the fit of the bandgap evolution with temperature. This high carbon doping has been observed in the literature, and the concentration can be adjusted by increasing the V/III ratio during epitaxy [START_REF] Longo | Controlled intrinsic carbon doping in MOVPE-grown GaAs layers by using TMGa and TBAs[END_REF]. Nevertheless, recent results obtained in our growth reactor not clearly seem to allow such a good control of the p-type doping level but this point will be further investigated with both higher growth temperatures and V/III ratio. However, a fallback solution is to adjust the p-type concentration by the use of TEGa (TriethylGallium) and TMGa together with TBAs. The use of TEGa with TBAs makes it possible to obtain low carbon concentrations of the order of 10 15 -10 16 cm -3 reproducibly. The simultaneous use of TEGa / TMGa ratio in variable proportions is therefore expected to allow a good control of the level of doping over a range from 10 14 cm -3 to 10 20 cm -3 [5]. These routes to a better control of the non-intentional carbon incorporation and the further p-type intentional doping will be addressed.

RESULTS AND DISCUSSION

Micro-scale GaAs crystals growth

The GaAs µ-crystal growth was carried out in a chemical beam epitaxy (CBE) system with the Epitaxial Lateral overgrowth on Tunnel Oxide from nano-seed (ELTOn) technique. TriMethylGallium (TMGa), TriEthylGallium (TEGa) and TertiaryButylArsine (TBAs) were used as gas sources. The (111)Si wafers were cleaned using the modified Shiraki chemical cleaning which results in a reproducible ultrathin Si02 layer with a thickness of 1.2 nm thick oxide layer as deduced from Auger electron spectrometry and HRTEM. The important next step is the opening of the nanoscale holes through the ultrathin silica layer prior to GaAs epitaxy.

To demonstrate the ability of the ELTOn growth for the perfect integration of micrometric size GaAs crystals, we first used a random patterning of the silica layer directly performed in situ in the growth reactor. For this purpose, the nucleation seeds must be created. This method has the advantage of easily providing nanoscale seeds around 50 nm in width, randomly spaced at the surface of the oxidized Si substrate [START_REF] Renard | High current density GaAs/Si rectifying heterojunction by defect free Epitaxial Lateral overgrowth on Tunnel Oxide from nano-seed[END_REF]. After this step, the GaAs epitaxy was initiated with a two steps procedure. The first step consists in introducing solely TBAs (during 2 min) at 430 °C. In the second step TMGa, respectively TEGa, were also introduced while the growth temperature was increased up to 575 °C. After, this nucleation step that tooks 2 min, GaAs epitaxial lateral overgrowth was then continued for 45 min. In this paper we investigate particularly the effect of the As/Ga (V/III) ratio on the doping of GaAs µ-crystals. For this purpose, several V/III ratios, denoted R, will be investigated. Such as 6, 10 and 14 for [TBAs]/[TMGa]. And 6, 10, 14, 18 and 22 for [TBAs]/[TEGa]. For the sake of clarity, the V/III ratios studied in this paper are not the same for TMGa and TEGa. In fact, the results given for TMGa correspond to a previous study in which we limited ourselves to R=14. Top SEM views of µ-scale GaAs crystals grown on silicon using the ELTOn Process for different V/III ratios with TBAs and TEGa as gas precursors are showed in Figure 2.

Micro-Photoluminescence (PL) experiments

µ-PL measurements were performed on a WITec alpha-300R confocal microscope in a backscattering configuration. Samples were under vacuum at 10 -3 mbar in a Linkam cryostat and probed at a requested temperature imposed by the temperature controller from 80K to 340 K, each 20 K. Samples were excited at 532nm through an Olympus 20x long working distance objective, with 9µW laser power (360nW/µm²). Light is collected throught a 200µm pinhole, and resolved with a 150 grooves/mm grating, with 750nm blaze wavelength. The detector is an Andor TEC cooled CCD bar at -70°C. For each spectrum, the camera thermal signal is removed.

Figures 3 and4 show PL spectra obtained at 300 K on GaAs/Si crystals made with the TMGa or TEGa precursor respectively, for the different R values indicated in the insert. The spectrum of an undoped GaAs wafer is also shown as a reference on each of the figures. We note that in the case of TMGa (Fig. 3), the PL peaks of the crystals have an energy position lower than or very close to that of the reference peak, whereas in the case of TEGa, the energy positions of the PL peaks of the crystals have an energy position higher than that of the reference. Figure 5 summarizes these trends by showing the evolution with R of the energetic position of these peaks as a function of the ratio R. In the case of GaAs, it has been shown that the energetic position of the main PL peak is practically equal to the direct gap [6], to within a few meV (around 4 meV at 300 K). In Figure 5, peak energies below the reference value may therefore reflect an apparent reduction in the gap due to Band Gap Narrowing in the case of a P-type, whereas peak energies above the reference value could be explained by the Moss-Burstein effect for a degenerate Ntype material. If this interpretation is correct, the different behavior of TMGa and TEGa precursors may be related to the amphoteric character of carbon with respect to GaAs doping, given the different dissociation properties of the methyl and ethyl radicals present in these precursors. provide a better answer to this problem, but this route is likely to be more tricky to implement since, with TBAs, it will require the use of three precursor gases. However, before exploring this avenue, it will be necessary to analyse in more detail our interpretation of the evolution with R of the main peak shown in Figure 5. To this end, we will exploit photoluminescence spectra measured at different temperatures in the range 100 K to 340 K. In particular, we will analyze the evolution of the characteristics of the main peak and the secondary peak which is clearly visible at 300 K in the case of TMGa in Figure 5. We only point out here that, in a certain range of lower temperatures, this peak becomes stronger and also becomes apparent in the case of TMGa. The detailed interpretation of the evolution of the characteristics of these peaks with temperature will be one of the follow-ups to this work.

