Tristan Gomez
email: tristan.gomez@univ-nantes.fr

Thomas Fréour
email: thomas.freour@chu-nantes.fr

Harold Mouchère
email: harold.mouchere@univ-nantes.fr

Comparison of attention models and post-hoc explanation methods for embryo stage identification: a case study ⋆

An important limitation to the development of AI-based solutions for In Vitro Fertilization (IVF) is the black-box nature of most state-of-the-art models, due to the complexity of deep learning architectures, which raises potential bias and fairness issues. The need for interpretable AI has risen not only in the IVF field but also in the deep learning community in general. This has started a trend in literature where authors focus on designing objective metrics to evaluate generic explanation methods. In this paper, we study the behavior of recently proposed objective faithfulness metrics applied to the problem of embryo stage identification. We benchmark attention models and post-hoc methods using metrics and further show empirically that (1) the metrics produce low overall agreement on the model ranking and (2) depending on the metric approach, either post-hoc methods or attention models are favored. We conclude with general remarks about the difficulty of defining faithfulness and the necessity of understanding its relationship with the type of approach that is favored.

Introduction

Infertility is a global health issue worldwide [START_REF] Inhorn | Infertility around the globe: new thinking on gender, reproductive technologies and global movements in the 21st century[END_REF]. One of the most common treatments for infertile couples is In Vitro Fertilization (IVF). This procedure notably consists of embryo culture for 2-6 days under controlled environmental conditions, leading to intrauterine transfer or freezing of embryos identified as having a good implantation potential by embryologists. To allow continuous monitoring of embryo development, Time-lapse imaging incubators (TLI) were first released in the IVF market around 2010. This time-lapse technology provides a dynamic overview of embryonic in vitro development by taking photographs of each embryo at regular intervals throughout its development. TLI appears to be the most promising solution to improve embryo quality assessment methods, and subsequently, the clinical efficiency of IVF [START_REF] Pribenszky | Time-lapse culture with morphokinetic embryo selection improves pregnancy and live birth chances and reduces early pregnancy loss: a meta-analysis[END_REF]. In particular, the unprecedented ⋆ Supported by Nantes Excellence Trajectory (NExT) high volume of high-quality images produced by TLI systems has already been leveraged using deep learning (DL) methods. Previous work has notably focused on designing models that automatically identify the embryo development stages, a crucial task to identify embryos with low live-birth potential [START_REF] Khan | Deep convolutional neural networks for human embryonic cell counting[END_REF][START_REF] Rad | Blastomere cell counting and centroid localization in microscopic images of human embryo[END_REF]. However, an important limitation to the development of AI-based solutions for IVF is the black-box nature of most state-of-the-art models, due to the complexity of deep learning architectures, which raises potential bias and fairness issues [START_REF] Afnan | Interpretable, not black-box, artificial intelligence should be used for embryo selection[END_REF]. The need for interpretable AI has risen not only in the IVF field but also in the deep learning community in general. This has started a trend in literature where authors confront users with models' decisions along with various explanations to determine which fits better the users' needs on a particular application [START_REF] Alqaraawi | Evaluating saliency map explanations for convolutional neural networks: A user study[END_REF][START_REF] Tsai | Evaluating Visual Explanations for Similarity-Based Recommendations: User Perception and Performance[END_REF][START_REF] Van Der Waa | Evaluating xai: A comparison of rule-based and example-based explanations[END_REF]. However, the financial cost and the difficulty of establishing a correct protocol make this approach difficult. Because of these issues, another trend focuses on designing objective metrics to evaluate generic explanation methods [START_REF] Petsiuk | Rise: Randomized input sampling for explanation of black-box models[END_REF][START_REF] Jung | Towards better explanations of class activation mapping[END_REF]5,[START_REF] Gomez | Metrics for saliency map evaluation of deep learning explanation methods[END_REF]. In this paper, we follow this trend and study the behavior of objective faithfulness metrics recently proposed applied to the problem of embryo stage identification. We apply faithfulness metrics to attention models and further show empirically that (1) the metrics produce low overall agreement on the model ranking and (2) depending on the metric approach, either post-hoc methods or attention models are favored.

First, we describe the attention models, post-hoc methods, and faithfulness metrics we use in this work. Secondly, we compute an extensive benchmark and study the behavior of the metrics on 9 saliency approaches, including 5 post-hoc methods and 4 attention models. We conclude with general remarks about the difficulty of defining faithfulness and the necessity of understanding its relationship with the type of approach that is favored.

Method

Saliency map generation approaches

We now describe the models and methods used to generate saliency maps. First, we list several attention models as they integrate the computation of a saliency map (called an attention map) that is used to guide the decision process. Secondly, we list generic post-hoc explanation methods that can generate saliency maps for a wide range of models and architectures without requiring model training.

Attention models. In this category, we include models featuring a spatial attention layer in their inference phase. Note that we do not include visual transformers [START_REF] Dosovitskiy | An image is worth 16x16 words: Transformers for image recognition at scale[END_REF] are they are difficult to interpret due to their non-local heads, as opposed to the local CNN kernels. Hu et al. [START_REF] Hu | See better before looking closer: Weakly supervised data augmentation network for fine-grained visual classification[END_REF] proposed a variant of the model originally proposed by Lin et al. [START_REF] Lin | Bilinear cnn models for fine-grained visual recognition[END_REF] with a convolutional module that processes the feature maps and outputs the attention map. In the rest of this paper, this model is designated as the Bilinear-CNN (B-CNN). Fukui et al. [START_REF] Fukui | Attention branch network: Learning of attention mechanism for visual explanation[END_REF] proposed another variant of Lin et al.'s architecture called Attention Branch Network (ABN) where the attention module is also trained to correctly predict the object class along with the regular classification head. This was applied to the problem of embryo quality prediction [START_REF] Sawada | Artificial intelligence with attention branch network and deep learning can predict live births by using time-lapse imaging of embryos after in vitro fertilisation[END_REF], a problem related to the one studied here.

Another line of work proposes to use prototypes to increase interpretability. Notably, Interpretability By Parts (IBP) [START_REF] Huang | Interpretable and accurate fine-grained recognition via region grouping[END_REF] is a model that improves upon Chen et al.'s work [START_REF] Chen | This looks like that: deep learning for interpretable image recognition[END_REF] by encouraging the model to generate prototypes presence probabilities close to 0 or 1, to generate more accurate attention maps and improve interpretability.

The last attention model included in this study is called Bilinear Representative Non-Parametric Attention (BR-NPA) [START_REF] Gomez | Br-npa: A non-parametric highresolution attention model to improve the interpretability of attention[END_REF] and proposes to generate attention maps to guide the model spatially without any dedicated parameter, contrary to the models mentioned above which feature either a convolution attention module or parametric prototypes.

