Emmanuel Doumard
email: emmanuel.doumard@irit.fr

Elodie Escriva
email: elodie.escriva@kaduceo.com

Jean-Baptiste Excoffier
email: jeanbaptiste.excoffier@kaduceo.com

Paul Monsarrat
email: paul.monsarrat@univ-tlse3.fr

Chantal Soulé-Dupuy
email: chantal.soule-dupuy@irit.fr

A comparative study of additive local explanation methods based on feature influences

Keywords: Explainable Artificial Intelligence (XAI), Prediction explanation, Machine learning

Local additive explanation methods are increasingly used to understand the predictions of complex Machine Learning (ML) models. The most used additive methods, SHAP and LIME, suffer from limitations that are rarely measured in the literature. This paper aims to measure these limitations on a wide range (304) of OpenML datasets, and also evaluate emergent coalitional-based methods to tackle the weaknesses of other methods. We illustrate and validate results on a specific medical dataset, SA-Heart. Our findings reveal that LIME and SHAP's approximations are particularly efficient in high dimension and generate intelligible global explanations, but they suffer from a lack of precision regarding local explanations. Coalitional-based methods are computationally expensive in high dimension, but offer higher quality local explanations. Finally, we present a roadmap summarizing our work by pointing out the most appropriate method depending on dataset dimensionality and user's objectives.

INTRODUCTION

Machine Learning (ML) represents a real revolution in various domains, such as finance, insurance, healthcare, biomedical. However, machine learning models give a prediction without necessarily being accompanied by an understandable explanation. These models, often referred as "black-boxes", raise the challenging question of how humans can understand the determinants of the prediction. Explainability is also more than a technological problem, it involves among other ethical, societal and legal issues. In healthcare, this may involve the professional being able to explain to the patient how the algorithm works and the criteria for the decision process. The results of ML models must therefore be expressed in a way that can be understood by domain-experts, like medical practitioners [START_REF] Amann | Explainability for artificial intelligence in healthcare: a multidisciplinary perspective[END_REF][START_REF] William K Diprose | Physician understanding, explainability, and trust in a hypothetical machine learning risk calculator[END_REF]. Since SHAP [START_REF] Scott | A Unified Approach to Interpreting Model Predictions[END_REF], machine learning experts show a very clear interest for the additive methods as a huge number of works using these methods are published each year. The additive methods include LIME [START_REF] Ribeiro | Why Should I Trust You?[END_REF], SHAP [START_REF] Scott | A Unified Approach to Interpreting Model Predictions[END_REF] and more recently the coalitional-based methods [START_REF] Ferrettini | Coalitional Strategies for Efficient Individual Prediction Explanation[END_REF]. The userfriendly representation of explanations, based on feature influences, allows domain and non-domain experts to better understand models predictions [START_REF] Molnar | A guide for making black box models explainable[END_REF]. Existing explanation methods are model-specific or model-agnostic depending on whether they can be applied to some or all types of machine learning models, with local or global explanations to understand either an individual prediction or the behaviour of the model as a whole. While these methods have been evaluated in a number of contexts, no in-depth evaluation is available for a rational choice of one technique over another. The objective of this work is to study the advantages and disadvantages of using each additive method to provide pertinent insights. In particular, we study the effects of the models used and the type of dataset considered on the feature influences (both at the instance and feature level).

The paper is organised as follows. Section 2 reviews the existing work, classifying and comparing explanation methods for tabular data. Section 3 describes the four additive methods to be compared in this paper. The experiments are presented in Section 4 where we study the explanation characteristics, the impact of the predictive model on explanation profiles, highlighting the behavior of explanation methods based on a practical medical use case. Conclusive lessons-learned are then detailed in Section 5.

RELATED WORKS

Few works [START_REF] Burkart | A Survey on the Explainability of Supervised Machine Learning[END_REF][START_REF] Vilone | Notions of explainability and evaluation approaches for explainable artificial intelligence[END_REF] exist in the literature to classify and categorize machine learning explanation methods. In [START_REF] Molnar | A guide for making black box models explainable[END_REF], a complete description of explanation approaches from literature is given. In particular, the authors explain their advantages and disadvantages, giving an overview of their limits. For example, even if the LIME and SHAP approaches are model-agnostic and humanfriendly, they suffer from no consideration of feature correlation and possible instability of the explanations. Another paper tackling the limits of the additive methods (LIME and SHAP) is presented in [START_REF] Slack | Fooling LIME and SHAP: Adversarial Attacks on Post hoc Explanation Methods[END_REF]. The paper shows that biased classifiers can fool explanation methods, whose problem is even more accentuated on LIME.

Comparative studies between local explanation methods are also available, such as [START_REF] Shawi | Interpretability in HealthCare A Comparative Study of Local Machine Learning Interpretability Techniques[END_REF][START_REF] Ferrettini | Coalitional Strategies for Efficient Individual Prediction Explanation[END_REF][START_REF] Man | The Best Way to Select Features? Comparing MDA, LIME, and SHAP[END_REF]. In [START_REF] Ferrettini | Coalitional Strategies for Efficient Individual Prediction Explanation[END_REF], a new additive method was proposed based on Shapley values and taking into account feature correlation. This method was compared with LIME and SHAP through computation time and accuracy score. For this last measure, the authors consider as baseline the complete method, computing all Shapley values with each possible coalition of features. The authors show that their proposal is competitive with the literature, both in accuracy and in computation time. In [START_REF] Man | The Best Way to Select Features? Comparing MDA, LIME, and SHAP[END_REF], LIME and SHAP are used in a context of feature selection and compared to a Mean Decrease Accuracy (MDA) approach. A stability measure indicates that a feature selection obtained with LIME or SHAP seems more stable than via MDA. In [START_REF] Shawi | Interpretability in HealthCare A Comparative Study of Local Machine Learning Interpretability Techniques[END_REF], the authors compare 6 local model-agnostic techniques using custom quantitative measures, such as similarity, bias detection, execution time, and trust. From these experiments, no single method stands out for all metrics and all data sets. Each ones have strengths and weaknesses based on the metrics used and choices between methods can only be made based on the users' goal and dataset.

The latest results are therefore indications for the absence of a single method that would provide the best explanations in all situations. However, none of these previous works clearly indicates in which situation a method should be preferred to another one. Consequently, our aim is to give the key factors to make an informed decision among the existing additive methods. As indicated in [17], evaluating explanations methods is very subjective and no consensus yet exists to propose relevant metrics. As all additive methods give an influence score for each feature, we propose to compare them based on these influences. From there, we want to analyse and compare the effects of different predictive models and dataset on these influence scores.

ADDITIVE METHODS TO COMPARE

Additive methods are described as explanation models that produce a vector of weights to represent the influence of each feature, the sum of which approximates the output of the original model. Explanations can be computed for a single instance, so for every instance of a data set, hence the term "local". In this section, we explore several existing methods that fit this definition. We focus on post-hoc methods that deliver their explanations for a given model already trained. Methods used in this study are all agnostic, meaning that they can be applied to any kind of machine learning model, except for the TreeSHAP method [START_REF] Scott M Lundberg | Consistent individualized feature attribution for tree ensembles[END_REF] that is designed specifically for tree-based models.

LIME

LIME method is a well-known local explanation method described in [START_REF] Ribeiro | Why Should I Trust You?[END_REF]. LIME uses explainable models to locally approximate a complex black-box model and, for each instance, explain the influence of each feature on the prediction. For each instance to be explained, LIME generates new data in a close neighborhood and computes the predictions of these new instances with the blackbox model. A regressor linear model, an interpretable model, is trained with the new dataset. This local model is then used to explain the prediction of the instance of interest in the form of a weight vector associating each feature with its influence on the prediction. A well-known limitation of LIME is the restrictive hypothesis on which LIME is based, such as local linearity and feature independence [START_REF] Garreau | Explaining the Explainer: A First Theoretical Analysis of LIME[END_REF][START_REF] Slack | Fooling LIME and SHAP: Adversarial Attacks on Post hoc Explanation Methods[END_REF]. Defining the locality around an instance of interest can also be a challenge, as the fit of the surrogate model has a significant impact on the accuracy of the explanations [START_REF] Laugel | Defining Locality for Surrogates in Post-hoc Interpretablity[END_REF] as well as their stability [START_REF] Elshawi | Interpretability in healthcare: A comparative study of local machine learning interpretability techniques[END_REF].

The full implementation of LIME is available on GitHub : https://github.com/marcotcr/lime.

