Loukas Kavouras
email: kavouras@athenarc.gr

Konstantinos Tsopelas
email: k.tsopelas@athenarc.gr

Giorgos Giannopoulos

Dimitris Sacharidis
email: dimitris.sacharidis@ulb.be

Eleni Psaroudaki
email: epsaroudaki@mail.ntua.gr

Nikolaos Theologitis
email: n.theologitis@athenarc.gr

Dimitrios Rontogiannis

Dimitris Fotakis
email: fotakis@cs.ntua.gr

Ioannis Z Emiris
email: emiris@athenarc.gr

Fairness Aware Counterfactuals for Subgroups

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

Introduction

Machine learning is now an integral part of decision-making processes across various domains, e.g., medical applications [START_REF] Kyrimi | Medical idioms for clinical bayesian network development[END_REF], employment [START_REF] Bogen | Help wanted: An examination of hiring algorithms, equity, and bias[END_REF][START_REF] Cohen | Efficient candidate screening under multiple tests and implications for fairness[END_REF], recommender systems [START_REF] Pitoura | Fairness in rankings and recommenders: Models, methods and research directions[END_REF], education [START_REF] Joshua R Loftus | Causal reasoning for algorithmic fairness[END_REF],

Preprint. Under review.

arXiv:2306.14978v1 [cs.LG] 26 Jun 2023

credit assessment [START_REF] Boer | Just in time: Personal temporal insights for altering model decisions[END_REF]. Its decisions affect our everyday life directly, and, if unjust or discriminative, could potentially harm our society [START_REF] Osonde | An intelligence in our image: The risks of bias and errors in artificial intelligence[END_REF]. Multiple examples of discrimination or bias towards specific population subgroups in such applications [START_REF] Tashea | Courts are using ai to sentence criminals. that must stop now[END_REF] create the need not only for explainable and interpretable machine learning that is more trustworthy [START_REF] Burkart | A survey on the explainability of supervised machine learning[END_REF], but also for auditing models in order to detect hidden bias for subgroups [START_REF] Inioluwa | Actionable auditing: Investigating the impact of publicly naming biased performance results of commercial ai products[END_REF].

Bias towards protected subgroups is most often detected by various notions of fairness of prediction, e.g., statistical parity, where all subgroups defined by a protected attribute should have the same probability of being assigned the positive (favorable) predicted class. These definitions capture the explicit bias reflected in the model's predictions. Nevertheless, an implicit form of bias is the difficulty for, or the burden [START_REF] Sharma | CERTIFAI: A common framework to provide explanations and analyse the fairness and robustness of black-box models[END_REF][START_REF] Kuratomi | Measuring the burden of (un) fairness using counterfactuals[END_REF] of, an individual (or a group thereof) to achieve recourse, i.e., perform the necessary actions to change their features so as to obtain the favorable outcome [START_REF] Gupta | Equalizing recourse across groups[END_REF][START_REF] Ustun | Actionable recourse in linear classification[END_REF]. Recourse provides explainability (i.e., a counterfactual explanation [START_REF] Wachter | Counterfactual explanations without opening the black box: Automated decisions and the gdpr[END_REF]) and actionability to an affected individual, and is a legal necessity in various domains, e.g., the Equal Credit Opportunity Act mandates that an individual can demand to learn the reasons for a loan denial. Fairness of recourse captures the notion that the protected subgroups should bear equal burden [START_REF] Gupta | Equalizing recourse across groups[END_REF][START_REF] Amir-Hossein | A survey of algorithmic recourse: contrastive explanations and consequential recommendations[END_REF][START_REF] Julius Von Kügelgen | On the fairness of causal algorithmic recourse[END_REF][START_REF] Kuratomi | Measuring the burden of (un) fairness using counterfactuals[END_REF].

To illustrate these notions, consider a company that supports its promotion decisions with an AI system that classifies employees as good candidates for promotion, the favorable positive class, based on various performance metrics, including their cycle time efficiency (CTE) and the annual contract value (ACV) for the projects they lead. Figure 1a draws ten employees from the negative predicted class as points in the CTE-ACV plane and also depicts the decision boundary of the classifier. Race is the protected attribute, and there are two protected subgroups with five employees each, depicted as circles and triangles. For each employee, the arrow depicts the best action to achieve recourse, i.e., to cross the decision boundary, and the number indicates the cost of the action, here simply computed as the distance to the boundary [START_REF] Gupta | Equalizing recourse across groups[END_REF]. For example, x 1 may increase their chances for promotion mostly by acquiring more high-valued projects, while x 2 mostly by increasing their efficiency. Burden is defined as the mean cost for a protected subgroup [START_REF] Sharma | CERTIFAI: A common framework to provide explanations and analyse the fairness and robustness of black-box models[END_REF]. For the protected race 0, the burden is 2, while it is 2.2 for race 1, indicating thus unfairness of recourse against race 1. In contrast, assuming there is an equal number of employees of each race in the company, the classifier satisfies fairness of prediction in terms of statistical parity (equal positive rate in the subgroups). While fairness of recourse is an important concept that captures a distinct notion of algorithmic bias, as also explained in [START_REF] Amir-Hossein | A survey of algorithmic recourse: contrastive explanations and consequential recommendations[END_REF][START_REF] Julius Von Kügelgen | On the fairness of causal algorithmic recourse[END_REF], we argue that it is much more nuanced than the mean cost of recourse (aka burden) considered in all prior work [START_REF] Gupta | Equalizing recourse across groups[END_REF][START_REF] Sharma | CERTIFAI: A common framework to provide explanations and analyse the fairness and robustness of black-box models[END_REF][START_REF] Julius Von Kügelgen | On the fairness of causal algorithmic recourse[END_REF][START_REF] Kuratomi | Measuring the burden of (un) fairness using counterfactuals[END_REF], and raise three issues.

First, the mean cost, which comprises a micro viewpoint of the problem, does not provide the complete picture of how the cost of recourse varies among individuals and may lead to contrasting conclusions. Consider again Figure 1a, and observe that for race 1 all but one employee can achieve recourse with cost at most 2, while an outlier achieves recourse with cost 6. It is this outlier that raises the mean cost for race 1 above that of race 0. For the two race subgroups, Figure 1b shows the cumulative distribution of cost, termed the effectiveness-cost distribution (ecd) in Section 2. These distributions allow the fairness auditor to inspect the tradeoff between cost and recourse, and define the appropriate fairness of recourse notion. For example, they may consider that actions with a cost of more than 2 are unrealistic (e.g., because they cannot be realized within some timeframe), and thus investigate how many employees can achieve recourse under this constraint; we refer to this as equal effectiveness within budget in Section 2.3. Under this notion, there is unfairness against race 0, as only 60% of race 0 employees (compared to 80% of race 1) can realistically achieve recourse.

There are several options to go beyond the mean cost. One is to consider fairness of recourse at the individual level, and compare an individual with their counterfactual counterpart had their protected attribute changed value [START_REF] Julius Von Kügelgen | On the fairness of causal algorithmic recourse[END_REF]. However, this approach is impractical as, similar to other causal-based definitions of fairness, e.g., [START_REF] Kusner | Counterfactual fairness[END_REF], it requires strong assumptions about the causal structure in the domain [START_REF] Amir-Hossein | Algorithmic recourse: from counterfactual explanations to interventions[END_REF]. In contrast, we argue that it's preferable to investigate fairness in subpopulations and inspect the trade-off between cost and recourse.

Second, there are many cases where the aforementioned micro-level aggregation of individuals' costs is not meaningful in auditing real-world systems. To account for this, we introduce a macro viewpoint where a group of individuals is considered as a whole, and an action is applied to and assessed collectively for all individuals in the group. An action represents an external horizontal intervention, such as an affirmative action in society, or focused measures in an organization that would change the attributes of some subpopulation (e.g., decrease tax or loan interest rates, increase productivity skills). In the macro viewpoint, the cost of recourse does not burden the individuals, but the external third party, e.g., the society or an organization. Moreover, the macro viewpoint offers a more intuitive way to audit a system for fairness of recourse, as it seeks to uncover systemic biases that apply to a large number of individuals.

To illustrate the macro viewpoint, consider the group within the shaded region in Figure 1a. In the micro viewpoint, each employee seeks recourse individually, and both race subgroups have the same distribution of costs. However, we can observe that race 0 employees, like x 1 achieve recourse by actions in the ACV direction, while race 1 employees, like x 2 in the orthogonal CTE direction. This becomes apparent when we take the macro viewpoint, and investigate the effect of action a 1 , depicted on the border of the shaded region, discovering that it is disproportionally effective on the race subgroups (leads to recourse for two-thirds of one subgroup but for none in the other). In this example, action a 1 might represent the effect of a training program to enhance productivity skills, and the macro viewpoint finds that it would perpetuate the existing burden of race 0 employees. Third, existing notions of fairness of recourse have an important practical limitation: they require a cost function that captures one's ability to modify one's attributes, whose definition may involve a learning process [START_REF] Rawal | Beyond individualized recourse: Interpretable and interactive summaries of actionable recourses[END_REF], or an adaptation of off-the-shelf functions by practitioners [START_REF] Ustun | Actionable recourse in linear classification[END_REF]. Even the idea of which attributes are actionable, in the sense that one can change (e.g., one cannot get younger), or "ethical" to suggest as actionable (e.g., change of marital status from married to divorced) hides many complications [START_REF] Venkatasubramanian | The philosophical basis of algorithmic recourse[END_REF].

Conclusions drawn about fairness of recourse crucially depend on the cost definition. Consider individuals x 1 , x 2 , and actions a 1 , a 2 , shown in Figure 1c. Observe that is hard to say which action is cheaper, as one needs to compare changes within and across very dissimilar attributes. Suppose the cost function indicates action a 1 is cheaper; both individuals achieve recourse with the same cost. However, if action a 2 is cheaper, only x 2 achieves recourse. Is the classifier fair?

To address this limitation, we propose definitions that are oblivious to the cost function. The idea is to compare the effectiveness of actions to the protected subgroups, rather than the cost of recourse for the subgroups. One way to define a cost-oblivious notion of fairness of recourse is the equal choice for recourse (see Section 2.3), which we illustrate using the example in Figure 1c. According to it, the classifier is unfair against x 1 , as x 1 has only one option, while x 2 has two options to achieve recourse among the set of actions {a 1 , a 2 }.

Contribution Our aim is to showcase that fairness of recourse is an important and distinct notion of algorithmic fairness with several facets not previously explored. We make a series of conceptual and technical contributions.

Conceptually, we distinguish between two different viewpoints. The micro viewpoint follows literature on recourse fairness [START_REF] Gupta | Equalizing recourse across groups[END_REF][START_REF] Amir-Hossein | A survey of algorithmic recourse: contrastive explanations and consequential recommendations[END_REF][START_REF] Julius Von Kügelgen | On the fairness of causal algorithmic recourse[END_REF][START_REF] Kuratomi | Measuring the burden of (un) fairness using counterfactuals[END_REF] in that each individual chooses the action that is cheaper for them, but revisits existing notions. It considers the trade-off between cost and recourse, and defines several novel notions that capture different aspects of it. The macro viewpoint considers how an action collectively affects a group of individuals, and quantifies its effectiveness. It allows the formulation of cost-oblivious notions of recourse fairness. It also leads to an alternative trade-off between cost and recourse that may reveal systemic forms of bias.