Post-hoc explanation methods. This category proposes generic methods that can be applied to any CNN with feature maps and does not require training the model, contrarily to the attention modules mentioned above. Class Activation Map (CAM) [START_REF] Zhou | Learning deep features for discriminative localization[END_REF] was proposed to visualize the areas that contributed the most to the prediction of one specific class. Grad-CAM [START_REF] Selvaraju | Grad-cam: Visual explanations from deep networks via gradient-based localization[END_REF] improves upon CAM and proposes to compute the gradients of the feature maps relatively to the class score to identify the maps that contributed the most to the decision. A weighted average of the maps is then computed to obtain the saliency map. The first method included here is Grad-CAM++ [5], which further replaces feature-wise gradients with pixel-wise positive partial derivatives to compute the saliency map. We also use Score-CAM [START_REF] Wang | Score-cam: Score-weighted visual explanations for convolutional neural networks[END_REF] and Ablation-CAM [START_REF] Desai | Ablation-cam: Visual explanations for deep convolutional network via gradient-free localization[END_REF] later proposed to remove gradients by evaluating each feature map's importance by masking the input image according to the activated areas of the map or by ablation analysis.

We also apply RISE [START_REF] Petsiuk | Rise: Randomized input sampling for explanation of black-box models[END_REF], a method that uses random masks to find the areas that impact the class score the most. The last method is a baseline that we call Activation Map (AM) which consists to visualize the average activation map by computing the mean of the last layer's feature maps.

Faithfulness metrics

The input image is a 3D tensor I ∈ R H×W ×3 and the saliency map is a 2D matrix S ∈ R H ′ ×W ′ with a lower resolution, H ′ < H and W ′ < W . We study the following faithfulness metrics.

Increase In Confidence (ICC) [5]. The IIC metrics measures how often the confidence of the model in the predicted class increases when highlighting the salient areas. First, the input image is masked with the explanation map as follows :

I m = norm(upsamp(S)) • I, (1)
where norm(S) is the min-max normalization function, defined as norm(S) = S-min(S) max(S)-min(S) , upsamp(S) is a function that upsample S to the resolution of I, and • is the element-wise product. The IIC metric is defined as:

IIC = 1 [c I <c Im ] , (2)
where c I is the score of the predicted class with I as input and c Im is the score of the same class with I m as input. The intuition is that a good saliency map S highlights areas such that when the non-salient areas are removed, the class score increases. Therefore, maximizing this metric corresponds to an improvement. Note that this metric is a binary value and is only useful when computing its mean value over a large number of images, as in section 4.

Average Drop (AD) [5]. Similar to IIC, this metric measures the average score drop when highlighting the salient areas. Using the same masking of the input image, the AD metric computes the relative score difference between the two images I and I m :

AD = max(0, c I -c Im ) c I (3)
Given that when highlighting the salient areas the class score is not supposed to decrease, minimizing this metric corresponds to an improvement. Note that the metric name implies computing an average value over several images, as is done in section 4. However, we did not include the averaging operator in the AD definition to keep notation simple and uniform throughout this section.

Average Drop in Deletion (ADD) [START_REF] Jung | Towards better explanations of class activation mapping[END_REF]. Jung et al. proposed a variant of the preceding metric which consists to mask the salient areas instead of the nonsalient areas. To do this, the image is masked with the inverse of the saliency map, which highlights the non-salient areas :

I 1-m = (1 -norm(upsamp(S))) • I (4)
The ADD metric is then defined as follows:

ADD = max(0, c I -c I1-m ) c I , (5)
where c I1-m is the score with I 1-m . Contrarily to AD, this metric removes the salient areas and the class score is expected to decrease, which means that maximizing this metric results in an improvement. Also, as for AD, we dropped the averaging operator to keep notation simple.

Deletion Area Under Curve (DAUC) [START_REF] Petsiuk | Rise: Randomized input sampling for explanation of black-box models[END_REF]. This metric evaluates the reliability of the saliency maps by progressively masking the image starting with the most important areas according to the saliency map and finishing with the least important. First, S is sorted and parsed from the highest element to its lowest element. At each element S i ′ j ′ , we mask the corresponding area of I by multiplying it by a mask M k ∈ R H×W , where

M k ij = 0, if i ′ r < i < i ′ (r + 1) and j ′ r < j < j ′ (r + 1) 1, otherwise, (6)
and r = H/H ′ = W/W ′ . After each masking operation, the model m runs an inference with the updated version of I, and the score of the initially predicted class is updated, producing a new score c k :

c k = m(I • k=k k=1 M k), (7)
where k ∈ {1, ..., H ′ × W ′ }. Secondly, once the whole image has been masked, the scores c k are normalized by dividing them by the maximum max k c k and then plotted as a function of the proportion p k of the image that is masked. The DAUC is finally obtained by computing the area under the curve (AUC) of this graph. The intuition behind this is that if a saliency map highlights the areas that are relevant to the decision, masking them will quickly result in a large decrease in the initially predicted class score, which in turn will minimize the AUC. Therefore, minimizing this metric corresponds to an improvement.

Insertion Area Under Curve (IAUC) [START_REF] Petsiuk | Rise: Randomized input sampling for explanation of black-box models[END_REF]. Instead of progressively masking the image, the IAUC metric starts from a blurred image and then progressively unblurs it by starting from the most important areas according to the saliency map. Similarly, if the areas highlighted by the map are relevant for predicting the correct category, the score of the corresponding class (obtained using the partially unblurred image) is supposed to increase rapidly. Maximizing this metric corresponds to an improvement.

Deletion Correlation (DC) [START_REF] Gomez | Metrics for saliency map evaluation of deep learning explanation methods[END_REF]. The DC metric also consists of gradually masking the input image by following the order suggested by the saliency map, but instead of computing the area under the class score/pixel rank curve, it computes the linear correlation of the class score variations and the saliency scores.

Once the scores c k have been computed, we compute the variation of the scores

v k = c k -c k+1 .
The DC metric is obtained by the linear correlation coefficient between the v k and the s k where s k is the saliency score of the area masked at step k. As it represents the correlation between the saliency of a pixel and its impact on the class score, maximizing this metric is an improvement.

Insertion Correlation (IC) [START_REF] Gomez | Metrics for saliency map evaluation of deep learning explanation methods[END_REF]. Similarly, the IC metric is inspired by IAUC and starts from a blurred image, and gradually reveals the image according to the saliency map. Once the image is totally revealed, the score variations are computed v k = c k+1 -c k and the linear correlation of the v k with the s k is computed. This correlation metric should also be maximized to correspond to an improvement.