Shapley Values (complete method)

To explain individual predictions, a method based on Shapley values is described in [START_REF] Štrumbelj | Towards a model independent method for explaining classification for individual instances[END_REF][START_REF] Strumbelj | An Efficient Explanation of Individual Classifications Using Game Theory[END_REF][START_REF] Štrumbelj | Explaining prediction models and individual predictions with feature contributions[END_REF]. Shapley values 'fairly' weight groups of features according to their relative importance to a defined gain [21]. In machine learning, the gain can be linked to the prediction made by the model. Influences of each feature are computed based on its impact on the prediction for each coalition of features. The explanation method based on Shapley values is called the complete method. All coalitions are evaluated with and without each feature and the change on the prediction is used to compute the influence of the feature. The complete method can be used as a baseline to compare other methods as it is an exhaustive method close to the original intuition behind feature influence [START_REF] Ferrettini | Coalitional Strategies for Efficient Individual Prediction Explanation[END_REF]. This method is however very expensive to compute, with an exponential complexity in relation to the number of features in the dataset.

Several more recent methods, including SHAP [START_REF] Scott | A Unified Approach to Interpreting Model Predictions[END_REF] and coalitional methods [START_REF] Ferrettini | Coalitional Strategies for Efficient Individual Prediction Explanation[END_REF], are based on Shapley values with the aim to solve limitations of the complete method.

SHAP

SHAP (SHapley Additive exPlanations) [START_REF] Scott | A Unified Approach to Interpreting Model Predictions[END_REF] method worked on improving computation time and explanation precision, especially for tree-based models [START_REF] Scott M Lundberg | Consistent individualized feature attribution for tree ensembles[END_REF]. It combines LIME [START_REF] Ribeiro | Why Should I Trust You?[END_REF] and Shapley values [START_REF] Štrumbelj | Explaining prediction models and individual predictions with feature contributions[END_REF], along with other methods from the literature [START_REF] Bach | On Pixel-Wise Explanations for Non-Linear Classifier Decisions by Layer-Wise Relevance Propagation[END_REF][START_REF] Datta | Algorithmic Transparency via Quantitative Input Influence: Theory and Experiments with Learning Systems[END_REF][START_REF] Lipovetsky | Analysis of regression in game theory approach[END_REF][START_REF] Shrikumar | Learning Important Features Through Propagating Activation Differences[END_REF], in a unique framework to produce local explanations. The main idea is to create perturbations to simulate the absence of a feature and to use a linear local model to approximate the change in the prediction, as in LIME. This avoids retraining the complex model without the feature of interest. Local explanations can be aggregated to explain the global behaviour of the model. Global and local explanations are then consistent with each other as they have the same foundation. SHAP includes an agnostic explainer, KernelSHAP, as well as model-specific explainers, such as TreeSHAP, LinearSHAP or DeepSHAP for tree-based models, linear models and deep models respectively. While commonly used in Machine Learning context [START_REF] Linardatos | Explainable ai: A review of machine learning interpretability methods[END_REF], SHAP still suffers from lack of precision [START_REF] Kumar | Problems with Shapley-value-based explanations as feature importance measures[END_REF][START_REF] Slack | Fooling LIME and SHAP: Adversarial Attacks on Post hoc Explanation Methods[END_REF] mostly due to their restrictive hypothesis (local linearity and feature independance) as with LIME. Moreover, computation time is still high for other models than tree-based models [START_REF] Van Den Broeck | On the tractability of SHAP explanations[END_REF].

The full implementation of SHAP is available in GitHub : https://github.com/slundberg/shap.

Coalitional-based method

Another agnostic explainer based on Shapley values, the coalitional method, was introduced to take into account the interdependence of features and solve some restrictions of SHAP. It uses grouping methods such as Principal Component Analysis (PCA), Spearman correlation factor (Spearman) and Variance Inflation Factor (VIF) to pre-compute groups of features for explanations [START_REF] Ferrettini | Coalitional Strategies for Efficient Individual Prediction Explanation[END_REF]. These groups are then used as coalitions to compute Shapley values as in the complete method. The influence of each feature is defined as its impact on the prediction only on the pre-computed groups of features, approximating the complete method and reducing the computational time. Grouping methods are defined with a parameter that changes the number and size of feature groups

EXPERIMENTS

In this section, we propose experiments comparing the explanation methods presented in the previous section. The goal is to identify the general behavior of each method and how this behavior eventually differs according to a predictive model (learned from data) and the dimensionality of the data (number of features).

Experimental protocol

All experiments are run on an Intel Xeon Gold 6230 processor with 125 GB of RAM using Python 3.9.7. All runs are performed on a single core of CPU for optimization and reproducibility. To compare explanation methods, we apply them to a wide range of 304 datasets available on OpenML (www.openml.org). Due to computational constraints of explanation methods, we only considered datasets with at most 13 features, and at most 10 000 instances. We also only considered classification tasks to use comparable predictive models and metrics. We describe the amount and size of datasets per number of features in Table 1.

As an explanation method needs a model to be applied to, we choose four widely used types of ML models for classification: Logistic Regression (LR), Support Vector Machines (SVM), Random Forests (RF) and Gradient Boosted Machines (GBM). For the first three, we use the implementation of Python library scikitlearn version 1.0.1. For GBM, we use the Python library XGBoost version 1.5. We use default values for models hyperparameters. For explanations methods, we use Python libraries shap 0.40 and lime 0.2.0.1.

Then, to be able to compare the explanations, we need to define metrics of interest. In Section 4.2, we present three metrics that we will use for this study.

Section 4.3 aims to compare the four additive methods introduced in Section 3. In particular, we use two distinct coalitionalbased methods: the Complete method, which serves as reference for an influence deviation measurement (second metric), and the Spearman method with a threshold of 25% of all groups of features. Regarding SHAP, we use the model-agnostic KernelSHAP on all datasets. As this method is very slow to execute if we use the whole dataset as background samples for permutations, we choose to follow SHAP's recommendation1 by doing a K-Means clustering on the input dataset, and then taking the centroids as background samples. We choose 𝐾 = 10 clusters for each dataset, thus naming the method KernelSHAP10. In addition, for the two tree-based predictive models XGBoost and Random Forests, we use the model-specific explainer TreeSHAP by two implementations. The first one determines SHAP values with background samples, similarly to KernelSHAP but optimised for tree-based methods. We use the whole dataset as background samples for this method. The second one approximates SHAP values by considering the trees structures, and does not need background samples in input, so we name it TreeSHAPapprox. Last, we consider LIME, which requires a number of perturbed samples to be created for explaining each instance. We choose to set this number to 100 samples for all datasets.

With similar methodology, Section 4.4 identifies the impact of the predictive model on specific explanation methods.

Lastly, we present in Section 4.5 a practical example of the different explanations methods applied to a specific dataset, SA-Heart. This dataset is chosen for its medical context (coronary heart diseases), a sufficient number of instances (462) and features [START_REF] Kumar | Problems with Shapley-value-based explanations as feature importance measures[END_REF] to train a coherent model and compute the explanations in acceptable computational times. The underlying idea is to illustrate the highlighted behaviors by taking a concrete example as it could be used by an end user (e.g. a physician).

Metrics of interest

Because of the subjective nature of explanations, there is no consensus on objective mathematical ways to evaluate the explanations. Therefore, to evaluate explanation methods performances and compare them over a high number of datasets, we define three different metrics that only need the influence values given by the method. The first one is the computational time per instance, which is the amount of time taken by a given method to compute the local influences of a whole dataset, divided by the number of instances in the dataset. The second one is a quantification of the average deviation of the influence given by a method from the Complete method (see Section 3). This error rate is defined as:

𝑒𝑟𝑟 (𝐼, 𝑋 ) = 1 𝑛 𝑛 𝑖=1 1 𝑝 𝑝 𝑘=1 𝐼 𝑘 (𝑋 𝑖 ) -𝐼 𝐶 𝑘 (𝑋 𝑖 )
where, for a given a dataset, 𝑛 the number of instances, 𝑝 the number of features, 𝑋 𝑖 the features vector for the instance 𝑖, 𝐼 𝑘 (𝑥) the influence of a feature 𝑘 for a given instance 𝑥, a given explanation method and a given machine learning model, and 𝐼 𝐶 𝑘 (𝑥) the influence given by the Complete method for the same model, same feature and same instance. The third metric evaluates the distribution of feature importance assigned by a given explanation. The raw value being not necessarily comparable between explanation methods, the cumulative importance proportion of features given by a method was considered. This metric shows whether an explanation method favours the attribution of great importance to a few features or, on the contrary, a more homogeneous distribution among a larger number of features. The importance of a feature is defined here as the mean absolute value of influence assigned to instances for such feature. For example, in a dataset with 2 features, if a method gives 80% of the importance to the most important feature (and so 20% to the second), it would have a cumulative importance proportion vector of [0, 0.8, 1]. We can then compute the (normalised) Area Under Curve (AUC) of such a vector 𝐶 with :

𝐴𝑈 𝐶 (𝐼, 𝑋 ) = 1 𝑝 𝑝-1 𝑖=0 𝐶 𝑖 + 𝐶 𝑖+1 2
where 𝐶 𝑖 is the total importance proportion taken by the 𝑖 most important features.