Technically, we propose an efficient, interpretable, model-agnostic, highly parametrizable framework termed FACTS (Fairness-Aware Countefactuals for Subgroups) to audit for fairness of recourse. FACTS is efficient in computing the effectiveness-cost distribution, which captures the trade-off between cost and recourse, in both the micro or macro viewpoint. The key idea is that instead of determining the best action for each individual independently (i.e., finding their nearest counterfactual explanation [START_REF] Wachter | Counterfactual explanations without opening the black box: Automated decisions and the gdpr[END_REF]), it enumerates the space of actions and determines how many and which individuals achieve recourse through each action. Furthermore, FACTS employs a systematic way to explore the feature space and discover any subspace such that recourse bias exists among the protected subgroups within. FACTS ranks the subspaces in decreasing order of recourse bias it detects, and for each provides an interpretable summary of its findings.

Related Work

We distinguish between fairness of predictions and fairness of recourse. The former aims to capture and quantify unfairness by comparing directly the model's predictions [START_REF] Mehrabi | A survey on bias and fairness in machine learning[END_REF][START_REF] Binns | Fairness in machine learning: Lessons from political philosophy[END_REF] at: the individual level, e.g., individual fairness [START_REF] Dwork | Fairness through awareness[END_REF], counterfactual/causal-based fairness [START_REF] Kusner | Counterfactual fairness[END_REF][START_REF] Kilbertus | Avoiding discrimination through causal reasoning[END_REF][START_REF] Joshua R Loftus | Causal reasoning for algorithmic fairness[END_REF]; and the group level, e.g., demographic parity [START_REF] Bilal Zafar | Fairness beyond disparate treatment & disparate impact: Learning classification without disparate mistreatment[END_REF], equal odd/opportunity [START_REF] Hardt | Equality of opportunity in supervised learning[END_REF].

Fairness of recourse is a more recent notion, related to counterfactual explanations [START_REF] Wachter | Counterfactual explanations without opening the black box: Automated decisions and the gdpr[END_REF], which explain a prediction for an individual (the factual) by presenting the "best" counterfactual that would result in the opposite prediction, offering thus recourse to the individual. Best, typically means the nearest counterfactual in terms of a distance metric in the feature space. Another perspective, which we adopt here, is to consider the action that transforms a factual into a counterfactual, and specify a cost function to quantify the effort required by an individual to perform an action. In the simplest case, the cost function can be the distance between factual and counterfactual, but it can also encode the feasibility of an action (e.g., it is impossible to decrease age) and the plausibility of a counterfactual (e.g., it is out-of-distribution). It is also possible to view actions as interventions that act on a structural causal model capturing cause-effect relationships among attributes [START_REF] Amir-Hossein | Algorithmic recourse: from counterfactual explanations to interventions[END_REF]. Hereafter, we adopt the most general definition, where a cost function is available, and assume that the best counterfactual explanation is the one that comes from the minimum cost action. Counterfactual explanations have been suggested as a mechanism to detect possible bias against protected subgroups, e.g., when they require changes in protected attributes [START_REF] Amir-Hossein | Model-agnostic counterfactual explanations for consequential decisions[END_REF].

Fairness of recourse, first introduced in [START_REF] Ustun | Actionable recourse in linear classification[END_REF] and formalized in [START_REF] Gupta | Equalizing recourse across groups[END_REF], is defined at the group level as the disparity of the mean cost to achieve recourse (called burden in subsequent works) among the protected subgroups. Fairness of recourse for an individual is when they require the same cost to achieve recourse in the actual world and in an imaginary world where they would have a different value in the protected attribute [START_REF] Julius Von Kügelgen | On the fairness of causal algorithmic recourse[END_REF]. This definition however only applies when a structural causal model of the world is available. Our work expands on these ideas and proposes alternate definitions that capture a macro and a micro viewpoint of fairness of recourse.

There is a line of work on auditing models for fairness of predictions at the subpopulation level [START_REF] Kearns | Preventing fairness gerrymandering: Auditing and learning for subgroup fairness[END_REF][START_REF] Kearns | An empirical study of rich subgroup fairness for machine learning[END_REF]. For example, [START_REF] Tramer | Fairtest: Discovering unwarranted associations in data-driven applications[END_REF] identifies subpopulations that show dependence between a performance measure and the protected attribute. [START_REF] Black | Fliptest: fairness testing via optimal transport[END_REF] determines whether people are harmed due to their membership in a specific group by examining a ranking of features that are most associated with the model's behavior. There is no equivalent work for fairness of recourse, although the need to consider the subpopulation is recognized in [START_REF] Amir-Hossein | Algorithmic recourse under imperfect causal knowledge: a probabilistic approach[END_REF] due to uncertainty in assumptions or to intentionally study fairness. Our work is the first that audits for fairness of recourse at the subpopulation level.

A final related line of work is global explainability. For example, recourse summaries [START_REF] Rawal | Beyond individualized recourse: Interpretable and interactive summaries of actionable recourses[END_REF][START_REF] Ley | Global counterfactual explanations: Investigations, implementations and improvements[END_REF] summarizes individual counterfactual explanations globally, and as the authors in [START_REF] Rawal | Beyond individualized recourse: Interpretable and interactive summaries of actionable recourses[END_REF] suggest can be used to manually audit for unfairness in subgroups of interest. [START_REF] Lakkaraju | Faithful and customizable explanations of black box models[END_REF] aims to explain how a model behaves in subspaces characterized by certain features of interest. [START_REF] Cornacchia | Auditing fairness under unawareness through counterfactual reasoning[END_REF] uses counterfactuals to unveil whether a black-box model, that already complies with the regulations that demand the omission of sensitive attributes, is still biased or not, by trying to find a relation between proxy features and bias. Our work is related to these methods in that we also compute counterfactual explanations for all instances, albeit in a more efficient manner, and with a specific goal.

Fairness of Recourse for Subgroups 2.1 Preliminaries

We consider a feature space X = X 1 × • • • × X n , where X n denotes the protected feature, which, for ease of presentation, takes two protected values {0, 1}. For an instance x ∈ X, we use the notation x.X i to refer to its value in feature X i .

We consider a binary classifier h : X → {-1, 1} where the positive outcome is the favorable. For a given h, we are concerned with a dataset D of adversely affected individuals, i.e., those who receive the unfavorable outcome. We prefer the term instance to refer to any point in the feature space X, and the term individual to refer to an instance from the dataset D.

We define an action a as a set of changes to feature values, e.g., a = {country → US, education-num → 12}. We denote as A the set of possible actions. An action a when applied to an individual (a factual instance) x results in a counterfactual instance x ′ = a(x). If the individual x was adversely affected (h(x) = -1) and the action results in a counterfactual that receives the desired outcome (h(a(x)) = 1), we say that action a offers recourse to the individual x and is thus effective. In line with the literature, we also refer to an effective action as a counterfactual explanation for individual x [START_REF] Wachter | Counterfactual explanations without opening the black box: Automated decisions and the gdpr[END_REF].

An action a incurs a cost to an individual x, which we denote as cost(a, x). The cost function captures both how feasible the action a is for the individual x, and how plausible the counterfactual a(x) is [START_REF] Amir-Hossein | A survey of algorithmic recourse: contrastive explanations and consequential recommendations[END_REF].

Given a set of actions A, we define the recourse cost rc(A, x) of an individual x as the minimum cost among effective actions if there is one, or otherwise some maximum cost represented as c ∞ :

rc(A, x) = min{cost(a, x)|a ∈ A : h(a(x)) = 1}, if ∃a ∈ A : h(a(x)) = 1; c ∞ , otherwise.
An effective action of minimum cost is also called a nearest counterfactual explanation [START_REF] Amir-Hossein | A survey of algorithmic recourse: contrastive explanations and consequential recommendations[END_REF].

We define a subspace X p ⊆ X using a predicate p, which is a conjunction of feature-level predicates of the form "feature-operator-value", e.g., the predicate p = (country = US) ∧ (education-num ≥ 9) defines instances from the US that have more than 9 years of education.

Given a predicate p, we define the subpopulation group G p ⊆ D as the set of affected individuals that satisfy p, i.e., G p = {x ∈ D|p(x)}. We further distinguish between the protected subgroups

G p,1 = {x ∈ D|p(x) ∧ x.X n = 1} and G p,0 = {x ∈ D|p(x) ∧ x.X n = 0}.
When the predicate p is understood, we may omit it in the designation of a group to simplify notation.

Effectiveness-Cost Trade-Off

For a specific action a, we naturally define its effectiveness (eff) for a group G, as the proportion of individuals from G that achieve recourse through a:

eff(a, G) = 1 |G| |{x ∈ G|h(a(x)) = 1}|.
We want to examine how recourse is achieved for the group G through a set of possible actions A.

We define the aggregate effectiveness (aeff) of A for G in two distinct ways.

In the micro viewpoint, the individuals in the group are considered independently, and each may choose the action that benefits itself the most. Concretely, we define the micro-effectiveness of set of actions A for group G as the proportion of individuals in G that can achieve recourse through some action in A, i.e.,:

aeff µ (A, G) = 1 |G| |{x ∈ G|∃a ∈ A, eff(a, x) = 1}|.
In the macro viewpoint, the group is considered as a whole, and an action is applied collectively to all individuals in the group. Concretely, we define the macro-effectiveness of set of actions A for group G as the largest proportion of individuals in G that can achieve recourse through the same action in A, i.e.,:

aeff M (A, G) = max a∈A 1 |G| |{x ∈ G| eff(a, x) = 1}|.
For a group G, actions A, and a cost budget c, we define the in-budget actions as the set of actions that cost at most c for any individual in G:

A c = {a ∈ A|∀x ∈ G, cost(a, x) ≤ c}.
We define the effectiveness-cost distribution (ecd) as the function that for a cost budget c returns the aggregate effectiveness possible with in-budget actions:

ecd(c; A, G) = aeff(A c , G).
We use ecd µ , ecd M to refer to the micro, macro viewpoints of aggregate effectiveness.

The value ecd(c; A, G) is the proportion of individuals in G that can achieve recourse through actions A with cost at most c. Therefore, the ecd function has an intuitive probabilistic interpretation. Consider the subspace X p determined by predicate p, and define the random variable C as the cost required by an instance x ∈ X p to achieve recourse. The function ecd(c; A, G p ) is the empirical cumulative distribution function of C using sample G p .

The inverse effectiveness-cost distribution function ecd -1 (ϕ; A, G) takes as input an effectiveness level ϕ ∈ [0, 1] and returns the minimum cost required so that ϕ|G| individuals achieve recourse.

