Thomas Guyet
email: thomas.guyet@irisa.fr

Semantic(s) of negative sequential patterns

Dans le domaine de la fouille de motifs, un motif séquentiel négatif exprime un comportement par une séquence d'événements devant survenir et par des événements qui doivent être absents. Par exemple, le motif a ¬b c décrit l'absence d'un événement b entre les occurrences des événements a et c. Dans cet article, nous mettons la lumière sur l'ambiguité de cette notation et nous identifions huit sémantiques possibles à la relation d'inclusion d'un motif dans une séquence. Ces sémantiques sont illustrées et nous les étudions formellement. Nous proposons ainsi des relations de dominance et d'équivalence entre ces sémantiques, et nous mettons en évidence de nouvelles propriétés d'anti-monotonie. Ces résultats pourraient être utilisés pour développer de nouveaux algorithmes efficaces pour la fouille de motifs séquentiels négatifs fréquents.

Introduction

Pattern mining consists in exploring a set of potential patterns to output all but only the most interesting ones. The set of potential patterns can be seen as a search space and the notion of pattern interestingness can be seen as a set of constraints.

The search space in which lies the potential patterns defines a pattern domain. A large number of pattern domains have been proposed in the pattern mining community. The most studied patterns domains [START_REF] Han | Advanced pattern mining[END_REF] are: itemsets, sequential patterns [START_REF] Mooney | Sequential pattern mining -approaches and algorithms[END_REF] and graph patterns. But a lot of variants from these patterns domains have been proposed. As we focus our attention on temporal data, we can mention: temporal patterns [START_REF] Laxman | A survey of temporal data mining[END_REF], episodes [START_REF] Mannila | Inkeri Verkamo: Discovery of frequent episodes in event sequences[END_REF] or chronicles [START_REF] Besnard | [END_REF][START_REF] Dauxais | Discriminant chronicles mining -application to care pathways analytics[END_REF].

The notion of interestingness studied in pattern mining often refers to the number of its occurrences in a database. A pattern is said to be interesting if it occurs frequently in a dataset of examples (frequent sequential pattern mining task). In practice, a pattern is frequent if its number of occurrences is above a user-defined threshold. Therefore, counting sequences in which the pattern occurs is of paramount importance. It is strongly related to the containment relation which decides whether a pattern occurs in a sequence or not. The support measure denotes the counting of sequences in a dataset D that contains the pattern p, denoted supp D (p).

The success of pattern mining techniques comes from an anti-monotonicity property of some support measures [START_REF] Agrawal | Fast Algorithms for Mining Association Rules in Large Databases[END_REF]. Intuitively, if p is not frequent, no "larger" pattern than p is frequent. The pattern mining trick is to prune the search space as soon as an unfrequent pattern as been found. The "is larger than" relation is a topological structure on the set of patterns. As soon as a support measure is antimonotonic on this topological structure, the frequent pattern mining trick can be used to efficiently prune the search space. Ideally, this structure is a lattice. In this case, the above strategy is complete and correct.

It is worth noticing that the support measure strongly contributes to the semantic of the interestingness of patterns. In fact, output patterns depend on their supports in the dataset(s) of examples. Different support measures have different outputs. For instance, in sequential pattern mining, counting occurrences considering gap constraints does not result in the same set of patterns. 1 In this work, we explore the domain of negative sequential patterns. Sequential patterns describe a sequence of items. The pattern occurs in a sequence while its items appear in the same order in a sequence. For instance, a pattern a b c is read as "a occurs and then b occurs and finally c occurs". Negative sequential patterns are sequential patterns with the specification of absent events. Intuitively, the syntax of a negative sequential pattern will be the following a ¬b c . This pattern is read as "a occurs and then c occurs, but b does not occurs in between".

The apparently intuitive notion of absent event appears to be complex. Few approaches explored what seems to be the same pattern domain [START_REF] Cao | e-NSP: Efficient negative sequential pattern mining[END_REF][START_REF] Chen | Discovering time-interval sequential patterns in se-9. A demo of such an algorithm is accessible[END_REF][START_REF] Gong | e-NSPFI: Efficient mining negative sequential pattern from both frequent and infrequent positive sequential patterns[END_REF][START_REF] Hsueh | Mining negative sequential patterns for ecommerce recommendations[END_REF][START_REF] Xu | E-msNSP: Efficient negative sequential patterns mining based on multiple minimum supports[END_REF][START_REF] Zheng | Negative-GSP: An efficient method for mining negative sequential patterns[END_REF]. They use the similar notation of negation (¬) but this unique syntax is hiding different semantics.

In this article, we bring the light on eight different semantics of negative sequential patterns and we study them formally: we introduce dominance and equivalence relations between these semantics, and we establish antimonotony results to thwart the idea that negative sequential patterns are not anti-monotonic.

Negative sequential patterns

In the sequel, [n] = {1, . . . , n} denotes the set of the first n strictly positive integers. Let I be the set of items (alphabet). An itemset A = {a 1 a 2 ... a m } ⊆ I is a set of items. A sequence s is a set of sequentially ordered itemsets s = s 1 s 2 ... s n : ∀i, j ∈ [n], i < j means that s i is located before s j in sequence s which starts by s 1 and finishes by s n .

Definition 1 (Negative sequential patterns (NSP)). A negative pattern p = p 1 ¬q 1 p 2 ¬q 2 . . . p n-1 ¬q n-1 p n is a finite sequence where p i ⊆ I \ ∅ for all i ∈ [n] and q i ⊆ I

for all i ∈ [n -1].
The length of a NSP, denoted | p| is n, its number of itemsets (negative or positive). p + = p 1 . . . p n is so-called the positive part of the NSP.

We denote by N the set of negative sequential patterns. It can be noticed that Definition 1 introduces a syntactic limitation on negative sequential patterns:

-a pattern cannot start neither finish by a negative pattern, -a pattern cannot have two successive negative itemsets.

Example 1 (Negative sequential pattern). This example illustrates notations of Definition 1. Let I = {a, b, c, d} and p = a ¬(bc) (ad) d ¬(ab) d . We have p 1 = {a}, p 2 = {ad}, p 3 = {d}, p 4 = {d} and q 1 = {bc}, q 2 = ∅, q 3 = {ab}. The length of p is | p| = 6 and p + = a (ad) d d .

Semantics of negative sequential patterns

The semantics of negative sequential patterns relies on negative containment: a sequence s supports pattern p (or p matches the sequence s) iff s contains a sub-sequence s such that every positive itemset of p is included in some itemset of s in the same order and for any negative itemset ¬q i of p, q i is not included in any itemset occurring in the sub-sequence of s located between the occurrence of the positive itemset preceding ¬q i in p and the occurrence of the positive itemset following ¬q i in p.

Definition 2 (Non inclusion). We introduce two operators relating two itemsets P ⊆ I \ ∅ and I ⊆ I:

-partial non inclusion: P I ⇔ ∃e ∈ P, e I -total non inclusion: P I ⇔ ∀e ∈ P, e I and, by convention, ∅ I and ∅ I for all I ⊆ I.

In the sequel we will denote the general form of itemset non inclusion by the symbol , meaning either or .

Intuitively, partial non inclusion considers the itemset as a disjunction of negative constraints, i.e. at least one of the item has to be absent, and total non-inclusion consider the itemset as a conjunction of negative constraints: all items have to be absent.