The embryo development stage dataset

The images used in this work are microscopic images of growing embryos between their first and fifth day of development in vitro. During these five days, the embryo grows through 16 successive development stages that we note sPB2, sPNa, sPNf, s2, s3, s4, s5, s6, s7, s8, s9+, sM, sSB, sB, sEB, and sHB. These stages are delimited by 16 events called tPB2, tPNa, tPNf, t2, t3, t4, t5, t6, t7, t8, t9+, tM, tSB, tB, tEB, tHB and defined by Ciray et al. [START_REF] Ciray | Proposed guidelines on the nomenclature and annotation of dynamic human embryo monitoring by a time-lapse user group[END_REF]. The timings of each event are valuable information to determine if the quality of the embryo is high enough to be transferred to the uterus [START_REF] Pribenszky | Time-lapse culture with morphokinetic embryo selection improves pregnancy and live birth chances and reduces early pregnancy loss: a meta-analysis[END_REF].

The raw dataset consists of 704 videos that we split equally into one training/validation set and one test set. From each set and each video, we extract one-third of the frames regularly spaced the video. The images are in grayscale with a 500 × 500 resolution. The total size of each set is 29843 images for the training/validation set and 28282 images for the test set.

We choose to model this problem as an image classification task. The models studied in this paper are trained to process an embryo image and infer which development stage the embryo is at. Note that the video recording starts at the tPB2 event which means that the first frames show the sPB2 stage. The class distribution and some samples are shown respectively in fig. 1

Implementation details

The backbone used for all networks is ResNet-50 [START_REF] He | Deep residual learning for image recognition[END_REF]

Results

In table 1 one can see the faithfulness evaluation of the explanation methods and the attention models on the test set. Given the size of the test set and the computation time of the metrics/approaches, we sample randomly 100 images from the test set and compute the average performance of each approach to each metric on these images. First, note that we also include the average accuracy per video of the trained models over the whole test set to highlight that all models have similar accuracies. This means that the differences in faithfulness observed can not be explained by different levels of accuracy but only by the approach (attention model/post-hoc method) or the metrics. Secondly, the faithfulness metrics do not agree on which model is the best. For example, depending on the metric, the most faithful explanation method is BR-NPA, RISE, Ablation-CAM, ABN, or Score-CAM and the least faithful is ABN, InterByParts, RISE, BR-NPA, or AM. We propose to use the Kendall-tau correlation coefficient [START_REF] Kendall | The treatment of ties in ranking problems[END_REF] to obtain a better idea of the disagreement between the metrics. This correlation coefficient indicates if two variables have a monotonic relationship with each other. When close from 1/-1, the variables have an increasing/decreasing relationship, and when close to 0, they are independent. We represent each metric using the 9 mean performances they have attributed to each approach. Let x and y be two such lists of performance for two metrics. We define a concordant pair, which is a pair of approaches A and B that are ranked the same in x and y, i.e. metrics x and y agree that A is better or worse than B. We also define a discordant pair, where x and y disagree on the ranking of A and B. Furthermore, we say that a pair is a tie according to x if they have the same ranking in x. Finally, the Kendall-tau correlation coefficient between two metrics x and y is computed as follows:

τ (x, y) = (P -Q) √ P + Q + U * √ P + Q + T , ( 8
)
where P is the number of concordant pairs, Q the number of discordant pairs, T the number of ties only in x, and U the number of ties only in y. If a tie occurs for the same pair in both x and y, it is not added to either T or U. The coefficient values obtained between each pair of metrics are shown in fig. 3. Overall, there is a low correlation value between metrics. This means that metrics globally disagree on the model ranking, with many metrics producing independent ranking (DC and IC have a close from 0 correlation with AD, ADD, IIC, and IAUC) and some metrics even ranking models in reversed order (AD and ADD, ADD and IIC, IC, and DC). However, some metrics offer similar rankings like AD with IIC or IAUC.

To obtain a clear representation of metrics' similarities with each other, we propose to use a dimension reduction algorithm like t-SNE [START_REF] Van Der Maaten | Visualizing data using t-sne[END_REF]. We represented the 9 values given by each of the metrics on each of the 100 images using the dimension reduction algorithm t-SNE. Note that we exclude the IIC metric as it is a binary metric and will not provide a meaningful ranking on a single image. We obtain 6 point clouds in a 9-dimensional space, where each cloud is composed of 100 points and represents one metric, for a total of 600 points. We use t-SNE to reduce the dimension of the 600 points at the same time and visualize the result on a single 2D plot. Note that an appropriate distance function has to be chosen, given that each point represents a ranking. Instead of the default Euclidean distance, we use the following distance function based on Kendall's tau correlation: Fig. 3: Kendall-tau correlation coefficient between the metrics.

D(x, y) = -log( 1 2 (τ (x, y) + 1)), (9)
where τ (•, •) is the Kendall's tau correlation. With this metric, we ensure a distance that goes from 0 to +∞ when the correlation τ (x, y) goes from 1 to -1. Furthermore, when τ = -1 we replace its value by -0.999 to prevent any numerical errors. 3 .

The plot in fig. 4 shows that the metrics are arranged in two groups: {DAUC, DC, ADD}, {IAUC, IC, AD}. This structure is not surprising given that DAUC, DC and ADD all consist to mask the important areas of the image, measure the score drop, and favor the model with the lowest class score once the mask(s) are applied. Similarly, IAUC, IC, and AD all highlight/reveal the important areas and favors models for which the class score is the highest possible once the important areas have been revealed. This is why we call 'Mask' and 'Highlight' the two groups of metrics obtained: {DAUC, DC, ADD} and {IAUC, IC, AD, IIC}. Note that we also include IIC in the Highlight group as it also consists to highlight the salient areas of the image and expects a score increase as a consequence. This demonstrates that the method used to measure the impact of an area (Highlight or Mask) of the image majorly impacts the ranking produced.

We propose now to identify the best approaches according to each group of metrics. We compute the average rank of an approach A on the Highlight and Fig. 4: t-SNE projection of the ranking given by metrics on the 100 images. The distance between points is computed using a metric based on Kendall's tau correlation.

Mask groups as follows:

r M ask (A) = r DC (A) + r DAU C (A) + r ADD (A) 3 r Highlight (A) = r IC (A) + r IAU C (A) + r AD (A) + r IIC (A) 4 , (10)
where r X (A) is the rank of A according to the metric X.

Table 2 shows the average ranks computed. One can see a link between metric type and approach type where the Mask group seems to favor the Attention models whereas the Highlight group favors the generic post-hoc methods.

At this point, we identify the most faithful approaches according to each metric group: BR-NPA and Score-CAM. Next, we propose to examine the maps they generate to determine which model seems the most reliable between BR-NPA and the CNN, using the Score-CAM algorithm to explain the CNN's decision. By reliable, we mean a model which uses biologically relevant features in the image.