Definitions of Subgroup Recourse Fairness

We define recourse fairness of classifier h for a group G by comparing the ecd functions of the protected subgroups G 0 , G 1 in different ways.

The first two definitions are cost-oblivious, and simply compare the aggregate effectiveness of a set of actions of the protected subgroups.

Equal Effectiveness This definition has a micro and a macro interpretation, and says that the classifier is fair if the same proportion of individuals in the protected subgroups can achieve recourse:

aeff(A, G 0 ) = aeff(A, G 1 ) .
Equal Choice for Recourse This definition has only a macro interpretation and claims that the classifier is fair if the protected subgroups can choose among the same number of sufficiently effective actions to achieve recourse, where sufficiently effective means the actions should work for at least 100ϕ% (for ϕ ∈ [0, 1]) of the subgroup:

|{a ∈ A| eff(a, G 0 ) ≥ ϕ}| = |{a ∈ A| eff(a, G 1 ) ≥ ϕ}|.
The next three definitions assume the cost function is known, and have both a micro and a macro interpretation.

Equal Effectiveness within Budget

The classifier is fair if the same proportion of individuals in the protected subgroups can achieve recourse with a cost at most c:

ecd(c; A, G 0 ) = ecd(c; A, G 1 ).

Equal Cost of Effectiveness

The classifier is fair if the minimum cost to achieve aggregate effectiveness of ϕ ∈ [0, 1] in the protected subgroups is equal:

ecd -1 (ϕ; A, G 0 ) = ecd -1 (ϕ; A, G 1 ).
Fair Effectiveness-Cost Trade-Off The classifier is fair if the protected subgroups have the same effectiveness-cost distribution, or equivalently for each cost budget c, their aggregate effectiveness is equal:

max c | ecd(c; A, G 0 ) -ecd(c; A, G 1 )| = 0.
The left-hand side represents the two-sample Kolmogorov-Smirnov statistic for the empirical cumulative distributions (ecd) of the protected subgroups. We say that the classifier is fair with confidence α if this statistic is less than -ln(α/2)

|Gp,0|+|Gp,1| 2|Gp,0||Gp,1| . The last definition takes a micro viewpoint and extends the notion of burden [START_REF] Sharma | CERTIFAI: A common framework to provide explanations and analyse the fairness and robustness of black-box models[END_REF] from literature to the case where not all individuals may achieve recourse. The mean recourse cost of a group G,

rc(A, G) = 1 |G| x∈G rc(A, x)
considers individuals that cannot achieve recourse through A and have a recourse cost of c ∞ . To exclude them, we denote as as G * the set of individuals of G that can achieve recourse through an action in A, i.e., G * = {x ∈ G|∃a ∈ A, h(a(x)) = 1}. Then the conditional mean recourse cost is the mean recourse cost among those that can achieve recourse:

rc * (A, G) = 1 |G * | x∈G * rc(A, x).
If G = G * , the definitions coincide with burden.

Equal (Conditional) Mean Recourse

The classifier is fair if the (conditional) mean recourse cost for the protected subgroups is the same:

rc * (A, G 0 ; A) = rc * (A, G 1 ).
Note that when the group G is the entire dataset of affected individuals, and all individuals can achieve recourse through A, this fairness notion coincides with fairness of burden [START_REF] Ustun | Actionable recourse in linear classification[END_REF][START_REF] Sharma | CERTIFAI: A common framework to provide explanations and analyse the fairness and robustness of black-box models[END_REF].

Fairness-aware Counterfactuals for Subgroups

This section presents FACTS (Fairness-aware Counterfactuals for Subgroups), a framework that implements both the micro and the macro viewpoint, and all respective fairness definitions provided in Section 2.3 to support auditing of the "difficulty to achieve recourse" in subgroups. The output of FACTS comprises population groups that are assigned (a) an unfairness score that captures the disparity between protected subgroups according to different fairness definitions and allows us to rank groups and (b) a user-intuitive, easily explainable counterfactual summary, which we term Comparative Subgroup Counterfactuals (CSC). Figure 2 presents an example result of FACTS derived from the adult dataset [START_REF] Bache | UCI machine learning repository[END_REF]. The "if clause" represents the subgroup G p , which contains all the affected individuals that satisfy the predicate p = (hours-per-week = FullTime) ∧ (marital-status = Married-civ-spouse) ∧ (occupation = Adm-clerical). The information below the predicate refers to the protected subgroups G p,0 and G p,1 which are the female and male individuals of G p respectively. With blue color, we highlight the percentage cov(G p,i ) = |G p,i |/|D i | which serves as an indicator of the size of the protected subgroup. The most important part of the representation is the actions applied that appear below each protected subgroup and are evaluated in terms of fairness metrics. In this example, the metric is the Equal Cost of Effectiveness with effectiveness threshold ϕ = 0.7. For G p,0 there is no action surpassing the threshold ϕ = 0.7, therefore we display a message accordingly. On the contrary, the action a = {hours-per-week → Overtime, occupation → Exec-managerial} has effectiveness 0.72 > ϕ for G p,1 , thus allowing a respective 72% of the male individuals of G p to achieve recourse. The unfairness score is "inf", since no recourse is achieved for the female subgroup. Method overview Deploying FACTS comprises three main steps: (a) Subgroup and action space generation, where frequent subgroups defined as frequent sets of feature-operator-value combinations in the affected population are derived and, respectively, frequent actions are collected in similar way from the unaffected population; (b) Counterfactual summaries generation, where proper matching of subgroups and actions is performed to produce valid counterfactual summaries for each subgroup and (c) CSC construction and fairness ranking, where each counterfactual summary is scored according to each definition from Section 2.3, producing a respective number of CSCs for each subgroup and allowing to produce separate subgroup rankings per definition. Next, we describe these steps in detail.

(a) Subgroup and action space generation Subgroups are generated by executing the fp-growth [START_REF] Han | Mining frequent patterns without candidate generation[END_REF] frequent itemset mining algorithm on D 0 and on D 1 resulting to the sets of subgroups G 0 and G 1 and then by computing the intersection G = G 0 G 1 . In our setting, an item is a feature-level predicate of the form "feature-operator-value" and, consequently, an itemset is a predicate p defining a subgroup G p . This step guarantees that the evaluation in terms of fairness will be performed between the common subgroups G of the protected populations. The set of all actions A is generated by executing fp-growth on the unaffected population to increase the chance of more effective actions and to reduce the computational complexity. The above process is parameterizable w.r.t. the selection of the protected attribute(s) and the minimum frequency threshold for obtaining candidate subgroups.

(b) Counterfactual summaries generation For each subgroup G p ∈ G, the following steps are performed: (i) Find valid actions, i.e., the actions in A that contain a subset of the features appearing in p and at least one different value in these features; (ii) For each valid action a compute eff(a, G p,0 ) and eff(a, G p,1 ). The aforementioned process extracts, for each subgroup G p , a subset V p of the actions A, with each action having exactly the same cost for all individuals of G p . Therefore, individuals of G p,0 and G p,1 are evaluated in terms of subgroup-level actions, with a fixed cost for all individuals of the subgroup, in contrast to methods that rely on aggregating the cost of individual counterfactuals. This approach provides a key advantage to our method in cases where the definition of the exact costs for actions is either difficult or ambiguous: a misguided or even completely erroneous attribution of a cost to an action will equally affect all individuals of the subgroup and only to the extent that the respective fairness definition allows it. In the setting of individual counterfactual cost aggregation, changes in the same sets of features could lead to highly varying action costs for different individuals within the same subgroup.

(c) CSC construction and fairness ranking At this stage, FACTS evaluates all definitions of Section 2.3 on all subgroups, producing an unfairness score per definition, per subgroup. In particular, each definition of Section 2.3 quantifies a different aspect of the difficulty for a protected subgroup to achieve recourse. This quantification directly translates to difficulty scores for the protected subgroups G p,0 and G p,1 of each subgroup G p , which we compare accordingly (computing the absolute difference between them) to arrive at the unfairness score of each G p based on the particular fairness metric.

The outcome of this process is the generation, for each fairness definition, of a ranked list of CSC representations, in decreasing order of their unfairness score. Apart from unfairness ranking, the CSC representations will allow users to intuitively understand unfairness by directly comparing differences in actions between the protected populations within a subgroup.

Experiments

This section presents the experimental evaluation of FACTS on the Adult dataset [START_REF] Bache | UCI machine learning repository[END_REF]. First, we briefly describe the experimental setting and then present and discuss Comparative Subgroup Counterfactuals for subgroups ranked as the most unfair according to various definitions of Section 2.3.

Experimental Setting

The first step was the dataset cleanup (e.g., removing missing values and duplicate features, creating bins for continuous features like age). The resulting dataset was split randomly with a 70:30 split ratio. It was used for the training of a logistic regression model (consequently used as the black-box model to audit). For the generation of the subgroups and the set of actions we used fp-growth with a 1% support threshold on the test set. We also implemented various cost functions, depending on the type of feature, i.e., categorical, ordinal, and numerical. A detailed description of the experimental setting, the models used, and the processes of our framework can be found in the supplementary material.

Unfair subgroups Table 1 presents three subgroups which were ranked at position 1 according to three different definitions: Equal Cost of Effectiveness (ϕ = 0.7), Equal Choice for Recourse (ϕ = 0.7) and Equal Cost of Effectiveness (ϕ = 0.3), meaning that these subgroups were detected to have the highest unfairness according to the respective definitions. For each subgroup, its rank, bias against, and unfairness score are provided for all definitions presented in the left-most column. When the unfairness score is 0, we display the value "Fair" in the rank column. Note that subgroups with exactly the same score w.r.t. a definition will receive the same rank. The CSC representations for the fairness metric that ranked the three subgroups of Table 1 at the first position are shown in Figure 3.

Subgroup 1 is ranked first (highly unfair) based on Equal Cost of Effectiveness with ϕ = 0.7, while it is ranked much lower or it is even considered as fair according to most of the remaining definitions (see the values of the "rank" column for Subgroup 1). The same pattern is observed for Subgroup 2 and Subgroup 3: they are ranked first based on Equal Choice for Recourse with ϕ = 0.7 and Equal Cost of Effectiveness with ϕ = 0.3 accordingly, but much lower according to the remaining definitions. This finding provides a strong indication of the utility of the different fairness definitions, i.e., the fact that they are able to capture different aspects of the difficulty in achieving recourse. 1What is more, these different aspects can easily be motivated by real-world auditing needs. For example, out of the aforementioned definitions, Equal Cost of Effectiveness with ϕ = 0.7 would be suitable in a scenario where a horizontal intervention to support a subpopulation needs to be performed, but a limited number of actions is affordable. In this case, the macro viewpoint demonstrated in the CSC Subgroup 1 (top result in Figure 3) serves exactly this purpose: one can easily derive single, group-level actions that can effectively achieve recourse for a desired percentage of the unfavored subpopulation. On the other hand, Equal Choice for Recourse (ϕ = 0.3), for which the CSC of a top result is shown in the middle of Figure 3, is mainly suitable for cases where assigning costs to actions might be cumbersome or even dangerous/unethical. This definition is oblivious to costs and measures bias based on the difference in the number of sufficiently effective actions to achieve recourse between the protected subgroups.

