Hadrien Lorenzo
email: hadrien.lorenzo@inria.fr

Jérôme Saracco
email: jerome.saracco@ensc.fr

Computational outlier detection methods in sliced inverse regression

Sliced inverse regression (SIR) focuses on the relationship between a dependent variable y and a p-dimensional explanatory variable x in a semiparametric regression model in which the link relies on an index x β and link function f . SIR allows to estimate the direction of β that forms the effective dimension reduction (EDR) space. Based on the estimated index, the link function f can then be nonparametrically estimated using kernel estimator. This two-step approach is sensitive to the presence of outliers in the data. The aim of this paper is to propose computational methods to detect outliers in that kind of single-index regression model. Three outlier detection methods are proposed and their numerical behaviors are illustrated on a simulated sample. To discriminate outliers from "normal" observations, they use IB (in-bags) or OOB (out-of-bags) prediction errors from subsampling or resampling approaches. These methods, implemented in R, are compared with each other in a simulation study. An application on a real data is also provided.

Introduction

On one hand, classical linear regression or more generally parametric regression have achieved resounding success in many real problems whose goal is to investigate the relationship between a response variable y ∈ R and a covariate x ∈ R p . However it can be argued that assuming specific structural constraints on the link function of y on x is too stringent. On the other hand, nonparametric regression is clearly a more flexible approach, but it is well-known that it typically suffers from the curse of dimensionality, i.e., a poor rate of convergence when the dimension p of x increases. To address these problems from purely parametric or nonparametric approaches, several authors studied single-index or multiple-index models. This kind of regression models can be viewed as an alternative semiparametric approach based on sufficient dimension reduction. So, in a dimension reduction setting, many authors suppose that x can be replaced by a linear combination of its components, β x, without losing information on the conditional distribution of y given x. One way to express this assumption is:

y ⊥ x | β x (1)
where the notation v 1 ⊥ v 2 | v 3 means that the random variable v 1 is independent of the random variable v 2 given any values for the random variable v 3 . One can write (1) as, for instance, the following single-index model with an additive error:

y = f (β x) + ε, (2)
where f is an unknown real-valued function, the distribution of ε is arbitrary and unknown, and ε ⊥ x. Since f is unknown, the p-dimensional parameter β is not totally identifiable, but the subspace spanned by β is identifiable. This subspace is referred to as the effective dimension reduction (EDR) subspace following Duan and Li [START_REF] Duan | Slicing regression: a link-free regression method[END_REF] in their original presentation of sliced inverse regression (SIR). Li [START_REF] Li | Sliced inverse regression for dimension reduction, with dicussion[END_REF] consider a multiple-index regression model. The Euclidean parameter β is now a p × K matrix: β = [β 1 , . . . , β K ] where the vectors β k are assumed to be linearly independent. The EDR subspace is then the K-dimensional linear subspace of R p spanned by the β k 's. Note that the dimension reduction is very useful in an exploratory stage of data analysis since model [START_REF] Auger | Algorithms for the Optimal Identification of Segment Neighborhoods[END_REF] relies on very few structural assumptions. For instance, it is not assumed that the indices act additively as often assumed in multiple-index models. It is likewise not necessary to assume that the error term is additive on mean (as for the model ( 2)), thus heteroscedastic model are potentially included in this modelling. Note also that sufficient dimension reduction of the regression leads to summary plot of y versus estimated indices which provides useful graphical modelling information.

In a second step, to study the relationship between the response variable and the few estimated indices, standard nonparametric approaches (such as kernel or spline smoothing) can be used. This stage usually involves additional assumptions such as an additive error term (as in model ( 2)) to get consistent properties of the corresponding estimate of the link function f .

In the statistical literature, different methods have been developed with the aim of estimating the EDR subspace. SIR, SIR-II, SIR α , SAVE (sliced average variance estimation) and pHd (principal Hessian directions) approaches are the most popular, see [START_REF] Azais | Optimal quantization applied to Sliced Inverse Regression[END_REF][START_REF] Cai | Online Sufficient Dimension Reduction Through Sliced Inverse Regression[END_REF][START_REF] Chavent | A sliced inverse regression approach for data stream[END_REF][START_REF] Chen | Can SIR be as popular as multiple linear regression[END_REF][START_REF] Cook | SAVE: A method for dimension reduction and graphics in regression[END_REF][START_REF] Gannoun | An asymptotic theory for SIR α method[END_REF][START_REF] Hsing | Nearest neighbor inverse regression[END_REF][START_REF] Jlassi | Variable importance assessment in sliced inverse regression for variable selection[END_REF][START_REF] Li | Sufficient dimension reduction. Methods and applications with R[END_REF][START_REF] Li | On principal Hessian directions for data visualization and dimension reduction: Another application of Stein's lemma[END_REF][START_REF] Li | Dimension reduction for multivariate response data[END_REF][START_REF] Li | Asymptotics for sliced average variance estimation[END_REF][START_REF] Saracco | An asymptotic theory for Sliced Inverse Regression[END_REF][START_REF] Saracco | Asymptotics for pooled marginal slicing estimator based on SIR α approach[END_REF][START_REF] Yin | Asymptotic distributions for dimension reduction in the SIR-II method[END_REF][START_REF] Zhu | On sliced inverse regression with large dimensional covariates[END_REF][START_REF] Zhu | On kernel method for sliced average variance estimation[END_REF] among others. The important question of the determination of the EDR space dimension in SIR and related methods has also been much studied, see for example [START_REF] Ferré | Determining the dimension in sliced inverse regression and related methods[END_REF][START_REF] Liquet | Application of the bootstrap approach to the choice of dimension and the α parameter in the SIR α method[END_REF].

SIR is known to be a relevant technique for the purpose of dimension reduction. Several properties of SIR have been extensively studied and numerous extensions have been already proposed. However little attention has been paid to sensitivity of SIR to outliers or to robustness aspects. Since SIR theory is based on conditional expectation and covariance matrix properties (see Sect. 2 for details), it is obvious that SIR can be severely influenced by outliers in the data, see [START_REF] Gather | A note on outlier sensitivity of Sliced Inverse Regression[END_REF] or [START_REF] Cook | Identifying Regression Outliers and Mixtures Graphically[END_REF] for instance. In [START_REF] Prendergast | Detecting influential observations on the SIR e.d.r. space[END_REF][START_REF] Prendergast | Implications of influence function analysis for sliced inverse regression and sliced average variance estimation[END_REF], the detection of influential observations on the estimation of the dimension reduction subspaces returned by SIR, SIR-II and SAVE have been studied using the notion of influence functions of single observations. However, the proposed empirical influence values are very sensitive to the choice of the number H of slices (introduced in the next section) in detecting influential observations, which makes this approach complicated to use in practice. Robust SIR methods were then developed and only focused on the estimation of the EDR space (regardless of the estimation of the link function f ). For example, in [START_REF] Chiancone | Student Sliced Inverse Regression[END_REF] the inverse regression formulation of SIR is therefore extended to non-Gaussian errors with heavy-tailed distributions (Student). The underlying Expectation-Maximization algorithm was tested in presence of outliers and provided good numerical results. [START_REF] Dong | Robust inverse regression for dimension reduction[END_REF] also mentioned that classical sufficient dimension reduction methods are sensitive to outliers present in predictors, and may not perform well when the distribution of the predictors is heavy-tailed. Two robust inverse regression methods which are insensitive to data contamination (weighted inverse regression estimation and sliced inverse median estimation) were then introduced and they demonstrated very interesting numerical performances in the presence of potential outliers. In the same spirit, [START_REF] Babos | Sliced inverse median difference regression[END_REF] proposed sliced inverse median difference regression to robustify SIR methodology at the presence of outliers. In [START_REF] Dikheel | Robust Sliced Inverse Regression[END_REF], robust SIR extensions were presented through robust estimates of the covariance matrix.

The goal of this paper is to propose computational methods to detect outliers in a single-index regression model, comprising EDR space estimation using SIR and link function estimation based on kernel smoothing. In practice, it is always interesting to detect outliers (rather than only developing robust methods), to isolate them and to understand why these observations are aberrant (wrong numerical values, unusual individuals, ...). Once the dataset has been cleaned, it is then possible to implement the usual methodology, SIR followed by a non-parametric estimation of f . In Sect. 2 a brief overview of usual SIR is given. Three outlier detection methods, named MONO, TTR and BOOT hereafter, are presented in Sect. 3. They use IB (in-bags) or OOB (out-of-bags) prediction errors from subsampling or resampling approaches in order to discriminate outliers from "normal" observations. These methods have been implemented in R. How these methodologies work is described on a simulated example in Sect. 4. Sect. 5 provides a more extensive simulation study that compares the numerical performances of the proposed methods. A real dataset is also used to illustrate these approaches in Sect. 6. Finally concluding remarks are given in Sect. 7.

A brief review of usual SIR

In order to estimate the EDR space, various methods based on the use of inverse regression are widely available in literature. In order for inverse regression to be useful in estimating the EDR space, some of them, like SIR or SAVE or principal Hessian direction, need additional conditions on the marginal distribution of the covariate x. In this paper, we focus the usual SIR approach which relies on the following linearity condition (LC) on x:

For all b ∈ R p , E [b x | β x] is linear in x β. (3)
Note that the LC is required to hold only for the true Euclidean parameter β. Since β is unknown, it is not possible in practice to verify a priori this assumption. Therefore we can assume that LC holds for all possible values of β, this is equivalent to assume an elliptical symmetry of the distribution of x: for instance the well-known multivariate normal distribution satisfies this condition. Finally, following [START_REF] Hall | On almost linearity of low-dimensional projections from highdimensional data[END_REF], the LC is not a severe restriction because this LC holds to a good approximation in many problems as the dimension p of the predictors increases. Interesting discussions on the LC can also be found in [START_REF] Chen | Can SIR be as popular as multiple linear regression[END_REF][START_REF] Li | Sufficient dimension reduction. Methods and applications with R[END_REF] for instance.

Let us now consider a monotone transformation T. Under model (1) and LC, [START_REF] Duan | Slicing regression: a link-free regression method[END_REF] showed that the centered inverse regression curve satisfies:

E[x | T(y)] -µ ∈ Span(Σβ), (4)
where µ := E[x] and Σ := V(x). Therefore the space spanned by the centered inverse curve, {E[x | T(y)] -E[x] : y ∈ Y} where Y is the support of response variable y, is a subspace of the EDR space, but it does not guarantee equality. A pathological model, often called symmetric dependent model, has been identified in the literature, and is model for which the centered inverse regression curve is degenerated. To solve this problem, specific methods (based on higher order inverse moments), such as SIR-II, SIR α or SAVE, have been developed. When the model is not pathological (which is often the case in pratice), the centered inverse regression curve can be used to recover the EDR space from (4). Indeed, a direct consequence of this result is that the covariance matrix of this curve,

Γ := V(E[x | T(y)]),
is degenerate in any direction Σ-orthogonal to β (i.e. to the β k 's for a multiple-index model). Therefore, the eigenvectors associated with the non null eigenvalues of Σ -1 Γ are EDR directions, which means that they span the EDR space E.