Figure 5 shows examples of saliency maps generated by BR-NPA and Score-CAM during 3 stages: sPNa (fig. 5a), s4 (fig. 5b) and sB (fig. 5c). In fig. 5a, BR-NPA focuses distinctly on the Pro-Nuclei (PN) (the nuclei of a sperm or an egg cell during the process of fertilization) during the sPNa stage whereas the focus of the CNN seems to be at an upper level as it highlights the whole embryo. The focus of BR-NPA's attention on the PN is biologically relevant because the PN are only visible during this stage. In fig. 5c one can see that BR-NPA focuses on the wall of the embryo during the sB phase. This stage marks the beginning of a new cell structure in the embryo where, instead of being arranged as a pack, cells start to specialize and notably some cells form the wall of the embryo. Therefore, focusing on the embryo's wall is also biologically relevant as it is a marker of this phase. On the other hand, Score-CAM produces lowresolution maps that can not highlight small details and focus on the embryo as a whole, a less biologically relevant level of focus given that it is visible during all stages. Globally, BR-NPA's saliency maps are sharp and precise and highlight biologically relevant elements of the image. However, this does not imply that BR-NPA focuses on more biologically relevant features than the CNN. Indeed, the blurriness of Score-CAM's saliency maps is probably due at least to the low resolution of the CNN's feature maps (14×14) compared to the resolution of BR-NPA's which is higher (56 × 56). This shows the interest in using high-resolution attention maps like BR-NPA maps to disambiguate the model's focus.

Discussion

Bastings et al. [START_REF] Bastings | The elephant in the interpretability room: Why use attention as explanation when we have saliency methods?[END_REF] argue that post-hoc methods should be privileged over attention models when it comes to faithfulness, as post-hoc methods take the whole computation path into account whereas attention maps only reflect input importance at one point in the computation. However, we showed here that depending on the way the map is evaluated (Highlight or Mask), attention models can provide superior faithfulness. Also, the intuitions behind the metrics discussed here should be examined. For example, the IIC metric suggests that the class score should increase once the salient parts of the image have been highlighted. However, it is unclear if such a phenomenon would happen in practice. Indeed, filling the masked areas with black pixels generates out-of-distribution samples that can induce an unexpected behavior of the model [START_REF] Gomez | Metrics for saliency map evaluation of deep learning explanation methods[END_REF]. Moreover, ICC suggests that the confidence is increased by the presence of salient areas and decreased by the presence of non-salient areas, but the non-salient areas could also play a neutral role and not affect the confidence much. More generally, the major difficulty in designing these metrics resides in measuring the impact of an area of the input image reliably. Measuring the impact of an area by masking it is difficult because the score variation depends on if the other areas of the image are masked or not. To reach a consensus on how this measure should be done, the community has first yet to understand how exactly the score variation depends on the masking of other areas. Moreover, the metrics studied here were all developed to quantify faithfulness but in practice adopt two distinct behavior and seems to favor different methods: attention models for the Mask group and post-hoc methods for the Highlight group. Therefore, there should also be future works on the definition of faithfulness and its relationship with the approach type (attention models or post-hoc methods).

Finally, the practical relevance of the faithfulness metrics should be evaluated in a user study. To what extent the reliability of the maps can improve, for example, the acceptability of the decision by a user remains an open question.

Conclusion

In this paper, we compared the faithfulness of the saliency maps generated by 9 different approaches, including 4 attention models and 5 post-hoc explanation methods. We showed low overall agreement between the metrics proposed in the literature and in particular demonstrated the tendency of metrics of the Highlight group to favor the post-hoc methods whereas metrics of the Mask group favor instead attention models. We then visualized the saliency maps generated by the two best approaches, namely Score-CAM and BR-NPA, and showed that the low resolution of Score-CAM limits the insights that can be obtained whereas BR-NPA's maps highlight biologically relevant features. Finally, we discussed the difficulty of measuring the impact of each image part reliably and future work.

  and fig.2.

Fig. 1 :

 1 Fig. 1: The class distribution of the dataset used. There are only 5 and 15 sHB samples respectively in the training/validation set and the test set as this stage usually occurs after the video recording has been stopped.

Fig. 2 :

 2 Fig. 2: Illustrations of the 16 development stages used.

Fig. 5 :

 5 Fig. 5: The saliency maps generated by the Score-CAM method and BR-NPA model (respectively middle and right columns) on stages sPNa (a), s4 (b), and sB (c).

Table 1 :

 1 Performance of the studied models and methods.

	Model	Viz. Method DAUC IAUC DC	IC	IIC AD ADD Accuracy
		AM	0.134 0.308 -0.174 0.075 0.19 0.397 0.325
	CNN	Grad-CAM++ 0.1162 0.333 -0.101 0.032 0.52 0.14 0.383
		RISE	0.113 0.457 -0.221 -0.077 0.5 0.137 0.436	71.0
		Score-CAM 0.1079 0.315 -0.123 0.081 0.52 0.108 0.362
		Ablation-CAM 0.0954 0.329 0.272 -0.071 0.57 0.111 0.328
	ABN	-	0.1464 0.249 -0.186 0.136 0.12 0.591 0.475	71.0
	InterByParts	-	0.0876 0.115 0.196 -0.255 0.08 0.901 0.879	71.3
	B-CNN	-	0.0772 0.221 -0.208 0.124 0.13 0.491 0.482	70.2
	BR-NPA	-	0.0709 0.261 0.185 -0.146 0.04 0.91 0.887	70.7

Table 2 :

 2 Average ranking of the methods according to the two metrics groups: Mask and Highlight. Attention models and post-hoc methods are respectively highlighted in orange and cyan.

		Mask			Highlight
	Average rank Approach	Type Average rank Approach	Type
	1.67	BR-NPA	Attention	2.75	Score-CAM Post-Hoc
	2.33	InterByParts Attention	3.0	Ablation-CAM Post-Hoc
	4.33	B-CNN	Attention	3.25	Grad-CAM++ Post-Hoc
	4.33	Ablation-CAM Post-Hoc	3.75	RISE	Post-Hoc
	5.67	Grad-CAM++ Post-Hoc	4.75	AM	Post-Hoc
	5.67	Score-CAM Post-Hoc	5.5	ABN	Attention
	6.67	ABN	Attention	5.5	B-CNN	Attention
	6.67	RISE	Post-Hoc	8.0	BR-NPA	Attention
	7.67	AM	Post-Hoc	8.5	InterByParts Attention

Note that we did not use the more recent UMAP algorithm[START_REF] Mcinnes | Umap: Uniform manifold approximation and projection[END_REF] because the only UMAP implementation available requires the custom distance function to be compiled with Numba, which is currently not possible with the Kendall's tau implementation of the Scikit-Learn python package[START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF].