Choosing one non inclusion interpretation or the other has consequences on extracted patterns as well as on pattern search. Let's illustrate this with following sequence dataset:

D =                    s 1 = (bc) f a s 2 = (bc) (c f ) a s 3 = (bc) (d f ) a s 4 = (bc) (e f ) a s 5 = (bc) (cde f ) a                    .
Table 1 compares the support of patterns under the two semantics of itemset non-inclusion. Let's consider pattern p 2 on sequence s 2 . Considering that the positive part of p 2 is in s 2 , p 2 occurs in the sequence iff (cd) (c f ). In case of total non inclusion, it is false that (cd) (c f ) because of c that occurs in (c f ), and thus p 2 does not occur in s 2 . But in case of a partial non inclusion, it is true that (cd) (c f ), because of d that does not occur in (c f ), and thus p 2 occurs in s 2 . Lemma 1.2 Let P, I ⊆ I be two itemsets:

P I =⇒ P I (1)
Now, we formulate the notions of sub-sequence, non inclusion and absence by means of the concept of embedding.

Table 1 -Lists of supported sequences in D by negative patterns ( p i ) i=1..4 under the total and partial non inclusion semantics. Each pattern has the shape a ¬q i b where q i are itemsets such that q i ⊂ q i+1 . partial total non inclusion non inclusion -

p 1 = b ¬c a {s 1 , s 3 , s 4 } {s 1 , s 3 , s 4 } p 2 = b ¬(cd) a {s 1 , s 2 , s 3 , s 4 } {s 1 , s 4 } p 3 = b ¬(cde) a {s 1 , s 2 , s 3 , s 4 } {s 1 } p 4 = b ¬(cdeg) a {s 1 ,
p i ⊆ s e i , ∀i ∈ [m] -q i s j , ∀ j ∈ [e i + 1, e i+1 -1] for all i ∈ [m -1] e = (e i ) i∈[m] ∈ [n] m is a strict-embedding of pattern p in sequence s iff: -p i ⊆ s e i , ∀i ∈ [m] -q i j∈[e i +1,e i+1 -1] s j for all i ∈ [m -1]
Intuitively, the constraint of a negative itemset q i is checked on the sequence's itemsets at positions in interval [e i + 1, e i+1 -1], i.e. between occurrences of the positive itemset surrounding the negative itemset in the pattern. The soft embedding considers individually each of the sequence's itemsets of [e i +1, e i+1 -1] while strict embedding consider them as a whole. Notation 1. Soft-embedding is denoted -embedding, and strict-embedding is denoted -embedding. One can notice that each sequence contains a unique occurrence of a d , the positive part of pattern p. Using soft-embedding and partial non inclusion ( := ), p occurs in s 1 , s 3 and s 4 but not in s 2 . Using strict-embedding and partial non-inclusion, p occurs in sequence s 3 and s 4 . Indeed, items b and c occur between occurrences of a and d in sequences 1 and 2. With total non inclusion ( := ) and either type of embeddings, the absence of an itemset is satisfied if any of its item is absent. As a consequence, p occurs only in sequence s 4 .

Lemma 2. If e is a -embedding, then e is a -embedding, whatever is the itemset non-inclusion ( ).

Lemma 3. e is a -embedding iff e is a -embedding when := .

Lemma 4. Let p = p 1 ¬q 1 . . . ¬q n-1 p n ∈ N s.t. |q i | ≤ 1 for all i ∈ [n -1]
, then e is a -embedding iff e is aembedding.

Lemma 4 shows that in the simple case of patterns with negative singleton only, strict and soft-embeddings are equivalent.

Lemma 5. Let p = p 1 ¬q 1 . . . ¬q n-1 p n ∈ N, if e is an embedding of pattern p in some sequence s, then e is an embedding of the positive sequential pattern p + in s.

Example 2 illustrates the impact of itemset noninclusion operator and of embedding type.

Another point that determines the semantics of negative containment concerns the multiple occurrences of some pattern in a sequence: should all or at least one occurrence(s) of the pattern positive part in the sequence satisfy the non inclusion constraints? Definition 5 (Negative pattern occurrence). Let s be a sequence, p be a negative sequential pattern, and p + the positive part of p. Let ∈ { , } be a itemset non-inclusion operator, and ∈ { , } correspond to the embedding strategy ( : soft-embedding, and : strict-embedding).

-Pattern p softly-occurs in sequence s, denoted p s, iff there exists at least one embedding of p in s. -Pattern p strictly-occurs in sequence s, denoted p s, iff for each embedding e of p + in s, e is also an embedding of p in s, and there exists at least one embedding e of p + .

Definition 5 allows for capturing two semantics for negative sequential patterns depending on the occurrences of the positive part:

strict occurrence: a negative pattern p occurs in a sequence s iff there exists at least one occurrence of the positive part of pattern p in sequence s and every such occurrence satisfies the negative constraints, soft occurrence: a negative pattern p occurs in a sequence s iff there exists at least one occurrence of the positive part of pattern p in sequence s and at least one of these occurrences satisfies the negative constraints. in s 1 so there is no difference for occurrences under the two semantics. But, it occurs fourth in s 2 with embeddings (1, 2, 5), [START_REF] Agrawal | Fast Algorithms for Mining Association Rules in Large Databases[END_REF][START_REF] Besnard | [END_REF][START_REF] Han | Advanced pattern mining[END_REF], [START_REF] Agrawal | Fast Algorithms for Mining Association Rules in Large Databases[END_REF][START_REF] Guyet | NegPSpan: efficient extraction of negative sequential patterns with embedding constraints[END_REF][START_REF] Han | Advanced pattern mining[END_REF] and [START_REF] Chen | Discovering time-interval sequential patterns in se-9. A demo of such an algorithm is accessible[END_REF][START_REF] Guyet | NegPSpan: efficient extraction of negative sequential patterns with embedding constraints[END_REF][START_REF] Han | Advanced pattern mining[END_REF]. The two first occurrences do not satisfy the negative constraint (¬c) while the two last occurrences do. Under the soft occurrence semantics, pattern p occurs in sequence s 2 whereas it does not under the strict occurrence semantics.

Lemma 6. Let p be a NSP and s a sequence,

p s =⇒ p s ( 2
)
where ∈ { , } and ∈ { , }.

Lemma 7. Let p be a NSP and s a sequence,

p s =⇒ p s (3)
where ∈ { , } and ∈ { , }

In this section, we shown that there are several semantics associated to negative patterns. This leads to eight different types of pattern occurrences. We denote Θ the set of considered pattern occurrence operators:

Θ =
, , , , , , ,

These operators allows to disambiguate the semantics of negative pattern containment. But, is there no useless distinctions between containment relations? Is there some equivalent containment relations in Θ? The next section answers these questions by introducing the notion of dominance between semantics. Then, we provide some results about the anti-mononicity of these containment relations.

Dominance and equivalence between containment relations

Definition 6 (Dominance). For all θ, θ ∈ Θ, θ dominates θ , denoted θ θ , iff pθs =⇒ pθ s for all p ∈ N and all sequence s.

The idea behind the dominance relation between two containment relations θ and θ is related to the sequences in which a pattern occurs. By definition, if θ θ then for any pattern p ∈ N, if p occurs in a sequence s according to the θ pattern containment relation, then it also occurs in s according to the θ pattern containment relation. In the context of pattern mining, such kinds of relation are useful to propose algorithms which could benefit from properties of a dominating containment relation to extract efficiently the patterns according to dominated containment relations. Notation 2. We denote by θ θ iff θ θ is false. Lemma 8. Dominance relation is a pre-order. Definition 7 (Equivalent containment relations). For all θ, θ ∈ Θ, θ is equivalent to θ , denoted θ ∼ θ iff θ θ and θ θ.

Lemma 9. ∼ is an equivalence relation on Θ.

Equivalent containment relations have the same semantic. The sets of sequences in which a given pattern occurs are the same and, reciprocally, the sets of negative patterns which occur in a sequence are the same considering these equivalent containment relations.

We now study the practical dominance relations we have between the elements of Θ.

Proposition 1. The following dominances between containment relations hold:

(4) (5) (6) (7)
and the following non-dominance statements hold:

(8) (9) [START_REF] Laxman | A survey of temporal data mining[END_REF] where ∈ , , ∈ { , } and ∈ { , }.