In the slicing step of SIR, the range of y is partitioned into H non-overlapping slices {s 1 , . . . , s H }. With such slicing, the covariance matrix Γ can be straightforwardly written as

Γ := H h=1 p h (m h -µ)(m h -µ)
where p h = P(y ∈ s h ) and

m h = E[x | y ∈ s h ].
Let us now consider a random sample {(x i , y i ), i = 1, . . . , n} generated from the single-index regression model [START_REF] Azais | Optimal quantization applied to Sliced Inverse Regression[END_REF]. By substituting the empirical versions of µ, Σ, p h and m h for their theoretical counterparts, we obtain an estimated basis of E spanned by the eigenvector bSIR associated with the largest eigenvalue of the estimate Σ -1 n Γ n of Σ -1 Γ where

Σ n = 1 n n i=1 (x i -xn )(x i -xn ) and Γ n = H h=1 ph,n ( mh,n -xn )( mh,n -xn ) , with xn = 1 n n i=1 x i , nh,n = n i=1 I [y i ∈s h ] , ph,n = nh, n n , mh,n = 1 nh, n
i ∈s h x i , the notation I [.] standing for indicator function. This approach is the one proposed by [START_REF] Duan | Slicing regression: a link-free regression method[END_REF][START_REF] Li | Sliced inverse regression for dimension reduction, with dicussion[END_REF] when they initially introduced the SIR approach. Since the early 1990s, the SIR method has been extensively studied by many authors, see for instance all the references mentioned in the introduction.

The link function f of model ( 2) can then be estimated by the usual kernel estimator (see for example [START_REF] Schimek | Smoothing and regression: approaches, computation, and application[END_REF]) based on the sample {(x i bSIR , y i ), i = 1, . . . , n} where the x i bSIR 's are the values of the estimated index. For a given value x 0 of x, the kernel estimation of f (β x 0 ) is given by fn

( b SIR x 0 ) = n i=1 K x i bSIR -x 0 bSIR h n y i n i=1 K x i bSIR -x 0 bSIR h n ,
where K is the kernel and h n is the bandwidth. The kernel is usually a positive symmetric weighting function with integral equal to 1. In the rest of the paper, the chosen kernel is the density of the normal distribution N (0, 1), called the Gaussian kernel. The bandwidth h n > 0 is called the smoothing parameter in kernel regression because it controls variance and bias of the estimator. This parameter must therefore be correctly tuned using cross-validation for instance:

h opt n = arg min h n >0 1 n n i=1 y i -f (-i) n ( b SIR x 0 ) 2 ,
where f (-i) n ( b SIR x 0 ) stands for the estimation of f (β x 0 ) based on the sample {(x j bSIR , y j ), j i}.

When the underlying regression model is a multiple-index model, the estimated EDR space is spanned by the eigenvectors associated with the largest K eigenvalues of the estimate Σ -1 n Γ n . Let B SIR be the p×K matrix of these K eigenvectors. The estimated indices x i BSIR 's are now K-dimensional and the kernel estimation of f (β x 0 ) is then based on multivariate kernel. For example K can be the density of the multivariate normal distribution N (0 K , I K ) where 0 K (resp. I K ) stands for the null vector of dimension K (resp. the identity matrix of order K), and the associated smoothing parameter h n can be K-dimensional. Another way is to consider the following kernel estimator

fn ( B SIR x 0 ) = n i=1 K | |x i BSIR -x 0 BSIR | | h n y i n i=1 K | |x i BSIR -x 0 BSIR | | h n ,
where ||.|| stands for a chosen norm in R K and the bandwidth h n is unidimensional.

Outlier detection methods in SIR

Three outlier detection methods for single-index regression model ( 2) are presented.

Let us consider a sample S = {(x i , y i ), i = 1, . . . , n} of n individuals among which some may be outliers.

For each of the three methods, the parameter β (more properly, the EDR direction b) is estimated by the usual SIR method (with the number of slices H = 10) and the link function f is estimated using the kernel estimator with the Gaussian kernel and the bandwidth tuned via cross-validation.

A naive method

This naive method relies on the following three steps. The errors considered are naturally the residuals: for i = 1, . . . , n, êi = y i -ŷi . The detection of potential outliers is simply based on the definition of outliers in the boxplot of the absolute error | êi |'s, i.e. the outliers correspond to individuals whose values are greater than the value of the 3rd quartile plus 1.5 times the inter-quartile interval. Note that, in the same spirit, the bootstrap histogram of "mean -trimmed mean" for a suitable trimming number was proposed by [START_REF] Singh | Bootlier-Plot: Bootstrap Based Outlier Detection Plot[END_REF] as a nonparametric graphical tool for detecting outlier(s) in a dataset. The bootlier-plot was introduced and it is shown that the multimodality in the bootlier plot is caused by outlier(s) in the sample.

This naive method is called MONO hereafter. The name MONO stands for a single use of the initial sample S and a single estimate of the underlying single-index model. In the numerical example of Sect. 4, Fig. 1 allows to visualize the position of the outliers in the corresponding boxplot.

TTR method

This method relies on training sample and test sample replications for evaluating the "stability" of the estimated model, hence the name TTR of the method for Training Test Replications.

The TTR approach works in two major steps. Let R be the number of replications chosen by the user. In practice R = 2000 is more than enough for reasonable sample sizes, i.e. n ≤ 500. Let α ∈ [0, 1] be the proportion of the sample which will constitute the test sample. In the rest of the paper, the parameter is fixed to α = 0.1, thus 90% of the sample S is used as the training sample S train and the remaining 10% of the sample S constitutes the test sample S test . Note that individuals are drawn with equal weight and without replacement. test , calculate the error of prediction of the response variable y as follows:

ê(r) i * = y i * -f (r) n ( b(r) SIR ) x i * ,
where the estimate f

(r) n (.) is based on the sample {(( b(r) SIR ) x i , y i ), i ∈ S (r) train }. S 2.
Evaluation of the error means.

For each i * = 1, . . . , n, calculate the associated error mean over the R replications (when the individual i * is present in the corresponding test sample):

e i * = R r=1 ê(r) i * I [i * ∈S (r ) test ] R r=1 I [i * ∈S (r ) test ] . S
3. Detection of the outliers via a change point detection.

The idea is to find a single change point position in the sequence of the errors' means {e (i * ) , i * = 1, . . . , n} ordered by decreasing values (where the subscript (i * ) enclosed in parentheses indicates the i * th order statistic of the sample). Indeed, if there are no outlier in the data, no change points should clearly appear in this sequence of ordered absolute mean errors. On the other hand, in the presence of outliers, the corresponding mean absolute errors should naturally be significantly larger than the errors associated with other individuals. Thus, looking for a single change point in mean and variance in this sequence should intuitively allow us to separate outliers from other observations. Many authors have proposed a single search method to detect change points.

Recently, [START_REF] Killick | Changepoint: An R Package for Changepoint Analysis[END_REF] have developed the R package changepoint that helps to detect the location of different change points. For single or multiple change point detection, the approach allows to estimate the points at which the statistical properties of a sequence of observations change. Within this package, several change in mean methods are available as well as methods focusing on detection of change in variance and methods searching a change in both mean and variance. Briefly, let us give an overview of the underlying approach. Let z 1:n = (z 1 , . . . , z n ) be the ordered sequence of the errors' means and τ i:m = (τ 1 , . . . , τ m ) the positions of the m change points (each change point position is between 1 and n -1, τ 0 = 0 and

τ m+1 = n). The idea is to minimize m+1 i=1 [C(z (τ i-1 +1):τ i )] + γg(m) (5)
where C is a cost function (for instance negative log-likelihood ratio test statistic) and γg(m) is a penalty to guard against over fitting. This package implements several algorithms to minimize (5): binary segmentation [START_REF] Edwards | Method for Cluster Analysis[END_REF], segment neighborhood [START_REF] Auger | Algorithms for the Optimal Identification of Segment Neighborhoods[END_REF] and pruned exact linear time (PELT) [START_REF] Killick | Optimal Detection of Changepoints with a Linear Computational Cost[END_REF]. Here the changepoint package is used to detect only one change point (m = 1) in mean and variance with the binary segmentation algorithm in the ordered sequence of means {e (i * ) , i * = 1, . . . , n}. In the numerical example of Sect. 4, Fig. 2 (top left) visualizes the position of the estimated single change point.

An individual associated with an ordered error's mean before the single change point position is then considered as an outlier.

Remark. In the associated R code, the bandwidth is tuned only once in step 1.c for the kernel estimation of each iteration. This "optimal" bandwidth is obtained via cross-validation using the whole sample of the y i 's versus the estimated indices x i bSIR . This is a reasonable choice if one assumes that there are no outlier in the x i 's and thus in the x i b's or in the x i b(r) 's. Note that the presence of visible outliers in the x i 's would have been detected in a preliminary step and the dataset would then have been cleaned. This choice of only one tuned bandwidth clearly saves calculation time for the TTR method. Finally, note also that, in each iteration of step 3 for the TTR method, it is easy to integrate an automatic optimal bandwidth selection in the R code. The same strategy is used for the BOOT method in step 1.c presented in the following section.

BOOT method

The MONO method deals with in-bag (IB) errors and the TTR method with outof-bag (OOB) errors. While MONO risks overfitting, TTR risks significance loss of statistical power (since the train sample is a subsample of the initial sample) but cannot quantify the impact of IB individuals. The current BOOT method uses IB errors in that objective.

Isolated individuals that are not outliers, in the plot of the estimated index versus the response y are usually hard to predict especially if those individuals are not in the training dataset. However, if any of those individuals are included in the training dataset, it has a beneficial effect on the built model. Indeed, those individuals are therefore better predicted while the non isolated individuals are still well predicted since those isolated individuals are in line with the regression model. For those isolated individuals, the OOB error is then high while the IB error is potentially low. They are denoted as "borderline" observations in the following. On the other hand, the "outliers" are always badly predicted with high IB and OOB errors and the "normal" individuals are always well predicted with low IB and OOB errors (see an illustration of these comments in Figure 4 that gives examples of those three types of observations).

The BOOT method is based on two simple decision rules to discriminate between these three types of individuals ("normal" observation, "borderline" observation,"outlier") using the IB error and its logarithmic transformation. This method relies on bootstrap samples of S. Let B be the number of bootstraps chosen by the user. In practice B = 2000 is more than enough for reasonable sample sizes, i.e. n ≤ 500. Note that individuals are drawn with equal weight and with replacement.

ê(b) i = y i -f (b) n ( b(b) SIR ) x i ,
where the estimate f

(b) n (.) is based on the sample {(( b(b) SIR ) x i , y i ), i ∈ S (b) }.
Note that, even if they are not used in Steps 2 and 3, the OBB errors (for all the individuals i S (b) ) have also been calculated since they are used in graphical representations (see Fig. 4).

S

2. Evaluation of the error means. For each i = 1, . . . , n, calculate the associated error mean over the B replications (when the individual i is present at least once in the corresponding bootstrap sample):

e (i) = B b=1 ê(b) i I [i such that n (b) i ≥1] B b=1 I [i such that n (b) i ≥1] . S
3. Detection of outliers and "bordeline" observations

The idea here is to first identify among the errors {e (i) , i = 1, . . . , n} those which are particularly high and which will naturally correspond to these "big" outliers. For this purpose, the log scale was used to detect these outliers. Then, in a second step, the usual scale is used in order to identify other possible "small" remaining outliers which are then called "borderline" observations.