CONCLUSIONS

We have presented an innovative growth strategy based on Epitaxial Lateral Overgrowth On Tunnel oxide (ELTOn) in order to monolithically integrate defect-free micrometric size (Al)GaAs crystals on silicon. Integrating these crystals in a regular array to have a quasi-complete covering of the Si substrate, without coalescence of the GaAs microcrystals, is therefore a very promising route to produce in a first instance an AlGaAs/Si tandem cell at an affordable cost for terrestrial PV applications.

A fine control of the residual doping with only TMGa or TEGa as the Ga precursors during the CBE growth could not be achieved in the present work. However we have shown that mixing TMGa and TEGa precursors could be a possible route to solve this problem, taking into account the amphoteric nature of the carbon with respect to GaAs doping, and the different dissociation properties of the methyl and ethyl radicals present in these precursors.

The results already achieved have permitted the identification of the bottlenecks and discuss the possible routes for III-V on silicon multijunctions solar cells as the tandem structure presented in this paper can be the building block for further extension to triple junctions.

Figure 1 :

 1 Figure 1: Schematic view of Epitaxial Lateral overgrowth on Tunnel Oxide from nanoseeds (ELTOn): the growth only begins inside the nanoholes performed in the thin silica layer covering the silicon wafer and then extend laterally on top of the SiO2 layer. The (tunnelling assisted) electrical connection between GaAs and the Si substrate thought thin SiO2 layer is indicated with the red bidirectional arrow.

Figure 2 :

 2 Therefore, in order to open nanoscale holes through the thin silica layer, the oxidized silicon substrate was fast ramped up 750 °C, under vacuum, and maintain at such temperature during 1 min 30 s. This results in pure Si nano-areas formed via the reduction of SiO2 into volatile SiO. Top SEM images of µ-scale GaAs crystal sgrown on silicon using the ELTOn Process for different V/III ratio with TBAs and TEGa as gas precursors. a) R = 6, b) R = 10, c) R = 18 and d) R = 22.

Figure 3 :

 3 Figure 3: PL spectra at 300 K of GaAs/Si crystals made with the TMGa precursor for the different R values indicated in the insert. The black line (Ref) is the spectrum of an undoped GaAs wafer.

Figure 4 :

 4 Figure 4 : PL spectra at 300 K of GaAs/Si crystals made with the TEGa precursor for the different R values indicated in the insert. The black line (Ref) is the spectrum of an undoped GaAs wafer.

Figure 5

 5 Figure 5 also shows that it appears difficult to finely control doping in GaAs by using only one or other of the precursors when growing with the CBE technique. The use of a mixture of the two precursors should, in principle, provide a better answer to this problem, but this route is likely to be more tricky to implement since, with TBAs, it will require the use of three precursor gases. However, before exploring this avenue, it will be necessary to analyse in more detail our interpretation of the evolution with R of the main peak shown in Figure5.

Figure 5 :

 5 Figure 5 : Evolution with R of the energetic position of the main peak of the spectra presented in Fig. 3 and 4 as a function of the ratio R. The upper (lower) curve is for GaAs/Si crystals made with the TEGa precursor (TMGa precursor) respectively.

4 ACKNOLEGMENTS

 4 This work was supported by the french Agence Nationale de la Recherche (ANR HELLO_PV project, Convention N° ANR-22-CE50-0011-01). The authors also thank the PIMENT platform of C2N and RENATECH for technological backup.


Alexandre Clausse

Guillaume Cabanac

Pascal Cuxac

Cyril Labbé

Mining tortured acronyms from the scientific literature

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Objective. The 'Problematic Paper Screener' (PPS, WCRI'22, https://doi.org/10.48550/arXiv.2210.04895) supports the human re-assessment of scientific articles flagged as suspicious. The 'tortured detector' tabulates 12k papers containing tortured phrases: established scientific concepts paraphrased with synonyms, such as 'butt-centric waterway' for 'anal canal.' Some acronyms are even tortured, such as 'Convolutional Brain Organisation (CNN)' for 'Convolutional Neural Network (CNN).' This abstract tackles the following task: discover and classify all acronyms from any given article: tortured or genuine.

Method. First, we built a test collection by sampling 75 tortured articles in open access in engineering. The first author visually studied each article to highlight and extract all acronyms, each one was labelled as tortured or not. Second, we designed heuristics to classify acronyms probing initials (mis)match. Third, we benchmarked the algorithm's output to compute recall and precision scores.

Results. The publicly released test collection (https://doi.org/10.5281/zenodo.10014634) contains 975 acronyms with 355 tortured and 53 suspects ones. Most tortured acronyms mismatch their initials (98%), genuine acronyms match their initials (97%), both may contain compounds and stop words. These statistics informed the designing of the classifier that yields 76% precision and 89% recall. Incremental failure analysis identifies false positives (e.g., examples given in brackets) and false negatives (i.e., matching acronyminitials yet irrelevant). As a positive consequence, we extended the list of tortured phrases used by the PPS with 190 new tortured phrases (4%). This extended list now informs the detection of papers featuring tortured acronyms.

Conclusion.

The 'acronym classifier' allows the screening of incoming manuscripts or peer-reviewed articles and spot questionable passages. It as agnostic as it requires no prior knowledge of tortured acronyms already found. Publishers may add this screener in their editorial workflow, scientific sleuths can use it to screen articles for post-publication review. We welcome developers to use the test collection (and its foreseeable extended versions) to benchmark their own implementations and compare to the baseline performance reported in this WCRI abstract.

Acknowledgements. GC and CL acknowledge the NanoBubbles project that has received Synergy grant funding from the European Research Council (ERC), within the European Union's Horizon 2020 program, grant agreement no. 951393.