Philippe Besnard
email: besnard@irit.fr

Marie-Odile Cordier
email: cordier@irisa.fr

Yves Moinard
email: yves.moinard@inria.fr

Arguments using ontological and causal knowledge

Nous décrivons l'utilisation d'un système logique de raisonnement à partir de données causales et ontologiques dans un cadre argumentatif. Les données consistent en liens causaux ({A1, • • • , An} cause B) et ontologiques (o1 est_un o2). Le système en déduit des liens explicatifs possibles ({A1, • • • , An} explique {B1, • • • , Bm}). Ces liens explicatifs servent ensuite de base à un système argumentatif qui fournit des explications possibles. Un exemple inspiré de la tempête Xynthia, laquelle a provoqué un trop grand nombre de victimes par rapport aux conditions purement météorologiques, illustre une utilisation de notre système.

Introduction and Motivation

Looking for explanations is a frequent operation, in various domains, from judiciary to mechanical fields. We consider the case where we have some precise (not necessarily exhaustive) description of some mechanism, or situation, and we are looking for explanations of some facts. The description contains logical formulas, plus some causal and ontological formulas (or links). Indeed, it is known that, while there are similarity between causation and implication, causation cannot be rendered by a simple logical implication. Also, confusing causation and co-occurrence could lead to undesirable relationships. This is why we use here a causal formalism where some causal links and ontological links are added to classical logical formulas. Then the causal formalism will produce various explanation links [START_REF] Besnard | Ontology-based inference for causal explanation[END_REF]. However, if the situation described is complex enough, this will result in a great number of possible explanations, and some argumentation is involved in order to get some reasons to choose between all these candidate explanations.

In this text, we will consider as an example a severe storm, called Xynthia, which made 26 deaths in a single group of houses in La Faute sur Mer, a village in Vendée during a night in February 2010. This was a severe storm, with strong winds, low pressure, but it had been forecast. Since the casualties were excessive with respect to the strength of the meteorological phenomenon, various investigations have been ordered. This showed that various factors combined their effects. The weather had its role, however, other factors had been involved : recent houses and a fire station had been constructed in an area known as being susceptible of getting submerged. Also, the state authorities did not realize that asking people to stay at home was inappropriate in case of flooding given the traditionally low Vendée houses.

In this paper, we define in section 2 an enriched causal model, built from a causal model and an ontological model. We then show in section 4 how explanations can be derived from this enriched causal model. We explain in section 5 the use of argumentation in order to deal with the great number of possible explanations obtained and conclude in section 6. The Xynthia example, introduced in section 3, is used as illustration throughout the article.

Enriched causal model = Causal model + ontological model

The model that is used to build tentative explanations and support argumentation, called the enriched causal model, is built from a causal model relating literals in causal links, and from an ontological model where classes of objects are related through specialization/generalization links.

The causal model

By a causal model [START_REF] Hugh | The Facts of Causation[END_REF], we mean a representation of a body of causal relationships to be used to generate arguments that display explanations for a given set of facts.

The basic case is that of a causal link "α causes β" where α and β are literals. In this basic case, α stands for the singleton {α} as the general case of a causal link is the form

{α 1 , α 2 , • • • , α n } causes β where {α 1 , α 2 , • • • α n } is a set (interpreted conjunctively) of literals.
Part of the causal model for our Xynthia example is given in Fig. 2 (each plain black arrow represents a causal link).

The ontological model

The literals

P (o 1 , o 2 , • • • , o k )
occurring in the causal model use some predicates P applied to classes of objects o i . The ontological model consists of specialization/generalization links between classes of objects

o 1 is-a -→ o 2 ,
where is-a -→ denotes the usual specialization link between classes. E.g., we have Hurri

is-a -→ SWind, House1FPA
is-a -→ HouseFPA and HouseFPA is-a -→ BFPA : a "hurricane" (Hurri) is a specialization of a "strong wing" (SWind), and the class of "low houses with one level only in the flood-prone area" (House1FPA) is a specialization of the class of "houses in the flood-prone area" (HouseFPA), which itself is a specialization of the class of "buildings in this area" (BFPA). A part of the ontological model for our Xynthia example is given in Fig. 1 (each white-headed arrow labelled with is-a represents an is-a -→ link).

The enriched causal model

The causal model is extended by resorting to the ontological model, and the result is called the enriched causal model. The enrichment lies in the so-called ontological deduction links (denoted -→ links among literals are generated by means of a single principle as follows. The predicates used in the causal model are annotated so that each of their parameters is sorted either as universal or as existential.

A universal parameter of a predicate "inherits" by specialization, meaning that if the predicate is true on this parameter then the predicate is also true for specializations of this parameter. The existential parameters of a predicate "inherits" by generalization, meaning that if a the predicate is true on this parameter, then the predicate is also true for generalizations of this parameter (cf above example of our owner of a sedan).

As another example, consider the unary predicate Flooded, where Flooded(o) means that class o (an area or a group of buildings) is submerged. Its unique parameter is taken to be "universal" so that if the literal Flooded(o) is true, then the literal Flooded(s) is also true whenever s is a specialization of o. The causal model is enriched by adding Flooded(o)

ded ont -→ Flooded(s) for each class s satisfying s is-a -→ o.
Let us now consider the unary predicate Occurs so that Occurs(Hurri) intuitively means : some hurricane occurs. Exactly as above "I own" predicate, this predicate is existential on its unique parameter. By means of the is-a -→ link Hurri -→ links in the case of unary predicates. Then, the general case is a natural, but intricate and thus ommitted here, generalization of the unary case.

Definition 1 Let us suppose that P rop1 ∃ and P rop2 ∀ are two unary predicates, of the "existential kind" for the first one, and of the "universal kind" for the second one. If in the ontology is the link class 1 is-a -→ class 2 , then the following two links are added to the enriched model :

P rop1 ∃ (class 1 ) ded ont -→ P rop1 ∃ (class 2 ) and P rop2 ∀ (class 2 ) ded ont -→ P rop2 ∀ (class 1 ).
In our formalism, causal and ontological links ded ont -→ entail classical implication :

{α 1 , • • • α n } causes α entails ( n i=1 α i ) → α. α ded ont -→ β entails α → β. (1)
Ordinary logical formulas are also allowed (for example for describing exclusions), which are added to the formulas coming from [START_REF] Besnard | Ontology-based inference for causal explanation[END_REF]. When resorting to explanations (see section 4 below), these ded ont -→ links are extended to sets of literals (links denoted by → ded ont set ) as follows :

Definition 2 Let Φ and Ψ be two sets of literals, we define

Φ → ded ont set Ψ, if for each ψ ∈ Ψ, there exists ϕ ∈ Φ such that ϕ ded ont -→ ψ (remind that ψ ded ont -→ ψ).
Notice that if a sedan is a kind of car (sedan is-a -→ car), and Own a predicate existential on its unique argument, then we get {Iown(sedan), IamHappy} → ded ont set {Iown(car), IamHappy}, and {Iown(sedan), IamHappy} → ded ont set {Iown(car)}.

The Xynthia example

From various enquiries, including one from the French parliament 1 and one from the Cours des Comptes 2 and many articles on the subject, we have many information about the phenomenon and its dramatic consequences. We have extracted a small part from all the information as an illustration of our approach.

The classes we consider in the ontological model and/or in the causal model are the following ones : Hurri, SWind, BFPA, House1FPA, HouseFPA, and BFPA have already been introduced in §2.2, together with a few is-a -→ links. Among the buildings in the flood-prone area, there is also a fire station FireSt (remind that we have also a group of houses HouseFPA, including a group of typical Vendée low houses with one level only House1FPA). We consider also three kinds of natural disasters (NatDis) : Hurri, together with "tsunami" (Tsun) and "flooding" (Flooding). As far as meteorological phenomena are concerned, we restrict ourselves to "Very low pressure" (VLPress), together with already seen Hurri and SWind, and finally we add 'high spring tide" (HSTide) to our list of classes.