3.a. The detection of potential outliers is based on the definition of outliers in the boxplot1 of the log e (i) 's. The corresponding observations are plotted in blue in Fig. 3 (on the left).

3.b. The detection of potential "borderline" observations is based on the definition of outliers in the boxplot of the e (i) 's, these "current outliers" are plotted with orange triangle in Fig. 3 (in the middle). The "borderline" observations are thus defined as the current detected outliers not identified as outliers in the previous step 3.a. (plotted with blue circle behind the orange triangle in this graphic).

The corresponding "borderline" observations are therefore those represented only in orange on the graphic in Fig. 3 (on the right).

Remark. In Step 3.a., the log transformation is used by default to detect the potential outliers. However, the relevant transformation of the considered errors, e (i) , i = 1, . . . , n, is probably not always log but that it may depend on the link function f itself and on the distribution of in the regression model (2).

A numerical example

Let us consider a simulated sample to clearly illustrate how the previous three outlier detection methods (MONO, TTR and BOOT) work. Note that steps 3 of the different methods (MONO, TTR and BOOT) are interchangeable with each other and thus they can be used after any of the error calculation steps (steps 1 and 2). In Sections 4 and 5, only the MONO, TTR and BOOT methods are compared with each other, not ideally all possible combinations. Thus, we are well aware that this will make it difficult to identify whether the success of the method is due mainly to the different error calculation processes (steps 1 and 2) or to the technique of detecting "abnormally large" errors (step 3).

Description of the simulated dataset

The following single-index regression model is used in all numerical studies in Sections 4 and 5:

y = (x β) 3 100 + , (6)
where β = (2, 2, 1, -2, -3, 0, . . . , 0) ∈ R p , x follows the p-dimensional uniform distribution on [-2; 2] p , and ∼ N (0, σ 2 = 0.25) is independent of x. In a first step, ñ = 200 observations {(x i , y i ), i = 1, . . . , ñ} are generated from model 6 with p = 5. Then in a second step, ñ = 10 new individuals are generated as follows: for i = ñ + 1, . . . , ñ + ñ,

• x i is drawn from the uniform distribution on [-2; 2] p , • y i is drawn (independently from x i ) from the uniform distribution on the support of the first ñ values of y.

These ñ new observations are then "potential" outliers for the model ( 6) since their y i 's are not linked to the x i 's via this model. Note that these observations are not outliers regarding the distribution of the x i 's (resp. of the y i 's). The term "potential" refers to the fact that an observation (x i , y i ) (for an i ∈ { ñ + 1, . . . , ñ + ñ} may be close, just by chance, to the "true" structure of the data (based on the underlying model ( 6)). The objective is to detect these potential ñ outliers in the sample S = {(x i , y i ), i = 1, . . . , n} where n = ñ + ñ and then to estimate as best as possible the relationship between y and x through the single-index x β.

Numerical results

In a first step, based on the available sample S = {(x i , y i ), i = 1, . . . , n}, the EDR direction b is estimated by bSIR using the usual SIR method (with the number of slices H = 10) and the link function f is estimated by fn (.) using the kernel estimator with the Gaussian kernel and the bandwidth tuned via cross-validation. The distance between the true EDR space and the estimated one is defined as

d 2 (E, E) = 1 - Trace(P E P E ) K ∈ [0, 1],
where P E = β(β β) -1 β (resp. P E ) is the orthogonal projector onto E (resp. E) with K the dimension of the EDR space (here K = 1 for a single-index model). The closer this distance is to zero, the better the estimation E of E. On the simulated sample, we have d 2 (E, E) = 0.0093. The corresponding MSE (mean squared error) defined as

MSE = 1 n n i=1 y i -fn (x i bSIR ) 2 is equal to MSE = 12.99.
Using the naive MONO method, 15 outliers have been detected, see Fig. 1 (left) for the boxplot of the absolute residual errors (with outliers in blue) and Fig. 1 (right) for the visualization of these outliers on the plot of the estimated indices x i bSIR versus the y i 's. Note that all the ñ = 10 generated outliers have been identified. Among these 15 detected outliers, 5 are false positive, however the individual 21 (at the top right of the plot of Fig. 1(right)) can be considered as an "extreme" observation. An "extreme" observation may obviously be detected as an outlier by the method because the nonparametric estimation of f by the kernel method is based on local smoothing. Thus since an "extreme" observation is too isolated in the plot of the estimated indices (x i bSIR , i = 1, . . . , n) versus the y i 's, its kernel prediction is difficult due to the lack of observations around it (this is the problem of data sparsity in nonparametric regression). Using the initial sample without these 15 outliers, the associated MSE is now equal to 0.24, and we have d 2 (E, E) = 0.00361. These two quantities clearly show the benefits of removing the detected outliers. Using the TTR method with R = 3000, 11 outliers have been detected, see Fig. 2 (top left) for the detection of the unique change point position in the sequence of the ordered errors' means, {e (i * ) , i * = 1, . . . , n}. Fig. 2 (top right) provides the visualization of these outliers on the plot of the estimated indices x i bSIR versus the y i 's. Among these 11 outliers, only 2 are false positive: observations 10 and 21 (at the bottom left and at the top right of the plot of Fig. 2 (top right)) can naturally be considered as "extreme" observations but they are still selected as outliers for the same reasons of nonparametric kernel estimation as those mentioned for the MONO method. Note also that observation 206 (in red) has not been detected as an outlier by TTR method, but its projection is very close to the "true data" (in black, i.e. that is those generated by the underlying model) and thus this observation is not really a significant outlier. Using the initial sample without these 11 outliers, the associated MSE is now equal to 0.29, and we have d 2 (E, E) = 0.00367. The benefits of removing these detected outliers is again very clear. Fig. 2 (bottom) provides the plot of the estimated indices x i bSIR versus the y i 's considering the dataset without the detected outliers. The kernel estimation of the link function (in blue) is superimposed on the plot. One can observe the very good fit of the data to the underlying model. Using the BOOT method with B = 3000, 9 out of the ñ = 10 outliers were detected, and 4 "bordeline" observations have been identified. Fig. 3 (right) provides the visualization of the outliers (in blue) and of the "borderline" observations (in orange) on the plot of the y i 's versus the estimated indices x i bSIR . The boxplot on the right allows to detect the outliers while the boxplot in the middle identifies the "borderline" observations. The individual 206 (simulated as an outlier) is here detected as a "bordeline" observation. Note that there is no false positive. Graphics in Fig. 4 • for observation 209 (which was detected as an outlier), the corresponding mean |e (209) | is clearly higher than the previous ones, no matter the number of times that observation is present in the bootstrap sample. Fig. 3 BOOT method description. The two graphics on the left correspond to the boxplots of the log mean absolute errors (defining outliers, in blue) and of the mean absolute errors (defining "borderline" observations, in orange). Selected outliers (in blue) and selected "borderline" observations (in orange) are showed on the plot of the estimated indices x i bSIR versus the y i 's (right graphic), the red points correspond to the ñ (true) potential outliers. The kernel estimations of the link function are superimposed for both the original dataset (in light blue) and the dataset without the detected outliers and " borderline" observations in dark blue).

Using the initial sample without these 9 outliers and 4 "bordeline" observations, the associated MSE is now equal to 0.32 and we have d 2 (E, E) = 0.00172, which highlights the high effectiveness of the BOOT method. Finally, let us remark that, in the computation of the mean descriptors, we chose to consider only the (absolute) error values for which each individual is represented at least once in the bootstrap sample as to prevail from selecting "extreme" observations, as discussed in the comments of the previous two methods.

Simulation results

In this simulation study, N = 100 replications of samples from model ( 6) have been generated with various values of the sample size ñ (= 100, 200, 300), various values of the dimension p (= 5, 20) of the covariate x, and two numbers of potential outliers ñ (= 3, 10). For each generated sample and each outlier detection method (MONO, TTR with R = 2000 and BOOT with B = 2000), the following quantities were calculated:

• the quality of the estimated EDR direction d 2 (E, E) where E is the estimated EDR space based on the complete sample (unique for all the three methods), • the MSE evaluated on the complete sample (unique for all the three methods),

• the number of detected outliers (and the number of "borderline" observations for the BOOT method), • the number of false positives, • the quality of the estimated EDR direction d 2 (E, E ) where E is the estimated EDR space based on the sample without the outliers (and the "borderline" observations) detected by the method , • the MSE evaluated on the sample without the detected outliers (and the "borderline" observations for BOOT method).

To visualize and easily compare all these indicators, boxplots were used. According to results available on Fig. 5, all the three methods allow to reduce the distance to the true model. All methods, and even the model based on the complete dataset (in yellow), naturally perform better if the size of the available sample ( ñ + ñ) increases.

If the number of outliers is ñ = 10, the model based on the complete dataset shows poorer results whatever the number ñ. Note that, for a given number ñ of outliers, the proportion of outliers naturally decreases as the sample size increases. BOOT seems to suffer from a large proportion of outliers only when the sample size is small. Fig. 6 shows the MSE's for all the proposed methods for p = 5 and ñ = 10. Other simulations have been conducted and results are not provided because of Fig. 5 Boxplots the quality measures of the estimated EDR space based on the d 2 (E, E ) values, for simulated datasets with p = 20 and ñ ∈ {3, 10}. "All" stands for the estimation of the EDR using the complete sample. redundancy in the associated comments. Errors are larger for the complete dataset model (in yellow) than for any of the three methods but tend to decrease as ñ increases and thus the proportion of outliers decreases. TTR seems to provide the best results for large sample sizes (and thus for low proportions of outliers), while BOOT shows larger errors, especially when ñ is small (and thus when the proportion of outliers is high). An explanation of the phenomenon is that MSE is computed on the sample without the outliers. In that context, the MONO and TTR methods that select extreme (or "borderline") observations as outliers tend to get smaller MSE. On the contrary, the BOOT approach does not exclude these "borderline" observations which are more difficult to predict correctly, leading to a larger MSE. The MSE descriptor must be interpreted with this remark in mind, as well as by taking into account the number of false positives of each method, which is done thanks to Fig. 7. Whatever the sets of parameters in Fig. 7, BOOT is the only method that seems to be able to select the true outliers without selecting too many false positives (i.e. individuals detected as outliers when they are not). BOOT seems to be the most efficient method by showing the lowest number of false positives for all ñ. The number of false positives stays somewhat constant over the sample size ñ for BOOT but increases with ñ for the other two methods. MONO and TTR methods seem to have a sensibility to ñ with an increase of the numbers of detected outliers and false positives as the sample size increases (and thus as the proportion of outliers decreases since their number ñ is fixed at 10).

A real data application

Daily measurements of meteorological variables and ozone concentration are available in the dataset "ozone" (Source: [START_REF] Cornillon | R for Statistics[END_REF]). More precisely, this dataset contains n = 112 daily measurements of meteorological variables (wind speed, temperature, rainfall, cloudiness) and ozone concentration recorded in Rennes (France) in summer 2001. In this study, an individual is a day. Eleven numerical variables are measured with no missing values:

• maxO3: maximum of daily ozone concentration measured in gr/m 3 ,

• T9, T12, T15: daily temperatures measured in degree Celsius at 9, 12 and 15h (called "temperature" variables hereafter), • Ne9, Ne12, Ne15: cloudiness measured at 9, 12 and 15h (called "cloudiness" variables hereafter), • Vx9, Vx12, Vx15: wind speed (E-W component) measured at 9, 12 and 15h (called "wind" variables hereafter), • maxO3v: maximum concentration of ozone measured the day before.