Adam Fisher

Arthur Levy

James Kratz

The significance of spatial temperature variations in large volume oven curing

The production of thermoset composite laminates in an oven or an autoclave involves considerable uncertainty, the majority of which comes from two main sources, the processing conditions, and the material itself. The complexity inherent in thermoset composite materials accommodates various unknown factors such as: initial degree of cure, resin composition, volume fractions and fibre alignment, all of which are liable to vary spatially within a part and from part-to-part [START_REF] Mesogitis | Uncertainty in the manufacturing of fibrous thermosetting composites: A review[END_REF]. Similarly, the processing conditions contain uncertainties, for example in temperature, pressure, tool thickness and localised heat transfer coefficients, as such the prescribed values are rarely those seen by the material. However, despite this, it is common practice to use deterministic cure kinetics models to predict the extent of a cure reaction [START_REF] Mesogitis | Heat transfer simulation of the cure of thermoplastic particle interleaf carbon fibre epoxy prepregs[END_REF]. These models assume values assigned to both the material properties and the processing conditions conform to a set of predetermined values.

In this study the validity of assuming a deterministic cure kinetics model was considered, addressing uncertainty in the cure kinetics parameters and the applied temperature values in the chosen model. The material considered during this study was HexPly® M21 particle interleaf toughened epoxy resin system. It has been found that a cure kinetics model that combines an nth order model and an autocatalytic model is capable of representing the cure reaction of this material [START_REF] Mesogitis | Heat transfer simulation of the cure of thermoplastic particle interleaf carbon fibre epoxy prepregs[END_REF]. The reaction rate constants have a diffusion term in addition to the conventional chemical component to capture the diffusion rate limitation phenomena.

The effect of the uncertainty of the inputs was assessed by considering its effect on the final degree of cure of the chosen material. To this end the deterministic cure kinetics model described above was used to propagate the uncertainty of the inputs to the final degree of cure, the Monte-Carlo method was used to produce a distribution of the output. The Monte-Carlo method required that the stochastic parameters were represented by distributions so a value could be sampled for each trial. To capture uncertainty in the cure kinetics parameters that was representative of the M21 material, the available 17 sets of dynamic DSC data [START_REF] Mesogitis | Heat transfer simulation of the cure of thermoplastic particle interleaf carbon fibre epoxy prepregs[END_REF] were fitted to the cure kinetics model, producing 17 values of each cure kinetics parameter. The relevant statistics of these values were then used to generate the distributions for each parameter. Sampled values from these distributions were then used as inputs to the cure kinetics model for each Monte-Carlo trial. It is noted that the DSC data was collected (i) by a single operator, (ii) using the same machine, (iii) on neat resin samples, and (iv) the material was from a single batch, thus eliminating four potential sources of uncertainty.

To determine which of the cure kinetics parameters had the most influence on the degree of cure, a sensitivity analysis was conducted. The sensitivity analysis showed that four of the cure kinetics parameters had a significant influence on the final degree of cure, the most influential of these were the two activation energies in the chemical component. The influence of two of the reaction orders was also above the remaining parameters but was less significant than the activation energies. In comparison with these four parameters the influence of the remaining parameters was negligible, thus in accordance with previous work, for the sake of efficiency they were treated as deterministic [START_REF] Mesogitis | Stochastic simulation of the influence of cure kinetics uncertainty on composites cure[END_REF].

Following on from Mesogitis et al [START_REF] Mesogitis | Stochastic simulation of the influence of cure kinetics uncertainty on composites cure[END_REF] who found that an activation energy and a reaction order followed a normal distribution, analysis was performed under the null hypothesis that the chosen parameters were normally distributed. A combination of visual tools and hypothesis testing was used to increase the reliability of the conclusion. Quantile-Quantile plots showed that in all cases the majority of the data points were close to the line representing a normal distribution. These observations were supported by results from Shapiro-Wilk normality tests, where all values were well within the 95% confidence interval that is standard for the test. It is noted that this result was achieved for the order of reaction m after an extreme value was removed. The inability to reject the null hypothesis was consistent with the literature thus it was decided to assume the parameters were normally distributed.

A total of 10,000 Monte-Carlo trials were performed to generate an accurate representation of the distribution of the degree of cure values caused by the stochastic cure kinetics parameters after 1 and 2 hours at 180°C. These values were chosen as the recommended cure cycle for the M21 material consists of a 2-hour 180°C dwell. The histogram in Figure 1 representing the degree of cure after an hour indicates the distribution is symmetric about the mean and looks to be normally distributed. The long tails indicate the potential for large variability in the degree of cure during the curing process. The physical consequences of this variability could be the initiation of residual stresses, and dimensional instabilities that could cause issues in the assembly process. After 2 hours the histogram depicts a significant reduction in variability with most of the degrees of cure exceeding 95%. In both cases the deterministic degree of cure is a good approximation of the stochastic averages, suggesting that as the degree of cure increases and the variability decrease the need for stochastic methods decreases. Analysis of thermocouple data from inside a range of ovens has shown that the amount of deviation from the programmed temperature (the temperature typically assumed in a deterministic model) is highly variable and can be large. The nature of the temperature variation depends on oven specific features, hence the oven-to-oven variability. In additions to temperature, there was also significant variability in the heat transfer coefficients calculated in each oven. Heat transfer coefficient is largely dependent on airflow, again this is influenced by oven specific features such as: the location of fans and exhausts, size and shape. All the ovens considered here were empty, however, under operating conditions the positioning of parts will obstruct the airflow, leading to variability between operations [START_REF] Humfeld | A Machine Learning Framework for Real-time Inverse Modeling and Multiobjective Process Optimization of Composites for Active Manufacturing Control[END_REF].

In this study attention was focused on a case study involving a 5m pre-production oven that was empty apart from thermocouples and the rig used to hold them. The spatial temperature variation was mapped using a 3x3 grid, at each node of the grid there was a lumped-mass calorimeter consisting of two thermocouples, one to measure air temperature, the other embedded in a stainless-steel rod allowing the heat transfer coefficient to be calculated. The temperature measurements were taken during an oven program that consisted of an initial 5°Cpm ramp from ambient temperature (25°C) to 120°C, a 30-minute hold at 120°C, a second ramp of 1°Cpm to 180°C and a 60-minute hold at 180°C.

The measured temperature distribution in this oven was characterised by a significant temperature gradient from front to back, the left-right temperature was more uniform, but variation was still present. The hottest region was at the front (closest to the door) and was the only region where the temperature closely followed the temperature program. In contrast, at the back of the oven where it was coldest, from the end of the first ramp the temperature was consistently below the prescribed temperature, with a peak difference of above 20°C in air temperature and above 50°C in the lumped mass. The largest temperature differences occurred in the lumped masses towards the end of the aggressive first ramp, where the lag caused by the higher heat capacity was most pronounced.