Another important observation comes again from Subgroup 1, where bias against the Male protected subgroup is detected, contrary to empirical findings from works employing more statistical-oriented approaches (e.g., [START_REF] Tramer | Fairtest: Discovering unwarranted associations in data-driven applications[END_REF]), where only subgroups with bias against the Female protected subgroup are reported. We deem this finding important, since it directly links to the problem of gerrymandering [START_REF] Kearns | Preventing fairness gerrymandering: Auditing and learning for subgroup fairness[END_REF], which consists in partitioning attributes in such way and granularity to mask bias in subgroups.

Our framework demonstrates robustness to this phenomenon, given that it can be properly configured to examine sufficiently small subgroups via the minimum frequency threshold described in Section 3.

Conclusion

In this work, we delve deeper into the difficulty (or burden) of achieving recourse, an implicit and less studied type of bias. In particular, we go beyond existing works, by introducing a framework of fairness notions and definitions that, among others, conceptualizes the distinction between the micro and macro viewpoint of the problem, allows the consideration of a subgroup as a whole when 1.

exploring recourses, and supports oblivious to action cost fairness auditing. We complement this framework with an efficient implementation that allows the detection and ranking of subgroups according to the introduced fairness definitions and produces intuitive, explainable subgroup representations in the form of counterfactual summaries. Our next steps include the extension of the set of fairness definitions, focusing on the comparison of the effectiveness-cost distribution; improvements on the exploration, filtering, and ranking of the subgroup representations, via considering hierarchical relations or high-coverage of subgroups; and the development of a fully-fledged auditing tool.

A Experimental Setting

Models To conduct our experiments, we have used the Logistic Regression2 classification model, where we use the default implementation of the python package scikit-learn3 . This model corresponds to the black box one that our framework audits in terms of fairness of recourse.

Train-Test Split For our experiments, all datasets are split into training and test sets with proportions 70% and 30%, respectively. Both shuffling of the data and stratification based on the labels were employed. Our results can be reproduced using the random seed value 131313 in the data split function (train_test_split 4 from the python package scikit-learn). FACTS is deployed solely on the test set.

Frequent Itemset Mining

The set of subgroups and the set of actions are generated by executing the fp-growth5 algorithm for frequent itemset mining. We used the implementation in the Python package mlxtend6 . We deploy fp-growth with support threshold 1%, i.e., we require the return of subgroups and actions with at least 1% frequency in the respective populations. Recall that subgroups are derived from the affected populations D 0 and D 1 and actions are derived from the unaffected population.

Effectiveness and Budgets As we have stated in Section 2 our main paper , the metrics Equal Choice for Recourse and Equal Cost of Effectiveness require the definition of a target effectiveness level ϕ, while the metric Equal Effectiveness within Budget requires the definition of a target cost level (or budget) c.

Regarding the metrics that require the definition of an effectiveness level ϕ, we used two different values arbitrarily, i.e., a relatively low effectiveness level of ϕ = 30% and a relatively high effectiveness level of ϕ = 70%.

For the estimation of budget-level values c we followed a more elaborate procedure. Specifically, 1. Compute the Equal Cost of Effectiveness (micro definition) with a target effectiveness level of ϕ = 50% to calculate, for all subgroups G, the minimum cost required to flip the prediction for at least 50% of both G 0 and G 1 . 2. Gather all such minimum costs of step 1 in an array. 3. Choose budget values as percentiles of this set of cost values. We have chosen the 30%, 60% and 90% percentiles arbitrarily.

Cost Functions Our implementation allows the user to define any cost function based on their domain knowledge and requirements. For evaluation and demonstration purposes, we implement an indicative set of cost functions, according to which, the cost of a change of a feature value v to the value v ′ is defined as follows:

1. Numerical features: |norm(v) -norm(v ′ )|, where norm is a function that normalizes values to [0, 1]. 2. Categorical features: 1 if v ̸ = v ′ , and 0 otherwise. 3. Ordinal features: |pos(v) -pos(v ′ )|,
where pos is a function that provides the order for each value.

Additionally to the above costs, the user is able to define a feature-specific weight that indicates the difficulty to change the given feature through an action. Thus, for each dataset, the cost of actions can be simply determined by specifying the numerical, categorical, and ordinal features, as well as the weights for each feature.

Feasibility Apart from the cost of actions, we also take care of some obvious unfeasible actions such as that the age and education features can not be reduced and actions should not lead to unknown or missing values.

Compute resources Experiments were run on commodity hardware (AMD Ryzen 5 5600H processor, 8GB RAM). On the software side, all experiments were run in an isolated conda environment using Python 3.9.16.

B Datasets Description

We have used four datasets in our experimental evaluation; the main paper presented results only on the first. For each dataset, we provide details about the preprocessing procedure, specify feature types, and list the cost feature weights applied.

B.1 Adult

We have generated CSCs in the Adult dataset7 using two different features as protected attributes, i.e., 'sex', and 'race'. The assessment of bias for each protected attribute is done separately. The results for 'sex' as the protected attribute are presented in the main paper. Before we present our results for race as the protected attribute, we briefly discuss the preprocessing procedures and feature weights used for the adult dataset.

Preprocessing We removed the features 'fnlwgt' and 'education' and any rows with unknown values. The 'hours-per-week' and 'age' features have been discretized into 5 bins each.

Features All features have been treated as categorical, except for 'capital-gain' and 'capital-loss', which are numeric, and 'education-num' and 'hours-per-week', which we treat as ordinal. The feature weights that we used for the cost function are presented in Table 2. We need to remind here that this comprises only an indicative weight assignment to serve our experimentation; the weight below try to capture the notion of how feasible/actionable it is to perform a change to a specific feature.

B.2 COMPAS

We have generated CSCs in the COMPAS dataset 8 for race as the protected attribute. Apart from our results, we provide some brief information regarding preprocessing procedures and the cost feature weights for the COMPAS dataset.

Preprocessing We discard the features 'age' and 'c_charge_desc'. The 'priors_count' feature has been discretized into 5 bins: [-0.1,1), [START_REF] Bache | UCI machine learning repository[END_REF][START_REF] Bogen | Help wanted: An examination of hiring algorithms, equity, and bias[END_REF] [5, 10) [START_REF] Gupta | Equalizing recourse across groups[END_REF][START_REF] Amir-Hossein | Algorithmic recourse: from counterfactual explanations to interventions[END_REF] and [START_REF] Amir-Hossein | Algorithmic recourse: from counterfactual explanations to interventions[END_REF][START_REF] Bilal Zafar | Fairness beyond disparate treatment & disparate impact: Learning classification without disparate mistreatment[END_REF], while trying to keep the frequencies of each bin approximately equal (the distribution of values is highly asymmetric so this is not possible with the direct use of e.g., pandas.qcut9 ).

Features We treat the features 'juv_fel_count', 'juv_misd_count', 'juv_other_count' as numerical and the rest as categorical. The feature weights used for the cost function are shown in Table 3.

B.3 SSL

We have generated CSCs in the SSL dataset 10 for race as the protected attribute. Before we move to our results, we discuss briefly preprocessing procedures and feature weights applied in the SSL dataset.

Preprocessing We remove all rows with missing values ('U' or 'X') from the dataset. We also discretize the feature 'PREDICTOR RAT TREND IN CRIMINAL ACTIVITY' into 6 bins. Finally, since the target labels are values between 0 and 500, we 'binarize' them by assuming values above 344 to be positively impacted and below 345 negatively impacted (following the principles used in [START_REF] Black | Fliptest: fairness testing via optimal transport[END_REF].

Features In this dataset, we treat all features as numerical (apart from the protected race feature).

The feature weights used for the cost function are presented in Table 4.

B.4 Ad Campaign

We have generated CSCs in the Ad Campaign dataset 11 for gender as the protected attribute.

Preprocessing We decided not to remove missing values, since they represent the vast majority of values for all features. However, we did not allow actions that lead to missing values in the CSCs representation.

Features In this dataset, we treat all features, apart from the protected one, as categorical. The feature weights used for the cost function are shown in Table 5.

C Additional Results

This section repeats the experiment described in the main paper, concerning the Adult dataset with 'gender' as the protected attribute (Section 4), in three other cases. Specifically, we provide three subgroups that were ranked first in terms of unfairness according to a metric, highlight why they were marked as unfair by our framework, and summarize their unfairness scores according to rest of the metrics.

C.1 Results for Adult with race as the protected attribute

We showcase three prevalent subgroups for which the rankings assigned by different fairness definitions truly yield different kinds of information. This is showcased in Table 6. We once again note that the results presented here are for 'race' as a protected attribute, while the corresponding results for 'gender' are presented in Section 4 of the main paper. In Figure 4 we present the Comparative Subgroup Counterfactual representation for the subgroups of Table 6 that corresponds to the fairness metric for which each subgroup presents the minimum rank.

These results are in line with the findings reported in the main paper (Section 4), on the same dataset (Adult), but on a different protected attribute (race instead of gender). Subgroups that are ranked first (highly unfair) with respect to a specific definition, are ranked much lower or even considered as fair according to most of the remaining definitions. This serves as an indication for the utility of the different fairness definitions, which is further strengthen by the diversity of the respective CSCs of Table 6. For example, the Subgroup 1 CSC (ranked first Equal Choice for Recourse (ϕ = 0.3)), demonstrates unfairness by contradicting a plethora of actions for the "White" protected subgroup, as opposed to much less actions for the the "Non-White" protected subgroup. For Subgroup 2, a much more concise representation is provided, tied to the respective definition (Equal Cost of Effectiveness (ϕ = 0.3)): no recourses are identified for the desired percentage of the "Non-White" unfavored population, as opposed to the "White" unfavored population.

C.2 Results for COMPAS

We present some ranking statistics for three interesting subgroups for all fairness definitions (Table 7). The Comparative Subgroup Counterfactuals for the same three subgroups are shown in Figure 5.

C.3 Results for SSL

In Table 8 we present a summary of the ranking statistics for three interesting subgroups. and their respective Comparative Subgroup Counterfactuals in Figure 6.

Subgroup 1 I f w o r k c l a s s = P r i v a t e , a g e = ( 3 4 . 0 , 4 1 . 0 ] , c a p i t a l -g a i n =0 , c a p i t a l -l o s s =0 , m a r i t a l -s t a t u s =Never -m a r r i e d , n a t i v ec o u n t r y = U n i t e d -S t a t e s , r e l a t i o n s h i p =Not -i n -f a m i l y : P r o t e c t e d S u b g r o u p = ' Non-White ' , 1.09% c o v e r e d Make W o r k c l a s = F e d e r a l -gov , a g e = ( 4 1 . 0 , 5 0 . 0 ] , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 36.84% . Make c a p i t a l -g a i n =15024 , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 100.00% . Make a g e = ( 4 1 . 0 , 5 0 . 0 ] , c a p i t a l -g a i n =15024 , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 100.