The initial objective is to explain the maximum of daily ozone concentration (the response variable y is thus maxO3) by the p = 10 variables available (T9, T12, T15, Ne9, Ne12, Ne15, Vx9, Vx12, Vx15, maxO3v). Hereafter, let x be the vector of these ten covariates. To do this, the semiparametric regression model (2), is used and the EDR space E = Span(β) is estimated by the usual SIR method (with the number of slices H = 10) while the link function f is estimated using the kernel estimator with the Gaussian kernel and the bandwidth tuned via cross-validation. Our aim is here to detect the presence or absence of outliers in this dataset. The proposed three outlier detection methods (MONO, TTR with R = 1000 and BOOT with B = 1000) are compared.

The naive MONO method does not detect outliers. The TTR method provides 9 outliers and the BOOT method identify 4 "borderline" observations and no outlier (see the corresponding plots in Fig. 8 respectively at the top left and at the top right). Among the 9 TTR's outliers, 4 of them are the BOOT's "borderline" observations. These 4 observations correspond to specific days in terms of road traffic, since these are days of major departures or returns from summer holidays in France. It is known that ozone pollution is also due to car traffic, but the built model is based only on weather data and does not take into account this important source of pollution. It is therefore quite natural that these 4 days correspond to individuals outside the model's standards. The 5 other specific TTR's outlier observations are closer to the scatterplot structure and they correspond to the days of early June, mid-June (music festival on the first day of summer), late July (end of a week) and mid September.

In order to improve the final model, the method introduced by [START_REF] Jlassi | Variable importance assessment in sliced inverse regression for variable selection[END_REF] for selecting the relevant variables based on variable importance is now applied on the sample without the outliers (TTR method) or the "borderline" observations (BOOT method). Only the following p * = 4 covariates are then selected: a temperature variable, T12, a cloudiness variable, Ne9, a wind variable, Vx9 and the maximum concentration of ozone measured the day before, maxO3v. This is not surprising since the 3 variables of temperature (resp. cloudiness, wind speed) are strongly correlated with each other. The corresponding EDR directions are very close: bTTR SIR = (0.778, -0.565, 0.258, 0.094) and bBOOT SIR = (0.660, -0.724, 0.175, 0.094) . Finally, for the outlier detection method TTR (resp. BOOT), the plot of the estimated indices x i bTTR SIR (resp. x i bBOOT SIR ) versus the y i 's for the corresponding samples without outliers (resp. "borderline" observations) and the associated estimated link function (solid blue curve) are provided in Fig. 8 (at the bottom, on the right, resp. on the right). These two graphics are very similar and show an increasing link between the estimated index and the response variable maxO3. Then, it is possible to interpret the coefficients of the estimated EDR direction bTTR SIR (or similarly bBOOT) SIR using their signs. The variable T12 (resp. Vx9 and maxO3v) has a positive coefficient which means that an increase in daily temperatures at 12h (resp. of the wind speed at 9h, or of maximum concentration of ozone measured the day before) implies an increase of the estimated index and this then implies (not surprisingly) an increase of maximum of daily ozone concentration. On the contrary, the variable Ne09 has a negative coefficient and then an increase of its values leads to a decrease in the maximum of daily ozone concentration, which is relevant from an air pollution point of view.

Concluding remarks and extensions

Three computational outlier detection approaches for sliced inverse regression have been presented. In this work, the original idea is to consider potential outliers that are outliers only in the SIR model and that are not detectable outliers by studying only their distribution in x or y. Thus considering the plot of the estimated indices versus the dependent variable, only outliers can appear in y. The case of outliers in x or in y are not considered here since the corresponding observations should be detectable as outliers in an early stage before the SIR modeling step, and the dataset should then be cleaned up accordingly. The MONO, TTR and BOOT approaches were implemented in R and the code is available on https://github.com/hlorenzo/outlierSIR. The philosophy of these approaches does not rely neither on the SIR method used in the first estimation step nor on the nonparametric regression used in the second estimation step. For example, instead of the usual SIR method, it is possible to use the SIR-II, SIR α or SAVE methods among others. Moreover, the proposed approaches are also easily generalizable to the multiple-index model framework, i.e. when the dimension of the EDR space is equal to K > 1. All SIR-related methods, as well as non-parametric regression methods (like multivariate kernels), work well in this framework . However the non-parametric regression methods might suffer from the well-known curse of dimensionality. Note that the choice of the dimension K of this EDR subspace should be then discussed. Finally, these outlier detection approaches can also be extended to a q-dimensional response variable y. Several authors developed SIR-based methods to estimate the EDR space that is common to the q components of the multivariate response variable, see for instance [START_REF] Barreda | Some extensions of multivariate sliced inverse regression[END_REF][START_REF] Coudret | A new sliced inverse regression method for multivariate response[END_REF][START_REF] Li | Dimension reduction for multivariate response data[END_REF][START_REF] Lue | Sliced inverse regression for multivariate response regression[END_REF][START_REF] Saracco | Asymptotics for pooled marginal slicing estimator based on SIR α approach[END_REF] among others. However, the concept of an outlier in this multivariate framework must be first clarified since it is not entirely natural.

S 1 . 2 . 3 .

 123 Estimation the EDR direction from the sample S. The usual SIR provides the estimate bSIR of b. The corresponding indices { b SIR x i , i = 1, . . . , n} are then calculated. S Estimation of the adjusted value f (β x i )'s.From the sample {( b SIR x i , y i ), i = 1, . . . , n, the adjusted values are obtained via the kernel estimator based on the Gaussian kernel and the bandwidth tuned via cross-validation. Let ŷi = fn ( b SIR x i ) for i = 1, . . . , n. S Evaluation of the error associated with the model estimation and outlier detection.

S 1 .

 1 For each replication r (with r = 1, . . . , R), 1.a. Split the initial sample S into a training sample S (r) train and a test sample S (r) test containing respectively (1 -α)% and α% of the individuals. 1.b. Using S (r) train , calculate the estimated EDR direction b(r) SIR and the associated indices {( b(r) SIR ) x i , i ∈ S (r) train }. 1.c. For all the individuals i * ∈ S (r)

S 1 .

 1 For b = 1, . . . , B, 1.a. Draw a bootstrap sample S (b) from the initial sample S. Let n (b) i denote the number of times the observation i is present in the bootstrap sample S (b) . 1.b. Using S (b) , calculate the corresponding estimated EDR direction b(b) SIR and the associated indices {( b(b) SIR ) x i , i ∈ S (b) }. 1.c. For all the individuals i ∈ S (b) , calculate the IB error of prediction of the response variable y as follows:

Fig. 1

 1 Fig.1MONO method description. Left graphic provides the boxplot of the absolute errors, the detected outliers are in blue. On the plot of the estimated indices x i bSIR versus the y i 's (right graphic), these outliers are also plotted in blue, red points correspond to the ñ (true) potential outliers. The kernel estimations of the link function are superimposed for both the original dataset (in light blue) and the dataset without the detected outliers (in dark blue).

Fig. 2

 2 Fig.2TTR method description. Top-left graphic shows the ordered means errors with the red vertical line providing the estimated single change point position. On the plot of the estimated indices x i bSIR versus the y i 's (top-right graphic), these outliers are also plotted in blue, red points correspond to the ñ (true) potential outliers. The kernel estimations of the link function are superimposed for both the original dataset (in light blue) and the dataset without the detected outliers (in dark blue).

  provide the plot of the n (b) i 's versus the |e (b) i |'s (for b = 1, . . . , B) for three individuals. The horizontal line on each plot represents the corresponding error mean |e (i) | over the B replications (when the individual i was present at least once in the corresponding bootstrap sample). One can observe that: • for observation 1 (which is a "normal" observation), the corresponding mean |e (1) | is low, • for observation 21 (which is characterized as a "borderline" observation"), the corresponding mean |e (21) | is intermediate. The model learns its position and modifies its tail, which explains the fall in error between n (b) (21) = 0 and n (b) (21) = 1,

Fig. 4

 4 Fig. 4 For the BOOT method, plots of the |e (b) (i) |'s versus the n (b) i 's (for b = 1, . . . , B) for three individuals: a "normal" individual (i = 1) on the left, a "borderline" individual (i = 21) in the middle, and an outlier (i = 209) on the right. Colored (resp. black, orange and blue) segments show their corresponding computed IB error means |e (i) | (for n (b) i ≥ 1). The last plot at the bottom provides a density estimation of the |e (i) |'s with these three individuals showed through colored vertical lines. Since the OOB errors (for n (b) i = 0) are not used, individual i = 21 (in orange) is not considered as outlier but as "borderline" observation.

Fig. 6

 6 Fig.[START_REF] Chavent | A sliced inverse regression approach for data stream[END_REF] Root MSE for the different methods. Visualizations of the errors for the complete dataset model (in yellow) and for the three outlier detection methods have been split in two graphics since scales are different. Here p = 5 and ñ = 10 have been detailed.

Fig. 7

 7 Fig. 7 Number of detected outliers (left column) and number of false positive detections (right column) for different values of p, ñ and ñ = 10.

Fig. 8

 8 Fig. 8 Study of Ozone dataset. Top-Left: Selected outliers with TTR method. Top-Right: Selected "borderline" observations with BOOT method. Bottom-Left: plot of the y i 's (values of maxO3) versus the final estimated indices based on the n * TTR = n -9 observations, i.e. removing the 9 selected outliers. Bottom-Right: plot of the y i 's (values of maxO3) versus the final estimated indices based on the n * BOOT = n -4, removing the 4 selected "borderline" observations. The corresponding estimated link functions (solid blue curve) are superimposed on the last two plots.

Acknowledgements Jérôme Saracco would like to sincerely thank Prof. Christine Thomas-Agnan for having "brought back in his luggage" the SIR method in Toulouse in the early 90s. His first research focused on contributions to SIR (master's thesis, doctoral thesis, first articles in international journals). Thank you for all the discussions I had with Christine throughout my career on many scientific subjects (non-parametric estimation, conditional quantiles, etc.) and many other subjects.


Keywords: Fluorine atom, Methane, Kinetics, Rate constant, Activation energy

Reaction F + CH 4  HF + CH 3 (1) has been studied using a discharge flow reactor combined with mass spectrometry in the temperature range 220 -960 K. The rate constant of the reaction was determined using both absolute and relative rate measurements: k 1 = (1.28 ± 0.04) × 10 -10 exp(-(219 ± 9)/T) cm 3 molecule -1 s -1 (the uncertainties represent precision at the 2 level, the estimated total uncertainty on k 1 being 15% at all temperatures). The kinetic data from the present work are discussed in comparison with previous measurements and theoretical calculations.