Proposition 1 gathers the results from the previous section. Each line expresses several relationships between pairs of containment relations. Equations 4-7 are dominances deduced from Lemmas 2, 3, 6 and 7. Equations 8-10 states the absence of dominance for which we can exhibit counterexamples. Figure 1 summarizes them. 3 In addition, many other dominance and non dominance relationships can be deduced from Proposition 1 using the transitivity of dominance (Lemma 8). Nonetheless, in this work, some relationships between pairs of containment relations can not be deduced from Proposition 1. For instance, the relationship between and is not determined by Proposition 1.

One interesting result of Proposition 1 is that there are two pairs of containment relations, , and , , whose elements are equivalent.

3. Assuming that two containment relations are not equals iff they are different, the dominance relation is a pre-order. The property is not central in the following. Figure 1 illustrates clearly that it is not a partial order. Indeed, two pairs of different containment relations are symmetrically dominated. We can finally point out that Lemma 4 adds a dominance relation when negative sequential patterns are restricted to have singleton negative itemsets. In this case, the equiv-

alent classes become: , , , , , and , .

Anti-monotonicity

Our question is now to know whether there are containment relations that have more interesting properties. In our original context of mining frequent negative sequential patterns, we investigate the anti-monotonicity properties.

According to Zheng et al. [START_REF] Zheng | Negative-GSP: An efficient method for mining negative sequential patterns[END_REF], "the APriori principle doesn't apply to negative sequential pattern". The "APriori principle" can be understood as the anti-monotonicity property. We will see that assertion is actually only partially true.

The anti-monotonicity makes sense only with a partial order on the set of NSPs. We first introduce different possible partial orders and then we introduce the antimonotonicity.

For sake of conciseness, the remaining of the section assumes that := . Thus, we can count on the antimonotonicity of the non inclusion of itemsets: q ⊆ q =⇒ q q for all q, q ⊆ I. The following results can be extended to the case := by reversing the inclusion relations for negatives in the partial orders.

Partial orders

Definition 8 introduces several relations between negative sequential patterns that are partial orders (see Proposition 2).

Definition 8 (NSP relations). Let p

= p 1 ¬q 1 p 2 ¬q 2 • • • p k-1 ¬q k-1 p k and p = p 1 ¬q 1 p 2 ¬q 2 • • • p k -1 ¬q k -1 p k be two NSPs. By definition, p • p iff k ≤ k and ∃(u i ) i∈[k] ∈ [k ] k s.t.: 1. ∀i ∈ [k], p i ⊆ p u i 2. ∀i ∈ [k -1], q i ⊆ j∈[u i ,u i+1 -1] q j 3. k = k =⇒ ∃ j ∈ [k], p j p j or q j q j 4. u i < u i+1 , for all i ∈ [k -1]
by definition, p p iff k ≤ k s.t.:

1. ∀i ∈ [k], p i ⊆ p i 2. ∀i ∈ [k -1], q i ⊆ q i 3. k = k =⇒ p k p k or ∃ j ∈ [k -1] s.t. q j q j
and, by definition, p + p iff k = k s.t.:

1. ∀i ∈ [k], p i = p i 2. ∀i ∈ [k -1], q i ⊆ q i 3. ∃ j ∈ [k -1] s.t. q j q j
The • relation can be seen as the "classical" inclusion relation between sequential patterns [START_REF] Mooney | Sequential pattern mining -approaches and algorithms[END_REF]. A NSP p is less specific than p iff p + is a subsequence of p + and negative constraints are satisfied. The principal difference with is that • permits to insert new positive itemsets in the middle of the sequence while permits only insertion of new positive itemsets at the end. 4, 5 Nonetheless, it is still possible to insert items to the positive itemsets. The + does not even permit such differences: each pair of positive itemsets must be equals to have comparable NSP.

Lemma 10. For all p, p ∈ N,

p + p =⇒ p p =⇒ p • p (11)
Proposition 2 (Strict partial orders). • , and + are partial orders on N.

We can notice that the third conditions in Definition 8 enforce the relations to be irreflexive. Removing these conditions enables to define non-strict partial orders.

Anti-monotonicity

Let us first define the anti-monotonicity property of a containment relation θ ∈ Θ considering a strict partial order ∈ { , • , + }.

Definition 9 (Anti-monotonicity on (N, )). Let θ ∈ Θ be a containment relation, θ is anti-monotonic on (N, ) iff for all p, p ∈ N and all sequence s:

p p =⇒ p θs =⇒ pθs
First of all, we provide an example showing that none of the containment relation is monotonic on (N, • ). Let p = b ¬c a , p = b ¬c d a and s = b e d c a . Then, we have p • p . 6 Nonetheless, for each θ ∈ Θ, p θs but it is false that pθs. In fact, the presence of the item d in the sequence changes the scope for checking the absence of c.

This example illustrates the case of Zheng et al. [START_REF] Zheng | Negative-GSP: An efficient method for mining negative sequential patterns[END_REF] to argue for the absence of anti-monotonic property for negative patterns. But, using partial orders that prevent from changing the scope for absent items enables to exhibit antimonotonicity properties.

Proposition 3.
is anti-monotonic on (N, ), where ∈ { , }.

Proposition 3 shows that using the order leads to have anti-monotonicity only for containment with softoccurrence, but not with strict-occurrence. Let us give a counterexample illustrating the problem with strictoccurrence. Let p = a ¬b c , p = a ¬b c d and s = a c d a b c . Then, we have p p . 7 Nonetheless, p s holds but it is false that p s. In fact, without the presence of the item d in the pattern, there are three possible embeddings of p in s. Considering each embedding must satisfy the negation of b, which is not the case, while it is sufficient to have only one embedding satisfying negations for . The previous example illustrates the problem while extending the pattern with additional itemsets. The same issue is encountered with the following example considering same length patterns but with an extended itemset. Let p = a ¬b c , p = a ¬b (cd) and s = a (cd) a b c . Then, we have p p . Nonetheless, p s holds but it is false that p s.

Proposition 4.

is anti-monotonic on (N, + ), where ∈ { , } and ∈ { , }.

We remind that this section presented the case of total non-inclusion ( := ) but similar results can be obtained with partial non-inclusion.

Come back to pattern mining

The definitions of pattern support, frequent pattern and pattern mining derive naturally from the notion of occurrence of a negative sequential pattern, no matter the choices for embedding (soft or strict), non inclusion (partial or total) and occurrences (soft or strict). However, these choices concerning the semantics of NSPs impact directly the number of frequent patterns (under the same minimal threshold constraint) and further the computation time. The stronger 6. In this case, we do not have p p nor p + p 7. In this case, we also have p • p but not p + p the negative constraints, the lesser the number of sequences containing a pattern, and the lesser the number of frequent patterns.

Definition 10 (Pattern supports). Let D = {s i } i∈[n] be a set of n sequences and p be a NSP. The support of p in D, denoted supp D θ (p) is the number of sequences of D in which p occurs according to the θ ∈ Θ containment relation. Notation 3. When there is no ambiguity on the dataset of sequences, supp D θ (p) is denoted supp θ (p). And, for sake of readability, the θ operators are represented as triplets.

It is clear that if a containment relation θ is dominated by another containment relation θ , then the Proposition 1 implies that the support of the pattern evaluated with θ is lower than the support of the pattern evaluated with θ . Thus, we have the following proposition.