Two kinds of alerts (Alert) may be given by the authorities, "Alert-Evacuate" (AlertEvac) and "Alert-StayAtHome" (AlertStay). We consider also an anemoter (Anemo) able to measure the wind strength and a fact asserting that "people stay at home" (PeopleStay).

1. http://www.assemblee-nationale.fr/13/rap-info/ i2697.asp 2. www.ccomptes.fr/Publications/Publications/ Les-enseignements-des-inondations-de-2010-sur -le-littoral-atlantique-Xynthia-et-dans-le-Var

The following predicates are introduced : (Flooded) and (Victims_I) applied to a group of building respectively meaning that "flooding" occurs over this group, and that there were "victims" in this group (I ∈ {1, 2, 3} is a degree of gravity, e.g. Victims_1, Victims_2 and Victims_3 respectively mean, in % of the population of the group : "a small number", "a significant number" and "a large number" of victims). OK means that its unique parameter is in a normal state. Occurs means that some fact has occurred (a strong wind, a disaster, . . .), Expected means that some fact is expected to occur.

All these predicates are "universal" on their unique parameter, except for the predicates Occurs and Expected which are "existential".

Note that negation of a literal is expressed by Neg as in Neg OK(Anemo) meaning that the anemometer is not in its normal behaviour state (the formula ¬(OK(Anemo) ∧ (NEG OK(anemo)) is thus added).

The classes and the ontological model are given in Figure 1 with the is-a -→ links represented as white-headed arrows labelled with (is-a).

BFPA (is-a) (is-a)

FireSt HouseFPA

(is-a) House1FPA AlertEvac (is-a) Alert AlertStay (is-a) (is-a) (is-a) Flooding Tsun Hurri NatDis (is-a) SWind (is-a)

Anemo

HSTide

PeopleStay

VLPress

FIGURE 1 -Ontological model for Xynthia

Part of the causal model is given in Figure 2 (remember, black-headed arrows represent causal links). It represents causal relations between [sets of] literals. It expresses that an alert occurs when a natural disaster is expected, or when a natural disaster occurs. Also, people stay at home if alerted to stay at home, and having one level home flooded results in many victims, and even more victims if the fire station itself is flooded,... From the ontological model and the causal model, the enriched causal model can be build, adding ded ont -→ links between literals when possible.

For instance, for our "existential" predicates, from Hurri

Explaining a literal from a [set of] literal[s]

Causal and ontological links allow us to infer explanation links. We want to exhibit candidate reasons that can explain a fact by means of at least one causal link. We disregard explanations that involve only links of the implicational kind. Here is how causal and ontological links are used in our formalism :

Let Φ denote a set of literals and β be a literal.

The basic case is that Φ causes β yields that β can be explained by Φ.

(

) 2
The general initial case involves two literals β = P rop(cl 2 ) and δ = P rop(cl 1 ) built on the same predicate P rop (eventual other parameters equal in these two literals) :

Φ causes β δ ded ont
-→ β yields that δ can be explained by Φ, provided δ is possible.

(

) 3
{δ} is the set of justifications for this explanation of δ by Φ.

We require also that the origin of the explanation is possible, thus the full set of justifications is here the set Φ ∪ {δ}, while in case of (2), this full set is Φ, even if no set is given explicitly. Indeed, it is always understood that the starting point of the explanation link must be possible, thus we sometimes omit it. Remind that from (1) we get ( ϕ∈Φ ϕ) → β and δ → β, thus adding β to the justification set is useless.

An explaining link Φ can explain α provided Ψ is possible where Ψ is a set of literals means that α can be explained by Φ provided the set Φ ∪ Ψ is possible : if adding Φ ∪ Ψ to the available data leads to inconsistency, then the explanation link cannot be used to explain δ by Φ.

In the figures, white-headed arrows represent "ontological links" : ded ont -→ links for bare arrows, and is-a -→ links for arrows with (is-a) mentioned, while dotted arrows represent explanation links (to be read can explain) produced by these rules, these arrows being sometimes labelled with the corresponding justification set. We want also that our explanation links can follow ded ont -→ links as follows, in the case where Φ = {ϕ} is a singleton (thus we note sometimes ϕ instead of {ϕ}) :

ϕ can explain δ provided Ψ is possible ϕ 0 ded ont -→ ϕ δ ded ont -→ δ 1 yields that ϕ 0 can explain δ 1 provided Ψ is possible (4)
{ϕ 0 } ∪ Ψ is the full justification set for this explanation of δ 1 by ϕ 0 . Again we get ϕ 0 → ϕ by (1), so we need not to mention ϕ here. Generalizing the case of (4) when Φ is a set of literals needs the use of → ded ont set :

Φ can explain δ provided Ψ is possible Φ 0 → ded ont set Φ δ ded ont -→ δ 1 yields that Φ 0 can explain δ 1 provided Ψ is possible (5)
Now, we also want that our explanation links are transitive, and this necessitates to be able to explain not only literals, but sets of literals. Thus we introduce explanation links among sets of literals, which extend our explanation links from sets of literals towards literals (since it is an extension, we can keep the same name explanation link) :

Explaining a set of literals from a set of literals

Definition 3 Let n be some natural integer, and, for i ∈ {1, 2, • • • , n} Φ, Ψ i be sets of literals and δ i be literals. If, for each i ∈ {1, 2, • • • , n}, we have Φ can explain δ i provided Ψ i is possible, then we define the following [set] explanation link

Φ can explain {δ i /i ∈ {1, 2, • • • , n}} provided n i=1 Ψ i is possible.
Again, such an explanation link applies only when its (full) justification set, here Φ ∪ n i=1 Ψ i , is possible (not contradicted by the data).

Notice that we do not want to explain Φ by Φ itself, and we extend this restriction to → ded ont set links : We do not want to explain Φ by Φ 0 is all we know is Φ 0 → ded ont set Φ. Indeed, this seems to be cheating about what an explanation is (we want some causal information to play a role).

However, in the line of ( 5), we want that explanations follow → ded ont set links, thus we introduce the following definition :

Definition 4 If we have Φ can explain ∆ provided Ψ is possible Φ 0 → ded ont set Φ ∆ → ded ont set ∆ 1
then we have

Φ 0 can explain ∆ 1 provided Ψ is possible
Again, the full justification set of the resulting explaining link is Φ 0 ∪ Ψ.

Our last definition of [set] explanation links concerns transitivity of explanations. This is a "weak" transitivity since the justifications are gathered, however, we will call this property "transitivity". We need to be able to omit in the resulting link the part of the intermediate set which is already explained in the first explanation giving rise to a transitive link :

Definition 5 If Φ can explain ∆ 1 ∪ ∆ 2 provided Ψ 1 is possible and Γ ∪ ∆ 2 can explain Θ provided Ψ 2 is possible, then Φ ∪ Γ can explain ∆ 1 ∪ ∆ 2 ∪ Θ provided Ψ 1 ∪ Ψ 2 is possible.
Again, the full justification set is

Φ ∪ Γ ∪ Ψ 1 ∪ Ψ 2 .

About explanation links and arguments

As a small example, let us suppose that in the causal model is the link reason causes P rop ∃ (class 2 ) and that in the ontology are the links :

class 1 is-a -→ class 2 class 1 is-a -→ class 3
Resulting are the next two links in the enriched model :

P rop ∃ (class 1 ) ded ont -→ P rop ∃ (class 2 ) P rop ∃ (class 1 ) ded ont -→ P rop ∃ (class 3 )
By pattern matching over the diagrams in Fig. 4 and5 [cf patterns [START_REF] Minh | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] and [START_REF] Halpern | Causes and Explanations : A Structural-Model Approach. Part I : Causes[END_REF], we get reason can explain P rop ∃ (class 3 ) through P rop ∃ (class 1 )

("justification").