C.4 Results for Ad Campaign

In Table 9 we present, as we did for the other datasets, the ranking results for 3 interesting subgroups, while in Figure 7, we show the respective Comparative Subgroup Counterfactuals for these subgroups.

Figure 1 :

 1 Figure 1: (a) An example of affected individuals, the decision boundary, actions, and a subpopulation (in the shaded region), depicted in the feature space; (b) Cumulative distribution of cost of recourse for the individuals in (a); (c) Comparison of two actions to achieve recourse for two individuals.

I

  f h o u r s -p e r -week= F u l l T i m e , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , o c c u p a t i o n =Adm-c l e r i c a l : P r o t e c t e d S u b g r o u p = ' Male ' , 1.87% c o v e r e d Make h o u r s -p e r -week= O v e r t i m e , o c c u p a t i o n =Exec -m a n a g e r i a l w i t h e f f e c t i v e n e s s 72.00% P r o t e c t e d S u b g r o u p = ' Female ' , 1.80% c o v e r e d No r e c o u r s e s f o r t h i s s u b g r o u p . B i a s a g a i n s t ' Female ' due t o E q u a l C o s t o f E f f e c t i v e n e s s ( t h r e s h o l d = 0 . 7 ) . U n f a i r n e s s s c o r e = i n f .

Figure 2 :

 2 Figure 2: CSC for a highly biased subgroup in terms of Equal Cost of Effectiveness with ϕ = 0.7.

Subgroup 1 I

 1 f a g e = ( 4 1 . 0 , 5 0 . 0 ] , m a r i t a l -s t a t u s =Never -m a r r i e d , r a c e =White , r e l a t i o n s h i p =Not -i n -f a m i l y : P r o t e c t e d S u b g r o u p = ' Male ' , 1.34% c o v e r e dNo r e c o u r s e s f o r t h i s s u b g r o u p . P r o t e c t e d S u b g r o u p = ' Female ' , 1.47% c o v e r e d Make m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 70.49% B i a s a g a i n s t ' Male ' due t o E q u a l C o s t o f E f f e c t i v e n e s s ( t h r e s h o l d = 0 . 7 ) . U n f a i r n e s s s c o r e = i n f . Subgroup 2 I f w o r k l a s s = P r i v a t e , h o u r s -p e r -week= F u l l T i m e , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , o c c u p a t i o n =Adm-c l e r i c a l , r a c e = White : P r o t e c t e d S u b g r o u p = ' Male ' , 1.04% c o v e r e d Make h o u r s -p e r -week=OverTime , o c c u p a t i o n =Exec -m a n a g e r i a l w i t h e f f e c t i v e n e s s 70.00% Make h o u r s -p e r -week=OverTime , o c c u p a t i o n = P r o f -s p e c i a l t y w i t h e f f e c t i v e n e s s 70.00% Make h o u r s -p e r -week= B r a i n D r a i n , o c c u p a t i o n =Exec -m a n a g e r i a l w i t h e f f e c t i v e n e s s 70.00% Make h o u r s -p e r -week= B r a i n D r a i n , o c c u p a t i o n = P r o f -s p e c i a l t y w i t h e f f e c t i v e n e s s 70.00% Make W o r k c l a s s = S e l f -emp-i n , o c c u p a t i o n =Exec -m a n a g e r i a l w i t h e f f e c t i v e n e s s 70.00% Make W o r k c l a s s = S e l f -emp-i n , h o u r s -p e r -week=OverTime , o c c u p a t i o n =Exec -m a n a g e r i a l w i t h e f f e c t i v e n e s s 80.00% Make W o r k c l a s s = S e l f -emp-i n , h o u r s -p e r -week=OverTime , o c c u p a t i o n = S a l e s w i t h e f f e c t i v e n e s s 70.00% Make W o r k c l a s s = S e l f -emp-i n , h o u r s -p e r -week= B r a i n D r a i n , o c c u p a t i o n =Exec -m a n a g e r i a l w i t h e f f e c t i v e n e s s 70.00% P r o t e c t e d S u b g r o u p = ' Female ' , 3.51% c o v e r e d Make W o r k c l a s s = S e l f -emp-i n , h o u r s -p e r -week=OverTime , o c c u p a t i o n =Exec -m a n a g e r i a l w i t h e f f e c t i v e n e s s 74.51% Make W o r k c l a s s = S e l f -emp-i n , h o u r s -p e r -week= B r a i n D r a i n , o c c u p a t i o n =Exec -m a n a g e r i a l w i t h e f f e c t i v e n e s s 74.51% B i a s a g a i n s t ' Female ' due t o E q u a l C h o i c e f o r R e c o u r s e ( t h r e s h o l d = 0 . 7 ) . U n f a i r n e s s s c o r e = 6 Subgroup 3 I f a g e = ( 4 1 . 0 , 5 0 . 0 ] , o c c u p a t i o n = S a l e s : P r o t e c t e d S u b g r o u p ' Male ' , 1.18% c o v e r e d Make o c c u p a t i o n C r a f t -r e p a i r w i t h e f f e c t i v e n e s s 0.00% Make o c c u p a t i o n =Adm-c l e r i c a l w i t h e f f e c t i v e n e s s 0.00% Make o c c u p a t i o n =Tech -s u p p o r t w i t h e f f e c t i v e n e s s 19.23% Make o c c u p a t i o n = P r o f -s p e c i a l t y w i t h e f f e c t i v e n e s s 28.21% Make o c c u p a t i o n =Exec -m a n a g e r i a l w i t h e f f e c t i v e n e s s 39.74% P r o t e c t e d S u b g r o u p ' Female ' , 1.56% c o v e r e d Make o c c u p a t i o n = C r a f t -r e p a i r w i t h e f f e c t i v e n e s s 0.00% Make o c c u p a t i o n =Adm-c l e r i c a l w i t h e f f e c t i v e n e s s 0.00% Make o c c u p a t i o n =Tech -s u p p o r t w i t h e f f e c t i v e n e s s 0.00% Make o c c u p a t i o n =Exec -m a n a g e r i a l w i t h e f f e c t i v e n e s s 6.94% Make o c c u p a t i o n = P r o f -s p e c i a l t y w i t h e f f e c t i v e n e s s 6.94% Make a g e = ( 5 0 . 0 , 9 0 . 0 ] w i t h e f f e c t i v e n e s s 6.94% Make a g e = ( 5 0 . 0 , 9 0 . 0 ] , o c c u p a t i o n = P r o f -s p e c i a l t y w i t h e f f e c t i v e n e s s 6.94% Make a g e = ( 5 0 . 0 , 9 0 . 0 ] , o c c u p a t i o n = C r a f t -r e p a i r w i t h e f f e c t i v e n e s s 6.94% Make a g e = ( 5 0 . 0 , 9 0 . 0 ] , o c c u p a t i o n =Adm-c l e r i c a l w i t h e f f e c t i v e n e s s 6.94% Make a g e = ( 5 0 . 0 , 9 0 . 0 ] , o c c u p a t i o n =Exec -m a n a g e r i a l w i t h e f f e c t i v e n e s s 6.94% B i a s a g a i n s t ' Female ' due t o E q u a l C o s t o f E f f e c t i v e n e s s ( t h r e s h o l d = 0 . 3 ) . U n f a i r n e s s s c o r e = i n f .

Figure 3 :

 3 Figure 3: Comparative Subgroup Counterfactuals for the subgroups of Table1.

  00% . P r o t e c t e d S u b g r o u p = ' White ' , 1.94% c o v e r e d Make a g e = ( 4 1 . 0 , 5 0 . 0 ] , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 45.14% . Make m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 40.00% . Make a g e = ( 5 0 . 0 , 9 0 . 0 ] , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 42.86% . Make W o r k c l a s s = L o c a l -gov , a g e = ( 4 1 . 0 , 5 0 . 0 ] , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 44.00% . Make W o r k c l a s s = L o c a l -gov , a g e = ( 4 1 . 0 , 5 0 . 0 ] , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 44.00% . Make W o r k c l a s = S e l f -emp-i n c , a g e = ( 4 1 . 0 , 5 0 . 0 ] , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 52.57% . Make W o r k c l a s s = S e l f -emp-i n c , a g e = ( 5 0 . 0 , 9 0 . 0 ] , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 49.71% . Make W o r k c l a s s = L o c a l -gov , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 36.00% . Make W o r k c l a s s = S e l f -emp-i n c , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 45.14% . Make W o r k c l a s s = F e d e r a l -gov , a g e = ( 4 1 . 0 , 5 0 . 0 ] , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 62.29% . Make W o r k c l a s s = S t a t e -gov , a g e = ( 4 1 . 0 , 5 0 . 0 ] , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 40.57% . Make c a p i t a l -g a i n =15024 , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 99.43% . Make W o r k c l a s s = L o c a l -gov , a g e = ( 5 0 . 0 , 9 0 . 0 ] , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 40.00% . Make a g e = ( 4 1 . 0 , 5 0 . 0 ] , c a p i t a l -g a i n =15024 , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e , r e l a t i o n s h i p = M a r r i e d w i t h e f f e c t i v e n e s s 100.00% . B i a s a g a i n s t ' Non-White ' due t o E q u a l C h o i c e f o r R e c o u r s e ( t h r e s h o l d = 0 . 3 ) . U n f a i r n e s s s c o r e = 1 0 .Subgroup 2 I f h o u r s -p e r -week = F u l l T i m e , n a t i v e -c o u n t r y = U n i t e d -S t a t e s , o c c u p a t i o n = Adm-c l e r i c a l , r e l a t i o n s h i p = M a r r i e d : P r o t e c t e d S u b g r o u p = ' Non-White ' , 1.66% c o v e r e d No r e c o u r s e s f o r t h i s s u b g r o u p . P r o t e c t e d S u b g r o u p = ' White ' , 1.66% c o v e r e d Make h o u r s -p e r -week = B r a i n D r a i n , o c c u p a t i o n = Exec -m a n a g e r i a l w i t h e f f e c t i v e n e s s 70.00% . B i a s a g a i n s t ' Non-White ' due t o E q u a l C o s t o f E f f e c t i v e n e s s ( t h r e s h o l d = 0 . 3 ) . U n f a i r n e s s s c o r e = i n f . Subgroup 3 I f h o u r s -p e r -week = P a r t T i m e , m a r i t a l -s t a t u s = D i v o r c e d , n a t i v e -c o u n t r y = U n i t e d -S t a t e s : P r o t e c t e d S u b g r o u p = ' Non-White ' , 1.15% c o v e r e d Make m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e w i t h e f f e c t i v e n e s s 0.00% . Make h o u r s -p e r -week=MidTime , m a r i t a l -s t a t u = M a r r i e d -c i v -s p o u s e w i t h e f f e c t i v e n e s s 0.00% . Make h o u r s -p e r -week= F u l l T i m e , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e w i t h e f f e c t i v e n e s s 0.00% . Make h o u r s -p e r -week=OverTime , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e w i t h e f f e c t i v e n e s s 0.00% . Make h o u r s -p e r -week=OverTime , m a r i t a l -s t a t u s =Never -m a r r i e d w i t h e f f e c t i v e n e s s 0.00% . Make h o u r s -p e r -week= B r a i n D r a i n , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e w i t h e f f e c t i v e n e s s 0.00% . P r o t e c t e d S u b g r o u p = ' White ' , 1.66% c o v e r e d Make m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e w i t h e f f e c t i v e n e s s 1.01% . Make h o u r s -p e r -week=MidTime , m a r i t a l -s t a t u = M a r r i e d -c i v -s p o u s e w i t h e f f e c t i v e n e s s 1.01% . Make h o u r s -p e r -week= F u l l T i m e , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e w i t h e f f e c t i v e n e s s 7.07% . Make h o u r s -p e r -week=OverTime , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e w i t h e f f e c t i v e n e s s 7.07% . Make h o u r s -p e r -week=OverTime , m a r i t a l -s t a t u s =Never -m a r r i e d w i t h e f f e c t i v e n e s s 15.15% . Make h o u r s -p e r -week= B r a i n D r a i n , m a r i t a l -s t a t u s = M a r r i e d -c i v -s p o u s e w i t h e f f e c t i v e n e s s 16.16% . B i a s a g a i n s t ' Non-White ' due t o E q u a l C o n d i t i o n a l Mean R e c o u r s e . U n f a i r n e s s s c o r e = i n f .