Proposition 5. For all pattern p ∈ N:

supp , , (p) ≤ supp , , ( p) (12)
supp , , ( p) ≤ supp , , ( p)

(13) supp , , ( p) ≤ supp , , ( p) (14)
supp , , (p) ≤ supp , , ( p) (15)
In addition, we can also deduce anti-monotonicity properties for support measures from the Propositions 3 and 4. Proposition 6. For all pairs of NSPs p, p ∈ N:

p p =⇒ supp , , ( p ) ≤ supp , , ( p) (16) p + p =⇒ supp , , (p ) ≤ supp , , ( p) (17)
Then, there is two ways to use these results to implement efficient frequent NSP mining algorithms. On the one hand, the results from Proposition 6 can be directly used to implement algorithms with efficient and correct strategies to prune the search space. 8 For containment relation, Equation 16 fully exploits the partial order to early prune a priori unfrequent patterns. For containment relation, the + partial order must be used to ensure the correctness of the algorithm (Equation 17). Unfortunalety, this partial order is less interesting than because it gives a priori information on less patterns than does. On the other hand, the support evaluated with is an upper bound for the support of (Equation 14). Thus, it is possible also to prune patterns accessible with partial order without losing the correctness of the pruning strategy.

Conclusion and perspectives

In this article, we explored the semantics of negation in sequential patterns. We gave eight possible semantics 8. The completeness of the algorithms requires to study how to traverse the search space. It is out of the scope of this article.

where the state of the art in sequential pattern mining did not notice these differences. We investigated the formal properties of these semantics: their respective relations (dominance and equivalence) and anti-monotonicity properties. These results may be used to develop new efficient algorithms to extract negative sequential patterns.

It is worth noticing that no semantics is "more" correct or relevant than another one. It depends on the information to be captured. Our objective is to give the opportunity to make an informed choice. Even if, in the context of pattern mining, the choice is constrained by computational considerations. With the three proposed partial orders, we have seen that interesting anti-monotonicity holds only for some semantics. Thus, dominance relations may be used to propose alternative search heuristics.

Hence, the first perspective of this work is to develop negative sequential pattern mining algorithms. This has been done for containment relations ≺ [START_REF] Guyet | NegPSpan: efficient extraction of negative sequential patterns with embedding constraints[END_REF]. 9 In this algorithm, we also introduced maxgap and maxspan constrains negative sequential patterns. The anti-mononicity properties still hold with these constraints. A step further, the framework presented in this article would make possible to propose a complete and correct algorithm to mine closed NSP.

Our second perspective is to provide a complete results on the dominance between containment relations. There are still few cases that are undetermined to achieve this objective. Further studies aim at having results on the quotient set Θ/ ∼.

Our third perspective is to propose new intuitive syntax(es) of containment relations. The main problem was the semantic overload of the negation symbol (¬) in the literature of negative sequential patterns. We are currently working on some proposals and plan to evaluate them. To be intuitive, these proposals are based on the equivalent classes we highlight in this work.

Proof of Lemma 2. Let e = (e i ) i∈[m] ∈ [n] m be aembedding of a NSP p = p 1 ¬q 1 . . . ¬q m-1 p m in a sequence s = s 1 . . . s n . For all positive itemsets p i , the definition of -embedding matches the one forembedding. For a negative itemset q i , let us start with := . By definition 4, q i j∈[e i +1,e i+1 -1] s j , and by Definition 2, ∃α ∈ q i , α

j∈[e i +1,e i+1 -1] s j . And then, ∃α ∈ q i , ∀ j ∈ [e i + 1, e i+1 -1], α s j . That is ∀ j ∈ [e i + 1, e i+1 -1], ∃α ∈ q i , α s j . This shows ∀ j ∈ [e i + 1, e i+1 -1], q i s j ( -embedding definition). It remains := . By definition 4, q i j∈[e i +1,e i+1 -1] s j , and by Definition 2, ∀α ∈ p i , α

j∈[e i +1,e i+1 -1] s j . And then, ∀α ∈ q i , ∀ j

∈ [e i + 1, e i+1 -1], α s j . That is ∀ j ∈ [e i + 1, e i+1 -1], ∀α ∈ q i , α s j . This shows ∀ j ∈ [e i + 1, e i+1 -1], q i s j .
Proof of Lemma 3. Let s = s 1 . . . s n be a sequence and p = p 1 ¬q 1 . . . ¬q m-1 p m be a negative sequential pattern. Lemma 2 shows that -embedding impliesembedding. It remains the implication to the left. Let e = (e i ) i∈[m] ∈ [n] m be a -embedding of pattern p in sequence s. Then, the definition matches the one forembedding for positives, p i . For negatives, q i , then ∀ j ∈ [e i + 1, e i+1 -1], q i s j , i.e. ∀ j ∈ [e i + 1, e i+1 -1], ∀α ∈ q i , α s j and then ∀α ∈ q i , ∀ j ∈ [e i + 1, e i+1 -1], α s j . It thus implies that ∀α ∈ p i , α

j∈[e i +1,e i+1 -1] s j , i.e. by definition, q i j∈[e i +1,e i+1 -1] s j .

Proof of Lemma 4. Let s = s 1 . . . s n be a sequence and p = p 1 ¬q 1 . . . ¬q m-1 p m be a NSP s.t. ∀i, |q i | ≤≤ 1. Let e = (e i ) i∈[m] ∈ [n] m be a -embedding of p in s then, by definition, 1) p i ⊆ s e i for all i ∈ [m] and 2) q i s j for all j ∈ [e i + 1, e i+1 -1]. In case |q i | = 0, there is no constraint. In case |q i | = 1, and 2) becomes q i s j for all j ∈ [e i + 1, e i+1 -1] whatever ∈ { , }. Hence, q i j∈[e i +1,e i+1 -1] s j i.e. q i j∈[e i +1,e i+1 -1] s j (no matter or ). As a consequence e is a -embedding of p.

Proof of Lemma 5. Let s = s 1 . . . s n be a sequence and p = p 1 ¬q 1 . . . ¬q m-1 p m ∈ N be a pattern. By definition 4, if e = (e i ) i∈[m] ∈ [n] m is an embedding of pattern p in sequence s then ∀i ∈ [l]: p i ⊆ s e i because p i is positive. According to definition 3, e is an embedding of the positive pattern p + .

Proof of Lemma 6. Let s = s 1 . . . s n be a sequence and p = p 1 ¬q 1 . . . ¬q m-1 p m ∈ N be a pattern s.t. p s. Then, there exists e an embedding of p + in s and, by definition, it is also an embedding of p in s. This means that p s.

Proof of Lemma 7. Let s = s 1 . . . s n be a sequence and p = p 1 ¬q 1 . . . ¬q m-1 p m ∈ N be a pattern. We start by considering relations between semantics at the embedding level, and then we will conclude at the pattern level.

Let's first assume that := . Then, for all ( , )embedding e = (e i ) i∈[m] of pattern p in sequence s. Hence, ∀i ∈ [m], p i ⊆ s e i and ∀i ∈ [m-1], ∀ j ∈ [e i +1, e i+1 -1], q i s j . According to eq. 1, we have q i s j . It comes that e is a ( , )-embedding.

Let's now assume that := . Then, let e = (e i ) i∈[m] be a ( , )-embedding of pattern p in sequence s. Hence, ∀i ∈ [m], p i ⊆ s e i and ∀i ∈ [m -1], q i j∈[e i +1,e i+1 -1] s j . In addition j∈[e i +1,e i+1 -1] s j ⊆ I, then, ∀i ∈ [m -1], q i j∈[e i +1,e i+1 -1] s j according to eq. 1. It comes that e is an ( , )-embedding.

Let's come back at the pattern level. if p s, in the two cases ( ∈ { , }). In the first case the existing ( , )embedding is a ( , )-embedding, and in the second case, all ( , )-embeddings are ( , )-embeddings. Therefore, we have that p s.

Proof of Lemma According to Lemma 2, a -embedding is a -embedding whatever the itemset non-inclusion operator. Then, we can conclude that p s =⇒ p s (Equality 4). In addition, Lemma 3 shows that a -embedding isembedding in case of total itemset non-inclusion. Then, we can conclude that p s =⇒ p s (Equality 5).