Importantly, it is assumed that the causal link is expressed on the appropriate level : in other words, should there be some doubts about the kind of objects (here class 2 ) that enjoy P rop ∃ due to reason, the causal link would be about another class. The proviso accompanying the explanation takes place at the level of justifications : the candidate explanation is worth inasmuch as P rop ∃ (class 1 ) is not contradicted. In particular it must be consistent with the reason causing P rop ∃ (class 3 ) [remind [START_REF] Besnard | Ontology-based inference for causal explanation[END_REF]].

Here is an example from Xynthia data. Consider the causal link

Expected(V LP ress) causes Expected(SW ind) together with the following ontological links Hurri

is-a -→ SW ind Hurri is-a -→ N atDis.
The links below can be obtained in the enriched model

Expected(Hurri) ded ont -→ Expected(SW ind) Expected(Hurri) ded ont -→ Expected(N atDis)
We get an argument to the effect that from the set of data

Θ =    Expected(V LP ress) causes Expected(SW ind) Expected(Hurri) ded ont -→ Expected(StrW ) Expected(Hurri) ded ont -→ Expected(N atDis)    we obtain : Θ yields that
Expected(V LP ress) can explain Expected(N atDis) provided Expected(Hurri) is possible.

The intuition is that, from these data, it is reasonable to explain Expected(N atDis), by Expected(V LP ress), provided Expected(Hurri) is possible (not contradicted by other data).

An example of compound explanations

Figure 6 displays an example from Xynthia (cf also Fig. 3) of a few possible explanations, represented by dotted lines, with their label such as 1 or 1a. The sets of literals, from which the explaining links start, are framed and numbered from (1) to [START_REF] Halpern | Causes and Explanations : A Structural-Model Approach. Part I : Causes[END_REF]. This shows transitivity of explanations at work : e.g. set 1 can explain Victims_1(BFPA) (explanation link labelled 1+1a+1b) uses explanation links 1, 1a and 1b.

Another example is set 5 can explain Victims_3(House1FPA) (explanation link 1+1a+2+3) : explanations 1, 1a, 2 and 3 are at work here, and link 2 uses links 1 and 1a together with Flooded(BFPA) ded ont -→ Flooded(House1FPA) while explanation 3 comes from links 1+1a together with Flooded(BFPA)

ded ont -→ Flooded(FireSt). (1) (3)

Argumentation

As just seen, the enriched causal graph allows us to infer explanations for assertions and these explanations might be used in an argumentative context [START_REF] Besnard | Elements of Argumentation[END_REF][START_REF] Minh | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF]. Let us first provide some motivation from our example.

A possible set of explanations for the flooded buildings is constituted by the bad weather conditions ("very low pressure" and "strong wind") together with "high spring tide" (see Fig. 2). Given this explanation (argument), it is possible to attack it by noticing : a strong wind is supposed to trigger the red alarm of my anemometer and I did not get any alarm. However, this counter-argument may itself be attacked by remarking that, in the case of a hurricane, that is a kind of strong wind, an anemometer is no longer operating, which can explain that a red alarm cannot be observed.

Let us see how to consider formally argumentation when relying on an enriched causal model and explanations as described in sections 2 and 4. Of course, we begin with introducing arguments, as follows.

Argument : That

"Φ can explain γ in view of Θ, provided δ is possible" is formalized here as an argument whenever (3) holds, that is : Θ yields that γ can be explained by Φ, provided δ is possible.

The components of an argument consist of : -Φ, the explanation, a set of literals.

γ, the statement being explained, a literal. -∆, the justification of the explanation (see Section 4), a set of literals. -Θ, the evidence, comprised of propositions (e.g., Φ → γ), causal links (e.g., Φ causes β), and ontological deduction links (e.g., δ ded ont -→ β).

Here is an illustration from the Xynthia event. From Fig. 2, that the BF P A buildings are flooded can be explained via the set of two causal links -The justification of the explanation is empty. -The evidence is Θ.

Counter-arguments

Generally speaking, an argument "Φ can explain γ in view of Θ, provided ∆ is possible" is challenged by any statement which questions 1. either Φ (e.g., an argument exhibiting an explanation for the negation of Φ) 2. or γ (e.g., an argument exhibiting an explanation for the negation of γ) 3. or ∆ (e.g., an argument exhibiting an explanation for the negation of ∆) 4. or any item in Θ (e.g., an argument exhibiting an explanation for the negation of Θ for some θ occurring in Θ) 5. or does so by refutation : using any of Φ, Θ, ∆ and γ to explain some falsehood. Such objections are counter-arguments (they have the form of an argument : they explain something -but what they explain contradicts something in the challenged argument).

Dispute. Let us consider the illustration at the start of this section : The argument (that the buildings in the flood-prone area are flooded can be explained, partly, by a strong wind) is under attack on the grounds that my anemometer did not turn red -indicating that no strong wind occurred. The latter is a counter-argument of type 5 in above list. Indeed, the statement to be explained by the counter-argument is the falsehood Red(anemo) (e.g., Green(Anemo) has been observed), using SW ind, i.e., an item in the explanation in the attacked argument. Remember, the attacked argument is Θ yields that F looded(BF P A) can be explained by    Occurs(V LP ress) Occurs(SW ind) Occurs(HST ide)    Taking Red(Anemo) to be a falsehood, the counterargument at hand is Θ' yields that Red(anemo) can be explained by Occurs(SW ind) OK(Anemo)

where -The explanation is Φ = {Occurs(SW ind), OK(Anemo)}. -The statement being explained is γ = Red(Anemo).

-The evidence is Θ = Occurs(SW ind) OK(Anemo) causes Red(anemo)

In our illustration, this counter-argument has in turn a counter-argument (of type 1.) explaining why the anemometer did not get red : i.e., explaining the negation of an item (OK(Anemo) is the item in question) in the explanation in the counter-argument. So, the counter-counterargument here is :

Θ" yields that ¬OK(Anemo) can be explained by Occurs(Hurri) where -The explanation is Φ = {Occurs(Hurri)}.

-The statement being explained is γ = ¬OK(Anemo). -The evidence is Θ = {Occurs(Hurri) causes ¬OK(Anemo)}

The dispute can extend to a counter-counter-counterargument and so on as the process iterates.

Conclusion

The aim of this work is to study the link between causes and explanations [START_REF] Halpern | Causes and Explanations : A Structural-Model Approach. Part I : Causes[END_REF][START_REF] Halpern | Causes and Explanations : A Structural-Model Approach. Part II : Explanations[END_REF], and to rely on explanations in an argumentative context [START_REF] Besnard | Elements of Argumentation[END_REF].

In a first part, we define explanations as resulting from both causal and ontological links. An enriched causal model is built from a causal model and an ontology, from which the explaining links are derived (cf e.g. Fig4 and (2). Our work differs from other approaches in the literature in that it strictly separates causality, ontology and explanations, while considering that ontology is key in generating sensible explanations from causal statements. Note however that some authors have already introduced ontology to be used for problem solving tasks as planning [7, Chapter 2] and more recently for diagnosis and repair [START_REF] Torta | Hypothesis discrimination with abstractions based on observation and action costs[END_REF]. We then argue that these causal explanations are interesting building blocks to be used in an argumentative context.

Although explanation and argumentation have long been identified as distinct processes [START_REF] Walton | Explanations and arguments based on practical reasoning[END_REF], it is also recognized that the distinction is a matter of context, hence they both play a role [START_REF] Scott Giboney | User acceptance of knowledge-based system recommendations : Explanations, arguments, and fit[END_REF] when it comes to eliciting an answer to a "why" question. This is exactly what is attempted in this paper, as we are providing "possible" explanations, that thus can be turned into arguments. The argument format has some advantages inasmuch as its uniformity allows us to express objection in an iterated way : "possible" explanations are challenged by counter-arguments that happen to represent rival, or incompatible, "possible" explanations. However, a lot remains to be done. Among others, comparing competing explanations according to minimality, preferences, and generally a host of criteria.