Introduction

This study reports an experimental investigation of the kinetics of the reaction of fluorine atoms with CH 4 : F + CH 4  HF + CH 3 [START_REF] Wallington | Hydrofluorocarbons and stratospheric ozone[END_REF] The role of this reaction in atmospheric chemistry consists in converting F atoms to stable species HF, which is removed from the stratosphere via deposition/rain-out. This is one of the main reasons why catalytic ozone destroying cycles involving F atoms are not efficient [START_REF] Wallington | Hydrofluorocarbons and stratospheric ozone[END_REF][START_REF] Ravishankara | Do Hydrofluorocarbons Destroy Stratospheric Ozone?[END_REF]. In the laboratory studies, reaction [START_REF] Wallington | Hydrofluorocarbons and stratospheric ozone[END_REF] is frequently used as a source of CH 3 radical, which is an important intermediate in atmospheric and combustion chemistry.

Consequently, the information on the kinetics of the reaction (1) in an extended temperature range covering atmospheric temperatures and those under combustion conditions is required. In addition, reaction of F atoms with CH 4 is very often used as a reference one in relative rate kinetic measurements. According to the NIST Chemical Kinetics Database [START_REF] Manion | NIST Chemical Kinetics Database[END_REF], the rate constants of the F-atom reactions with about 50 different molecules

were measured relative to the rate constant of reaction [START_REF] Wallington | Hydrofluorocarbons and stratospheric ozone[END_REF]. Obviously, accurate data on the rate constant of the reference reaction is essential to place the relative measurements on an absolute basis.

Reaction of F atoms with CH 4 , being a prototypical example of a polyatomic reaction, was a subject of numerous theoretical and dynamic studies ( [START_REF] Wang | Quantum Dynamics Study of the F + CH 4 → HF + CH 3 Reaction on an Ab Initio Potential Energy Surface[END_REF][START_REF] Espinosa-García | New Analytical Potential Energy Surface for the F( 2 P) + CH 4 Hydrogen Abstraction Reaction: Kinetics and Dynamics[END_REF] and refs therein).

Surprisingly, despite the considerable practical and theoretical interest in this reaction, the reaction rate constant is not well established, particularly in terms of its temperature dependence. Only three measurements of the temperature dependence of the reaction rate constant are available in the literature [START_REF] Wagner | Reaction of F Atoms with Methane[END_REF][START_REF] Moore | Rates of Processes Initiated by Pulsed Laser Production of F Atoms in the Presence of HCl, CH 4 , And CF 3 H[END_REF][START_REF] Persky | Kinetics of the F + CH 4 Reaction in the Temperature Range 184-406 K[END_REF]. Two of them reported significantly different activation energies (1.15±0.35 at T = 253-352 K [START_REF] Wagner | Reaction of F Atoms with Methane[END_REF] and 0.43±0.12 kcal mol -1 at T = 184-406 K [START_REF] Persky | Kinetics of the F + CH 4 Reaction in the Temperature Range 184-406 K[END_REF], updated in [START_REF] Persky | Kinetics of the Reactions F+H 2 S and F+D 2 S at 298 K[END_REF]), while in the third one (T = 139-294 K), a negative temperature dependence of k 1 was observed [START_REF] Moore | Rates of Processes Initiated by Pulsed Laser Production of F Atoms in the Presence of HCl, CH 4 , And CF 3 H[END_REF]. There are no experimental data for the reaction rate constant at temperatures above 406 K.

The purpose of the present work was to determine the rate constant of the title reaction in a wide temperature range, T = 220-960 K, in order to reduce the existing uncertainties on k 1 and provide a new extended and reliable data set for practical use and an experimental basis for validation and refining theoretical calculations.

Experimental

The title reaction was studied in a discharge-flow system, using modulated molecular beam mass spectrometry with electron impact ionization (25-30 eV energy) to monitor the species involved (Figs. S1a-S1c, Supplementary Material). The apparatus and experimental approach were similar to those used in previous studies of the elementary reactions of fluorine atoms in this group [START_REF] Bedjanian | Temperature-Dependent Rate Constant for the Reaction of F Atoms with HNO 3[END_REF][START_REF] Bedjanian | Reaction F + C 2 H 4 : Rate Constant and Yields of the Reaction Products as a Function of Temperature over 298-950 K[END_REF][START_REF] Bedjanian | Kinetics and Products of the Reactions of Fluorine Atoms with ClNO and Br 2 from 295 to 950 K[END_REF][START_REF] Bedjanian | Kinetic and Mechanistic Study of the F Atom Reaction with Nitrous Acid[END_REF]. Two flow reactors were used in this study. The first one, used at low temperatures (220 -326 K), was Halocarbon wax-coated Pyrex tube (45 cm length and 2.4 cm i.d.) surrounded by a jacket for circulation of the thermostated fluid (water or ethanol). The reactor used at high temperatures (T = 299 -960 K) was an electrically heated Quartz tube (45 cm length and 2.5 cm i.d.) [START_REF] Morin | Experimental Study of the Reactions of OH Radicals with Propane, n-Pentane, and n-Heptane over a Wide Temperature Range[END_REF].

Fluorine atoms were produced in the microwave discharge of trace amounts of F 2 in He in an alumina tube (more than 90% of F 2 , monitored by mass spectrometry, was found to be dissociated). F-atoms were detected indirectly at m/z = 98/100 (FBr + ) by using their chemical conversion in a rapid reaction with an excess of Br 2 (added 6 cm upstream of the sampling cone of the mass spectrometer, Figs. S1a-S1c):

F + Br 2  Br + FBr (2)
k 2 = (1.28 ± 0.20)  10 -10 cm 3 molecule -1 s -1 (T = 299 -940 K) [START_REF] Bedjanian | Kinetics and Products of the Reactions of Fluorine Atoms with ClNO and Br 2 from 295 to 950 K[END_REF] This reaction was also used for the determination of the absolute concentrations of F-atoms The purities of the gases used were as follows: He >99.9995% (Alphagaz), passed through liquid nitrogen trap; CH 4 ≥ 99.995% (Alphagaz); F 2 , 5% in helium (Alphagaz); Br 2 >99.99% (Aldrich); D 2 >99.7% (Alphagaz).

Results and Discussion

Rate constant of reaction (1).

Rate constant of the title reaction was determined using two different methods: absolute one, where k 1 was measured from F-atom decays monitored in an excess of CH 4 , and a relative rate method employing reactions of F atoms with Br 2 and D 2 as a reference. Values of k 1 ' obtained from these plots were corrected for axial and radial diffusion of Fatom in the carrier gas (corrections generally less than 10%) [START_REF] Kaufman | Kinetics of Elementary Radical Reactions in the Gas Phase[END_REF]. In Figs. 1 andS3 (Supplementary Material), the pseudo-first order rate constants observed at different temperatures in the reactor are plotted against concentration of CH 4 . 1. The combined uncertainty on the measurements of the rate constants was estimated to be ≤ 15 %, including (in quadrature) statistical error (within a few percent, Table 1) and those on the measurements of the flows (5%), pressure (3%), temperature (1%) and absolute concentration of CH 4 (< 10%).

Absolute measurements of k

T (K) (a) [CH 4 ] (10 12 molecule cm -3 ) k 1 (± 2) (10 -11 cm 3 molecule -1 s -1 )
Reactor surface One can note (Fig. 1 and Table 1) that similar, within experimental uncertainty, data were obtained at room temperature (297-299 K) in two reactors with different surfaces (Quartz and Halocarbon wax coating). In general, in this work, we did not observe any signs of heterogeneous complications that often accompany kinetic measurements in flow reactors. The potential impact on the measurements of k 1 of the secondary reaction,

F + CH 3  products (3),
is expected to be insignificant, considering the relatively low initial concentrations of fluorine atoms used in the experiments. This seems to be supported by the experimental evidence of the invariance of k 1 ' with respect to [F] 0 , observed at T = 280 K (Fig. S3, Supplementary Material).

Relative measurements of k 1 with reaction F+Br 2 as a reference. These measurements were carried out in an uncoated quartz reactor (Fig. S1b) at three temperatures: T = 310, 400

and 660 K and consisted in a monitoring of FBr formed upon fast titration of F atoms by a mixture of CH 4 and Br 2 :

F + CH 4  HF + CH 3 (1)
F + Br 2  FBr + Br (2)
The fraction of the initial concentration of F atoms, [F] 0 , transformed to FBr in reaction 2, is:

FBr F
Rearrangement of this expression provides: Reaction time was around 0.02 s, [Br 2 ] = (0.9-1.1)×10 13 molecule cm -3 , the ranges of concentrations of CH 4 are given in Table 2.

1 CH 4 (

Table 2

Experimental conditions and results of the relative measurements of k 1 with reaction F+Br 2 as a reference. The experimental data are shown in Fig. 2. In accordance with Equation I, the slopes of the straight lines in Fig. 2 provide the values of k 1 /k 2 at respective temperatures. Final values of k 1 (Table 2) were calculated with k 2 = (1.28 ± 0.20)  10 -10 cm 3 molecule -1 s -1 (T = 299 -940 K), determined in a recent study from this group [START_REF] Bedjanian | Kinetics and Products of the Reactions of Fluorine Atoms with ClNO and Br 2 from 295 to 950 K[END_REF]. Relative measurements of k 1 with reaction F+D 2 as a reference. The procedure, applied in these experiments, was similar to that described above. The fraction of the initial concentration of F atoms, [F] 0 , transformed to DF, is:

T (K) [CH 4 ] (a) [CH 4 ]/[Br 2 ] k 1 /k 2 (± 2) k 1 (b) 310
DF F ,
or after rearrangement:

1 CH 4 (II)
The observed experimental data are shown in Fig. 3. The final values of k 1 /k 4 at different temperatures, determined, in accordance with equation II, from the slopes of the linear dependencies in Fig. 3, are given in Table 3.

Table 3

Experimental conditions and results of the relative measurements of k 1 with reaction F+D 2 as a reference. c Units of 10 -11 cm 3 molecule -1 s -1 , estimated total uncertainty on k 1 is nearly 20% and includes that on the reference reaction.

T (K) (a) [CH 4 ] (b) [D 2 ] (b) k 1 /k 4 (± 2) k 1 (c)

Comparison with previous studies.

The results obtained in the present work for the temperature dependence of k 1 are shown in Fig. 4 in line with all the temperature dependent data for k 1 available in the literature. The most recent room temperature measurements of the rate constant [START_REF] Wörsdörfer | Bimolecular Reactions of Fluorine Atoms with CH 3 I and CH 2 I 2[END_REF][START_REF] Pagsberg | UV Spectrum and Kinetics of Hydroxymethyl Radicals[END_REF][START_REF] Clyne | Kinetics and Detection Of F( 2 P) Atoms in a Discharge Flow System[END_REF][START_REF] Fasano | Rate Determination for F + CH 4 by Real-Time Competitive Kinetics[END_REF] are in the range (5.7-6.6)10 -11 cm 3 molecule -1 s -1 (T = 298 K), i.e. in very good agreement with the present data (Table 1 and Fig. 4). Wagner et al. [START_REF] Wagner | Reaction of F Atoms with Methane[END_REF] used mass spectrometry to monitor the species involved in the F + CH 4 reaction system, where F atoms were generated in the reaction of NF 2 with excess nitrogen atoms. The rate constant of reaction (1) was determined at three temperatures (253, 286 and 352 K) both from F atom decays and from the kinetics of HF and HCN formed in reactions ( 1) and ( 5), respectively:

F + CH 4  HF + CH 3 (1)
N + CH 3  HCN + products (5)
The original rate constant data shown in Fig. 4 were kindly provided by Cornelius Zetzsch.