Let now gives some counterexamples for known nondominance relationships. A counterexample for Equation Proof of Lemma 10. We start with the implication p + p =⇒ p p . Let p, p ∈ N s.t. p + p . By definition, k = k and 1. ∀i ∈ [k], p i = p i , 2. ∀i ∈ [k -1], q i ⊆ q i and 3. ∃ j ∈ [k -1] s.t. q i q i . A particular case of 1. is that ∀i ∈ [k], p i ⊆ p i . In addition, third condition of is obtained easily from 3. adding a disjunctive condition. Hence, p p .

We now prove the second implication: p p =⇒ p • p . Let p, p ∈ N s.t. p p . Let's now define the sequence u i such that u i = i for all i ∈ [k]. By construction, we have that u i < u i+1 , for all i ∈ [k -1] (4.). In addition, by definition of , we have that ∀i ∈ [k], p i ⊆ p i = p u i , and ∀i ∈

[k -1], q i ⊆ q i = j∈[i,(i+1)-1] q j = j∈[u i ,u i+1 -1] q j . Assuming k = k , then p k p k or ∃ j ∈ [k -1] s.t. q j q j . If p k p k the third condition of • is satisfied (with j = k).
Otherwise, it is also satisfied with the j of the definition of .

Proof of Proposition 2. We begin with the + relation. We first remind that + is a strict partial order iff the three following conditions hold: Then, for all i ∈ [k], p i = p i = p i and for all i ∈ [k -1], q i ⊆ q i ⊆ q i (k = k = k ). Finally, it is not possible to have q i = q i for all i ∈ [k -1]. In fact, if we have these equalities, then we would have q i = q i and q i = q i for all i ∈ [k -1] because q i ⊆ q i ⊆ q i . But, it is not possible according to 3. Therefore, we have that p + p . Asymmetry. Let p, p ∈ N s.t. p + p . Then, according to 2. and 3., there exists j ∈ [k -1] s.t. q j q j . And then, it is not possible to have q j ⊆ q j . As a consequence, we can not have p + p. We now prove that is a strict partial order. Irreflexive. Let's assume that ∃ ∈ N s.t. p p. Then, because k = k and that it is not possible that p k p k , then the third condition implies that ∃ j ∈ [k -1] s.t. q j q j , which is also absurd.

i ∈ [k], p i ⊆ p i ⊆ p i and for all i ∈ [k -1], q i ⊆ q i ⊆ q i (k ≤ k ≤ k ). Finally, if k = k , then k = k = k . Assuming that p k = p k and p k = p k then p k = p k . Assuming that p k p k or p k p k , then ∃ j ∈ [k -1]
s.t. q j q j or q j q j , and hence q j q j . Then, we have that p p . Asymmetry. Let p, p ∈ N s.t. p p . Then, if k < k we can not have p p. Assuming that k = k and that p k = p k for the same reason as the asymmetry of + , it is not possible to have q j q j . If p k p k the, p k p k (according to 1.) and then it is not possible to have p j ⊆ p j . As a consequence, we can not have p p.

We now prove that • is a strict partial order. Irreflexive. Let's assume that ∃ ∈ N s.t. p p. Then, the unique strictly incremental (auto-)mapping is u i = i, for all i ∈ [k]. Then, the third condition implies that ∃ j ∈ [k -1] s.t. q j q j or p j p j , which is absurd. Then

) i ∈ [k ] [ k] such that w i = v u i for all i ∈ [k]. Then, for all i ∈ [k], p i ⊆ p u i ⊆ p v u i = p w i ; q i ⊆ j∈[u i ,u i+1 -1] q j = j∈[u i ,u i+1 -1] l∈[v j ,v j+1 -1] q l . The union of the q l in the intervals [v j , v j+1 -1] for j ∈ [u i , u i+1 -1] can be sum up as an union on the interval [v u i , v (u i+1 -1)+1 -1] = [v u i , v u i+1 -1] = [w i , w i+1 -1] because intervals are contiguous. Then, q i ⊆ j∈[w i ,w i+1 -1] q j . Fi- nally, if k = k , then k = k = k and then it exists j ∈ [k],
s.t. p j p j ⊆ p j or q j q j ⊆ q j . Thus, p j p j or q j q j . As a consequence, we have p • p . Asymmetry. Let p, p ∈ N s.t. p • p . Then, if k < k we can not have p • p. Assuming that k = k (and thus u i = i for all i ∈ [k]), we have that it exists j ∈ [k] s.t. p j p j or q j q j . If p j p j then, according to 1. p j p j , it is not possible to have p j ⊆ p j . If q j q j , then, according to 2. q j j∈[u i ,u i+1 -1] q j = q j . Thus, it is not possible to have q j ⊆ q j = j∈[u i ,u i+1 -1] q j . As a consequence, we can not have p • p.

Proof of Proposition 3. We start this proof by a small result about the anti-monotonicity of . Let P, Q ∈ I be two itemsets s.t. P ⊆ Q, and I ∈ I another itemset. Then, Q I =⇒ P I. In fact, Q I implies that for all e ∈ Q, e I, and because P ⊆ Q, we also have that e ∈ P, e I.

Let p = p 1 ¬q 1 . . . ¬q m-1 p m ∈ N and p = p 1 ¬q 1 . . . ¬q m -1 p m ∈ N be two NSP s.t. p p . We first show that an ( , )-embedding of p in a sequence s, denoted e = (e i ) i∈[m ] , induces an ( , )embedding of p. By Definition 4, we have p i ⊆ s e i , ∀i ∈ [m ] and q i s j , for all j ∈ [e i + 1, e i+1 -1] and for all i ∈ [m -1]. On the other side, p p implies that p i ⊆ p i for all i ∈ [m]. Then, because m ≤ m ( p p ), we have that p i ⊆ s e i for all i ∈ [m]. In addition, p p also implies that q i ⊆ q i for all i ∈ [m -1] and thus, by anti-monotonicity of (and q i s j ), we have q i s j for all j ∈ [e i + 1, e i+1 -1] and for all i ∈ [m -1]. In conclusion, we have that e = (e i ) i∈[m] is an ( , )-embedding of p.

We now show that an ( , )-embedding of p in a sequence s, denoted e = (e i ) i∈[m ] , induces an ( , )embedding of p. By Definition 4, we have p i ⊆ s e i , ∀i ∈ [m ] and q i j∈[e i +1,e i+1 -1] s j , for all i ∈ [m -1]. On the other side, p p implies that p i ⊆ p i for all i ∈ [m]. Then, because m ≤ m (p p ), we have that p i ⊆ s e i for all i ∈ [m]. In addition, p p also implies that q i ⊆ q i for all i ∈ [m -1], by anti-monotonicity of , we have q i j∈[e i +1,e i+1 -1] s j , for all i ∈ [m -1]. In conclusion, we have that e = (e i ) i∈[m] is an ( , )-embedding of p.

Proof of Proposition 4. Let p = p 1 ¬q 1 . . . ¬q k-1 p k ∈ N and p = p 1 ¬q 1 . . . ¬q k -1 p k ∈ N be two NSP s.t. p + p . Thus, we have that k = k .

Similarly to the proof of Proposition 3, we can show that any ( , )-embedding of p in s induces an ( , )embedding of p in s. This enables to conclude that is anti-monotonic on (N, + ).