We have designed a system in answer set programming that implements most of the explicative proposal introduced above. We plan to include it in an argumentative framework and think it will a good basis for a really practical system, able to manage with a as rich and tricky example as the full Xinthia example.

  SWind, we obtain the following ded ont -→ link : Occurs(Hurri) ded ont -→ Occurs(SWind). Let us provide the formal definition of ded ont

FIGURE 2 -

 2 FIGURE 2 -A part of the causal model for Xynthia

Figure 3

 3 Figure 3 represents a part of the enriched causal model, where the (unlabelled) white-headed arrows represent the ded ont-→ links and each black-headed arrow represents a causal link (from a literal or, in case of forked entry, form a set of literals).

FIGURE 3 -

 3 FIGURE 3 -Numerous victims in low and flood-prone houses 4 Explanations

FIGURE 4 -

 4 FIGURE 4 -The explanation link from (3) when Φ = {ϕ}

  FIGURE 5 -Explanation links follow ded ont -→ links [cf (4)]

OccursFIGURE 6 -

 6 FIGURE 6 -A few explanations for victims

  Θ = { Occurs(F looding) causes F looded(BF P A), F looding) } More precisely, using the basic case (2) twice, we obtain that F looded(BF P A) can be explained by the set of literals is Θ yields that γ can be explained by Φ, provided ∆ is possible where -The explanation is Φ.-The statement being explained is γ = F looded(BF P A).


Francois Bruno

Jean-Baptiste Laudereau

Max Lesaffre

Nicolas Verrier

Michael Atlan

Phase-sensitive narrowband heterodyne holography

We report on amplitude and phase imaging of out-of-plane sinusoidal surface vibration at nanometer scales with a heterodyne holographic interferometer. The originality of the proposed method is to make use of a multiplexed local oscillator to address several optical sidebands into the temporal bandwidth of a sensor array. This process is called coherent frequency-division multiplexing. It enables simultaneous recording and pixel-to-pixel division of sideband holograms, which permits quantitative wide-field mapping of optical phase modulation depths. Additionally, a linear frequency chirp ensures the retrieval of the local mechanical phase shift of the vibration with respect to the excitation signal. The proposed approach is validated by quantitative motion characterization of the lamellophone of a musical box, behaving as a group of harmonic oscillators, under weak sinusoidal excitation. Images of the vibration amplitude versus excitation frequency show the resonance of the nanometric flexural response of one individual cantilever, at which a phase hop is measured.

Imaging nanometric optical path length modulations is of interest in non-destructive testing of micro electromechanical systems [START_REF] Bosseboeuf | Characterization of the static and dynamic behaviour of m (o) ems by optical techniques: status and trends[END_REF][START_REF] Rembe | Laser-scanning confocal vibrometer microscope: Theory and experiments[END_REF][START_REF] Kokkonen | Scanning heterodyne laser interferometer for phase-sensitive absoluteamplitude measurements of surface vibrations[END_REF][START_REF] Bramhavar | Superheterodyne detection of laser generated acoustic waves[END_REF]. Among the approaches used for single-point vibration spectra measurements, laser Doppler schemes are commonly found; they enable non-contact out-of-plane vibration monitoring [START_REF] Rl Whitman | Probing of acoustic surface perturbations by coherent light[END_REF][START_REF] De La Rue | Acoustic surface wave amplitude and phase measurements using laser probes[END_REF][START_REF] Stegeman | Optical probing of surface waves and surface wave devices[END_REF]. In these measurement schemes, the resolution of the minimum measurable vibration amplitude is limited by the detection sensitivity [START_REF] Rembe | Laser-scanning confocal vibrometer microscope: Theory and experiments[END_REF][START_REF] Kokkonen | Scanning heterodyne laser interferometer for phase-sensitive absoluteamplitude measurements of surface vibrations[END_REF][START_REF] Rl Whitman | Probing of acoustic surface perturbations by coherent light[END_REF][START_REF] Monchalin | Heterodyne interferometric laser probe to measure continuous ultrasonic displacements[END_REF][START_REF] Wagner | Theoretical noise-limited sensitivity of classical interferometry[END_REF][START_REF] Royer | Mesures optiques de déplacements d'amplitude 10ˆ-4 à 10ˆ2 angström. application aux ondes élastiques[END_REF]. These methods exhibit high reliability for wideband [START_REF] Royer | Mesures optiques de déplacements d'amplitude 10ˆ-4 à 10ˆ2 angström. application aux ondes élastiques[END_REF][START_REF] Royer | Optical probing of the mechanical impulse response of a transducer[END_REF][START_REF] Jia | Laser interferometric detection of ultrasonic waves propagating inside a transparent solid[END_REF][START_REF] Royer | Measurement of piezoelectric constants using an optical heterodyne interferometer[END_REF] and narrowband [START_REF] Bramhavar | Superheterodyne detection of laser generated acoustic waves[END_REF][START_REF] De La Rue | Acoustic surface wave amplitude and phase measurements using laser probes[END_REF][START_REF] Monchalin | Heterodyne interferometric laser probe to measure continuous ultrasonic displacements[END_REF] measurements. Narrowband measurements facilitate high frequency monitoring without the need for specialized electronics [START_REF] Kokkonen | Scanning heterodyne laser interferometer for phase-sensitive absoluteamplitude measurements of surface vibrations[END_REF][START_REF] Bramhavar | Superheterodyne detection of laser generated acoustic waves[END_REF], while wideband measurements allow for transient signal analysis [START_REF] Royer | Optical probing of the mechanical impulse response of a transducer[END_REF]. Nevertheless, imaging is a tedious and time-consuming process because laser probe scanning is required. To circumvent scanning issues, wide-field heterodyne optical detection schemes with dedicated sensor arrays were introduced recently [START_REF] Kimachi | Real-time heterodyne speckle pattern interferometry using the correlation image sensor[END_REF][START_REF] Patel | Widefield heterodyne interferometry using a custom cmos modulated light camera[END_REF]. Another approach involves time-averaged holographic interferometry, which permits high sensitivity measurements [START_REF] Powell | Interferometric vibration analysis by wavefront reconstruction[END_REF][START_REF] Aleksoff | Time average holography extended[END_REF][START_REF] Stetson | Effects of beam modulation on fringe loci and localization in time-average hologram interferometry[END_REF][START_REF] Levitt | Mechanical vibrations: mapping their phase with hologram interferometry[END_REF][START_REF] Ueda | Signal-to-noise ratio and smallest detectable vibration amplitude in frequency-translated holography: an analysis[END_REF][START_REF] Picart | Timeaveraged digital holography[END_REF]. This method enables narrowband measurements of the optical pathlength modulation depth when the exposure time of the recorded interferogram is much longer than the vibration period [START_REF] Aleksoff | Time average holography extended[END_REF][START_REF] Psota | Measurement of piezoelectric transformer vibrations by digital holography[END_REF][START_REF] Verrier | Absolute measurement of small-amplitude vibrations by time-averaged heterodyne holography with a dual local oscillator[END_REF]. With this approach, the optical phase retardation of the recorded scattered field beating against a reference optical field can also be exploited for time-resolved (wideband) determination of the object motion [START_REF] Pedrini | High-speed digital holographic interferometry for vibration measurement[END_REF][START_REF] Perez-Lopez | Very high speed cw digital holographic interferometry[END_REF]. Nevertheless, typical vibration frequencies of interest in non-destructive monitoring are still much higher than camera frame rates, and hence the time-averaging condition holds. In this regime, the phase shift with respect to the excitation signal is usually retrieved from phase-stepped stroboscopic schemes [START_REF] Bosseboeuf | Characterization of the static and dynamic behaviour of m (o) ems by optical techniques: status and trends[END_REF][START_REF] Løkberg | Vibration phase mapping using electronic speckle pattern interferometry[END_REF][START_REF] Petitgrand | 3d measurement of micromechanical devices vibration mode shapes with a stroboscopic interferometric microscope[END_REF][START_REF] Leval | Full-field vibrometry with digital fresnel holography[END_REF][START_REF] Verrier | Phase-resolved heterodyne holographic vibrometry with a strobe local oscillator[END_REF]. Quantitative imaging of optical path length modulation and phase retardation in stationary regimes, at the time scale of the frame exposure, may thus find applications.