As this cumulative sum is sorted by construction from most important to least important features, this value is bound between 0.5 and 1. A value of 0.5 means that the explanation method gives the same importance to all features, while a value of 1 means that the explanation method gives non-zero influences only to a single feature, explaining the model's predictions with a single feature.

Additive methods comparison

We show in Figure 1 the evolution of the execution time of each method for each predictive model, averaged over datasets that share the same number of features. LIME, having a linear complexity with the number of features, is computationally expensive compared to other methods in low dimension (few features), but is less expensive than coalitional-based methods and KernelSHAP in higher dimensions. LIME also seems to have very low interdataset time variability, resulting in smaller error bars on the graph. Coalitional-based methods show an exponential complexity with the number of features, having high execution time in high dimension, but have a similar execution time with other methods in low dimension. Spearman method execution time seems naturally correlated to the Complete method execution time, taking a fraction of the time (roughly 25%) of the Complete method. KernelSHAP, despite a limitation on the amount of background samples, has a high execution time in high dimension, comparable to coalitional-based methods for non-tree based methods. For tree-based methods, KernelSHAP is slower in low dimension, but faster in high dimension than coalitional-based methods. Last, tree-based explainers seem to have constant execution time per instance no matter the number of features, and the approximate tree path dependent version of TreeSHAP has the lowest execution time per instance.

Regarding the second metric, Figure 2 shows the average absolute difference in influence between each method and the Complete method (reference). First, we can see that overall, the more features there are in a dataset, the closest (measured by the second metric) the influences are to the Complete method. This is probably due to the fact that usually, the more features there are, the less influence amplitude each individual feature has in the prediction. We also note that no matter the model, common methods are ranked in the same way. In low dimension (less than Figure 1: Execution time of each method per instance, averaged by number of features, for each model 6 features), KernelSHAP is the closest to the Complete method, followed by Spearman, while LIME is the the farthest. In higher dimensions, Spearman becomes more precise than KernelSHAP. TreeSHAP (both the approximate and the data dependent version) is more precise than KernelSHAP, but still less precise than Spearman in high dimensions. Note that the approximate version of TreeSHAP is not showed on the graph for XGBoost because its implementation forces its SHAP values to be in log odds instead of probabilities, making it impossible to compare to other methods. Finally, we show in Figure 3 an example of the graphical representation of the cumulative feature importance proportion. The figure shows the averaging of the cumulative importance proportion of the most-important features for the 37 datasets having 10 features. This way, for each predictive model and for each method, we obtain a curve from which we compute the third metric: the AUC of the curve. We see on the figure that some methods present steeper curves than others. For example, with Logistic Regression and SVM, LIME gives less proportion of the total importance to the few first most-important features, compared to coalitional-based and SHAP methods. For tree-based models, we see that SHAP, no matter the method, gives much more importance to the first few most-important features than the other methods. According to the method for computing AUC illustrated in Figure 3, we represent the average values of AUC for datasets from 2 to 13 features for each ML model and explanation method in Figure 4. For all models, we can see that SHAP methods tend to produce influences with a higher AUC compared to other methods. This means that SHAP methods tend to assign most of the feature importance to fewer most-important features, while other methods tend to distribute the feature importance more uniformly over all features. The two coalitional-based methods seem to generate similar AUCs for the features importance. Finally, LIME tends to produce influences with lower AUCs for non-tree-based methods, while it produces AUCs closer to the coalitional-based methods for tree-based methods.

Machine Learning models explanations comparison

We show in Figure 5 the computational time per instance needed to compute the explanations of each predictive model, for each explanation method. We can see that LIME's execution time has almost no intermodel variability: the computation time per instance is the same no matter the model. For the other methods, the ranking of the method's computational performances according to the model is roughly the same, from slowest to fastest: Random Forests, XGBoost, SVM and Logistic Regression. SVM has overall higher variability, presenting steeper curves and higher error bars. SVM even presents outlying results when applied to KernelSHAP in higher dimensions. Overall, we do not observe specific behavior This may be related to the fact that TreeSHAPapprox only considers tree structures, as Random Forests tree structures are simpler than XGBoost's. In general, the faster a model is to train and predict values and the simpler it is, the faster the explanations are to compute, no matter the method, We present in Figure 6 the mean absolute difference between each method applied to each model and the Complete method applied to each model. The figure does not present the results for TreeSHAPapprox because the only relevant model for this method is Random Forests, there is no other model to compare the results with.

For the three model-agnostic methods (LIME, KernelSHAP and Spearman), the Logistic Regression and SVM models generate the most precise explanations compared to the Complete method on the same models. We can see that the explanations based on Logistic Regression are usually more precise than SVM's, especially in low dimensions. XGBoost explanations are less precise than Random Forest's, except for the Spearman method (similar results observed). Overall, it seems that the simpler the model, the more precise it is in regards to the Complete method. We observe that for LIME and KernelSHAP, there is no significant difference between the AUC of the model's explanations. However, for the coalitional-based methods, we can see a clear separation between tree-based methods and non tree-based methods: the latter have higher AUC than the others. This means that, when using coalitional-based methods, one should be aware that different models may yield a different importance distribution over the features. For the tree-specific methods, we can see that XGBoost generates explanations with slightly higher AUCs than Random Forests on average.

Example on a medical dataset

Amongst the OpenML datasets previously studied, we choose a medical dataset, SA-Heart, to compare the explanations given by the different additive methods on an example. This way, we aim to both illustrate and validate the conclusions of the previous sections regarding explanation methods characteristics. We also aim to highlight practical differences that we can see on the influences of different methods for the same model and dataset.

SA-Heart is a dataset extracted from a larger database of South-Africans detailed in a 1983 study [START_REF] Rossouw | Coronary risk factor screening in three rural communities-the CORIS baseline study[END_REF]. The extracted dataset is a After model training, the different explanatory profiles obtained between the different methods of explanation are compared. By considering a reflection on the end-user side, the health care practitioners, explanatory profiles should be used 1) at the population level (global explanations), for example to highlight high-risk patient profiles, develop new prevention programs, develop new physio-pathological hypotheses but also 2) at the instance level (local explanations), for personalized medicine.

For conciseness in this paper, we limit the analysis to a single machine learning model. We choose Random Forests, as every explanation method that we benchmark is applicable to it. We present the results with SVM, Logistic Regression and XGBoost models in supplementary data.

To compare the explanations of the different additive methods, we look at global explanations given by each method. We use SHAP-like representations to visualize global explanations by aggregating local explanations on the same representation. This way, we build different figures. The first one, in Figure 8, represents a global explanation of the predictive model, given by each explanation method, by plotting the explanation profile of each feature on a separated line. For each method, the features are sorted in decreasing feature importance, the top one being the most contributing feature on average, while the bottom one being the least contributing feature on average. For each feature, each dot represents an individual from the dataset, its color representing the value of the associated feature. Its position on the x-axis represents the contribution of the feature to the prediction of this individual, and overlapping dots are jittered on the y-axis.

We can see that most of the features have similar ranking among the different methods: tobacco and age are the two most important features except for the Spearman method which ranks age 5 th . On the opposite side, alcohol, spb, and type-A are always in the 4 least important features. These features have also similar explanation profiles. Conversely, some other features exhibit more marked difference depending on the methods. The most important difference is observed on the binary feature family history of heart disease. This feature is assigned fairly low importance by the coalitional-based method, relatively high importance (3rd most important feature) by SHAP methods, and very high importance by LIME (most important feature). Obesity and adiposity have also different influences depending on the method: obesity is ranked second least contributing by LIME and SHAP, but more important by the coalitional-based methods. It is important to note that obesity and adiposity are highly correlated (Pearson's correlation r=0.72). We hypothesize that it may be the reason for such differences. Overall, the three SHAP methods give similar explanations and have almost identical ranking of the features. From a global perspective, we can also see that SHAP and LIME present a more homogeneous "gradient" of colors for the explanations, where coalitional-based methods present mixed up colors in the explanations. This means that LIME and SHAP's explanations are more locally monotonic, in the sense that the influence value of a feature for an individual is more locally correlated to the value of the feature for LIME and SHAP than it is for coalitional-based methods.