Although these largely scattered data show a steeper temperature dependence, in absolute terms they agree well with the present work, at least at T = 253 and 286 K. Moore and Smith [START_REF] Moore | Rates of Processes Initiated by Pulsed Laser Production of F Atoms in the Presence of HCl, CH 4 , And CF 3 H[END_REF] derived the rate constant of reaction (1) by monitoring the chemiluminescence of the reaction product, HF, formed upon pulsed laser photolysis of F 2 in the presence of CH 4 . The measurements were carried out at four temperatures (294, 218, 179, and 139 K) and shown rather negative temperature dependence of the rate constant: the reported values of k 1 were (5.3± 0.3)10 -11 at T = 294 and 218 K increasing to 7.4 and 8.010 -11 cm 3 molecule -1 s -1 at T = 179 and 139 K (not shown in Fig. 4), respectively. The k 1 data at T = 294 and 218 K agree with the present measurements in the range of the reported experimental uncertainties (Fig. 4). One can note an excellent agreement between the present data and those reported by Persky [START_REF] Persky | Kinetics of the F + CH 4 Reaction in the Temperature Range 184-406 K[END_REF][START_REF] Persky | Kinetics of the Reactions F+H 2 S and F+D 2 S at 298 K[END_REF]. The comparison could be limited to this statement; however, further analysis shows that the perfect agreement between two studies seems to be fortuitous. Persky [START_REF] Persky | Kinetics of the F + CH 4 Reaction in the Temperature Range 184-406 K[END_REF] using a discharge-flow-mass spectrometric technique measured the temperature dependence of k 1 by a relative rate method using reaction (4) as a reference one. To place the rate constant of reaction (1) on an absolute basis, a rather complicated approach was applied. First, the measured k 1 /k 4 ratios were combined with the data available for the kinetic isotope effect, k 6 /k 4 , to calculate the k 1 /k 6 ratios, which were subsequently multiplied by the recommended on that date values for k 6 to finally determine the absolute values of

k 1 : F + H 2  HF + H (6)
In a recent study from this group, we reported a new self-consistent kinetic dataset for the reactions of F atoms with H 2 and D 2 [START_REF] Bedjanian | Rate Constants for the Reactions of F Atoms with H 2 and D 2 over the Temperature Range 220-960 K[END_REF]. The rate constants of the reactions of F atoms with H 2 and D 2 have been directly measured in a wide temperature range T = 220 -960 K. In addition, the kinetic isotope effect, which was determined independently, was found to be in excellent agreement (within a few percent) with the ratio of k 1 to k 2 calculated with the Arrhenius expressions for k 1 and k 2 determined in the study [START_REF] Bedjanian | Rate Constants for the Reactions of F Atoms with H 2 and D 2 over the Temperature Range 220-960 K[END_REF]. The original data of Persky on k 1 /k 4 ratio [START_REF] Persky | Kinetics of the F + CH 4 Reaction in the Temperature Range 184-406 K[END_REF], when combined with k 4 = (8.47± 0.42)10 -11 exp(-(670 ± 16)/T) cm 3 molecule -1 s -1 [START_REF] Bedjanian | Rate Constants for the Reactions of F Atoms with H 2 and D 2 over the Temperature Range 220-960 K[END_REF], provide the absolute values of k 1 (gray filled squares in Fig. 4) which are by a factor of 1.3-1.4 lower than the current measurements. In fact, one of the reasons for the seemingly good agreement between the present data for k 1 and the data reported by Persky [START_REF] Persky | Kinetics of the F + CH 4 Reaction in the Temperature Range 184-406 K[END_REF][START_REF] Persky | Kinetics of the Reactions F+H 2 S and F+D 2 S at 298 K[END_REF], where the reaction rate constant was calculated as Figure 4 shows also the results of some theoretical calculations of k 1 . Espinosa-Garcia et al. [START_REF] Espinosa-García | New Analytical Potential Energy Surface for the F( 2 P) + CH 4 Hydrogen Abstraction Reaction: Kinetics and Dynamics[END_REF] developed a semiempirical potential energy surface (PES) for the F + CH 4 reaction and using variational transition-state theory calculated the thermal rate constants over the temperature range 180-500 K. The calculations are in very good agreement with experiments. However, it should be noted that the developed PES was calibrated to reproduce the experimental data of Persky [START_REF] Persky | Kinetics of the F + CH 4 Reaction in the Temperature Range 184-406 K[END_REF][START_REF] Persky | Kinetics of the Reactions F+H 2 S and F+D 2 S at 298 K[END_REF]. The calculations of Wang and Czakó [START_REF] Wang | Quantum Dynamics Study of the F + CH 4 → HF + CH 3 Reaction on an Ab Initio Potential Energy Surface[END_REF],

k 1 = k 1 /k 4 × k 4 /k 6 × k 6 ,
realized on an ab initio potential energy surface, are also in fair agreement with the experimental data. Both theoretical calculations predict some curvature in the Arrhenius plot. Although this effect was not clearly observed in the current experiments, the present data do not exclude the possible, however, statistically insignificant in the range of the experimental uncertainty, curvature of the temperature dependence of k 1 .

In conclusion, the rate constant of the title reaction was measured in a wide temperature range, T = 220 -960 K (for the first time at temperatures above 406 K). The measurements provide a new reliable dataset for practical use (atmospheric modeling, laboratory studies) and an experimental basis for further theoretical developments.

  considering their stoichiometric conversion to FBr in excess of Br 2 : [F] = [FBr] = [Br 2 ]. The absolute concentrations of all stable species (CH 4 , Br 2 , F 2 , D 2 ) in the reactor were calculated from their flow rates obtained from the measurements of the pressure drop of their mixtures in He stored in reservoirs of known volume.

1 .

 1 Both reactants were introduced into the reactor through a movable injector: F-atoms through Halocarbon wax-coated central tube and CH 4 through external output of the injector as shown in Fig. S1a. For all experiments, pseudo-first-order conditions were maintained with CH 4 in excess over fluorine atoms. Initial concentration of fluorine atoms in these experiments was in the range (1 -4)×10 11 molecule cm -3 ; the flow velocity in the reactor was in the range (2145-2965) cm s -1 . The F-atom decays followed equation [F] = [F] 0 ×exp(-k 1 '×t), where [F] and [F] 0 are F-atom concentrations at time t and zero and k 1 ' = k 1 ×[CH 4 ]+k w is the pseudo-first-order rate constant with k w representing the heterogeneous loss of F atoms. Some typical temporal profiles of F-atoms measured at T = 440 K with different concentrations of CH 4 are shown in Fig. S2 (Supplementary Material).

Fig. 1 .

 1 Fig. 1. Pseudo-first order rate constant, k 1 ' = k 1 [CH 4 ]+k w , as a function of the CH 4 concentration at different temperatures.

k 1 ' (s - 1 )

 11 Consumption of the excess reactant, CH 4 , although insignificant, was taken into account through use of the average concentration of CH 4 in the reaction zone, which was calculatedas [CH 4 ]=[CH 4 ] 0 -[F]×(k 1 '-k w )/k 1 '. The corrections on the methane concentration were quite small (within a few percent) in the most kinetic runs, however reached up to 15% in a few experiments with the lowest concentrations of CH 4 . The slopes of the straight lines in Figs. 1 and S3 provide the values of k 1 at respective temperatures. All the results of the absolute measurements of k 1 at different temperatures are shown in Table

  I) At a constant concentration of Br 2 , k 1 /k 2 can be determined as a slope of the linear dependence of ([F] 0 /[FBr] -1) on the [CH 4 ]/[Br 2 ] ratio. The measurements of the absolute concentrations of F atoms were not necessary in these experiments, since FBr was monitored in the presence of Br 2 in both CH 4 -free system, corresponding to [F] 0 , and in the presence of CH 4 in the reactor, corresponding to the fraction of [F] 0 reacted with Br 2 .

Fig. 2 .

 2 Fig. 2. Yield of FBr from titration of F atoms by Br 2 + CH 4 mixtures at different temperatures. Partially shown error bars represent typical uncertainties of the measurements ( 5%).

1 )F

 1 F atoms were titrated by a mixture of CH 4 and D 2 (Fig. S1c) and the yield of DF (monitored at m/z = 21, DF + )was measured as a function of the [CH 4 ]/[D 2 ] ratio: F + CH 4  HF + CH 3 (

Fig. 3 .

 3 Fig. 3. Yield of DF from titration of F atoms by D 2 + CH 4 mixtures at different temperatures. Partially shown error bars represent typical uncertainties of the measurements ( 5%).

Fig. 4 .

 4 Fig. 4. Summary of the data on the temperature dependence of the rate constant of reaction (1): AM, absolute measurements; RM/Br 2 and RM/D 2 , relative rate method with F+Br 2 and F+D 2

  can be the compensating effect of the first two terms, k 1 /k 4 and k 4 /k 6 . Indeed, the k 1 /k 4 ratios measured in the present work, k 1 /k 4 = 1.47×exp(454/T) (T = 230-720 K), are by approximately 20% higher than those reported by Persky, k 1 /k 4 = 1.21×exp(420/T) (T = 184-406 K)[START_REF] Persky | Kinetics of the F + CH 4 Reaction in the Temperature Range 184-406 K[END_REF] (Fig.S4, Supplementary Material). On the other hand, the values of k 4 /k 6 = 0.96×exp(-186/T) (T = 184-406 K)[START_REF] Persky | Kinetic Isotope Effects in the Reaction of Fluorine Atoms with Molecular Hydrogen. I H 2 /D 2 Kinetic Isotope Effect[END_REF], used by Persky in the calculations of k 1[START_REF] Persky | Kinetics of the F + CH 4 Reaction in the Temperature Range 184-406 K[END_REF][START_REF] Persky | Kinetics of the Reactions F+H 2 S and F+D 2 S at 298 K[END_REF] are significantly higher than recently reported, k 4 /k 6 = 0.63×exp(-131/T) (T = 220-960 K)[START_REF] Bedjanian | Rate Constants for the Reactions of F Atoms with H 2 and D 2 over the Temperature Range 220-960 K[END_REF].Concerning the results of the current study, one can note an excellent agreement between the absolute and relative (with two reference reactions) measurements of k 1 . The unweighted exponential fit to the present data (solid line in Fig.4) yields the following Arrhenius expression: k 1 = (1.28 ± 0.04) × 10 -10 exp(-(219 ± 9)/T) cm 3 molecule -1 s -1 at T = 220-960 K, where the cited uncertainties are 2 statistical ones.

Table 1

 1 Experimental Conditions and Results of the Absolute Measurements of the Rate Constant of Reaction (1).

  Units of 10 -11 cm 3 molecule -1 s -1 , estimated total uncertainty on k 1 is nearly 20% and includes that on the reference reaction.