In this letter, we report on a heterodyne holographic arrangement used for amplitude and phase imaging of optical path length modulation in time-averaged recording conditions, without strobe light. The proposed arrangement consists of a Mach-Zehnder holographic interferometer in an off-axis and frequency-shifting configuration [START_REF] Verrier | Absolute measurement of small-amplitude vibrations by time-averaged heterodyne holography with a dual local oscillator[END_REF][START_REF] Atlan | Spatiotemporal heterodyne de-tection[END_REF], with a multiplexed local oscillator (Fig. 1). Frequency-tunable narrowband detection is achieved in time-averaged heterodyne detection conditions. Time-averaged holograms of a phase-modulated optical field are recorded with an exposure time much longer than the modulation period. Optical heterodyning is performed with a frequency-shifted optical local oscillator, a frequency-conversion process aimed at shifting a given radiofrequency (RF) optical sideband in the sensor bandwidth. Optical local oscillator multiplexing is realized by the coherent addition of RF signals driving an acousto-optic modulator in the reference beam channel. Coherent frequency-division multiplexing is a technique by which the total available bandwidth is divided into non-overlapping frequency sub-bands carrying separate signals. Addressing simultaneously distinct domains of the spectral bandwidth of holograms was performed with success for angular [START_REF] Paturzo | Investigation of angular multiplexing and demultiplexing of digital holograms recorded in microscope configuration[END_REF], wavelength [START_REF] Kiire | Color digital holography using a single monochromatic imaging sensor[END_REF], space-division [START_REF] Tahara | Single-shot dualillumination phase unwrapping using a single wavelength[END_REF], and sideband [START_REF] Verrier | Absolute measurement of small-amplitude vibrations by time-averaged heterodyne holography with a dual local oscillator[END_REF] multiplexing schemes. Coherent frequency-division multiplexing in heterodyne holography offers the opportunity to measure complex-valued maps of the RF spectral components of the optical radiation field with accuracy and sensitivity, as long as they can be isolated in sub-bands of the sensor's temporal bandwidth. In the reported experiment, high-quality factor flexural responses at audio frequencies of a lamellophone (the metallic comb of cantilevers from a musical box) excited in a sympathetic resonance regime are sought. Sympathetic vibration is a harmonic phenomenon wherein a vibratory body responds to external excitation in the vicinity of a resonance frequency.

The optical setup used to perform non-destructive testing of the cantilevers' out-of-plane flexural responses at nanometric scales is sketched in Fig. 1. It is designed to monitor quantitatively steady-state vibration amplitudes and to detect mechanical phase jumps with respect to the excitation signal, by interferometric detection of an optical field scattered by the lamellophone, referred to as the object field, beating against a reference optical field. The out-of-plane motion of the cantilevers provokes an optical path length modulation of the backscattered object field. Hence the temporal part of the object field E undergoes a sinusoidal phase modulation φ(t) = 4πz(t)/λ. The equation of the out-of-plane motion z(t) of a cantilever, modeled as a damped harmonic oscillator of spring constant k, mass m, damping coefficient c, driven sinusoidally by the force F (t) = F 0 sin(ωt) provided by a piezo-electric actuator is

m ∂ 2 z ∂t 2 + c ∂z ∂t + kz = F (t) (1)
We seek a steady-state solution at the excitation frequency with an induced phase change of ψ with respect to the excitation F (t), as a result of viscous damping

z(t) = z 0 sin (ωt + ψ) (2)
Its amplitude, z 0 , is proportional to the driving force

z 0 = F 0 mω (2ω 0 ζ) 2 + 1 ω 2 ω 2 0 -ω 2 2 -1/2 (3)
Two intermediate variables were introduced : the angular resonance frequency ω 0 of the undamped oscillator, and the dimensionless damping ratio ζ. They satisfy the relationships k = mω 2 0 and c = 2mω 0 ζ. The phase shift of the sinusoidal response with respect to the driving force is

ψ = arctan 2ωω 0 ζ ω 2 -ω 2 0 ( 4
)
The resonance frequencies ω r = ω 0 1 -2ζ 2 of each cantilever are almost equal to the resonant frequencies ω 0 of the undamped system. The monochromatic optical field of amplitude E, oscillating at the angular frequency ω L , backscattered by the object in vibration can be written

E = E exp [iω L t + iφ(t)] (5)
The phase-modulated field can be decomposed on a basis of Bessel functions of the first kind J n (φ 0 ), via the Jacobi-Anger identity

E = E exp(iω L t) n J n (φ 0 ) exp [in (ωt + ψ)] (6)
where the quantity

E n = EJ n (φ 0 ) exp(inψ) (7)
is the complex weight of the optical sideband of order n, and φ 0 = 4πz 0 /λ is the modulation depth of the optical phase of the radiation backscattered by the cantilever.

The local amplitude z 0 of the out-of-plane vibration and phase retardation ψ with respect to the sinusoidal driving force can be derived from the complex weight of two optical sidebands with Eq. 7 and the first-order Taylor developments of J 0 (φ 0 ) and J 1 (φ 0 ), near φ 0 = 0, for z 0 ≪ λ

E 1 E 0 ≈ 2π λ z 0 exp(iψ) (8)
The holographic interferometer is used for the detection of RF phase modulation sidebands of an object field E in reflective geometry, beating against a LO field E LO , as reported in [START_REF] Verrier | Absolute measurement of small-amplitude vibrations by time-averaged heterodyne holography with a dual local oscillator[END_REF]. Its purpose is to enable a robust quantitative measurement of the nanometric modulation amplitudes of cantilever resonances, as well as the associated phase shift with respect to the sinusoidal excitation signal. To do so, the adopted strategy is to separate the sensor bandwidth [-ω S /2, ω S /2[ into three nonoverlapping frequency sub-bands, centered at δω 1 , δω 2 , and δω 3 , each of which is used to carry a separate optical modulation sideband. The LO signal is composed of the sum of three phase-locked RF signals, to yield an optical LO field of the form

E LO = E LO exp(iω L t) 3 n=1 α n exp(i∆ω n t) (9)
where ∆ω n = (n -2)ω -δω n is the frequency shift and

α n E LO is

I(t) = |E(t) + E LO (t)|

2 impinging on the sensor array involves a diffraction calculation performed with a numerical Fresnel transform. Its implementation in off-axis holographic conditions [START_REF] Cuche | Spatial for zero-order and twin-image elimination in digital off-axis holography[END_REF] yields complex-valued timeaveraged [START_REF] Powell | Interferometric vibration analysis by wavefront reconstruction[END_REF][START_REF] Aleksoff | Time average holography extended[END_REF][START_REF] Stetson | Effects of beam modulation on fringe loci and localization in time-average hologram interferometry[END_REF][START_REF] Levitt | Mechanical vibrations: mapping their phase with hologram interferometry[END_REF] holograms I(t) from which the off-axis region

H(t) = E(t)E * LO (t) (10)
is the support of the object-reference fields cross-term. The demodulation process (Eq. 12) from N of consecutive recordings of finite exposure times acts as a bandpass filter of width ω S /N in the temporal frequency domain. Only the contributions within the sensor temporal bandwidth, between the Nyquist frequencies ±ω S /2, have to be taken into account in H(t), which can hence be expressed as jH 1 à j jH 0 à j jH à à1 j FIG. 3: Magnitude of the FFT-spectrum (Eq. 12 and Eq. 13), where three modulation sidebands are addressed.