To put into perspective the significance of the measured temperature variation, the cure kinetics model was run with the air temperature measured at each calorimeter as the applied temperature. The results followed a similar pattern to the temperature, with the front measurements producing values closest to the expected degree of cure, the middle values do not deviate massively, but the rear values do. The peak discrepancy of nearly 20% represents a part with significant degree of cure gradients that would likely cause many issues including those mentioned above.

The temperature in the lump masses is dependent on the local heat transfer coefficient. Unlike the temperature measurements, the calculated heat transfer coefficients at any time varied little throughout the oven suggesting uniform airflow. The exception being at the central rear node where heat transfer coefficient was consistently greater, this was attributed to its proximity to the exhaust of the oven, demonstrating the potential effect of an oven specific feature.

To compare the influence of the oven temperature variability with the uncertainty in the cure kinetics parameters on the degree of cure, a sensitivity analysis was conducted. After a brief period at the beginning of the cycle where the temperature is low, the influence of the temperature exceeds any of the cure kinetics parameters and its peak reduced sensitivity value of 40% is more than double that of any of the cure kinetics parameters. This analysis clearly shows that the variability of the temperature in this oven has significantly more influence on the cure process than the uncertainty in the cure kinetics parameters.

A more general approach was taken to see what deviation from the programmed temperature cycle would produce an equivalent effect on the degree of cure as the uncertainty in the cure kinetics parameters and oven temperature. Treating the programmed temperature as the mean, the temperature was shifted downwards by a standard deviation assuming a range of coefficients of variation (COV) from 2% to 10%. The degree of cure under each of these cycles was predicted using the cure kinetics model and compared with those computed for the uncertain cure kinetics parameters and the lowest air temperature measurement from the oven. The 95% confidence interval (±1.96sd assuming normality) for the uncertain cure kinetics parameters just covers the 2% COV curve, whereas the maximum deviation from the set temperature in the 5m oven is equivalent to a notional oven with a temperature COV of 8%.

The above analysis has shown that the measured variability in the oven temperature had a much greater influence on the degree of cure than the uncertainties in the cure kinetics parameters. Furthermore, due to the well documented [START_REF] Yousefi | Kinetic studies of thermoset cure reactions: a review[END_REF] sensitivity of the cure process to temperature and the much greater influence exhibited, it has been shown that far less variability would be necessary before an equivalent influence on the degree of cure is realised. This suggests in cases where oven temperature variability is likely to be high, such as within large ovens, efforts to reduce the uncertainty in the final degree of cure of thermoset composite parts would be more effectively spent reducing oven temperature variability than in attempting to measure and account for the many sources of uncertainty in the material.

Figure 1 .

 1 Figure 1. Histograms of final degree of after 60 (left) and 120 (right) minutes at 180⁰C


Sihem Mallek

Nicolas Daclin

Vincent Chapurlat

Towards a Conceptualisation of Interoperability Requirements

Keywords: Interoperability requirements, compatibility, interoperation, reversibility, verification, enterprise

Enterprises that aim to work together want, prior to any effective collaboration, to know if they are able to interoperate. On the one hand, this induces to be able to define the particular needs having to be taken into account in order to demonstrate if an enterprise can be or must be interoperable. On the other hand, it requires techniques and approaches allowing to formalise these needs as a set of unambiguous and, as formal as possible, requirements called here interoperability requirements. Finally, verification techniques can be used to detect how and where some of these requirements cannot be satisfied. This allows to highlight interoperability problems. This paper focuses on the two first phases and describes an approach to define and to formalise interoperability needs into interoperability requirements. These requirements are decomposed on three classes named compatibility, interoperation and reversibility requirements.

Introduction

Collaboration between enterprises, to fulfill a common project, remains difficult and can be considered as efficient if these enterprises are really able to interact. In this context, the development of interoperability has become an important issue, in today's globalized world, to facilitate, optimize and improve these interactions. Basically, interoperability is defined in [START_REF]A compilation of IEEE standard computer glossaries. Standard computer dictionary[END_REF] as "the ability of two or more systems or components to exchange information and to use the information that has been exchanged". Therefore, interoperability can be seen as a set of needs that enterprises have to satisfy. However, these needs remain difficult to specify exhaustively considering the enterprise nature, organisation, skills etc. Thus, the description of an interoperability need can be ambiguous, not completely coherent and not complete. This is due to their specification based often on complex expression in natural language. As a consequence, to obtain an unambiguous and clear description of the interoperability needs, a formalisation of these needs is necessary. On the one hand, this formalisation allows to clarify and to structure each need as a set of requirements represented in a model of requirements. These requirements are done by the use of a limited set of concepts and terms coming from natural langage. On the other hand, this formalisation allows to become able to prove each need. This proof can be based on several techniques (expertise, behavioural simulation, formal approaches...). The here research work proposes that, formal approaches such as verification techniques will be used to verify if these requirements are satisfied by enterprises. This verification is performed thanks to requirements model applyed on collaborative model of enterprises. Moreover, to make this verification possible, formalisation of requirements must be made by (1) using formal languages such as mathematical language, and (2) an enrichment of the modelling languages used to describe the enterprises and their collaborative processes. The here proposed research focuses on interoperability requirements specification and verification. It allows:

• to characterise what are the needs of an enterprise which want to improve its interoperability and its ability to interact. The goal is to determine these needs by using different approaches such as literature analysis and questionnaire to conduct interviews. • to formalise these needs in the form of a set of unambiguous, and as formal as possible, requirements called "interoperability requirements". The main objective of this formalisation is to provide a requirement model allowing to define and structure interoperability requirements as proposed in different approaches such as [START_REF]REGAL: Requirements Engineering Guide for All[END_REF], [START_REF] Van Lamsweerde | The KAOS Project: Knowledge Acquisition in Automated Specification of Software[END_REF]. • to use this formalisation in order to verify (via formal verification techniques) if the enterprise components respect or not these requirements. This verification allows to highlight interoperability problems. The goal is to adapt existing techniques of verification. This adaptation has to consider (1) the model of interoperability requirements ; (2) existing modelling languages commonly used in enterprises and collaborative process modeling domain [START_REF] Petit | UEML Thematic Network -Contract n°: IST -2001-34229 -Work Package 1 Report -Enterprise Modelling State of the Art[END_REF]. This paper is structured as follows. Section 2 introduces interoperability needs with the presentation of different relevant works on the domain. An interoperability requirements model is presented and explained in section 3. To demonstrate the usefulness of the interoperability requirements model, a case study is briefly presented in section 4. Finally, section 5 presents a conclusion and outlines perspectives.