The anti-monotonicity of requires that each embedding of p + in s satisfies the negations. Let us assume that p s, then there exists an embedding (e i ) i∈[k] of p . (e i ) i∈[k] is also an embedding of p + (Lemma 5). According to 1. and because k = k , p + = p+, and then (e i ) i∈[k] is an embedding of p in s. Thus, we shown that there is at least one embedding of p + in s. If is not anti-monotonic, then there exists an embedding (e i ) i∈[k] of p + such that ∃ j ∈ [k] and l ∈ [e j + 1, e j+1 -1], s.t. it is false that q j s l (∃α ∈ q j , α ∈ s l ). According to 2. q j ⊆ q j , and thus it is false q j s l . Nonetheless, (e i ) i∈[k] is also an embedding of p + . And p s, it implies that q j s l . There is a contradiction, thus is anti-monotonic. The anti-monotonicity of requires that each embedding of p + in s satisfies the negations. Let us assume that p s, then there exists an embedding (e i ) i∈[k] of p . (e i ) i∈ [k] is also an embedding of p + (Lemma 5). According to 1. and because k = k , p + = p+, and then (e i ) i∈[k] is an embedding of p in s. Thus, we shown that there is at least one embedding of p + in s. If is not anti-monotonic, then there exists an embedding (e i ) i∈[k] of p + such that ∃ j ∈ [k], s.t. it is false that q j l∈[e j +1,e j+1 -1] s l . According to 2. q j ⊆ q j , and thus it is false q j l∈[e j +1,e j+1 -1] s l . Nonetheless, (e i ) i∈[k] is also an embedding of p + . And p s, it implies that q j l∈[e j +1,e j+1 -1] s l . There is a contradiction, thus is anti-monotonic.

Example 2 (

 2 Itemset absence semantics). Let p = a ¬(bc) d be a pattern and four sequences: Sequence s 1 = a c b e d s 2 = a (bc) e d s 3 = a b e d s 4 = a e d

Example 3 (

 3 Strict vs soft occurrence semantics). Let p = a b ¬c d be a pattern, s 1 = a b e d and s 2 = a b c a d e b d be two sequences. p + = a b d occurs once

(Figure 1 -

 1 Figure 1 -Summary of the dominance relation between relations of Θ. An arrow shows that the operator at the destination is dominated by the operator at the origin. Unidirectional arrows indicates that the dominance relation holds in one direction but not in the other direction.

8

  is p = a ¬(bc) d and s = a b d . We have that p s but p s is false. Similarly, a counterexample for Equation 9 is p = a ¬b c and s = a c b c , and a counterexample for Equation 10 is p = a ¬(bc) d and s = a b c d .

  s 2 , s 3 , s 4 , s 5 } {s 1 } Definition 3 (Positive pattern embedding). Let s = s 1 . . . s n be a sequence and p = p 1 . . . p m be a (positive) sequential pattern. e = (e i ) i∈[m] ∈ [n] m is an embedding of pattern p in sequence s iff ∀i ∈ [m], p i ⊆ s e i and e i < e i+1 for all i ∈ [m -1]. Let s = s 1 . . . s n be a sequence and p = p 1 ¬q 1 . . . ¬q m-1 p m be a negative sequential pattern. e = (e i ) i∈[m] ∈ [n] m is a soft-embedding of pattern p in sequence s iff:

	Definition 4 (Strict and soft embeddings of negative pat-
	terns).

  8. A pre-order is a reflexive, transitive binary relation. The reflexivity of the relation comes with Definition 6. Let θ, θ , θ ∈ Θ be three dominance relations s.t. θ θ and θ θ . Then, for all p ∈ N and sequence s: pθs =⇒ pθ s and pθ s =⇒ pθ s. Hence, we have, pθs =⇒ pθ s, i.e. θ θ . Proof of Lemma 9. Let θ, θ ∈ Θ, by reflexivity of we have that ∼ is reflexive. By definition (θ θ ∧ θ θ), ∼ is symmetric. And , ∼ is also transitive. Let θ, θ , θ ∈ Θ be three dominance relation s.t. θ ∼ θ and θ ∼ θ then,

	θ θ , θ	θ , θ	θ and θ	θ . Hence, by transitivity
	of , θ θ and θ	θ, θ ∼ θ .
	Proof of Proposition 1. Let p ∈ N and s a sequence.
	According to Lemma 6, p	s =⇒ p	s. Thus we
	obtain Equality 6 by Definition 6. is immediately deduced
	from .
	According to Lemma 7, p	s =⇒ p	s. Thus we
	obtain Equality 7 by Definition 6.

  1. ∀p ∈ N, not p + p (irreflexive), 2. ∀p, p , p ∈ N, p + p and p + p =⇒ p + p (transitivity), 3. ∀p, p ∈ N, p + p =⇒ not p + p (asymmetry) Irreflexive. Let's assume that ∃ ∈ N s.t. p + p. Then, because k = k the third condition implies that ∃ j ∈ [k -1] s.t. q j q j , which is absurd. Then + is irreflexive. Transitivity. Let p, p , p ∈ N s.t. p + p and p + p .

  • is irreflexive. Transitivity. Let p, p , p ∈ N s.t. p • p and p • p . We denotes by (u i ) i ∈ [k ] k and (v i ) i ∈ [k ] k the respective mapping, and we define (w i

  Proof of Proposition 5. Let θ, theta ∈ Θ, then θ θ =⇒ supp θ ( p) ≤ supp θ ( p) for all p ∈ N (by Definition 6 of the dominance relation). Thus, Proposition 5 comes immediately with Proposition 1.Proof of Proposition 6. Let p, p ∈ N be two negative sequential patterns such that p p . According to Proposition 3, p s =⇒ p s for all s. Thus, supp , , ( p) ≤ supp , , ( p).If p + p . According to Proposition 4, p s =⇒ p s for all s. Thus, supp , , ( p) ≤ supp , , ( p).

All proofs are provided in Appendix A.

In sequential pattern mining, it is called a backward-extension of the patterns.

We remind that, by Definition 1, p i ∅ and that we never have two successive negative itemsets in a NSP.


Dominique Knittel
email: knittel@unistra.fr

Hamid Makich

Mohammed Nouari

Milling Diagnosis using Machine Learning Approaches

The manufacturing industries more and more pay close attention to artificial intelligence (AI). For example, smart monitoring and diagnosis, real time evaluation and optimization of the whole production and raw materials management can be improved by using machine learning and big data tools. In this work, a smart milling diagnosis has been developed for composite sandwich structures based on honeycomb core. The use of such material has grown considerably in recent years, especially in the aeronautic, aerospace, sporting and automotive industries. But the precise milling of such material presents many difficulties. The objective of this work is to develop a data-driven industrial surface quality diagnosis for the milling of honeycomb material, by using supervised machine learning methods. In this approach cutting forces are online measured in order to predict the resulting surface flatness. The developed diagnosis tool can also be applied to the milling of other materials (metal, polymer, …).

Introduction

The manufacturing industries more and more pay close attention to artificial intelligence (AI). For example, smart monitoring and diagnosis, real time evaluation and optimization of the whole production and raw materials management can be improved by using machine learning and big data tools [START_REF] Beskri | Systèmes de surveillance automatique en usinage: Moyens et méthodes (Automatic monitoring systems in machining: tools and methods)[END_REF]. An accurate milling process implies a high quality of the obtained material surface (roughness, flatness) [START_REF]Usinage: Tournage Fraisage (IFPM courses : Machining: Milling Turning)[END_REF]. With the involvement of AI-based algorithms, milling process is expected to be more accurate during complex operations.

T. Mikołajczyk et al. developed an Artificial Neuronal Network (ANN) for tool-life prediction in machining with a high level of accuracy, especially in the range of high wear levels, which meets the industrial requirements [START_REF] Mikołajczyk | Predicting tool life in turning operations using neural networks and image processing[END_REF]. D. Pimenov et al. evaluated and predicted the surface's roughness through artificial intelligence algorithms (random forest, standard Multilayer perceptron) [START_REF] Pimenov | Artificial intelligence for automatic prediction of required surface roughness by monitoring wear on face mill teeth[END_REF]: in their investigation the obtained performance depends on the parameters contained in the dataset. M. Correa et al. compared the performances of Bayesian networks (BN) and artificial neural networks for quality detection in a machining process [START_REF] Correa | Comparison of Bayesian networks and artificial neural networks for quality detection in a machining process[END_REF]. Even ANN models are often used to predict surface quality in machining processes, they preferred BNs for their significant representation capability and for the fast model building.