H(t) = 3 n=1 E n-2 E * LO α n exp(iδω n t) ( 11
)
where the quantities δω n satisfy -ω S /2 < δω n < ω S /2.

The FFT-spectrum of N consecutive off-axis holograms

H (ω n ) = N p=1 H(2πp/ω S ) exp (-2inpπ/N ) ( 12
)
exhibits three peaks at the frequencies δω 1 , δω 2 , and δω 3 (Fig. 3). Their complex values yield holograms Hm , of rank m = -1, 0, 1, whose magnitude is proportional the modulation depth at the frequency δω m+2

Hm = H (δω m+2 ) = A E m ( 13
)
where A ∝ E * LO α m is a complex constant. Thus, Hm is a measure of the complex weight E m . For small-amplitude vibrations, a quantitative map of the local out-of-plane vibration amplitude z 0 can be assessed from the ratio of the first-order sideband H1 and the non-shifted light component H0

z 0 ≈ λ 2π H1 / H0 ( 14
)
In the same manner, a map of the local mechanical phase retardation ψ with respect to the excitation signal can be assessed from H1 , and H0

ψ -ψ 0 = Arg H1 / H0 ( 15
)
where ψ 0 (t) = δω 3 t -δω 2 t. Relationships 14 and 15 are derived from Eq. 8, and Eq. 13. One can also make use of the third recorded band H-1 and the property

J -1 (φ 0 ) = -J 1 (φ 0 ) to derive z 0 ≈ λ/(2π) H-1 / H0 .
To perform the measurement, an optical LO consisting of the combination of three frequency components shifted by ∆ω 1 , ∆ω 2 , and ∆ω 3 (Eq. 9) was realized. Those shifts were chosen to yield non-symmetrical (non-opposed) algebraic modulation frequencies δω 1 , δω 2 , and δω 3 in the detector bandwidth in order to avoid spurious aliasing or cross-talk effects. To do so, we made use of phase-locked RF signals generated by frequency-synthesizers at the carrier frequency ω C , ω C + δω ′ 1 , ω + δω ′ 2 , and ω C + δω ′ 3 , where the shifts used to generate the three RF bands were δω ′ 1 /(2π) = -1.25 Hz, δω ′ 2 /(2π) = 6.25 Hz, δω ′ 3 /(2π) = -2.5 Hz (see Tab. I). As sketched in Fig. 1, those signals were mixed and combined to drive a set of acousto-optic modulators (AA Opto Electronic) designed to operate around ω C /(2π) = 80 MHz, from which opposite diffraction orders were selected. The resulting modulation frequencies of the holograms (Eq. 13) were δω

1 = δω ′ 1 + δω ′ 2 , δω 2 = δω ′ 3 , δω 3 = δω ′ 1 -δω ′ 2 .
The three sideband components H-1 , H0 , and H1 , encoded at 5 Hz, -2.5 Hz, and -7.5 Hz respectively, are visible on the magnitude of the discrete Fourier spectrum (Eq. 13) reported in Fig. 3. Spectral maps of the vibration amplitude z 0 (ω) were calculated from a set of sequential measurements at each frequency. We took 3000 sequences of N = 8 raw interferograms I sampled at a frame rate of ω S /(2π) = 20 Hz, for excitation frequencies ω/(2π) ranging from 0 Hz to 3 kHz, in 1 Hz steps (supply voltage of the piezo electric actuator : 10 mV). The first resonance frequencies of the lamellophone's cantilevers are observed between 500 Hz and 2500 Hz (Media 1). Plots of the vibration amplitude of the 1st, 5th, and 17th cantilevers versus the excitation frequency ω/(2π) are reported in Fig. 4. In particular, individual resonances of the 1st, 5th, and 17th cantilevers were identified around 541 Hz (Fig. 4(a (Eq. 9) were chirped as well via ω(t) throughout the acquisition. During the time lapse T , a sequence of 248 × 8 = 1984 raw interferograms I was steadily recorded. In that manner, ψ 0 remained constant. Amplitude z 0 (ω) and phase ψ(ω) maps at each frequency ω were calculated from the demodulation of N = 8 consecutive raw frames, reported in media 2. The complex values of these maps were averaged over the spatial extent of the first cantilever and reported in Fig. 5 (blue lines). In agreement with Eq. 4, for non-zero viscosity, a phase shift of ∼ π is observed at the resonance frequency ω r ∼ 541.3 Hz. The theoretical resonance lines in Fig. 5 were determined for a resonance frequency of the undamped oscillator ω 0 /(2π) = 541.3 Hz and a dimensionless damping ratio ζ = 2 × 10 -5 , for which Eq. 3 and Eq. 4 best fit the experimentally-measured vibration amplitude.

In conclusion, we performed optical path length modulation imaging by time-averaged heterodyne holography in off-axis and frequency-shifting conditions. To do so, modulation sidebands were acquired simultaneously 2(b-f). The points were obtained from a sequential measurement, the lines were obtained with the linear chirp (Eq. 16). The theoretical resonance lines in gray were determined from Eq. 3 and Eq. 4.

through coherent frequency-division multiplexing. Additionally, a linear chirp of the excitation and the detection frequencies allowed to map the local mechanical phase retardation of a resonant cantilever with respect to the excitation signal without stroboscopy. The proposed method enabled robust and quantitative narrowband imaging of out-of plane vibration amplitude and phase.

FIG. 1 :

 1 FIG.1: Experimental set-up.

FIG. 2 :

 2 FIG. 2: Magnitude of a statically-scattered light hologram | H0| (a). Magnitude of sideband holograms | H-1| (b) and | H1| (c). Amplitude map of the flexural response z0 of the first cantilever for the first resonance at ω/(2π) = 541 Hz (d). Phase images ψ calculated from 128 raw interferograms around 540.9 Hz (e) and 541.8 Hz (f), in the neighborhood of the first resonance. Movies of the amplitude and phase maps of the out-of-plane vibration are reported in media 1 and 2.

3 - 2 . 5 2 - 7 . 5

 325275 ), inset), 1006 Hz (Fig. 4(b), inset), and 2211 Hz (Fig. 4(c), inset). A finer screening around the resonance of the first cantilever was performed, from 248 sequences of N = 8 raw interferograms, for excitation frequencies ranging from 536 Hz to 546 Hz. The values of z 0 (ω), spatially averaged over the extent of the resonant cantilever, arrowed in Fig. 2(b), are marked as points in the plot reported in Fig. 5(a). But ψ 0 (t) is randomized from the measurement of one frequency point to the next. Hence the phase retardation ψ cannot be retrieved. To prevent reference phase randomization between measurements, a chirp of the frequency ω was performed. It consisted of a linear drift with time t of the excitation frequency from ω I /(2π) = 536 Hz to ω F /(2π) = 546 Hz in T = 99.2 s ω(t) = ω I + (ω F -ω I ) t/T (16) Two out of the three components of the optical LO Hz δω3 = δω ′ 1 -δω ′ Hz ∆ω1 = -ω -δω1 ∆ω2 = δω2 ∆ω3 = ω -δω3

FIG. 4 :

 4 FIG. 4: Vibration amplitude z0 versus excitation frequency ω/(2π), averaged over the 1st (a), 5th (b), and 17th cantilevers. Insets: vibration amplitude maps at 541 Hz (a), 1006 Hz (b), and 2211 Hz (c).