Interoperability Needs and Related Works

Enterprises face many challenges related to the lack of interoperability such as loss of money due to the necessary efforts for an effective collaboration. Thus, the development of interoperability is became a crucial question for enterprises that want to gain competitiveness in a globalized environment. As a consequence, several works have been developed from the last years in order to reply to the needs of interoperability expressed by enterprises.

For instance, numerous maturity models (LISI, LCIM, OIM…) [START_REF]Levels of Information Systems Interoperability (LISI)[END_REF], [START_REF] Tolk | The Levels of Conceptual Interoperability Model[END_REF], [START_REF] Clark | Organisational Interoperability Maturity Model for C2[END_REF] have been developed. These models allow systems to know its ability to interoperate (at a given moment) with regards to different interoperability levels. Some of these models give recommendations that allow a given system to evolve from its own interoperability level toward full interoperability.

In [START_REF] Daclin | Methodology for enterprise interoperability[END_REF] a methodology is proposed to measure interoperability of enterprises. This measure includes two measures such as compatibility measurement and interoperation performance measurement. To perform the measure of compatibility, a matrix of compatibility is used to inform enterprises on the attendance (or not) of interoperability problems. To perform measure of interoperation performance, some criteria such as cost of interoperation, time of interoperation and quality of interoperation are measured.

Furthermore, in [START_REF] Blanc | A graph based approach for interoperability evaluation[END_REF], characterisation of interoperability is proposed to evaluate performance of enterprise interoperability. This characterisation identifies requirements as "rules of interoperability". Five rules of interoperability are identified. However, these rules are generic and hardly difficult to exploit (e.g. rule 1 explains that a feedback loop is required between two interoperating systems).

To structure and capitalize these different works, these ones are integrated in an enterprise interoperability framework developed in [START_REF] Interop | Enterprise Interoperability-Framework and knowledge corpus -Final report[END_REF]. This framework considers two fundamental aspects to treat interoperability needs: the "interoperability barriers" and "interoperability concerns". The goal is to achieve an efficient collaboration by solving interoperability barriers (conceptual, organisational and technological) with regards to interoperability concerns (data, services, processes and business).

Based on these different works, this research work proposes and provides a model of interoperability requirements, a structure and a method of work (such as REGAL [START_REF]REGAL: Requirements Engineering Guide for All[END_REF]) that enables a user to be guided -with a minimum of omission -in the expression of its needs.

Interoperability Requirements Model

As mentioned before, interoperability can be seen as a set of needs (often ambiguous and complex) and furthermore as requirements (more formal, clear and unambiguous) that enterprises must satisfy to be interoperable. A requirement can be defined as "a statement that specifies a function, ability or a characteristic that a product or a system must satisfy in a given context" [START_REF] Scucanec | A day in the life of a verification requirement[END_REF].

Generally speaking, interoperability is related to the concept of compatibility. Compatibility means to harmonize enterprises (method, organisation, tool...) together so that heterogeneous information exchanged can be understood and exploited by each ones without interfacing effort. However, compatibility is not sufficient to formulate interoperability requirements. In fact, during the collaboration between two enterprises, many problems, other than compatibility, can occur. For instance, two enterprises use the same language to code their data, but one of them is unable to send its data. In this case, "two interoperable systems are compatible, but two compatible systems are not necessarily interoperable" [START_REF] Kasunic | Measuring systems interoperability: challenges and opportunities. Software engineering measurement and analysis initiative[END_REF]. This statement leads to consider a second kind of interoperability requirements called "interoperation requirements". These requirements are related to the interaction between enterprises during the collaboration. It means that enterprises can work efficiently together when interacting.

Moreover, after working enterprises wish to retrieve their performance and autonomy while remaining efficient. For instance, after collaboration, one of these enterprises may lose performance that it had before this collaboration. In this case, it is question to be sure that a given enterprise will reach again, its own performance after the end of collaboration. Thus, interoperability considers another concept such as the reversibility [START_REF] Interop | Enterprise Interoperability-Framework and knowledge corpus -Final report[END_REF]. Reversibility means that enterprises can return to their original state and retrieve their autonomy at the end of collaboration with regards to its own performance and including positive and/or negative variation that accepted. As a consequence, requirements related to the reversibility of enterprises are also required.

In summary, interoperability requirements can be decomposed in three categories such as: compatibility requirements, interoperation requirements and reversibility requirements. For instance, these three categories are fully in agreement with the virtual enterprise life cycle (creation, operation and dissolution) where the implementation of interoperability plays a preponderant part [START_REF] Camarinha-Matos | Infrastructure developments for agile virtual enterprises[END_REF]. The relationship which consistently relates these three categories with the VE life cycle is highlighted in figure 1. Thus, two enterprises which aim to work together must, firstly create the collaboration, then collaborate efficiently, and finally stop the collaboration. The creation phase takes place before collaboration when a business opportunity is detected. The need is to know if the future collaboration can take place or not in good condition. Compatibility requirement are deployed and verified (e.g. during the collaboration, if a document from a given format is received by a person who dispose of an earlier version, he cannot read this document). Through the runtime phase, a good interoperation is required for an efficient collaboration (e.g. if a product is received later than required in an enterprise, it can retard the continued collaboration). The dissolution phase takes place after the collaboration where reversibility of enterprises is required to return to their original state. Furthermore, the set of requirements in each category (compatibility, interoperation and reversibility) must be represented clearly and in an unambiguous way. For this reason, the principle of a requirement model as a causal tree is developed. The concept of causal tree allows to represent the refinement of interoperability requirements by a causal relation linking an abstract requirement to more precise requirements with logical function. Thus, the causal tree allows with the refinement to reduce the residual ambiguity of requirements that may exist for each level as shown in figure 2 and presented in [START_REF] Chapurlat | Interoperability constraints and requirements formal modelling and checking framework[END_REF]. This research work proposes a causal tree to represent refinement of interoperability requirements. The first node represents the first interoperability requirement that means enterprises aim to be interoperable. This first requirement can be refined by the three categories of interoperability requirements: compatibility, interoperation and reversibility. Requirements of each category must be refined by concepts related to the INTEROP framework (interoperability concerns and interoperability barriers) [START_REF] Interop | Enterprise Interoperability-Framework and knowledge corpus -Final report[END_REF]. Therefore, each category can be refined by interoperability concerns which are refined by interoperability barriers as shown in figure 3.

node(i.

Fig. 3. The Interoperability requirements causal tree

The definitions and concepts related to the three categories of interoperability requirements are presented in following sections.

Compatibility Requirements

A compatibility requirement is defined as "a statement that specifies a function, ability or a characteristic, independent of time and related to interoperability barriers for each interoperability concerns, that enterprise must satisfy before collaboration effectiveness".