The second visualization that we present are Partial Dependence Plots (PDP). PDPs focus on the relationship between a feature and the influence of this feature on the model's prediction by plotting each pair of feature value and influence value on a 2-dimensional axis. We compare the PDPs of several important features in Figure 9.

Looking at the PDPs for the age feature, we show that LIME seems to form clusters of points around specific cut-off age values. To a lesser extent, this phenomenon can also be seen on the other SHAP methods. Conversely, coalitional-based methods have similar PDPs, and do not seem to find such cut-offs. However, it seems to be a special behavior of the explanation at specific ages. For example, subjects around 50 years have a marked lower contribution of this feature to the prediction of the presence of coronary heart disease than people even slightly younger or older. This may hint at an over-fitting of the machine learning model that would not have been captured by the other explanation methods. The explanation of the tobacco feature also largely differs among explanation methods. Where all the methods agree on attributing a low value to non-smoking individuals, the evolution of the contribution varies with the quantity of tobacco. Once again, LIME and SHAP explanations seem to find a cut-off value for tobacco consumption, of around 7 and 9 respectively, while coalitional-based methods capture a non-monotonic, more complex relationship.

We also look at adiposity PDPs. Once again, the three SHAP explanations are close to each other. Interestingly, they capture a non-monotonic relationship between the feature and the outcome, giving people around 30% of adiposity a higher influence for this feature (in absolute value) than people close to this value. This relationship seems to be captured in a lesser extent by coalitional-based methods, but not captured at all by LIME. We also note that the Complete and Spearman influences are more scattered, which means that more variance exists amongst subjects of the same adiposity for these methods than for the others.

Lastly, looking at obesity PDPs, LIME and SHAP methods find a negative relationship between obesity and the chd prediction. This seems counter intuitive, as obesity is a strong known comorbidity factor of heart diseases. As previously mentioned, obesity and adiposity are strongly correlated (r=0.72), and it may be the reason for such observation. Furthermore, we have mentioned in section 3 that SHAP works under the hypothesis that features are independent, but with such correlation, it is very unlikely that obesity and adiposity are independent. To better understand the relationship between these two features, as found by the methods, we plot in Figure 10 the influence values of adiposity and obesity given by each method.

The Complete and Spearman methods seems to find a positive correlation between the influences of the two features: when an individual is assigned a high influence value for obesity, a high influence value for adiposity is usually assigned, and conversely. We can even distinguish two clusters of individuals: one for individuals that have a high influence value for both features, and one for individuals that have a low influence value for both features. Such pattern is not found by LIME or SHAP, thus confirming the lack of ability of these methods to consider dependent features.

On a more global scale, we see that LIME and SHAP produce explanations that are easier to read at a first glance compared to Complete and Spearman explanations. However, LIME and SHAP seem to capture different cut-offs and relationships, and it is hard to confirm such values without further biological knowledge. Coalitional-based methods seems to produce explanations that are harder to read on a global scale, but more precise at an individual level and able to take into account the dependencies between features. PDPs for all features are available at https://github.com/ EmmanuelDoumard/local_explanation_comparative_study. We use all the results presented in this paper to show a simplified roadmap in the form of a decision tree in Figure 11 with However, we warn the user about the loss of precision induced by such method approximations.

LESSONS-LEARNED FOR THE USE OF ADDITIVE LOCAL METHODS

Finally, we show that SHAP and LIME can make important approximations in some cases, and that coalitional-based methods cannot be executed in reasonable time in high dimension. This leaves an empty space for high dimension precise explanations that is not yet addressed to our knowledge.

CONCLUSION AND PERSPECTIVES

In this paper we performed a practical analysis of several local explainability methods for tabular data. Our findings indicate that there is not a single method that is the most appropriate for every usage. Such usages include the need of a high precision for local explanations or on the contrary the need of explanations that can be aggregated to produce a better and clearer global understanding, while taking into account the complexity level of data especially concerning the high dimension case. Therefore, this thorough analysis allowed to identify strengths and limitations of each method along with practical recommendations on which method is most suitable for the use case of the user. The Complete is of course the most accurate but suffer for very long computational time. Nevertheless, Coalitional based methods allow an acceptable computational time while maintaining a strong precision of explanations. On the contrary, LIME and SHAP methods offer a more intelligible global view of feature effects. The greatest problem arises when high dimension (i.e., high number of features) is involved, as it is often the case in statistics and Machine Learning. In this case, the exponential complexity of Coalitional-based methods make them too long to compute. Indeed, the worst case scenario is the need for high precision local explanations in high dimension since there is a clear lack of methods addressing this problem in the current literature. However, it is still possible to have local explanations with limited quality in high dimension, with the level of quality mostly depending on the time available for the user to generate such explanations. It is thus a very interesting future axis of work to benchmark the performances, in terms of precision of local explanations, of every local explainability method in a high dimension context under the constraint of a time limit. This would add value to our recommendations by filling out the 'high-precision in high-dimension' gap identified in our study. It would also be interesting to look into other machine learning models, especially deep neural networks which are more and more used. The very high complexity of this type of models hints at a different behavior for the explanation methods, but also an increase in computation time.

Figure 2 :

 2 Figure 2: Mean absolute difference of each method with the Complete, averaged by number of features, for each model

Figure 3 :

 3 Figure 3: Most-important features cumulative importance proportion by method, for each model. Only influences computed on datasets with 10 features are shown.

Figure 4 :

 4 Figure 4: AUC of each method, averaged by number of features, for each model

Figure 5 :

 5 Figure 5: Execution time of each model per instance, averaged by number of features, for each method

Figure 6 :

 6 Figure 6: Mean absolute difference of each method with the Complete, averaged by number of features, for each model Finally, regarding the AUC, we present all the results in Figure 7.We observe that for LIME and KernelSHAP, there is no significant difference between the AUC of the model's explanations. However, for the coalitional-based methods, we can see a clear separation between tree-based methods and non tree-based methods: the latter have higher AUC than the others. This means that, when using coalitional-based methods, one should be aware that different models may yield a different importance distribution over the features. For the tree-specific methods, we can see that XGBoost generates explanations with slightly higher AUCs than Random Forests on average.

Figure 7 :

 7 Figure 7: AUC of each model, averaged by number of features, for each method

Figure 8 :Figure 9 :

 89 Figure 8: Summary plots of each method on the SA-Heart dataset

Figure 10 :

 10 Figure 10: Influence value of adiposity against the influence value of obesity

Figure 11 :

 11 Figure 11: Roadmap for the most appropriate use of methods

Table 1 :

 1

		Number of datasets Number of instances
	Number		Min Max	Mean
	of features
	1	5 130 9100	3079
	2	21	52 5456	901
	3	43	60 9989	1729
	4	23	96 8641	1016
	5	35	62 7129	941
	6	27	51 9517	949
	7	33	54 4052	499
	8	32	52 8192	1473
	9	23	52 1473	484
	10	37	57 5473	712
	11	8	66 4898	942
	12	12 123 8192	1175
	13	5 178	506	293
	Total	304	51 9989	1035

Datasets description

in order to prioritise a low computational time or an higher accuracy. As for SHAP, local explanations can be aggregated into global explanations with a common foundation to study global and local behavior of the model.

The full implementation of Coalitional-based method is available on GitHub : https://github.com/kaduceo/coalitional_explanation_methods.

Table 2

 2

	summarizes advantages and drawbacks of each method
	studied in this paper. Overall, we highlight the fact that coalitional-
	based methods should be better at producing precise local expla-
	nations while SHAP should be better at producing coherent and
	easily interpretable global explanations. It is also confirmed by
	the fact that SHAP tends to assign more importance to few fea-
	tures than other methods, producing global explanations that are
	easier to read, but potentially hiding other features contributions
	and inter-dependences. Technically, KernelSHAP gives access to
	hyper-parameters to balance between execution time and expla-
	nation precision, but they are less accessible than Spearman's

and LIME's parameters. Indeed, without extensive KernelSHAP knowledge or documentation readout, users can easily miss on these parameters.