FIG. 5 :

 5 FIG.5: Vibration amplitude z0 (a) and phase ψ (b) averaged over the first cantilever, arrowed in Fig.2(b), versus excitation frequency ω/(2π). Insets : retrieved vibration amplitude and phase maps in the neighborhood of the resonance, reported in Fig.2(b-f). The points were obtained from a sequential measurement, the lines were obtained with the linear chirp (Eq. 16). The theoretical resonance lines in gray were determined from Eq. 3 and Eq. 4.

TABLE I :

 I Frequency shifts used in the experiments.

We gratefully acknowledge support from Fondation Pierre-Gilles de Gennes (FPGG014), Agence Nationale de la Recherche (ANR-09-JCJC-0113, ANR-11-EMMA-046), région Île-de-France (C'Nano, AIMA), and the "investments for the future" program (LabEx WIFI: ANR-10-LABX-24, ANR-10-IDEX-0001-02 PSL*).


Dreyfus

Towards the selection of patients requiring ICD implantation by automatic classification from Holter monitoring indices

Towards the selection of patients requiring ICD implantation by automatic classification from Holter monitoring indices

Cappelaere CH 1, 2 , Dubois R 3 , Roussel P 2 , Baumann O 1 , Amblard A 1 , Dreyfus G 2 1 Sorin CRM SAS, Clamart, France 2 SIGnal processing and MAchine learning (SIGMA) lab, ESPCI ParisTech, Paris, France 3 IHU LIRYC, Bordeaux, France

Abstract

The purpose of this study is to optimize the selection of prophylactic cardioverter defibrillator implantation candidates. Currently, the main criterion for implantation is a low Left Ventricular Ejection Fraction (LVEF) whose specificity is relatively poor.

We designed two classifiers aimed to predict, from long term ECG recordings (Holter), whether a low-LVEF patient is likely or not to undergo ventricular arrhythmia in the next six months. One classifier is a single hidden layer neural network whose variables are the most relevant features extracted from Holter recordings, and the other classifier has a structure that capitalizes on the physiological decomposition of the arrhythmogenic factors into three disjoint groups: the myocardial substrate, the triggers and the autonomic nervous system (ANS). In this ad hoc network, the features were assigned to each group; one neural network classifier per group was designed and its complexity was optimized. The outputs of the classifiers were fed to a single neuron that provided the required probability estimate. The latter was thresholded for final discrimination A dataset composed of 186 pre-implantation 30-mn Holter recordings of patients equipped with an implantable cardioverter defibrillator (ICD) in primary prevention was used in order to design and test this classifier. 44 out of 186 patients underwent at least one treated ventricular arrhythmia during the six-month follow-up period. Performances of the designed classifier were evaluated using a cross-test strategy that consists in splitting the database into several combinations of a training set and a test set. The average arrhythmia prediction performances of the ad-hoc classifier are NPV = 77% ± 13% and PPV = 31% ± 19% (Negative Predictive Value ± std, Positive Predictive Value ± std).

Introduction

Sudden Cardiac Death (SCD) is an unexpected death caused by loss of heart function that occurs in a short time period (generally within one hour of symptom onset) in a person with known or unknown cardiac disease. Most SCDs are caused by a fast, erratic and disorganized propagation of impulses in the ventricles, named ventricular fibrillation. When it occurs, the heart is unable to pump blood anymore and death will occur within minutes, if left untreated by electrical shock(s).

Randomized clinical trials (MADIT II [START_REF] Moss | Multicenter Automatic Defibrillator Implantation Trial II (MADIT II): Design and Clinical Protocol[END_REF], SCD-HeFT [START_REF] Bardy | Amiodarone or an implantable cardioverter-defibrillator for congestive heart failure[END_REF]) have highlighted the benefits of prophylactic ICD implantations for SCD high-risk patients (post-Myocardial Infarction (MI) patients and Heart Failure (HF) patients with reduced LVEF).

However, according to [START_REF] Gehi | Primary Prophylaxis With the Implantable Cardioverter-Defibrillator: The Need for Improved Risk Stratification[END_REF], 81% of the patients have not received any therapy (appropriate or not) from their ICD over the 5-year follow-up period in SCD-HeFT. Beyond the economic issue caused by seemingly unnecessary implantations, are the health issues due to the peri-and postoperative complications. Thus, the selection of prophylactic ICD-implantation candidates must be improved.

In this study we propose the construction of a specific nonlinear classifier that relies on prior knowledge of the arrythmogenic factors, and uses the most relevant descriptors obtained from long-term ECG records (Holter) to identify patients who will undergo ventricular fibrillation in the next 6 months, hence are actually in need of a prophylactic ICD implantation.

2.

Materials and Methods

Population study

One hundred and eighty-six patients (age 67±11 yrs, 163 males) with history of myocardial infarction and/or with heart failure and left ventricular dysfunctions (LVEF < 30%) have undergone a 30-mn Holter recording before being equipped with an ICD in primary prevention. During a six-month follow-up period, 44 out of 186 patients underwent at least one ventricular arrhythmia requiring a therapy deliverance from the ICD.

We divided the database into two groups: the positive group is composed of the 44 records that led to treated ventricular event in the next six months, and the negative group composed of the other 142 records.

Feature grouping and selection

Most of the known rhythmological and morphological parameters available from a Holter recording (such as the descriptors of the Heart Rate Variability, of the QT segment, of the QRS complex, etc.) are computed for each record of the database, resulting in a set of more than seventy candidate features.

These parameters describe different components which are implied in the arrhythmia genesis, so that they can be grouped by arrhythmogenic factors.

Feature grouping

The principal electrophysiological mechanism involved in the ventricular tachyarrhythmia genesis arises from the myocardial substrate, which refers to areas of fibrosis and ventricular dilatation. However, the substrate alone is not capable of originating tachyarrhythmia. The participation of trigger elements (the most common one is premature ventricular contraction (PVC)) is usually necessary. Additionally, the autonomic nervous system (ANS) interacts with the substrate and the triggers to cause electrical instability and leading to fatal arrhythmias, such as VF. Coumel schematized the interaction in the form of a triangle, each angle of which refers to one of the three factors (the myocardial substrate, the trigger elements and the autonomic nervous system) involved in the tachyarrhythmia genesis [START_REF] Coumel | The management of clinical arrhythmias. An overview on invasive versus non-invasive electrophysiology[END_REF].

The structure of the second classifier that we have designed takes that prior physiological knowledge into account by processing separately the features pertaining to the three factors. Thus, the morphological features of the QRS complex, of the ST segment and of the T-wave, which describe the myocardial tissue state and the electrical conductivity condition, are grouped in the substrate hub; the occurrences of PVC and other rhythmic events are grouped in the triggers hub and the descriptors of the Heart Rate Variability and of the Heart Rate Turbulence, which characterize the autonomic regulation of the heart rate, are grouped in the ANS hub.

Nevertheless, due to the small amount of data compared to the large number of candidate features, any statistical model might be overly sensitive to noise or variance in the training data, and fail to estimate the underlying distribution from which the data were drawn.

In other words, the model might overfit the training data. Overfitting usually leads to poor generalization capabilities of the classifier, i.e. to loss of accuracy on test (out-of-sample) data. In order to limit overfitting, a strategy of feature selection is proposed in the next section.

Feature selection

Within each hub, the most relevant features for a classification are selected by the random probe method [START_REF] Stoppiglia | Ranking a random feature for variable and feature selection[END_REF]. This method ranks candidate features in order of decreasing relevance to predict ventricular arrhythmia, using Gram-Schmidt orthogonalisation. The originality of the random probe method lies in the addition of a pseudorandom variable (the probe) to the set of candidate features; its realizations are ranked just as all other candidate features. This results in an estimation of the risk ρ of selecting a candidate feature although it might rank worse than an irrelevant variable, as a function of the number of selected candidate features.