Compatibility requirements are derived from the enterprise interoperability framework. They are concerned by the compatibility of interoperability barriers which includes: conceptual (syntax and semantic), technological (platform, communication protocol) and organisational barrier (persons, organisational structure) at each level of enterprise. It is to note that compatibility requirements remain independent of time and have a static aspect.

Thus, a set of compatibility requirements is established taking into consideration these concepts. The following figure represents a set of compatibility requirements structured in the causal tree.

Fig. 4 Compatibility requirements structured into a causal tree An example of compatibility requirement at the level of services and related to organizational barrier is described as: "Services must have a clear and defined responsible", in order to know who is responsible for doing what.

To help enterprises to see where their compatibility problems are, verification of compatibility requirements can be performed. From this verification, enterprise will be informed of their compatibility problems in agreement with the three barriers of interoperability and the four levels where interoperability take place in enterprises. After, enterprises will have the opportunity to solve their problems.

Interoperation Requirements

An interoperation requirement is defined as "a statement that specifies a function, ability or a characteristic, dependent of time and related to the performance of the interaction, that enterprise must satisfy during the collaboration".

In the runtime phase of collaboration, enterprises want to know if the interoperation is efficient. The performance of the interoperation can be measured by some performance criteria such as: the cost, the quality and the time of the interoperation as mentioned in [START_REF] Daclin | Methodology for enterprise interoperability[END_REF]. For instance, time criterion can be related to the duration of the interoperation and the quality criterion can be related to the quantity received. It is to note that interoperation requirements are dependent of time and have a dynamic aspect.

As a consequence, a set of interoperation requirements is established according to these concepts and structured in the causal tree as shown in the following figure. An example of interoperation requirements at the level of data related to technological barrier for quality performance can be described as: "for each data received, a receipt must be returned" to be sure of the good reception of data.

To help enterprises to see if the interoperation is efficient, verification of interoperation requirements can be performed. From this verification, enterprise will be informed of their interoperation problems in according to their performance criteria. After, enterprises will have the opportunity to solve their interoperation problems.

Reversibility Requirements

A reversibility requirement is defined as "a statement that specify functions, abilities or characteristics related to the capacity of enterprises to retrieve their autonomy and to return to their original state (in terms of their own performance) that they must satisfy after collaboration".

During collaboration, enterprises working together are involved in a collaborative public process. This public process can implicate a part of (or whole) activity of an enterprise. On the one hand, enterprises must be able to ensure their own performance related to its private process and, on the other hand the performance related to the public process as shown in Fig 6 . Therefore, as mentioned above, usually after collaboration, enterprises want to retrieve their original own performance. This means that, if the implementation of interoperability leads to adaptations or modifications (working methods, tools, organisational structure ...) in enterprises, they must be able to return to their original state at the end of the collaboration. Therefore, enterprises want to know if their performance after the collaboration is in accordance with their performance before the collaboration including variations (e.g. enterprises can save time production) which can be admitted by enterprises. In this case, enterprises are reversible and can resume their former activities. An example of reversibility requirements at the level of service related to organisational barrier for quality criterion can be given as: "the cost of a given service after the collaboration corresponds to the cost before collaboration including variations".

To help enterprises to see if they are able to retrieve their own performance, verification of reversibility requirements can be performed.

In order to illustrate the usefulness of the proposed interoperability requirements model, a case study is presented in the next section.

Case Study

The collaborative process issued from [START_REF]Implementation methodology -Blue Print -Demo TelCo[END_REF] represents the collaboration between a retailer and a provider. The sales department of the provider want to collaborate with external and independent retailers in charge of delivering products to customers. The different activities and flows between partners (retailer and provider) are shown in figure 8. The goal is to detect if compatibility, interoperation and reversibility problems can occur, thanks to the interoperability requirements model. Before the retailer sends its order, several compatibility requirements have to be verified. As far as organisational aspect is concerned, one requirement that must be verified could be: "A service must have a clear and defined responsible". Indeed, if this requirement is not verified, it may dread mistakes during the transmission of orders, that could lead to a deterioration of collaboration performance (loss of time to convey the right order to the right person...).

Then, during the collaboration, it is necessary to verify some interoperation requirements. One requirement considered here can be: "For each data received, a receipt must be returned". For instance, the reception of invoices has to induce, in all cases, the generation of an acknowledgement by the retailer.

When the collaboration is over, it is necessary to verify that the collaboration has not deeply impacted behaviour, organisation, and practices of partners. Thus, verification of reversibility requirements is required. A reversibility requirement considered here, could be: "Internal functioning rules(e.g. logistics rules) of a partner P 1 must remain the same all along the collaboration and not be modified suddenly and under the control of another partner P n , excepted if this modification are proposed and validated by P 1 itself". The verification of this requirement allows to detect if new (or deeply modified) logistics rules used by sales department (P 1 ) during the activity named 'to check available quantities' emerge independently of the sales department.

Conclusion

The current economic context forces enterprises to work in collaboration to improve their performances. Therefore, formalisation, structuration and verification of interoperability requirements to help enterprises to find their interoperability problems can be a solution to improve this collaboration. This paper focused on the conceptualisation of interoperability requirements. Three categories of interoperability requirements were identified and defined: compatibility, interoperation and reversibility requirements.

To help enterprises to find their interoperability problems, analysis of interoperability requirements can be performed. The provable requirements which can have a static or a dynamic aspect will be analysed by formal verification techniques. There are two techniques related to the types of the provable interoperability requirements (static or dynamic). The first technique allows to verify static requirements as shown in [START_REF] Chapurlat | Interoperability constraints and requirements formal modelling and checking framework[END_REF] based on Conceptual Graphs. The second technique allows to verify dynamic requirements using model checker (Uppaal, SMV…) [START_REF] Behrmann | A tutorial on Uppaal[END_REF]. The requirements that are not provable will be analysed by experts. Future works are related to the formalisation of interoperability requirements into properties with a formal language to verify them later with formal verification techniques.

Fig. 1 .

 1 Fig. 1. The three interoperability requirement categories related to the VE life cycle [adapted from 13]

Fig. 2 .

 2 Fig. 2. Requirements refinement structured in a causal tree

Fig. 5 .

 5 Fig. 5. Interoperation requirements structured into a causal tree

Fig. 6

 6 Fig. 6 Reversibility of enterprises A set of reversibility requirements according to the performance of the enterprise measured by cost, quality and time, are structured in the causal tree as shown in the following figure.

Fig. 7 .

 7 Fig. 7. Reversibility requirements structured into a causal tree

Fig. 8 .

 8 Fig. 8. Collaborative process between provider/retailer