The work of C. Zhang et al. [START_REF] Zhang | Tool Condition Monitoring and Remaining Useful Life Prognostic Based on aWireless Sensor in Dry Milling Operations[END_REF] focused on monitoring the condition and life of the cutting tool in dry milling environment. From de-noised vibration signal they extract some relevant features such as the root mean square, the skewness and the kurtosis in both time and time-frequency domain. Based on Neuro-Fuzzy Network (NFN), they implemented a tool wear prediction model which performs the best, with the smallest Mean Squared Error (MSE) and Mean Absolute Percentage Error (MAPE) compared with Back Propagation Neural Network (BPNN) and Radial Basis Function Network (RBFN) algorithms. Z. Rui et al. [START_REF] Rui | Machine Health Monitoring Using Local Feature-Based Gated Recurrent Unit Networks[END_REF] implemented a hybrid approach combining handcrafted feature design with automatic feature learning for machine health monitoring: local feature-based gated recurrent unit (LFGRU) networks. By comparison with some other methods such as the Support Vector Machine (SVM), the k-nearest neighbor (kNN), they verified the effectiveness and robustness of the proposed LFGRU model for tool wear prediction. D. Wu et al. [START_REF] Wu | Cloud-Based Machine Learning for Predictive Analytics: Tool Wear Prediction in Milling[END_REF] have worked on cloud-based machine learning for tool wear prediction in milling. The research was about the development of a novel approach for machinery prognostics using a cloudbased random forest algorithm. Their experimental result have shown that despite the fact that random Forests give the best accuracy for large dataset, parallel random forest algorithm has the best ratio training time/accuracy. Future more, they will predict tool wear with other machine learning algorithms such as support vector machines as well as to make a comparison with their actual algorithms.

For machining result prediction, similar algorithms could be used but the recurrent problem is how to increase the accuracy of those algorithms. K. Javed. et al. [START_REF] Javed | Enabling Health Monitoring Approach Based on Vibration Data for Accurate Prognostics[END_REF] have worked on an enabling health monitoring approach based on vibration data for accurate prognostics. They have shown that prognostic efficiency is closely related to the extracted features and by the same way proposed a method for enabling features that can lead to simple and accurate prognostics. K. Durmus [START_REF] Karayel | Prediction and control of surface roughness in CNC lathe using artificial neural network[END_REF], by using neuronal networks, worked on the prediction and the control of surface roughness in CNC lathe using artificial neural network. His study has concluded that artificial neural network (ANN) can produce an accurate relationship between cutting parameters and surface roughness. Based on the ANN training model, he could find the best machining parameters for obtaining a desired surface roughness.

By also using neuronal artificial neural network M. Azlan [START_REF] Mohd Zain | Prediction of surface roughness in the end milling machining using Artificial Neural Network[END_REF] has developed a surface roughness prediction models for end milling machining, in the logic to find the best ANN network structure for surface roughness prediction.

Another approach consists to measure and analyze the drive power (for example by current measuring) [START_REF] Yu | Artificial intelligence for automatic prediction of required surface roughness by monitoring wear on face mill teeth[END_REF], which is not applicable in our experiment. In this paper few artificial intelligence methods are tested: random forest (RF), standard Multilayer perceptrons (MLP), Regression Trees, and radial-based functions.

In our work, a smart milling diagnosis has been developed for composite sandwich structures based on honey-comb core. The use of such material has grown considerably in recent years, especially in the aeronautic, aerospace, sporting and automotive industries. Recent development projects for Airbus A380 or Boeing 787 confirm the in-creased use of the honeycomb material. But the precise milling of such material presents many difficulties.

The objective of this work is to develop a data-driven industrial surface quality diagnosis for the milling of honeycomb material, by using supervised machine learning methods. Therefore, cutting forces are online measured in order to predict the resulting surface flatness.

Description of the Experiments

Workpiece material and tools

The workpiece material studied in this investigation is Nomex® honeycomb cores with thin cell walls. It is produced from aramid fiber dipped in phenolic resin (Fig. 1). The honeycomb cores consist of continuous corrugated ribbons of thin foil bonded together in the longitudinal direction. The aim of such a process is to create a structure allowing lightness and stiffness together thanks to the hexagonal geometry of formed cells. Figure 1 illustrates the geometric characteristics of the honeycomb core. The use of honeycomb material in sandwich composite is limited by the fragility of each wall of the honeycomb, which influences the quality of obtained surfaces after machining [START_REF] Kindinger | Lightweight structural cores[END_REF][START_REF] Gay | Matériaux composites (Composite materials)[END_REF][START_REF] Carl | Three-dimensional honeycomb core machining apparatus and method[END_REF].

The Nomex® honeycomb machining presents several defects related to its composite nature (uncut fiber, tearing of the walls), the cutting conditions and to the alveolar geometry of the structure which causes vibration on the different components of the cutting effort [START_REF] Jaafar | A 3D FE modeling of machining process of Nomex® honeycomb core: influence of the cell structure behaviour and specific tool geometry[END_REF].

It is clear that the use of ordinary cutting tools and also the mechanical and geometrical characteristics of honeycomb cores will have a crucial effect on machinability and on the quality of the resulting surface [START_REF] Mendoza | Development of a new milling cutter for aluminum honeycomb[END_REF]. In fact, ordinary cutting tools for machining honey-comb core produce generally tearing of fibers and delamination of cell structures. Subsequently, these cause a reduction of bond strength between the skin and the honey-comb core, and thus a weaker joint for composite sandwich structures.

In our study, the used milling cutter is provided from our industry partner, the EVATEC Tools Company. As shown in figure 2, the used EVATEC tool is a combined specific tool with two parts designed to surfacing/dressing machining operation. The first part is a cutter body made of high speed steel with 16 mm in diameter and having ten helixes with chip breaker. This tool part is designated by Hogger. The second part is a circular cut-ting blade made of tungsten carbide with a diameter of 18.3 mm and having a rake angle of 22° and a flank angle of 2.5°. These two parts are mechanically linked to each other with a clamping screw.

Milling experiments

All experimental milling tests illustrated in this paper were carried out on a three-axis vertical machining center Realmeca® RV-8.

For assessing the performance of the machining process of Nomex® honeycomb core we monitored and measured the cutting forces generated during cutting, by using the Kistler dynamometer model 9129AA. The Kistler table is mounted below the Nomex sample in order to measure the three components of the machining force as shown in figure 3. During the measurements, the x-axis of the dynamometer is aligned with the feed direction of the milling machine and the longitudinal direction of the workpiece (parallel to core ribbons and the direction of honeycomb double wall). The three orthogonal components of machining force (Fx, Fy and Fz) were measured according to figure 3 Table 1 : Milling experiment conditions Two main modes of surface damage are observed (Figure 4): uncut aramid fibers along the machined surface and tearing of the walls. The appearance of the uncut fibers is a machining defect specific to the composite material which depends on the type of the fibers and their orientation. The tearing of Nomex® paper, linked to the cellular appearance of the honeycomb structure, occurs under the shear loading effect [START_REF] Mohd Zain | Prediction of surface roughness in the end milling machining using Artificial Neural Network[END_REF][START_REF] Codjo | Milling diagnosis using machine learning techniques toward Industry 4[END_REF]. Uncut fibers are observed on Figure 4 -a and-c. It is well known that the surface quality is of high importance for the use of the Nomex ® honeycomb in sandwich materials. The machining defects cause a reduction of bond strength between the skin and the honeycomb core, and thus a weaker joint for composite sandwich structures. The case (b) represents the best milling result

Measured signals

Many milling experiences have been made in our study. For example, figure 5 shows the milling forces measured for honeycomb at 2000 rpm spindle speed and 3000 mm/min feed rate. Cutting forces are in the order of a few Newtons, they do not exceed 60 Newtons. Generally, the force in vertical direction (Fz) is quite small, thus, it is advised that to keeping vertical forces small in milling composite due to the delamination issue. In our case, the vertical cutting force component is greater than other forces components which can be attributed to the mechanical properties of the honeycomb structure where the honeycomb structure is characterized by a better out-of-plane compression behavior than its tensile and shear strength. The evolution of cutting forces shows significant oscillations. These oscillations are caused by vacuum in the cells of the honeycomb and the angle between the cutting direction and the honeycomb cell wall direction.