Table 2 :

 2 Summary table of advantages and drawbacks of each method

KernelSHAP documentation includes recommendation to use K-Means algorithm to speed up computation time https://shap-lrjball.readthedocs.io/en/latest/generated/ shap.KernelExplainer.html

ACKNOWLEDGMENTS

This study has been partially supported through the grant EUR CARe N°ANR-18-EURE-0003 in the framework of the Programme des Investissements d'Avenir.

We also thank the French National Association for Research and Technology (ANRT) and Kaduceo company for providing us with PhD grants (no. 2020/0964).


Y Becherini

A Djannati-Ataï

M Punch

K Bernlöhr

S Ehlert

J Masbou

E Moulin

M Paz Arribas

HESS-II reconstruction strategy and performance in the low-energy (20-150 GeV) domain

Keywords: Event reconstruction, angular resolution, energy resolution, effective area PACS: <Replace PACS numbers; choose

In mid-2009 a notable upgrade of the H.E.S.S. telescope system will take place: a new telescope with a 600 m 2 mirror area and very-high-resolution camera (0.07 • ) will be positioned at the centre of the present configuration, with the aim of lowering the threshold and enhance its sensitivity in the 100 GeV to several TeV energy range. HESS-II will permit the investigation of the lower energy γ-ray spectra in various cosmic accelerators, giving information on the origin of the γ-rays observed, and will detect AGNs with a redshift greater than 0.2 (being less affected by absorption by Extragalactic Background Light -EBL -in this energy range) and will search for new classes of very high energy γ-ray emitters (pulsars, microquasars, GRB, and dark matter candidates).

INTRODUCTION

By the end of 2009 the H.E.S.S. experiment will enter its Phase II and at that time the data taking configuration will consist of the four current telescopes (here called T1, T2, T3 and T4) plus the new very large HESS-II telescope (here called T5, see [START_REF] Deil | II -Telescope Structure, Mirror and Drive System[END_REF] at this conference for a detailed description of its structure). Since a new observational window is going to be opened in the range from 20 GeV to about 100 GeV with the largest ground-based Cerenkov telescope ever built, it is important to develop a powerful analysis in order to optimise the γ-ray detection in this energy range. We present here a first evaluation of the angular and energy resolution, as well as the effective detection area of this new configuration in the lowestenergy regime, focusing our attention on the pure mono T5 events. We have developed a shower reconstruction strategy for the events hitting the very large telescope alone. This consists of several consecutive steps: first of all we clean the images and evaluate the shower direction and core position (when possible) (see Sec. 3), then, knowing the basic parameters of the images and of the showers we can optimise and apply a cut for hadron rejection and a cut on the shape of the images (see Sec. 4), finally we estimate the event energy with the sample of selected events with a Neural Network (NN) approach (see Sec. 5).

THREE-ENERGY DOMAIN ANALYSIS

The intertelescope trigger system of the HESS-II configuration will operate in a hybrid mode, allowing to trigger on three classes of events in parallel: at very low energies on purely mono-telescope events of the very large telescope T5 (Mono domain), at mid energies on combined events with an image from the T5 and one from the smaller ones (T1 or T2 or T3 or T4) often with rudimentary information in the latter (semi-Stereo domain), and at even higher energies on current H.E.S.S. -Phase I type events with additional rich information in the central telescope (full-Stereo domain). Monte Carlo Simulations and trigger conditions. For the evaluation of the low energy performance of the upgraded system we simulated gamma and proton showers between 20 and 150 GeV at a zenith angle of 18 • assuming an optimal optical efficiency in the five telescopes. The γ-ray source is simulated on the optical axis, so the source is projected at the centre of the cameras, and the simulations are carried out over a sufficiently large radius (500 m) from the centre of the array. The local trig- ger configuration used in this analysis (see Tab. 1) can be summarised as follows: for T1, ..., T4 we required a pixel threshold of 2.5 p.e. and a minimum number of pixels of 4, while for T5 we raised these values to 3.5 and 5 to allow a better online rejection of low energy hadrons. The event is then kept only if it has at least one telescope satisfying the local trigger condition. The effective detection area at trigger level (see Fig. 3, upper continuous line) shows that up to about 90 GeV, events are mostly seen by the T5 telescope alone and that the detection maximum for the pure T5 events is reached at an energy of about 50 GeV. In the following analysis, we consider as pure Mono events all the showers triggering only the T5 telescope. As soon as even a little information is available from another telescope, the events are considered as stereo (since stereo reconstruction is more efficient).

SHOWER RECONSTRUCTION

Images are first of all cleaned with the following filtering rule: a pixel is accepted if it has at least a charge of 7 p.e. and some neighbouring pixels having a charge greater than 5 p.e. To reconstruct the shower parameters with the filtered pixels, we use the Hillas algorithm [2] which gives the moments of the images in each telescope. The Hillas parameters used are the length and the width of the major axis of a fitted ellipse, the angular distance between the centre of the camera and the barycentre of the image called nominal distance, the orientation angle between the fitted ellipse's axis and the source direction. Stereo events. In the case of multi-telescope events, the direction and the core position are easily inferred by the intersection of the axes of the telescope images. For this kind of event, an additional reconstruction algorithm (called Model3D ) is available, see [3]. The Model3D reconstruction algorithm is based on a 3-dimensional model of the shower in the atmosphere having the advantage of giving as a result some basic parameters of the air shower, as the length, the width, the altitude, and the number of photons generated by the shower. Mono events. For pure Mono-telescope events, the shower direction has to be estimated from the measured parameters using relations derived from Monte Carlo simulations. The distance between the barycentre of the image and the source position along the shower major axis can be parametrized taking into account the image length and charge for example. What is not trivial to estimate in the Mono case is the shower core position: the same image could be given by a nearby low energy event or by a distant high energy event. The very large telescope will also have the advantage of having a completely new electronics which could reach a sampling frequency of up to 2 GHz, i.e. a timing resolution of 0.5 ns. The arrival times of the photons on the pixels add information about the structure of the shower that can be used in a more sophisticated reconstruction procedure. In particular, the pixel times could help to evaluate the impact parameter of the event. A reconstruction strategy using the pixel timing is under development.

ANALYSIS CUTS FOR THE MONO REGIME

Hadron rejection cut. The Hillas parameters toghether with other robust parameters of the event can be used for the definition of a hadron rejection cut. The optimisation of the background rejection cut for the pure T5 Mono events has been performed with the Fisher algorithm implemented in the multi-variate analysis package TMVA [4]. We choose to train the algorithm with the nominal distance, the Hillas length and width, and the total cleaned charge in the event. The Fisher response to the sample of simulated gammas and protons is shown in Fig. 1. For this study we chose to use a cut at a value FIGURE 2. Angular resolution for the different HESS-II detection regimes: the upper continuous line shows the curve for the pure T5 Mono events after image filtering, while the lower continuous line shows the curve after the background rejection cut (cut 1) and the nominal distance cut (cut 2). The remaining curves are obtained in the case of the semi-Stereo plus full-Stereo events at filtering and reconstruction level (the upper dotted line is for the Hillas case while the lower dotted line is for the Model3D case) and no analysis cuts were applied; the final stereo angular resolution is expected to be much better. of 0.1 (called cut 1 ), where the maximum quality factor (S/ √ B, being S the signal and B the background) is reached. Image shape cut. An additional cut on the nominal distance (cut 2 ) is required in order to reject the events giving images at the border of the camera, and to reject the events hitting the telescope too close to the source direction, which give non-elliptical images. We save only the events giving images at a nominal distance greater than > 0.45 • and smaller than < 1 • for this case where the source is at the camera centre. Cut on the angular resolution. The final cut (cut 3 ) is a selection of the well-reconstructed events based on the angular resolution, which will be introduced in Sec. 5. A summary of the analysis cuts for the pure Mono regime is given in Tab. 2.

ANGULAR RESOLUTION AND EFFECTIVE AREA

With the events passing the above-mentioned cuts 1 and 2, we estimated the expected angular resolution and effective area for the pure Mono regime. Angular resolution. The angular resolution, defined as 68% containment radius for all the pure T5 Mono events after image filtering and reconstruction, is shown in Fig. 2 with the upper continuous line, while the resulting curve after the background rejection cut and the nominal distance cut is shown with the lower continuous line. We also estimated the angular resolution of the hybrid and stereo events after reconstruction with the Hillas (upper dotted line) and the Model3D (lower dotted line) algorithms without any cuts, for comparison. The angular resolution for the pure T5 events after cuts (1 + 2) is of the order of 0.25 • . Effective area. For the calculation of the effective area, another quality cut on the squared angular deviation between the source position and the reconstructed direction θ 2 (cut 3 ) is applied: we select all the events having a θ 2 < 0.13 deg 2 . effective area resulting after the three cuts given in Tab. 2 is shown in Fig. 3. The area in the T5 Mono range after cuts is shown with the lower continuous curve: it has a maximum around 40 GeV then it drops dramatically at about 80 GeV where the semi-Stereo and full-Stereo detection regimes take over.