		0.19-5.61	0.19-5.19	0.496±0.007	6.34
	400	0.18-4.41	0.19-4.75	0.561±0.006	7.18
	660	0.20-3.64	0.22-4.17	0.741±0.005	9.49

a Units of 10 13 molecule cm -3 . b

Acknowledgement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

This work (AM) / 220-960 This work (RM/Br 2 ) / 310-660 This work (RM/D 2 ) / 230-720 Espinosa-Garcia et al. [5] / 180-500 / Theory Wang & Czakó [4] / 184-406 / Theory

Declaration of interest statement

No conflicts of Interest.

Appendix A. Supplementary data

Supplementary data related to this article can be found at ...


Yuri Bedjanian
email: yuri.bedjanian@cnrs-orleans.fr

Rate Constant of the Reaction of OH radicals with HBr over the Temperature Range 235-960 K

The kinetics of the reaction of hydroxyl radical with HBr, important in atmospheric and combustion chemistry, has been studied in a discharge flow reactor combined with an electron impact ionization quadrupole mass spectrometer in the temperature range 235 -960 K. The rate constant of the reaction OH + HBr  H 2 O + Br (1) was determined using both relative rate method (using reaction of OH with Br 2 as a reference) and absolute measurements, monitoring the kinetics of OH consumption under pseudo-first order conditions in excess of HBr. The observed U-shaped temperature dependence of k 1 is well represented by the sum of two exponential functions: k 1 = 2.53×10 -11 exp(-364/T) + 2.79×10 -13 exp(784/T) cm 3 molecule - 1 s -1 (with estimated conservative uncertainty of 15% at all temperatures). This expression for k 1 recommended for T = 240-960 K combined with that from previous low temperature studies, k 1 = 1.06×10 -11 (T/298) -0.9 cm 3 molecule -1 s -1 at T = 23-240 K, allows to describe the temperature behavior of the rate constant over an extended temperature range 23-960 K. The current direct measurements of k 1 at temperatures above 500 K, the only ones to date, provide an experimental dataset for use in combustion and volcanic plume modelling and an experimental basis to test theoretical calculations.

INTRODUCTION

Reaction of OH radicals with HBr OH + HBr

 H 2 O + Br (1) is an important atmospheric reaction playing a central role in bromine partitioning between its active forms and reservoirs through transformation of the reservoir species HBr to Br-atom. [START_REF] Wayne | Chemistry of Atmospheres[END_REF] Reaction (1) has been extensively studied, both experimentally and theoretically, over the past decades. Currently, for the temperature range 230-360 K, there is a consensus on the slight negative temperature dependence and on the absolute value of the reaction rate constant (k 1 ). 2- 5 Kinetic studies carried out at ultra-low temperatures (down to 23 K), although somewhat scattered, have revealed a strong inverse temperature dependence of k 1 below 200 K. [START_REF] Mullen | Temperature Dependence and Kinetic Isotope Effects for the Oh + b Reaction and /D Isotopic Va iants at Low Tempe atu es (53-135 K) Measu ed Using a Pulsed Supersonic Laval Nozzle Flow Reactor[END_REF][START_REF] Jaramillo | A Consensus View of the Temperature Dependence of the Gas Phase Reaction: Oh + b → H2o + Br[END_REF][START_REF] Jaramillo | Temperature-Dependent Kinetic Isotope Effects in the Gas-Phase Reaction: Oh + b[END_REF][START_REF] Atkinson | Low-Temperature Kinetic Behavior of the imolecula Reaction Oh + b (76-242 K)[END_REF][START_REF] Sims | Ultra-Low Temperature Kinetics of Neutral-Neutral Reactions: New Experimental and Theoretical Results for Oh+Hbr between 295 and 23 K[END_REF] Reaction ( 1) is also important in combustion chemistry being an important step in flame inhibition mechanisms when using bromine-containing compounds as flame-retardants. [START_REF] Burgess | A Chemical Kinetic Mechanism for 2-Bromo-3,3,3-Trifluoropropene (2-Btp) Flame Inhibition[END_REF][START_REF] Covaci | Novel Brominated Flame Retardants: A Review of Their Analysis, Environmental Fate and Behaviour[END_REF][START_REF] Westbrook | Numerical Modeling of Flame Inhibition by Cf3br[END_REF] Another potential implication of the reaction at high temperatures is its involvement in the hot near-source volcanic plume chemistry. [START_REF] Roberts | Reaction Rates Control High-Temperature Chemistry of Volcanic Gases in Air[END_REF] Surprisingly, despite the potential practical interest in the reaction at high temperatures, there are no experimental data for the reaction rate constant at temperatures higher than 460 K. Recent theoretical work on the OH + HBr reaction was also mainly focused on temperatures below 500 K, refs and only a few evaluations [START_REF] Burgess | A Chemical Kinetic Mechanism for 2-Bromo-3,3,3-Trifluoropropene (2-Btp) Flame Inhibition[END_REF] and theoretical efforts [START_REF] Góger | Flame Inhibition Chemistry: Rate Coefficients of the Reactions of Hbr with Ch3 and Oh Radicals at High Temperatures Determined by Quasiclassical Trajectory Calculations[END_REF][START_REF] Ree | Dependence of the Four-Atom Reaction b + Oh → Br + H2o on Temperatures between 20 and 2000 K[END_REF] have been undertaken to define the reaction rate constant at elevated temperatures. Although all these works predict a similar small positive temperature dependence of k 1 at T > 500 K, the absolute values of k 1 differ by more than a factor of 2.

The objective of the present study consisted in the measurements of the rate constant of reaction (1) in the temperature range extended to ca. 1000 K in order to provide new experimental data for combustion modelling and validation of the theoretical calculations.

EXPERIMENTAL

Kinetic measurements have been conducted at nearly 2 Torr total pressure of Helium using a conventional fast-flow reactor combined with a molecular beam sampling electron-impact ionization mass spectrometer operated at 25-30 eV energy. [START_REF] Bedjanian | Temperature-Dependent Kinetic Study of the Reaction of Hydroxyl Radicals with Hydroxyacetone[END_REF][START_REF] Bedjanian | Temperature-Dependent Rate Constant for the Reaction of Hydroxyl Radical with 3-Hydroxy-3-Methyl-2-Butanone[END_REF][START_REF] Morin | Experimental Study of the Reactions of Oh Radicals with Propane, N-Pentane, and N-Heptane over a Wide Temperature Range[END_REF] Two different flow reactors were used depending on the temperature of the measurements. High temperature reactor, used at T = 320 -960 K, consisted of an electrically heated Quartz tube (45 cm length and 2.5 cm i.d.) with water-cooled attachments (Figure S1a, Supporting Information (SI)). [START_REF] Morin | Experimental Study of the Reactions of Oh Radicals with Propane, N-Pentane, and N-Heptane over a Wide Temperature Range[END_REF] Low temperature reactor (Figure S1b, SI), used at T = 235 -325 K, consisted of a Pyrex tube (45 cm length and 2.4 cm i.d.) surrounded by a jacket through which a thermostated ethanol was circulated. The internal surface of the low temperature reactor was coated with Halocarbon wax in order to reduce the heterogeneous loss of OH radicals.

OH radicals were produced inside the movable injector in a rapid reaction of hydrogen atoms with excess NO 2 ([NO 2 ] = (1.0-1.5)×10 13 molecule cm -3 ):

H + NO 2  OH + NO (2)
k 2 = (1.47 ± 0.26)  10 -10 cm 3 molecule -1 s -1 (T = 195 -2000). [START_REF] Su | Rate Constants, 1100 ≤ T ≤ 2000 K, fo + No2 → Oh + No Using Two Shock Tube Techniques: Compa ison of Theo y to Expe iment †[END_REF] H atoms, in turn, were generated from the microwave discharge in highly diluted H 2 /He mixtures (Figures S1a andS1b). Prior to their sampling to the mass spectrometer, hydroxyl radicals were converted to HOBr in the fast reaction with Br 2 ([Br 2 ] = (4 -6)×10 13 molecule cm -3 , added in the end of the reactor 6 cm upstream of the sampling cone as shown in Figure S1b), OH + Br 2  HOBr + Br (3) k 3 = 2.1610 -11 exp(207/T) cm 3 molecule -1 s -1 (T = 220 -950), [START_REF] Bedjanian | Temperature-Dependent Rate Constant for the Reaction of Hydroxyl Radical with 3-Hydroxy-3-Methyl-2-Butanone[END_REF] and were finally detected at m/z = 96/98 (HOBr + ). Reaction . [START_REF] Bedjanian | Kinetic Study of the Reactions of Br 2 with Oh and Od[END_REF][START_REF] Bedjanian | Kinetic Study of Oh + Oh and Od + Od Reactions[END_REF] HBr was transported to the reactor by flow of the bath gas (He) over HBr solution in water. The absolute calibration of the mass spectrometer for HBr was realized using chemical conversion of H-atom to HBr in reaction with excess Br 2 :

H + Br 2  HBr + Br (4)
In this way, the concentration of HBr formed in reaction (4) was related to the fraction of Br 2 consumed. The absolute concentrations of other stable species (H 2 , NO 2 , Br 2 ) in the reactor were calculated from their flow rates obtained from the measurements of the pressure drop in their manometrically prepared mixtures with He stored in calibrated volume flasks. All species were detected by mass spectrometry at their parent peaks: m/z= 160 (Br 2 + ), 96/98 (HOBr + ), 46 (NO 2 + ), 80/82 (HBr + ).

The purities of the gases used were as follows: He >99.9995% (Alphagaz), passed through liquid nitrogen trap; H 2 > 99.998% (Alphagaz); Br 2 >99.99% (Aldrich); NO 2 > 99% (Alphagaz); HBr, 48 wt. % in H 2 O > 99.99% (Sigma-Aldrich). The concentration of Br 2 impurity in HBr, measured by mass spectrometry, was less than 0.1%.

RESULTS AND DISCUSSION

Absolute Measurements of k 1 .

The absolute measurements of k 1 were carried out under pseudo-first order conditions monitoring consumption of hydroxyl radicals ([OH] 0 = (1 -3)×10 11 molecule cm -3 ) in excess of HBr (the ranges of the initial concentrations of HBr used are shown in Table 1). All the results of the absolute measurements of k 1 at different temperatures are shown in Table 1. The combined uncertainty on the measurements of the rate constants was estimated to be of nearly 15 %, including (in quadrature) statistical error (within a few percent, Table 1) and those on the measurements of the flows (5%), pressure (3%), temperature (1%) and absolute concentration of HBr (≤ 10%).

Relative Measurements

3.3.