Figure 4 :

 4 Figure 4: Example of three Comparative Subgroup Counterfactuals in Adult (protected attribute race); ref. Table 6

Figure 7 :

 7 Figure 7: Example of three Comparative Subgroup Counterfactuals in Ad Campaign; ref. Table 9

Table 1 :

 1 Unfair subgroups identified in the Adult dataset.

			Subgroup 1			Subgroup 2			Subgroup 3
		rank bias against unfairness score	rank	bias against unfairness score rank bias against unfairness score
	Equal Effectiveness	2950	Male	0.11	10063	Female	0.0004	275	Female	0.32
	Equal Choice for Recourse (ϕ = 0.3)	Fair	-	0	12	Female	2	Fair	-	0
	Equal Choice for Recourse (ϕ = 0.7)	6	Male	1	1	Female	6	Fair	-	0
	Equal Effectiveness within Budget (c = 5)	Fair	-	0	2806	Female	0.056	70	Female	0.3
	Equal Effectiveness within Budget (c = 10) 2350	Male	0.11	8518	Female	0.0004	226	Female	0.3
	Equal Effectiveness within Budget (c = 18) 2675	Male	0.11	9222	Female	0.0004	272	Female	0.3
	Equal Cost of Effectiveness (ϕ = 0.3)	Fair	-	0	Fair	-	0	1	Female	inf
	Equal Cost of Effectiveness (ϕ = 0.7)	1	Male	inf	12	Female	2	Fair	-	0
	Fair Effectiveness-Cost Trade-Off	4065	Male	0.11	3579	Female	0.13	306	Female	0.32
	Equal (Conditional) Mean Recourse	Fair	-	0	3145	Female	0.35	Fair	-	0

Table 2 :

 2 Cost Feature Weights for Adult

	feature name	weight value feature name	weight value
	native-country	4	Workclass	2
	marital-status	5	hours-per-week	2
	relationship	5	capital-gain	1
	age	10	capital-loss	1
	occupation	4	education-num	3

Table 3 :

 3 Cost Feature Weights for COMPAS

	feature name	weight value
	age_cat	10
	juv_fel_count	1
	juv_fel_count	1
	juv_other_count	1
	priors_count	1
	c_charge_degree	1

Table 4 :

 4 Cost Feature Weights for SSL

	feature name	weight value
	PREDICTOR RAT AGE AT LATEST ARREST	10
	PREDICTOR RAT VICTIM SHOOTING INCIDENTS	1
	PREDICTOR RAT VICTIM BATTERY OR ASSAULT	1
	PREDICTOR RAT ARRESTS VIOLENT OFFENSES	1
	PREDICTOR RAT GANG AFFILIATION	1
	PREDICTOR RAT NARCOTIC ARRESTS	1
	PREDICTOR RAT TREND IN CRIMINAL ACTIVITY	1
	PREDICTOR RAT UUW ARRESTS	1

Table 5 :

 5 Cost Feature Weights for Ad Campaign

	feature name	weight value
	religion	5
	politics	2
	parents	3
	age	10
	income	3
	area	2
	college_educated	3
	homeowner	1

Table 6 :

 6 Example of three unfair subgroups in Adult (protected attribute race)

			Subgroup 1			Subgroup 2			Subgroup 3
		rank	bias against unfairness score	rank	bias against unfairness score	rank	bias against unfairness score
	Equal Effectiveness	Fair	Fair	0.0	3047.0 Non-White	0.115	1682.0 Non-White	0.162
	Equal Choice for Recourse (ϕ = 0.3)	1	Non-White	10.0	10.0	Non-White	1.0	Fair	Fair	0.0
	Equal Choice for Recourse (ϕ = 0.7)	Fair	Fair	0.0	Fair	Fair	0.0	Fair	Fair	0.0
	Equal Effectiveness within Budget (c = 1.15)	Fair	Fair	0.0	Fair	Fair	0.0	Fair	Fair	0.0
	Equal Effectiveness within Budget (c = 10.0) 303.0	Non-White	0.242	2201.0 Non-White	0.115	4035.0 Non-White	0.071
	Equal Effectiveness within Budget (c = 21.0)	Fair	Fair	0.0	2978.0 Non-White	0.115	1663.0 Non-White	0.162
	Equal Cost of Effectiveness (ϕ = 0.3)	18.0	Non-White	0.15	1	Non-White	inf	Fair	Fair	0.0
	Equal Cost of Effectiveness (ϕ = 0.7)	Fair	Fair	0.0	Fair	Fair	0.0	Fair	Fair	0.0
	Fair Effectiveness-Cost Trade-Off	909.0	Non-White	0.242	4597.0 Non-White	0.115	2644.0 Non-White	0.162
	Equal (Conditional) Mean Recourse	5897.0	White	0.021	5309.0	White	0.047	1	Non-White	inf

Additional examples, as well as statistics measuring this pattern on a significantly larger sample, are included in the supplementary material to further support this finding.

https://scikit-learn.org/stable/modules/generated/sklearn.linear_model. LogisticRegression.html

https://scikit-learn.org/stable/index.html

https://scikit-learn.org/stable/modules/generated/sklearn.model_selection.train_ test_split.html

https://rasbt.github.io/mlxtend/user_guide/frequent_patterns/fpgrowth/

https://github.com/rasbt/mlxtend

https://raw.githubusercontent.com/columbia/fairtest/master/data/adult/adult.csv

https://aif360.readthedocs.io/en/latest/modules/generated/aif360.sklearn.datasets.fetch_compas.html

https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.qcut.html

https://raw.githubusercontent.com/samuel-yeom/fliptest/master/exact-ot/chicago-ssl-clean.csv

https://developer.ibm.com/exchanges/data/all/bias-in-advertising/

Acknowledgments and Disclosure of Funding

The authors were partially supported by the EU's Horizon Programme call, under Grant Agreement No. 101070568 (AutoFair)

D Comparison of Fairness Metrics

The goal of this section is to answer the question: "How different are the fairness of recourse metrics". To answer it, we consider all subgroups and compare how they rank in terms of unfairness according to 12 distinct metrics. The results justify our claim in the main paper that the fairness metrics capture different aspects of recourse unfairness. For each dataset and protected attribute, we provide: (a) the ranking analysis table, and (b) the aggregated rankings table.

The first column of the ranking analysis table shows the number of the most unfair subgroups per metric, i.e., how many ties are in rank 1. Depending on the unit of the unfairness score being compared between the protected subgroups (namely: cost, effectiveness, or number of actions), the number of ties can vary greatly. Therefore, we expect to have virtually no ties when comparing effectiveness percentages and to have many ties when comparing costs. The second and third columns show the number of subgroups where we observe bias in one direction (e.g., against males) and the opposite (e.g., against females) among the top 10% most unfair subgroups.

The aggregated rankings table is used as evidence that different fairness metrics capture different types of recourse unfairness. Each row concerns the subgroups that are the most unfair (i.e., tied at rank 1) according to each fairness metric. The values in the row indicate the average percentile ranks of these subgroups (i.e., what percentage of subgroups are more unfair) when ranked according to the other fairness metrics, shown as columns. Concretely, the value v of each cell i, j of this table is computed as follows:

1. We collect all subgroups of the fairness metric appearing in row i that are ranked first (the most biased) due to this metric.

2. We compute the average ranking a of these subgroups in the fairness metric appearing in column j.

3. We divide a with the largest ranking tier of the fairness metric of column j to arrive at v.

Each non-diagonal value of this table represents the relative ranking based on the specific metric of the column for all the subgroups that are ranked first in the metric of the respective row (all diagonal values of this table are left empty). A relative ranking of v in a specific metric m means that the most unfair subgroups of another metric are ranked lower on average (thus are fairer) for metric m.

D.1 Comparison on Adult for protected attribute gender

The number of affected individuals in the test set for the adult dataset is 10,205. We first split the affected individuals on the set of affected males D 1 and the set of affected females D 0 . The number of subgroups formed by running fp-growth with support threshold 1% on D 1 and on D 0 and computing their intersection is 12,880. Our fairness metrics will evaluate and rank these subgroups based on the actions applied.

Tables 10 and11 present the ranking analysis and the aggregated rankings respectively on the gender attribute, on the Adult dataset. Next, we briefly discuss the findings from these two tables; similar findings stand for the respective tables of the other datasets, thus we omit the respective discussion.

It is evident from Table 10 that the different ways to produce ranking scores by different definitions can lead to considerable differences in ties, i.e., the number of subgroups receiving the same rank (here only rank 1 is depicted). The "Top 10%" columns demonstrate interesting statistics on the protected subgroup for which bias is identified: while it is expected that mostly bias against "Female" will be identified, subgroups with reverse bias (bias against "Male") are identified, indicating robustness to gerrymandering, as hinted in Section 4 of the main paper.

D.2 Comparison on Adult for protected attribute race

The number of affected individuals in the test set for the adult dataset is 10,205. We first split the affected individuals on the set of affected whites D 1 and the set of affected non-whites D 0 . The number of subgroups formed by running fp-growth with support threshold 1% on D 1 and on D 0 and computing their intersection is 16,621. Our fairness metrics will evaluate and rank these subgroups based on the actions applied.

D.3 Comparison on COMPAS

The number of affected individuals in the test set for the COMPAS dataset is 745. We first split the affected individuals on the set of affected caucasians D 1 and the set of affected african-americans D 0 .