In the upper dotted and lower dotted lines we show the curves for the Hillas and Model3D (pre-cuts).

SHOWER ENERGY EVALUATION

A dedicated study for the evaluation of the energy of the pure T5 events has been carried out using a NN approach: we used the module called MultiLayerPerceptron developed in the ROOT framework [5]. We have trained our NN with the events passing the three cuts summarised in Tab. 2 and with the same parameters as used for the multivariate analysis described in Sec. 4. The architecture of our network (see Fig. 4) is given by 4 input nodes, 2 hidden layers composed by 4 nodes each and one output node (4 : 4 : 4 : 1). The resulting energy resolution defined as ∆E/E is shown in Fig. 5 in the upper continuous line, while the bias in the evaluation of the energy defined as < E > /E is shown in lower continuous line. The two curves are both obtained in the case of the overtraining check. The bias in the evaluation of the energy was found to be essentially due to the events having a large impact parameter: these events globally tend to lower the energy reconstruction performance. The dotted lines on Fig. 5 represent the energy resolution and the bias for the events having an impact parameter smaller than about 130 m.

CONCLUSIONS

From our preliminary study of the performance of the HESS-II telescope we found the angular resolution for the pure T5 events to be of the order of 0.25 • in the [30

Cut number Value

Cut 1 Fisher response > 0.1 Cut 2 0.45 GeV, 90 GeV] energy range after the hadron rejection and image shape cuts. The effective area after the additional cut on the angular resolution is found to be 4 × 10 4 m 2 at the maximum value of 40 GeV. Our current evaluation of the energy resolution varies from 40% to 20% as a function of the energy, while the bias spans from +40% to -40%: a more detailed study is needed in order to suppress this effect. At the time of writing we already know that we can improve the overall performance of the analysis, by adding the timing information and by further optimising the neural network layers and nodes. The results we have obtained for this pure T5 Mono analysis can be considered as a pessimistic case or a preliminary version of the final analysis.

• < nominal distance < 1 • Cut 3 θ 2 < 0.

FIGURE 1 .

 1 FIGURE 1. Signal S (upper line) and background B (lower line) efficiencies for the background rejection cut. The background rejection cut is obtained with a Fisher multi-variate analysis. The quality factor defined as S/ √ B having a maximum at a Fisher response value of 0.1 is also shown.

FIGURE 3 .

 3 FIGURE 3. Effective area given in m 2 for the different HESS-II detection regimes: at filter level (upper continuous line) and after cuts (lower continuous line) for the pure T5 events; at filter level for the semi-Stereo and full-Stereo events for two different reconstruction algorithms, the upper dotted line represents the results obtained with the Hillas algorithm, while the lower dotted line represents the Model3D resulting curve, see text.

FIGURE 4 .

 4 FIGURE 4. Neural network used for the evaluation of the energy for the pure Mono events seen only by T5. It consists of 4 input nodes, 2 hidden layers holding 4 nodes each and one output node.

FIGURE 5 .

 5 FIGURE 5. resolution (upper continuous line) and bias curve (lower continuous line) obtained with a neural network (see text) in the case of the pure T5 Mono events. The dotted curves represent the resolution and the bias for the events having an impact parameter smaller than 130 m.

TABLE 1 .

 1 Local trigger parameters applied in the analysis presented in this paper for the four H.E.S.S. telescopes (T1, T2, T3, T4) and for the upcoming HESS-II telescope (T5). An event is then accepted if it triggered at least one telescope.

	Parameter	T1, T2, T3, T4 T5
	pixel threshold (p.e.)	2.5	3.5
	minimum number of pixels	4	5

TABLE 2 .

 2 Summary of the quality cuts applied for the pure T5 analysis. Cut 1 has been optimised for background rejection, cut 2 to reject the incomplete or round images, cut 3 to select the well recontructed events.


L Maurin

Non-intrusive pipe internal pressure measurement  Fluid flow with massive fluid flow change (from 14 m 3 /h to zero) & temperature changes III -Water pressure loop (Ø 2" Sch. 80 section) 21 CEA, DES, BEARN 2 facility, Saclay, FR 2020 bending mitigation with 2 FBG-based sensors ( 2×2 = 4 FBGs) in ring configuration  maximum pressure error roughly reduced by a factor of two vs. single sensor measurement, from -8.6 bar to -5 bar