Comparison with Previous Data. Figure 3 summarizes the results of the present measurements of k 1 in comparison with those from three previous temperature dependence studies realized in overlapping temperature ranges. [START_REF] Bedjanian | Kinetic Study of the Reactions of Oh and Od with Hbr and Dbr[END_REF][START_REF] Ravishankara | Absolute Rate Constant for the Reaction Oh + b → 2o +[END_REF][START_REF] Khamaganov | Study of the Reactions of Oh with Hcl, Hbr, and Hi between 298 K and 460 K[END_REF] Concerning the results from this study, one can note an excellent agreement between absolute and relative measurements of the reaction rate constant. In the range of the reported uncertainties on the measurements of k 1 , the current data (T=235-960 K) match very well those from previous study from this group (T=230-360 K), [START_REF] Bedjanian | Kinetic Study of the Reactions of Oh and Od with Hbr and Dbr[END_REF] as well as the laser flash photolysis-resonance fluorescence measurements of Ravishankara et al. (T=249-416 K) [START_REF] Ravishankara | Absolute Rate Constant for the Reaction Oh + b → 2o +[END_REF] and recently reported results of discharge flow-electron paramagnetic resonance study by Khamaganov et al. (T = 298, 370 and 460 K). [START_REF] Khamaganov | Study of the Reactions of Oh with Hcl, Hbr, and Hi between 298 K and 460 K[END_REF] Three most recent room temperature measurements of the reaction rate constant (reporting the values of k 1 between 0.92 and 1.12×10 -11 cm 3 molecule -1 s -1 ), [START_REF] Jourdain | Epr Determination of Absolute Rate Constants for the Reactions of H and Oh Radicals with Hydrogen Bromide[END_REF][START_REF] Cannon | A Time-Resolved Lif Study of the Kinetics of Oh(Ν = 0) and Oh(Ν = 1) with cl and b[END_REF][START_REF] Ravishankara | The Oh+Hbr Reaction Revisited[END_REF] in turn, are in good agreement with the results of the temperature dependent studies. Solid line in Figure 3 represents the fit to the present data with sum of two exponential functions: k 1 = 2.53×10 -11 exp(-364/T) + 2.79×10 -13 exp(784/T) cm 3 molecule -1 s -1 Considering that, the present results are in excellent agreement with previous temperature dependent measurements of k 1 , this expression can be recommended to use in the temperature range 230 -960 K with conservative independent of temperature uncertainty of 15%.

Interestingly, the extrapolation of the above expression for k 1 to T = 1925 K (see Figure 3) leads to k 1 = (2.14±0.32)×10 -11 cm 3 molecule -1 s -1 , which is in a reasonable agreement with k 1 = 2.66×10 -11 cm 3 molecule -1 s -1 , reported by Wilson et al. in their study on flame inhibition by HBr at T=(1875-1975) K. [START_REF] Wilson | Flame Inhibition by Halogen Compounds[END_REF] One can note surprisingly excellent agreement between the present experimental data and the results of theoretical calculations of k 1 by Ree et al. [START_REF] Ree | Dependence of the Four-Atom Reaction b + Oh → Br + H2o on Temperatures between 20 and 2000 K[END_REF] in the whole temperature range of the present work. Ree et al. [START_REF] Ree | Dependence of the Four-Atom Reaction b + Oh → Br + H2o on Temperatures between 20 and 2000 K[END_REF] used a quasiclassical trajectory method (QCT) to calculate the temperature dependence of reaction (1) and shown that below room temperature, the reaction proceeds through formation of a collision complex with enhanced contribution of the H atom tunneling at low temperatures. At higher temperatures, according to the authors, the OH + HBr reaction occurs primarily through a direct H-atom abstraction mechanism. The expression recommended for the reaction rate constant at T = 20-2000 K reflected the proposed mechanism: k 1 = 0.35×10 -11 (1+300/T)exp(19/T) + 2.0×10 -11 (T/300) 0.1 exp(-626/T) + 0.15×10 -11 (in cm 3 molecule -1 s -1 units), with the first and second terms representing primarily the complex and direct reaction paths at low and high temperatures, respectively.

All the available temperature dependent measurements of k 1 are shown in Figure 4 in line with some other theoretical calculations and evaluations of the rate constant covering high temperatures. In the work of Burgess et al., [START_REF] Burgess | A Chemical Kinetic Mechanism for 2-Bromo-3,3,3-Trifluoropropene (2-Btp) Flame Inhibition[END_REF] reporting a detailed chemical kinetic mechanism of the flame inhibition by the fire-suppressant 2-bromo-3,3,3-trifluoropropene, the following expression for the rate constant of reaction (1), k 1 = 3.4×10 -13 T 0.58 exp(168/T) cm 3 molecule - 1 s -1 , was proposed for use at flame temperatures. This expression reproduces quite well the temperature dependence of k 1 measured in the current study between 600 and 1000 K, with the absolute values of k 1 by 20% higher compared with the experimental data. Similarly, the QCT calculations by Góger et al. [START_REF] Góger | Flame Inhibition Chemistry: Rate Coefficients of the Reactions of Hbr with Ch3 and Oh Radicals at High Temperatures Determined by Quasiclassical Trajectory Calculations[END_REF] predict a small positive temperature dependence of k 1 at high temperatures, but in absolute value, they are by a factor of  1.8 lower at T> 600 K than the current experimental results. Experimental data for k 1 obtained at ultralow temperatures using supersonic Laval nozzle flow reactors with monitoring of OH by laser induced fluorescence method [START_REF] Mullen | Temperature Dependence and Kinetic Isotope Effects for the Oh + b Reaction and /D Isotopic Va iants at Low Tempe atu es (53-135 K) Measu ed Using a Pulsed Supersonic Laval Nozzle Flow Reactor[END_REF][START_REF] Jaramillo | A Consensus View of the Temperature Dependence of the Gas Phase Reaction: Oh + b → H2o + Br[END_REF][START_REF] Jaramillo | Temperature-Dependent Kinetic Isotope Effects in the Gas-Phase Reaction: Oh + b[END_REF][START_REF] Atkinson | Low-Temperature Kinetic Behavior of the imolecula Reaction Oh + b (76-242 K)[END_REF][START_REF] Sims | Ultra-Low Temperature Kinetics of Neutral-Neutral Reactions: New Experimental and Theoretical Results for Oh+Hbr between 295 and 23 K[END_REF] were analyzed by Mullen and Smith 6 who recommended k 1 = (1.06±0.02)×10 -11 (T/298) (-0.9±0.11) cm 3 molecule -1 s -1 in the temperature range 23-360 K. This expression obtained from the fit to the experimental data at low temperatures and that from the present work describing behavior of k 1 at high temperatures give identical (within 1%) results for k 1 at temperatures 240-270 K.

On this basis, we recommend the following expressions for the rate constant of reaction (1): k 1 = 1.06×10 -11 (T/298) -0.9 cm 3 molecule -1 s -1 (23 -240 K), k 1 = 2.53×10 -11 exp(-364/T) + 2.79×10 -13 exp(784/T) cm 3 molecule -1 s -1 (T = 240-960 K), which fairly well describe the temperature behavior of the rate constant over an extended temperature range 23-960 K (blue and green lines in Figure 4, respectively).

TOC Graphic

( 3 )

 3 in excess of Br 2 was also used for the determination of the absolute concentrations of OH radicals by linking [OH] to the consumed fraction of [Br 2 ]: [OH] = [HOBr] = [Br 2 ]

  The flow velocity in the reactor was in the range (1750-2920) cm s -1 . The kinetics of OH consumption in excess of HBr were treated with an exponential function, [OH] = [OH] 0 ×exp(k 1 '×t), where k 1 ' = k 1 ×[HBr]+k w is the pseudo-first-order rate constant and k w corresponds to the rate of OH loss on the walls of the flow reactor. Examples of the kinetic runs observed at T = 700 K are shown in Figure S2 (SI). Examples of the typical dependencies of the pseudofirst order rate constant, k 1 ' = k 1 ×[HBr]+k w , on concentration of HBr at different temperatures are shown in Figure1. The consumption of the excess reactant, HBr, monitored by mass spectrometry, was observed to be less than 10% in all the kinetic runs. In all cases, the average concentration of HBr along the reaction zone was used in the treatment of the experimental data. All measured values of k 1 ' were corrected for the effects of axial and radial diffusion of OH radicals.[START_REF] Kaufman | Kinetics of Elementary Radical Reactions in the Gas Phase[END_REF] Typically, the corrections were <10 % and up to 18% in a few kinetic runs. The values of k 1 at respective temperatures were determined from the slopes of the straight lines like those shown in Figure1.

Figure 1 .

 1 Figure 1. Plots of pseudo-first-order rate constant k 1 ′ versus concentration of HBr at different temperatures. Partially shown error bars represent typical uncertainties (≤5%) on the determination of k 1 ′.

The experimental data observed in relative measurements of k 1

 1 are shown in Figure 2. In accordance with Equation I, the slopes of the straight lines in Figure 2 provide the values of k 1 /k 3 at respective temperatures. Final values of k 1 (Table2) were calculated with k 3 = 2.1610 -11 exp(207/T) cm 3 molecule -1 s -1 , recommended in a recent study from this group in the temperature range T = 220-950 K.[START_REF] Bedjanian | Temperature-Dependent Rate Constant for the Reaction of Hydroxyl Radical with 3-Hydroxy-3-Methyl-2-Butanone[END_REF]

Figure 2 .

 2 Figure 2. Yield of HOBr from titration of OH radicals by Br 2 + HBr mixtures at different temperatures. Partially shown error bars represent typical uncertainties of the measurements ( 5%).

Figure 3 .

 3 Figure 3. Summary of the data on the temperature dependence of the rate constant of reaction (1) in the temperature range of the present work.

Figure 4 .

 4 Figure 4. Summary of the temperature dependent measurements of the rate constant of reaction (1). The error bars are omitted for clarity.

Table 1 . Summary of the absolute measurements of k 1

 1

	T (K) a	[HBr] (10 13 molecule cm -3 )	k 1 (± 2) (10 -11 cm 3 molecule -1 s -1 )	Reactor surface b
	235	0.25-4.60	1.31±0.03	HW
	250	0.30-4.08	1.23±0.02	HW
	265	0.34-4.78	1.21±0.02	HW
	280	0.32-4.06	1.17±0.03	HW
	300	0.29-4.55	1.10±0.02	HW

a 8-13 kinetic runs at each temperature.

b HW: halocarbon wax; Q: Quartz.

Table 2 . Experimental Conditions and Results of the Relative Measurements of k 1

 2

			of k 1 . In this series of experiments, the rate constant of
	reaction (1) was measured relative to that of OH radicals with Br 2 . The experiments consisted
	in a titration of hydroxyl radicals by a mixture of HBr and Br 2 and the measurements of HOBr
	yield as a function of the [HBr]/[Br 2 ] ratio:
	OH + HBr		H 2 O + Br	(1)
	OH + Br 2 	HOBr + Br	(3)

ratio. The measurements of the absolute concentrations of OH were not necessary in these experiments, since HOBr was monitored in the presence of Br 2 in both HBr -free system, corresponding to [OH] 0 , and in the presence of HBr in the reactor, corresponding to the fraction of [OH] 0 reacted with Br 2 . Reaction time was around 0.02 s, [OH] 0 = (0.5-1.0)×10 11 , [Br 2 ]  9×10 12 molecule cm -3 , the ranges of HBr concentrations are given in Table

2

.

a Units of 10 13 molecule cm -3 .

b Units of 10 -11 cm 3 molecule -1 s -1 , estimated total uncertainty on k 1 is nearly 20% and includes the uncertainty on the rate constant of the reference reaction.

CONCLUSIONS

The rate constant of the reaction of OH radicals with HBr has been measured using a discharge flow reactor combined with an electron impact ionization mass spectrometer at T = 235 -960 K: k 1 = 2.53×10 -11 exp(-364/T) + 2.79×10 -13 exp(784/T) cm 3 molecule -1 s -1 . This expression combined with that from previous low temperature measurements, k 1 = 1.06×10 -11 (T/298) -0.9 cm 3 molecule -