Classifier design

We propose nonlinear classifiers that output an estimation of the probability for the patient to have a serious ventricular arrhythmia during the next six months.

These nonlinear classifiers are neural networks, all neurons of which have a sigmoid transfer function and the inputs of which are the features described in the previous section. In order to estimate the probability for a patient, given the inputs, to belong to the positive group [START_REF] Modha | A learning law for density estimation[END_REF], the samples were assigned the label 1 if the patients belonged to the positive group (i.e. had a treated ventricular arrhythmia), and 0 otherwise. Training was performed by gradient descent [START_REF] Qian | On the momentum term in gradient descent learning algorithms[END_REF] followed by a BFGS [START_REF] Gill | Practical Optimization[END_REF] optimization of the least squares cost function with weight decay term [START_REF] Mackay | A practical Bayesian framework for backpropagation networks[END_REF]. The class imbalance problem was alleviated by multiplicating the records of arrhythmic patients in the training database.

The optimal complexity was found by K-fold crossvalidation, whereby the training/validation set is split into K homogenous and disjoint subsets, trainings are performed on K-1 subsets, and the mean squared error of the resulting models on the examples of the last ("validation") subset are computed; the procedure is iterated K times, so that each example is in a validation subset once and only once. The cross-validation score is the average of the K smallest validation mean squared errors. The complexity of the models that result in the smallest cross-validation score is selected.

After completion of complexity selection, the performances of the classifier and their variability are estimated as follows: the whole database is split into K' homogenous and disjoint subsets, classifiers are trained on K'-1 subsets, the model that has the smallest training error is selected, and is applied to the data of the remaining ("estimation") subset; this procedure is iterated K' times in order to use each example once and only once in the estimation set. The K' classification performances are averaged and their standard deviation is computed.

Conventional neural network classifier

As a reference, a single hidden layer neural network classifier was designed, the inputs of which were the eighteen features selected as described in section 2.2.2 (Figure 1).

Ad hoc neural network classifier

In order to capitalize on prior knowledge, an ad hoc network was designed by grouping the variables as described in the section 2.2.1. Each group of factors undergoes a distinct nonlinear transformation, whose results are fed to a non-linear neuron that provides an estimate of the risk pertaining to the patient (Figure 2). The first step was to build one neural network per arrhythmogenic factor, with one hidden layer and one output neuron. The optimal complexity for each of these three subnetworks is found by cross-validation.

Finally, the output of each subnetwork is fed to a nonlinear neuron. The overall network is trained, the parameters of the subnetworks, obtained in the previous step, are taken as initial values, and the classifier that has the best performance on the training/validation set is selected.

Results

The most relevant features (with a risk ρ ≤ 10%) for discrimination between patients likely or not to undergo ventricular tachyarrhythmia in the next 6 months are listed, per hub, in Table 1. Couple of PAC † averaged over the signal duration ‡ heart rate variability index is the percentage of averaged differences between two successive complexes.

These features were available for 107 patients out of 186.

The selected structure for the conventional single hidden layer neural network had one hidden neuron in its hidden layer.

Concerning the ad hoc network, with three hubs of inputs, it was found that the most appropriate nonlinear transformation, for each subnetwork, was performed by a single neuron with sigmoid output. In other words, the subnetworks shown on Figure 1 had no hidden neuron, i.e. performed a linear separation between the classes.

The performance of the conventional classifier, estimated as described in section 2.3 with K' = 10 resulted in an average reduction of the ICD-implantation of 52% (std. 19%). The averaged negative predictive value (NPV) assessed on cross-test was 68% (std. 13%); it is the ratio of the number of patients who are correctly classified as not requiring an ICD-implantation to the number of patients who were classified as not requiring it. The averaged positive predictive value (PPV), which is the proportion of patients rightly classified as needing an ICD-implantation among the patients classified as requiring it., was 25% (std. 20%).

Likewise, the estimated performances obtained by the ad hoc classifier were an averaged reduction of the ICD-implantations of 59% (std. 15%), with an average NPV of 77% (std. 13%) and a PPV of 31% (std. 19%).

In other words, let us build, for each classifier, an overall confusion matrix by summing the ten matrices produced in the performance estimation procedure. With this "theoretical" classifier, based on the performances of the conventional classifier, only 52 out of the 107 patients would be implanted, resulting in a 51.4% reduction of the number of implantations. Among patients for whom implantation is recommended by the classifier, 13 actually require it, resulting in a PPV of 25%, and among the 55 patients for whom implantation is rejected, 39 do not require it, resulting in a NPV of 71% (Table 2).

In the same situation, a "theoretical" classifier based on the performances of the ad hoc classifier would recommend a reduction of ICD implantations by 58.9% Among the 44 patients for whom an implantation is recommended, 14 really need it, thus the PPV is equal to 32%. Among the 63 patients for whom the implantation is considered as unrequired, 48 do actually not need it, which reflects a NPV of 76% (Table 3).

Discussion

In this study, the objective was to reduce the rate of ICD-implantation with more than 90% of negative predictive value and at least 20% of positive predictive value. The reduction of ICD implantations made possible by both of the designed classifiers is noteworthy and the desired PPV is obtained in both cases. Furthermore, in this experimental case, the contribution of the decomposition of the inputs according to their arrhythmogenic participation provides a slight improvement (p-value = 0.0547) but is not statistically significant. Nonetheless, the aim in NPV remains out of reach considering the limitations we have to face.

The database contained few examples and was unbalanced, making the learning of the training set features difficult, regardless the complexity of the network.

Another limitation of the database is the length of the recordings, which was only 30 minutes, at any time in the daytime; it was thus impossible to study some time periods that would be of interest, such as the hour before awakening. Furthermore, it has been impossible to calculate some descriptors in the usual way (for example, some descriptors of the HRV are commonly averaged on 24 hours) and a temporal analysis of the descriptor variations was unfeasible.

Therefore, repeating the same process of classifier construction on a sizable database of 24-hr preimplantation Holter recordings of patients equipped with an ICD in primary prevention seems to be mandatory.

Conclusion

Improving prophylactic ICD-implantation candidate selection by automatic classification from ECG features may be possible. Nevertheless, to reach this aim, getting more suitable and larger databases is essential to decrease the number of False Negatives, hence increase the negative predictive value.

Figure 1 .

 1 Figure 1. Single hidden layer neural network classifier.

Figure 2 .

 2 Figure 2. Classifier composed of three subnetworks linked to an output nonlinear neuron.

Table 1 .

 1 Most relevant features for classification.

		QRS residuum †
		QRS-T angle †
	Myocardial Substrate	T-Wave residuum † QT end † QT apex †
		QT slope †
		ST elevation / depression †
		Minimum heart rate †
	Autonomic Nervous System	Mean RR interval † SDANN Poincaré Plot Analysis SD2 Heart rate variability index ‡
		Turbulence Onset
		Ventricular bigeminy
	Trigger elements	Ventricular trigeminy Non-sustained ventricular tachycardia Premature Atrial Contraction (PAC)

Table 2 .

 2 Performances of the conventional classifier.

	Overall confusion	"Theoretical"	Designed
	matrix		classifier	classifiers
	39	39	NPV = 71%	NPV = 68±13%
	16	13	PPV = 25%	PPV = 25±20%
	Correctly classified patients	5.1 ± 2.0

Table 3 .

 3 Performances of the ad hoc classifier.

	Overall confusion	"Theoretical"	Designed
	matrix		classifier	classifiers
	48	30	NPV = 76%	NPV = 77±13%
	15	14	PPV = 32%	PPV = 31±19%
	Correctly classified patients	6.2 ± 1.5 *
	* conventional vs ad hoc classifiers: p-value = 0.0547