---

  Why a non-intrusive pressure measurement ?  Non-intrusive pressure measurement and temperature compensation background -Non-intrusive pressure measurement -Background -Temperature cross-sensitivity compensation based on the same proof body -Background  Non-intrusive pressure measurement -Underlying principle and core theory -Underlying principle leading to temperature cross-sensitivity compensation -The infinitely long and straight pipe under hydrostatic pressure -The FBG transducer -Some orders of magnitude -Non-intrusive pressure ∆𝑃 𝑖𝑛𝑡 measurement with direction-sensitive strain transducers -Additional effects mitigation & application to other measurement techniques  Experimental validations -From simple to more severe conditions -I -Mechanically isolated pipe section -II -Air pressure loop -III -Water pressure loop (NPP primary coolant circuit)  Hot-air gun experiment to estimate the non-intrusive pressure measurement robustness  No fluid flow, no temperature change  No fluid flow, but temperature changes due to heat conduction  Fluid flow with massive fluid flow change (from 14 m 3 /h to zero) & temperature changes No more pipe structural health damage due to sensor installation -Does not require a hole per sensor  Measurement provided from pipe external surface, or even remotely -Sensors can be installed and removed on demand, without any downtime, nor requalification  No more contact with the fluid to monitor -The pipe acts like the membrane of the traditional, but intrusive, sensor  But any undesired mechanical action on the pipe may introduce a measurement bias  Mitigation solution to be implemented to compensate for the effects of pure bending  A crucial measurement for the oil & gas industry -First to prevent hydrate-plugs formation under specific {P, T} conditions (offshore) -Second to reduce the risk of pipe failure, and subsequent environmental consequences  Non-intrusive pressure sensors development mainly driven by the oil & gas industry -Previous attempts mainly based on ring sensors (unsteady pressure), surface strain measurements or acoustic waves  Still suffer from shortcomings, with in general poor temperature cross-sensitivity compensation -Only one solution (Ekechukwu, 2021) based on both DAS and Raman DTS led to satisfactory results  New solution applicable to any fluid transportation if pipe cross-section is circular -Temperature cross-sensitivity on pressure measurement intrinsically compensated / cancelled Why a non-intrusive pressure measurement ? 5 e.g.: pressure equipment directive 97/23/EC from EU 4 th of March, 2022 CEA L. Maurin, N. Roussel & G. Laffont -Original paper available at: https://www.frontiersin.org/articles/10.3389/fsens.2022.835140 Non-intrusive pipe internal pressure measurement Non-intrusive pressure measurement -Background 7 Magne et al, 2005, FBG strain rosette surface strain  temperature compensation issue  thin-shell theory approximation Meiring et al, 2016, FBG placed helically around the pipe surface strain  temperature compensation issue Ekechukwu et al, 2021, DAS & Raman DTS low-frequency DAS components  questionable if the fluid does not flow Diodati, 1986, ultrasonics acoustic signal amplitude  dependency on fluid compressibility Zhou et al, 2016, ultrasonics propagation time of multiple longitudinal waves  temperature dependency requiring calibration  thin-shell theory approximation Non-intrusive pressure measurement Underlying principle and core theory 9 6 -11 MARCH 2022, NURETH19, BRUSSELS, BELGIUM 19 th International Topical Meeting on Nuclear Reactor Thermal Hydraulics EUROPEAN NUCLEAR SOCIETY 15:20-15:50, the 4 th of March, 2022 -NURETH19 Workshop -Fiber-optic sensors for thermal hydraulic measurements modified from Fanchon, "Guide de mécanique", Pipe behaviour under pressure well-known for decades -Formal models available -Materials' strength recipes (Roark 1938)  averaged values over pipe sections, thin vs. thick models -Continuum mechanics in the elastic range (Love 1927, Timoshenko 1934)  full & accurate solutions derived from Airy's potential -Temperature dependency to be taken into account  Exhibits different hoop vs. longitudinal stress sensitivities under hydrostatic pressure -Balance of pipe wall internal forces vs. forces resulting from internal pressure 𝑃 𝑖𝑛𝑡 (closed pipe hypothesis) Underlying principle leading to temperature cross-sensitivity compensation 10 In the elastic range, leads roughly to the same ratio between the average hoop 𝜺 𝜽𝜽 and the longitudinal 𝜺 𝒛𝒛 mechanical strains, BUT …  Poisson's transverse effect & pipe thickness do interfere significantly  Thin-shell model from material's strength approximation is not accurate enough to properly describe strains distribution within the pipe wall and on its surfaces  Provides a clue for intrinsic temperature cross-sensitivity compensation with orientation-sensitive strain transducers by simple subtraction of 2 raw measurements  Requires a complete thermomechanical & accurate model  Expected better efficiency than the sole Poisson's transverse effect (Maurin et al, 2007)  FBG transducers are good candidates for such implementation (𝜎 𝜃𝜃 and 𝜎 𝑧𝑧 are resp. the hoop & longitudinal cross-sectional average stresses) hoop: 2𝜎 𝜃𝜃 . 𝑒. 𝑏 ≃ 𝑃 𝑖𝑛𝑡 . 2R. b (cylindrical cross-section balance) longitudinal: 𝜎 𝑧𝑧 . 𝑒. 2𝜋𝑅 ≃ 𝑃 𝑖𝑛𝑡 . π𝑅 2 (circular cross-section balance) Hooke's law: 𝝈 = 2𝜇 𝜺 𝑚𝑒𝑐 + 𝜆 Tr 𝜺 𝑚𝑒𝑐 𝑰 -Linearized Green-Lagrange strain tensor 𝜺 approximation:  Problem formulated in terms of displacements -𝑢 = 𝑢 𝜎 + 𝑢 𝑇 : total displacement 𝑢 (sum of mechanical 𝑢 𝜎 and thermal 𝑢 𝑇 displacements) -Homogeneous temperature in the pipe wall & no shear stress hypotheses  the temperature does not generate any additional mechanical strain  Fundamental equation of dynamics for deformable bodies div 𝝈 + 𝜌 Ԧ 𝑓 -Ԧ 𝛾 = 0 ⇒ div 𝝈 ≃ 0 (static conditions) The infinitely long and straight pipe under hydrostatic pressure 11 set of partial differential equations formal solution valid anywhere in the pipe wall & pipe surfaces 𝑲 𝟏 , 𝑳 𝟏 , 𝑳 𝟐 are integration parameters depending on: • pipe thermomechanical properties, • pipe inner 𝑟 0 𝑖𝑛𝑡 & outer 𝑟 0 𝑒𝑥𝑡 radiuses, • longitudinal pulling force Ԧ 𝐹, • inner 𝑃 𝑖𝑛𝑡 & outer 𝑃 𝑒𝑥𝑡 hydrostatic pressures, • surface temperature 𝑇. 𝐾 1 , 𝐿 1 , 𝐿 2 𝝈 = ℎ 𝐾 1 , 𝐿 1 , 𝐿 2 𝑢 = 𝑓 𝐾 1 , 𝐿 1 , 𝐿 2  linear relationship with hydrostatic pressures  quadratic relationship with temperature displacement strain tensor stress tensor 4 th of March, 2022 CEA L. Maurin, N. Roussel & G. Laffont -Original paper available at: https://www.frontiersin.org/articles/10.3389/fsens.2022.835140 Non-intrusive pipe internal pressure measurement  Temperature & strain measurements with the FBG transducer -𝜆 𝐵 = 2𝑛 𝑒𝑓𝑓 𝛬 ⇒ 𝑑𝜆 𝐵 𝜆 𝐵 = 𝜅 𝑇 + 𝜅 𝜀 . 𝛥𝛼 . 𝑑𝑇 + 𝜅 𝜀 . 𝑑𝜀 𝑚𝑒𝑐 -κ ε . Δα : additional temperature sensitivity when attached to pipe surface  Detached from the structure: 𝚫𝝀 𝑩 = 𝟏 𝒑𝒎 ⟷ 𝟎. 𝟏°𝑪 ⟷ 𝟏 Τ 𝝁𝒎 𝒎  Once attached to pipe surface: 𝚫𝝀 𝑩 = 𝟏 𝒑𝒎 ⟷ 𝟎. 𝟎𝟒°𝑪 ⟷ 𝟏 Τ 𝝁𝒎 𝒎 -Typically 2.3 times more sensitive to temperature -Typical mechanical strain amplitude of a Ø 4′′ NPS Sch. 160 metallic pipe under hydrostatic pressure  ∆𝑷 𝒊𝒏𝒕 = 𝟏 𝒃𝒂𝒓 ⇒ ∆𝜺 𝒎𝒆𝒄 ≃ 𝟏 Τ 𝝁𝒎 𝒎 ⇒ 𝜹𝑷 𝒊𝒏𝒕 = 𝟏 𝒃𝒂𝒓 resolution ↔ 𝟎. 𝟎𝟒°𝑪 max temperature discrepancy  Temperature cross-sensitivity compensation is critical to reach 𝜹𝑷 𝒊𝒏𝒕 = 𝟏 𝒃𝒂𝒓 resolution  FBG temperature is mainly weighted by the pipe thermal effusivity -Temperature discrepancy between 2 transducers attached to the pipe surface remains very small -The subtraction between two raw measurements can therefore be assimilated to a purely mechanical information, in direct relationship with the internal pressure in the elastic range  𝜆 1 and 𝜆 2 are the Bragg wavelengths of the FBG transducers attached to the pipe external surface The FBG transducer -Some orders of magnitude 12 temperature dependency mechanical strain dependency from Ferdinand, "Capteurs à fibres optiques à réseaux de Bragg", "Techniques de l'Ingénieur", vol. R 6 735 v2, 2018 (@ 1550 𝑛𝑚) ∆𝑃 𝑖𝑛𝑡 ≃ 𝑓 ∆𝜀 Δ𝜆 1 -Δ𝜆 2 𝑓 ∆𝜀 : linear function (typical performance target)  Non-intrusive pressure measurement on a straight and closed pipe section -Internal pressure 𝚫𝑷 𝒊𝒏𝒕 and surface temperature 𝚫𝑻 variations are two independent information  Temperature variation Δ𝑇 also based on pipe mechanical properties (Young's modulus 𝐸 & Poisson's ratio 𝜈) -Time delayed information (due to heat diffusion) to be considered with limited confidence  Bending mitigation by simple averaging -Requires 𝑁 ≥ 2 sensors evenly distributed in ring position around the pipe Non-intrusive pipe internal pressure measurement Non-intrusive pressure ∆𝑷 𝒊𝒏𝒕 measurement with direction-sensitive strain transducers 13 𝑁 sensors evenly arranged in ring position around the pipe 1 sensor = 1 pair of FBGs 𝜑 1 ≠ 𝜑 2 mod 𝜋  The subtraction compensation process leads to a purely mechanical information -Temperature influence on pressure measurement is intrinsically cancelled  Transducers sensitivity to temperature must be equal -Any additional effect, different from purely mechanical, should also be optimally mitigated  e.g.: nuclear radiations influence on FBGs (responsible for Bragg wavelength shifts) The temperature ∆𝑇 information should therefore be considered as a global influence parameter  Including temperature, but also any additional non-mechanical influence on the transducers  Application to other direction-sensitive surface strain or length measurement techniques -The longitudinal mechanical strain variation ∆𝜀 is nothing else but a relative length variation -Every measurement technique capable to measure length variations ∆𝐿 on pipe surface should also be able to perform non-intrusive pressure measurement -Examples of candidate measurement techniques:  Ultrasonics with PZT transducers  Non-contact cross-correlation technique with video cameras for remote pressure monitoring (if sensitivity OK with regards to strain ranges) strain variation definition) example of PZT transducers configuration distributed on pipe surface & dedicated to non-intrusive temperature self-compensated pressure measurement 4 th of March, 2022 CEA L. Maurin, N. Roussel & G. Laffont -Original paper available at: https://www.frontiersin.org/articles/10.3389/fsens.2022.835140 FROM RESEARCH TO INDUSTRY CEA -www.cea.fr Experimental validations -From simple to more severe conditions 15 6 -11 MARCH 2022, NURETH19, BRUSSELS, BELGIUM 19 th International Topical Meeting on Nuclear Reactor Thermal Hydraulics EUROPEAN NUCLEAR SOCIETY 15:20-15:50, the 4 th of March, 2022 -NURETH19 Workshop -Fiber-optic sensors for thermal hydraulic measurements 4 th of March, 2022 CEA L. Maurin, N. Roussel & G. Laffont -Original paper available at: https://www.frontiersin.org/articles/10.3389/fsens.2022.835140 Non-intrusive pipe internal pressure measurement  Pipe section filled with pressurized water -No external force applied on the pipe -Room temperature -Inflating pressure up to 300 bar -Pipe length greater than 5 times its diameter  FBG transducers in the middle of the pipe section to be free from side effects I -Mechanically isolated pipe section 16 one non-intrusive FBG-based pressure sensor made of two FBG transducers in hoop and longitudinal configurations (best sensitivity configuration, also insensitive to torsion) non-intrusive pressure measurement error better than 0.54% Full Scale at 2σ after calibration -5 Hz measurement rate 𝐴 𝑐𝑎𝑙 : calibration scale factor (i.e.: the gauge factor) 𝐵 𝑐𝑎𝑙 : calibration pressure offset measurement performed without delay (in real-time)  Air pressure loop including several bends -FBG transducers in the middle of a 5 m long pipe section -Complex pipe layout with many bends -Air flow rate steps up to 300 Rosemount™ air flow rate 𝑄: univocal relationship between pressure drop and air flow rate measurements averaged from 5 kHz to 2.5 Hz  BEARN 2 water pressure loop (up to 320°C -150 bar -14 m 3 /h) -FBG transducers installed on Ø 2" NPS Sch. 80 and Ø 4" NPS Sch. 160 pipe sections -Non-intrusive sensors made of pairs of FBG transducers in hoop & longitudinal configurations  Rings configurations to mitigate bending effects  Typical orders of magnitude (see also page 12) unpublished result due to unavailable hot-air gun temperature measurement data (Frontiers open data policy requirement) Non-intrusive pipe internal pressure measurement  Hot-air gun experiment to estimate the non-intrusive pressure measurement robustness Measurements performed on Ø 4" NPS Sch. 160 metallic pipe section -Hot-air temperature estimated greater than 100°C to generate asymmetrical temperature distribution around the pipe (hot spot) -Typically 5 cm distance between FBG transducers in hoop & longitudinal configurations  Temperature difference between FBG transducers estimated greater than 40°C III -Water pressure loop (Ø 4" NPS Sch. 160 section) No fluid flow, no temperature change  No fluid flow, but temperature changes due to heat conduction -Closed instrumented section -Fluid flowing in the other part of the loop III -Water pressure loop (Ø 4" NPS Sch. 160 section) performed during important flow ∆𝑄 and temperature ∆𝑇 changes • FBG pressure average error during calibration close to 2 bar (2 σ) • similar FBG surface temperature evolution than reference sensor massive flow change, combined with fluid flow direction changes (bends), may generate significant bending forces Ԧ 𝐹 on the pipe ∆𝑻 𝒎𝒂𝒙 ≃ 𝟑𝟖°𝑪 -∆𝑸 ≃ 𝟏𝟒 Τ 𝒎 𝟑 𝒉 4 th of March, 2022 CEA L. Maurin, N. Roussel & G. Laffont -Original paper available at: https://www.frontiersin.org/articles/10.3389/fsens.2022.835140 FROM RESEARCH TO INDUSTRY CEA -www.cea.fr Conclusion 22 6 -11 MARCH 2022, NURETH19, BRUSSELS, BELGIUM 19 th International Topical Meeting on Nuclear Reactor Thermal Hydraulics EUROPEAN NUCLEAR SOCIETY 15:20-15:50, the 4 th of March, 2022 -NURETH19 Workshop -Fiber-optic sensors for thermal hydraulic measurements 4 th of March, 2022 CEA L. Maurin, N. Roussel & G. Laffont -Original paper available at: https://www.frontiersin.org/articles/10.3389/fsens.2022.835140 Non-intrusive pipe internal pressure measurement  Optimal temperature cross-sensitivity compensation is a key parameter for a reliable pressure measurement The closed pipe with circular cross-section provides an intrinsic mean to compensate for the transducers temperature cross-sensitivity with the same proof body -The subtraction of two true relative raw measurements of direction-sensitive strain transducers is a purely mechanical information  Demonstrated by the linearized formal model for pipes with circular cross-section -Can probably be extended to other pipe cross-section shapes, but much more complicated to establish a dedicated formal model for demonstration…  In the elastic range, this information is proportional to the pipe internal pressure variation ∆𝑃 𝑖𝑛𝑡 -The pipe surface temperature variation Δ𝑇 is an additional, but time delayed (due to heat diffusion in the pipe wall) information provided by the formal model, independently from the pressure information  Any additional non-mechanical effect can theoretically be intrinsically compensated e.g.: nuclear radiations on Bragg wavelengths for FBG transducers (but additional work necessary…)  Can be extended to any direction-sensitive surface strain or distance measurement -Ultrasonics with PZT transducers -Non contact cross-correlation techniques (remote sensing)  Opens the way to new kinds of temperature self-compensated hydrostatic pressure sensors, based on a closed proof body with circular cross-section -From tens to hundreds of bar in pressure ranges (depending of proof body mechanical properties and dimensions) Conclusion 23 example of new kind of temperature self-compensated hydrostatic pressure sensor for external pressure 𝑃 𝑒𝑥𝑡 monitoring 4 th of March, 2022 CEA L. Maurin, N. Roussel & G. Laffont -Original paper available at: https://www.frontiersin.org/articles/10.3389/fsens.2022.835140 Non-intrusive pipe internal pressure measurement  M. Daniel Mabily from Technip Energies for the kind permission to publish the data and the results from the pressure test held in Marseille in July 2019  M. Kévin Cruz from CEA DES IRESNE for the pressure test held on Mercure facility in Cadarache in October 2019  M. Franck Ruffet from CEA DES DM2S for his availability during the pressure tests held on BEARN 2 facility in Saclay in 2020  The FSH public fund which permitted to initiate the work on the formal model  M. Marc Baqué and TotalEnergies which co-funded this research topic for a solution based on a different principle, but enabled us to benefit from the upgraded BEARN 2 pressure loop facility for this sensor  Prof. Patrice Mégret from the University of Mons for the invitation to NURETH19 workshop