The number of subgroups formed by running fp-growth with support threshold 1% on D 1 and on D 0 and computing their intersection is 995. Our fairness metrics will evaluate and rank these subgroups based on the actions applied.

D.4 Comparison on SSL

The number of affected individuals in the test set for the SSL dataset is 11,343. We first split the affected individuals on the set of affected blacks D 1 and the set of affected whites D 0 based on the race attribute (appears with the name RACE CODE CD in the dataset). The number of subgroups formed by running fp-growth with support threshold 1% on D 1 and on D 0 and computing their intersection is 6,551. Our fairness metrics will evaluate and rank these subgroups based on the actions applied.

D.5 Comparison on Ad Campaign

The number of affected individuals in the test set for the Ad campaign dataset is 273,773. We first split the affected individuals on the set of affected males D 1 and the set of affected females D 0 based on the gender attribute. The number of subgroups formed by running fp-growth with support threshold 1% on D 1 and on D 0 and computing their intersection is 1,432. Our fairness metrics will evaluate and rank these subgroups based on the actions applied.


G Pereira

Dos Santos Lima

S Ait-Amar

G Velu

P Frezel

Study of new ecological magnet wires performances during thermal aging tests

The principal purpose of this work is to investigate the performance of new technologies of electrical insulation (Solgel material top coated with an extrusion layer) intended for winding machines wires. New configurations of coating wire produced by an eco-friendly process, with less environmental impact than conventional ones, are tested and compared. The working method is based on measurements of samples that are produced in accordance with international standards to characterize the electrical and thermal properties of the insulated wires. The performance analysis of insulations is composed of a sequence of dielectric tests and thermal aging tests. This latter test is realized after validation of the basic electrical and mechanical tests. During thermal aging, the parallel capacitance, tangent delta, and Partial Discharge Inception Voltage (PDIV) are measured to investigate the evolution of those dielectric parameters and to compare dielectric behaviour between different configurations of insulating materials. The results of the tests show that an extrusion resin with fillers had a longer lifetime compared with non-filled resins. The concentration of sol-gel influenced the lifetime of insulated wire. This study shows some indicators of thermal improvement, but the percentage of sol-gel used is yet very low and therefore, far from the main objective of using fully sustainable products. The sol-gel formulation tested in higher concentrations had a lower performance of basic mechanical properties required for winding wires. However, the extrusion process for magnet wire insulation is possible and gives good results with pure stable thermoplastic polymer and mineral fillers on extrusion resins.

I. INTRODUCTION

Sustainable mobility is widely developed, electrification of cars, planes, and all means of transport as well as reducing the consumption of polluting products in manufacturing sectors are current issues. On the other hand, the electrical industrial sector is seeking to increase efficiency by reducing the mass of electrical machines and devices [START_REF] Sili | Study of the impact of the aeronautical environment on the Paschen curve and the aging of polyimide under partial discharge[END_REF]. In the environmental footprint of electrical motors [START_REF] Babicz | Dielectric study of flat anodized insulations for electrical machine design[END_REF], the manufacturing process of magnet wire requires high levels of energy consumption and pollution particles liberated on the environment [START_REF] Trauner | Case study: development of national emissions standards for hazardous air pollutants for magnet wire manufacturing[END_REF]. With the increase of electrification, the wires manufacturers use tend to be increased. Research and development of new technology or new products are necessary to find a compromise between increasing performance and reducing pollutant processes.

A magnet wire requires a compromise of several properties, such as physical, mechanical, thermal, and electrical. For developing a process and choosing materials, the physical, mechanical, and thermal properties are all essential for the machine winding application. Several standards define the specifications of magnet wires for windings taking into account these properties [START_REF]Specifications For Particular Types Of Winding Wires -Part 0-1: General Requirements -Enameled Round Copper Wire[END_REF]. This study is focused on thermal and dielectric properties for different configurations of insulation. Beyond the development of magnet wire using less enamelling pollutant process, this study consists of analysis between experimental dielectric results and its interpretation based on literature.

II. LITERATURE BASIS

The dielectrics properties behaviour of Electrical Insulation Systems (EIS) changes under temperature constraints. Several works have explained these phenomena [START_REF] Zhou | Advanced polymer dielectrics for high temperature capacitive energy storage[END_REF][START_REF] Kita | Dielectric and Insulation Properties of Polyimide-based Boehmite Nanocomposite Material[END_REF][START_REF] Savin | Turn-to-turn capacitance variations correlated to PDIV for AC motors monitoring[END_REF][START_REF] Savin | Aging effects on the AC motor windings: A correlation between the variation of turn-to-turn capacitance and the PDIV[END_REF][START_REF] Korcak | Thermal Accelerated Aging Methods for Magnet Wire: A Review[END_REF][START_REF] Werynski | Proposition of a new method for in-service monitoring of the aging of stator winding insulation in AC motors[END_REF][START_REF] Stone | Chapter 15: Off-Line Rotor and Stator Winding Tests[END_REF][START_REF] Madonna | Insulation Capacitance as Diagnostic Marker for Thermally Aged, Low Voltage Electrical Machines[END_REF][START_REF] Huan | Advanced polymeric dielectrics for high energy density applications[END_REF].

A. Relative Permittivity and Parallel Capacitance

The impact of temperature on the capacitance of insulators is still under investigation. However, several studies show that the capacitance of insulators increases with temperature [START_REF] Savin | Turn-to-turn capacitance variations correlated to PDIV for AC motors monitoring[END_REF][START_REF] Savin | Aging effects on the AC motor windings: A correlation between the variation of turn-to-turn capacitance and the PDIV[END_REF][START_REF] Korcak | Thermal Accelerated Aging Methods for Magnet Wire: A Review[END_REF][START_REF] Werynski | Proposition of a new method for in-service monitoring of the aging of stator winding insulation in AC motors[END_REF]. The parallel capacitance is commonly used to represent the inter-turn insulation and depends on the relative permittivity. Its evolution at low frequency is related to a phenomenon of conduction DC which increases with the temperature [START_REF] Kita | Dielectric and Insulation Properties of Polyimide-based Boehmite Nanocomposite Material[END_REF]. The phenomenon of augmentation of parallel capacitance can be principally explained by the variation of the dielectric constant caused by temperature [START_REF] Madonna | Insulation Capacitance as Diagnostic Marker for Thermally Aged, Low Voltage Electrical Machines[END_REF].

B. Partial Discharge Inception Voltage

Past works show us a decreasing behavior of Partial Discharge Inception Voltage (PDIV) as a function of temperature [START_REF] Sili | Study of the impact of the aeronautical environment on the Paschen curve and the aging of polyimide under partial discharge[END_REF], [START_REF] Savin | Turn-to-turn capacitance variations correlated to PDIV for AC motors monitoring[END_REF][START_REF] Savin | Aging effects on the AC motor windings: A correlation between the variation of turn-to-turn capacitance and the PDIV[END_REF]. Recent work using twisted pairs as samples obtained a PDIV decreasing with temperature, justified by the increased field strength between the air wedge gaps due to the increase in the complex relative permittivity of the insulation layer at high temperature [START_REF] Kita | Dielectric and Insulation Properties of Polyimide-based Boehmite Nanocomposite Material[END_REF].

C. Dissipation Factor

Thermal aging is one of the deterioration processes which causes an increase in dielectric losses [START_REF] Stone | Chapter 15: Off-Line Rotor and Stator Winding Tests[END_REF]. Two causes explain the behavior of the dissipation factor with the temperature, the molecular conditions inside the material, and the conductivity influence in high temperature.

-The dipole relaxation is a phenomenon that contributes to the dielectric loss factor (Tan δ) in polar dielectrics. The dipoles orient themselves to a greater or lesser extent in response to the electric field. In the solid, the dipoles suffer friction with the polymer matrix, so this is an obstacle to the reorientation of the dipoles in response to the applied field. The reorientation process has a time constant or several time constants. If the dipole oscillates 45° to the phase of the electric field, the loss induced by the dipole is maximized, resulting in a peak in the dielectric loss called the "dispersion peak". As the temperature increases, the viscosity of the polymer tends to decrease, resulting in a reduction in the reorientation time constant and an increase in the frequency of the dispersion peak [START_REF] Huan | Advanced polymeric dielectrics for high energy density applications[END_REF].

-At high temperatures, conduction can contribute to dielectric losses in a non-negligible way. The loss tangent includes a term σ(E, T)/εω, where the function σ(E, T) corresponds to the electric field and temperature-dependent conductivity [START_REF] Kita | Dielectric and Insulation Properties of Polyimide-based Boehmite Nanocomposite Material[END_REF].

III. ENAMELLING AND EXTRUSION PROCESS

The main steps of the process of insulation application around a copper conductor are: 1) Annealing oven: The process starts with the annealing furnace, where a copper conductor is cleaned in an atmosphere without oxygen and is softened, which is important to acquire the necessary flexibility to support the winding operations and warranties the best adhesion of the primer layer on copper with an etched surface. 2) Varnish application: an insulation layer in a solvent phase is applied around the conductor by enamelling dies. 3) Curing: Once the varnish is applied to the copper surface, the wire passes in an oven for the curing process. Around 70% of the solvents from the varnishes are extracted and the dry extract remains on the surface of the wire. 4) The evaporated solvents are passed through a catalyst where they are burnt off and the heat from the catalyst is reused by the curing oven temperature control. 5) Steps 2 and 3 are iterated to reach the targeted thickness.

This enamelling process is repeated several times until reaches the insulation thickness. In this study, an enamelling process is used for a small part of the insulation thickness (between 20% and 30% of the thickness). Most of the insulation thickness is made by only one pass by the extrusion machine, resulting in a resin coating. The final configuration obtained in this process is represented in Fig. 1, the copper conductor with a first coat composed of enamel thin layers and a final extruded layer.

IV. MATERIALS INSULATORS

To increase the thermal resistance of an insulator, it is important to choose the materials classed for higher temperature operation. The high-temperature resistant insulation supports a temperature operation above 175°C [START_REF] Hicyilmaz | Applications of Polyimide Coatings: A Review[END_REF]. They are characterized by having high thermal stability, high dielectric strength, and good adhesion to the cooper [START_REF] Hergenrother | Perspectives in the Development of High-Temperature Polymers[END_REF].

The conventional materials used for wire insulation for hightemperature applications are Polyestherimide, Polyamideimide, and polyimide [START_REF] Li | High-Temperature Dielectric Materials for Electrical Energy Storage[END_REF]. Polyimide (PI) is known for its thermal resistance, its thermal class can reach 240°C for 25µm thickness. Polyamide-imide (PAI) combines the good properties of PI and good processability, its thermal index is around 220°C. The PAI is frequently used on topcoats in electrical insulations of wires with a Polyestherimide (PEI) bottom coat. The PEI is a modified version of PI, with lower thermal stability due to its flexible ether linkage; its thermal class is 200°C.