15:20-15:50, the 4 th of March, 2022 -NURETH19 Workshop -Fiber-optic sensors for thermal hydraulic measurements 6 6 -11 MARCH 2022, NURETH19, BRUSSELS, BELGIUM 19 th International Topical Meeting on Nuclear Reactor Thermal Hydraulics EUROPEAN NUCLEAR SOCIETY

Non-intrusive pipe internal pressure measurement Temperature cross-sensitivity compensation based on the same proof body -Background 8 Chen et al, 2003, Hi-Bi FBG Schlumberger™ quartz pressure sensor -Donzier et al, 2000 from "Techniques de l'Ingénieur", vol. E 3 093 ("Capteurs microélectroniques") Pressure stability better than 0.01% Full Scale / Year Udd, 1995, overlaid FBGs at different wavelengths in birefringent optical fiber Schroeder et al, 1997 (Schlumberger™) FBG in birefringent optical fiber FBGS™ pressure sensor based on Hi-Bi FBG https://fbgs.com/solutions/pressure-sensing/

th of March, 2022 CEA L. Maurin, N. Roussel & G. Laffont -Original paper available at: https://www.frontiersin.org/articles/10.3389/fsens.2022.835140

Technip Energies, Marseille, FR July 2019 4 th of March, 2022 CEA L. Maurin, N. Roussel & G. Laffont -Original paper available at: https://www.frontiersin.org/articles/10.3389/fsens.2022.835140

6 -11 MARCH 2022, NURETH19, BRUSSELS, BELGIUM 19 th International Topical Meeting on Nuclear Reactor Thermal Hydraulics EUROPEAN NUCLEAR SOCIETY

Acknowledgements Additional information -Data availability Acknowledgements

 This topic is published in Frontiers in Sensors, "Recent Advances in Optical Fiber Sensors" https://www.frontiersin.org/research-topics/23029/recent-advances-in-optical-fiber-sensors -Original paper available at: https://www.frontiersin.org/articles/10.3389/fsens.2022.835140