Materials used on the bottom layer of the insulation were applied by the classic enamelling process. As an extrusion layer, a thermoplastic polymer is used for the topcoat. It's a crystalline resin that has high heat resistance [START_REF] Ito | Development of High Voltage Wire for New Structure Motor in Full Hybrid Vehicle[END_REF]. Precedent works with mineral fillers have shown an improvement in mechanical properties [START_REF] Wang | Microstructure and Fracture Behavior of Polypropylene/Barium Sulfate Composites[END_REF][START_REF] Amestoy | Crystallization Behavior and Mechanical Properties of Poly(ε-Caprolactone) Reinforced with Barium Sulfate Submicron Particles[END_REF]. The mineral fillers are also implemented in an extruded layer of some configurations tested.

Contents of silica-based sol-gel were implemented in the second layer of PAI varnish (enamel) before the extrusion layer. The sol-gel process contributes to improving the enamel wire properties, and better properties of breakdown voltage after thermal degradation [START_REF] Morikawa | Enameled Wire Having Polyimide-Silica Hybrid Insulation Layer Prepared by Sol-Gel Process[END_REF]. Recent works using silica particles have shown improvement in glass transition temperature [START_REF] Yoo | Surge-Resistant Nanocomposite Enameled Wire Using Silica Nanoparticles with Binary Chemical Compositions on the Surface[END_REF].

V. RESULTS AND DISCUSSIONS

Different insulations configurations on a copper conductor of 0,95 mm diameter are tested. The PEI and PAI polymers have been used as primer insulators of 25µm of diameter increase.

Thermal aging tests are carried out accordingly to the specifications of the IEC 60172 standard [START_REF]Test procedure for the determination of the temperature index of enamelled and tape wrapped winding wires[END_REF]. The aging temperature is 250°C and the cycle duration is 72h. Eleven twisted pairs samples are prepared following the standard specifications [START_REF]Winding wires -Test methods -Part 5: Electrical properties[END_REF]. The proof test voltage used is 1000V, due to the increase of diameter (insulation coat). The lifetimes obtained by this experimentation are presented in Table I.

TABLE I AGING TEST OF ENAMEL-EXTRUDED WIRES AT 250°C

The lifetimes indicate that the extruded layer has a relevant influence on the thermal resistance of the insulation configuration during the aging test. The wire containing micro fillers on extruded layer has resisted longer than those that are not filled.

Following the first indications, new configurations have been exploited using the mineral micro fillers and adding sol-

Wires insulation composition

Lifetime (h) PEI + extruded layer of thermoplastic polymer 348 PAI + extruded layer of thermoplastic polymer 420 PEI + extruded layer of thermoplastic polymer with mineral micro fillers 492 PAI + extruded layer of thermoplastic polymer with mineral micro fillers gel to the primer enamel. We used the sol-gel products on the enamel layer to verify their influence on the improvement of thermal resistance during thermal aging.

Table II shows the results of an aging test at a higher temperature, 300°C. This test has a group of eleven samples of the most performant wire of the precedent aging, comparing it with a group of eleven samples of a new wire insulated with sol-gel contents on the enamel PAI layer and filled resin on extrusion. In this case, it's possible to observe the higher performance of the group of samples insulated with sol-gel contents, which failed the proof test voltage after 7 thermal cycles (the duration per cycle is 24h), and the samples insulated with PAI + thermoplastic filled insulation failed after 3 cycles at 300°C. This higher performance is also verified in a 250°C aging test as presented in the last column of Table II.

TABLE II AGING TEST OF ENAMEL-EXTRUDED WIRES WITH AND WITHOUT SOL-GEL

The sol-gel insulated wire had been observed during the 250°C aging test, including classic wire samples, and PI enamel with a thermoplastic polymer extruded insulated samples. Eleven twisted-pair samples had been made for each insulation. The three insulations are described below: A. Wire insulated by the first layer of PAI enamelled, a second enamel layer of a mixed content of PAI and solgel in a concentration of 18%, and a topcoat of thermoplastic polymer on the extrusion layer. B. Wire insulated by a first layer of PI enamelled and an extruded layer of thermoplastic polymer. The proof test voltage was 1200 V for each sample, accordingly with its greater thickness compared with the others. C. A commercial wire insulated by enamel layers of PEI and PAI. The graphic in Fig. 2 shows the dielectric behaviour of each group of samples. Parallel capacitance and dissipation factors are measured by using an Agilent 4294A Precision Impedance Analyzer. The measurement frequency is 10 kHz to have the precision indicated on the accuracy chart of the equipment. The measurement of PDIV is achieved accordingly with the standard IEC 60270 [START_REF]High-voltage test techniques -Partial discharge measurements[END_REF]. These measurements are achieved before each thermal cycle and the proof test voltage is applied using the same device.

Despite the deviation between sample measurements indicated graphically in Fig. 2, the commercial wire C shows an increase in parallel capacitance, which can be associated with accelerated aging of most parts of the samples compared with the other configurations of insulators. The classic wire may be observed in a longer aging test to affirm the increase of parallel capacitance expected in accord with the literature 1 Result issue of Table I previously exhibited [7-8] [10]. And wire B has stable values of parallel capacitance during the aging test, resulting from the high thermal stability characteristics of the PI in the first coat. Wire A had a variation in parallel capacitance values. That can be influenced by the sol-gel contents on the primer layer, in which the relative permittivity is not known. To understand this phenomenon, more tests are being done with sol-gel contents without PAI polymer to evaluate the permittivity and conductivity of the material.

The PDIV had been decreased during the aging for all insulators as expected. The extruded wires have higher values of PDIV compared with the commercial one, it's kept until its end life. The insulation B presented higher values of PDIV during the test.

The dissipation factor had different behaviours between the insulators. The most resistant insulators had stables values, and the commercial one got a visible increase just before the last proof test voltage, meaning that the dielectric parameters were already degraded. This can also be related to its low thermal resistance compared with the others. The high deviation for the last point of dissipation factor measurement of the commercial wire happens because part of the samples was degraded (dissipation factor > 1,0). The degradation of the dissipation factor coincides with the end of life when more than 50% failed during the last proof test voltage.

The curves finish at the end of the thermal aging when more than 50% of samples insulators broke down during the proof test. Wire A got a higher duration of resistance on thermal aging, which indicates that the sol-gel and filled resin influenced the improvement of the resistance on this test. resistance during the aging test. The sol-gel mixed with polymer insulation varnish, shows an increase in the thermal resistance on the aging test. The results obtained in this study show the positive influence of the sol-gel which doesn't need hazardous solvents for polymerization (sol-gel is wateralcohol based). This type of insulation can be an eco-friendly candidate to be used as insulation of enamelled wires. On the other hand, the basic mechanical properties of the wires are assured only when the concentration is very low (18%), and the polymer with organic solvent is still being required for that.

Although the enamel-extruded technology studied is far from being free of polymer enamel coat, it represents a possibility of reducing the enamel process and a greener solution for magnet wire manufacturers. Despite this possibility, many tests must be executed to confirm a replacement of the conventional enamelling process. We envisage defining the thermal index for the enamel-extruded wires developed, by thermal aging tests. Many other properties are still to be validated as required by the magnet wire standards (such as chemical resistance, abrasion resistance, and more).

The thermal class improvement depends on the thermal resistance of the insulators during the aging test. The commercial wire having a thermal class of 210°C, indicates that the enamel-extruded has better performance so a greater or equivalent thermal class.

We are testing different fillers to improve this thermal resistance. New formulations of sol-gel will be studied soon with the aim of increasing the concentration of sol-gel on the primer coat and evaluating if the properties for winding applications will be conserved.

Fig. 1 .

 1 Fig. 1. Enamel-extruded wire

Fig. 2 .

 2 Fig. 2. Dielectric parameters collected between cycles of 250°C thermal aging CONCLUSIONS AND PERSPECTIVES Different polymers have been tested on primer layer of enamelled wire to observe their influence on the thermal

ACKNOWLEDGMENT

This research study is a part of the Hi-Ecowire Project, financed by INTERREG European founds. This project is piloted by Materia Nova which also developpes the sol-gel formulation and, GII which pilots the enamel-extrusion process supplying the insulated wires analyzed in this work. The authors are grateful to all the partners of Hi-Ecowire and the Laboratoire Systèmes Electrotechniques et Environnement (LSEE) for the contributions and materials which allow the advancement of this research project.


A laminated busbar (LBB) manufacturing today is faced with multiple challenges :

• The need to increase service life of the product; • Limit the appearance of partial discharges; • Reduce the cost and energy consumption.

• Endure constraints required by the futur SiC converters (voltage, frequency,…) • Deal with extreme environmental conditions.

• Introduction of micro & nano inorganic fillers (BaSO4, SiO2, Al2O3, TiO2) to improve performances in : thermal resistance, creepage discharge performance, corona resistance … • Use of an electret layer (varnish with BaTiO3 ferroic filler) in to significantly limit the electric field intensity. • Combine UV curing with thermal curing using a thermal curing agent.

• Characterization of a laminated busbar performance in a simulated aeronautical environment.

Currently, the varnish insulation is capable of withstanding 180°C continuously, thanks to various formula adaptations and the introduction of Silane coupling agent that serves as adhesion promoter.

1: Univ. Artois, UR 4025, Laboratoire Systèmes Electrotechniques et Environnement (LSEE), Béthune, F-62400, France 2: Amphenol AUXEL -ZI rue de la Barre Gondecourt, 59147, France

Oussama Rahmouni The coating on the edges is particularly important due the high concentration of the electric field in these areas. This coating has been significantly improved thanks to the use of thin layers and a thixotropic agent in the formula.

The breakdown voltage tests performed on the dielectric are rather promising : •9.5 kV AC for 100µm on the surface.

•2.5 kV AC for100 µm on the edges.

RESULTS

PROSPECTIVES

The process consists of depositing multiple very thin layers (~ 10 microns) of varnish using a micro spray gun and curing with UV light after each layer.

few to no air bubbles are encapsulated in the varnish.

high dielectric strength and high PDIV.

The insulation the critical areas of the busbar needs further improvement much lower electric field intensity in the air in these areas. higher PDIV and a longer life span. An alternative manufacturing process (patent application filed) is to use additive manufacturing, which consists of 3D printing successive layers using SLA -LCD technology on each face of the conductor. Well suitable for flat busbar (example : conductors for power planar transformer):

-complete encapsulation, including lateral sides (creepage distance, compacity -bubble-free (partial discharge)

  necessary adaptation of laminated busbars to the development of new high power switching devices operating in extreme conditions Provide a high level of reliability during the whole exploitation time of the product INTRODUCTION VARNISH INSULATED BUSBAR (VIB)

Innovative insulation technology for SiC Power Electronics

  1,2 , Sonia Ait Amar 1 , Stéphane Duchesne 1 , Jean-François Wecxsteen 2

		No silane	With Silane
	Copper conductor	Air
		Triple junction
	PET film
	Glue
	Copper conductor

OBJECTIVES

VIB is used for the first time in laminated busbars. This new technology is based on existing varnish insulation in electrical machines.