Figure 6 shows the obtained evolution of the surface quality (flatness) for various combinations of cutting conditions (spindle speed and feed rate). The defect of shape is higher for low speeds. Thus, for high feed rates that exceed the 1500 mm/min, the unevenness exceeds 500 µm which characterizes the severe tearing of the honeycomb walls. Given the low level of cutting forces, the quality of the obtained machined surface allows to establish criteria for determining the machinability of the honeycomb structures. The appearance of the uncut fibers is a machining defect specific to the composite material which depends on the type of the fibers and their orientation. The tearing of Nomex® paper, linked to the cellular appearance of the honeycomb structure, occurs under the effect of shear loading [START_REF] Correa | Comparison of Bayesian networks and artificial neural networks for quality detection in a machining process[END_REF][START_REF] Tchoutouo | Adhesiveless honeycomb sandwich structure with carbon graphite prepreg for primary structural application: a comparative study to the use of adhesive film[END_REF].

Alternatively a surface response [START_REF] Myers | Response surface methodology[END_REF] could have been built in order to predict the milling surface quality. But close milling parameters (such as splindle speed, feed rate, depth of cut) can lead to different results, depending on the material, the quality of the machining tool, etc. Therefore, in our approach supervised machine learning techniques (with labeled measurements for the model training) are used. These tools need the construction of features associated with the measurements.

Milling diagnosis using machine learning techniques

Machine learning techniques can be separated mainly in two categories [START_REF]Mastering Machine Learning A Step-by-Step Guide with MATLAB[END_REF][START_REF] Shalev-Shwartz | Understanding Machine Learning: From Theory to Algorithms[END_REF]:

-Unsupervised approaches: based only on input data (data are unlabeled). The goal is to find groups and structures in the data set, in order to classify new observations (measurements) into the different groups.

-Supervised approaches: based on input and output data (Now the data are labelled.

The raw data (measurements) are firstly filtered, with low pass filters in order to eliminate high frequency noises, and labeled ("obtained signals for good surface quality", "obtained signals for bad surface quality"). Then the features are calculated offline or online.

All the experiments are then split into two groups: 75% for the machine learning model training, 25% for the obtained model evaluation also called test phase in the literature (another percentage can be chosen, for example 60% -40%, depending on the number of experiments). This can be made randomly, but the ratio "good surface quality" and "bad surface quality" must be kept in each group.

Features calculation

The features are calculated in the time domain and the frequency domain [6,[START_REF] Rion | Prediction of the adhesive fillet size for skin to honeycomb core bonding in ultra-light sandwich structures[END_REF] from the raw signal represented on figure 7, in steady state behaviour. In fact, transient zones (that means when the cutting tool entries or exits the honeycomb core) are not taken into account. Secondly another 19 features are calculated in frequency domain in a similar way for the measured milling force signal. Therefore, the Fast Fourier transform (FFT) of the signal has been calculated.

All the calculated features (in time and frequency domains) are normalized and stored in a table whose lines and columns respectively represent the experimental number (also called instance) and the associated feature values. The description of the used features are indicated in [START_REF] Codjo | Honeycomb Core Milling Diagnosis using Machine Learning in the Industry 4.0 Framework[END_REF] The reduction of the features is then be made by using PCA (Principal Component Analysis) [START_REF] Vidal | Generalized Principal Component Analysis[END_REF].

Labeled data

From the evaluation of the effect of the cutting parameters on surface flatness result, we defined two classes of surface quality applied to the output data of each observation (see

Applied supervised learning algorithms

In this work, several classification algorithms have been implemented in the Matlab software environment [START_REF] Wei | Numerical control machine tool fault diagnosis using hybrid stationary subspace analysis and least squares support vector machine with a single sensor[END_REF][START_REF] Zhang | Tool Condition Monitoring and Remaining Useful Life Prognostic Based on aWireless Sensor in Dry Milling Operations[END_REF]: k-nearest neighbor (kNN), Decision trees (DT), Support Vector Machine (SVM). The different machine learning algorithms (with their adapted tuning parameters) are applied to the normalized labeled training data set (75% of the total experiments). The obtained trained models are then tested on the labeled test data set (25% of the total experiments). The objective is to find again the labels of the test data set: table 3 We used some news experimental data set in order to evaluate the performance of the trained model. The goal is to predict online (during milling) the surface quality. The results are presented here for the trained model by using the linear SVM classifier algorithm: The class B was the best predicted class. Despite the fact that linear SVM algorithm lost in performance for data set with large predictors (i.e. large number of features), it has been an accurate algorithm with a good prediction rate and the lowest training time.

Conclusion

The milling's performance is qualified by evaluating the roughness or the flatness of the resulted surface. In this work, different supervised machine learning algorithms have been implemented and compared. To do this, features were firstly calculated from measured milling forces and then each Artificial Intelligence (AI) based model has been trained by the labeled set of features. From the prediction results, SVM algorithm seems to be a good efficient diagnosis algorithm in this application of honeycomb material milling. The developed diagnosis approach can also be applied to the milling of other material.

Figure 1 :

 1 Nomex® honeycomb cores and the main geometrical characteristics

Figure 2 :

 2 Figure 2: Milling cutter used for Nomex® honeycomb core ''CZ10''.

  using the Kistler table.

Figure 3 :

 3 Figure 3: Experimental test setup

Figure 4 :

 4 Figure 4: Obtained honeycomb machining surfaces. The case (b) represents the best milling result

Figure 5 :

 5 Figure 5: Milling force measurements for 2000 rpm spindle speed and 3000 mm/min feed rate: (a) during all process; (b) during 0.2s (zoom)

Figure 6 :

 6 Figure 6: Effect of cutting parameters on surface flatness

Figure 7 :

 7 Figure 7: Measured milling force in time domain: (a) total data plot, (b) signal during steady-state phase

  true positive rate; FN: false negative rate; FP: false positive rate; TN: true negative rate)

table 2 )

 2

	Label Flatness (µm) Qualitative value
	'A'	0 -600	Best surface quality
	'B'	600 -…	Worst surface quality

Table 2 :

 2 Label table for the experimental observations

Table 3 :

 3 shows the obtained accuracy result of each algorithm. Prediction error for the normalized data set

	Algorithms	Accuracy
	KNN	83.4%
	KNN k=2	81.3%
	Weighted KNN k=2	83.4%
	Chebychev KNN k=2	87.5%
	Tree	99%
	Pruned tree	66.67%
	Linear SVM	83.4%
	Gaussian SVM	66.67%

Table 4 :

 4 Performance of the prediction using SVM classifier


Yuzhu Wang

Roger Serra

Pierre Argoul

Influence of Gaussian Signal Distribution Error on Random Vibration Fatigue Calculation

Keywords:

In the study of random vibration problems, Gaussian vibration and non-Gaussian vibrations are usually classified according to the excitation signal. The skewness and kurtosis are usually used to distinguish. Here we discuss a non-strict Gaussian signal, which is the error that exists in skewness and kurtosis and usually unavoidable in actual experiments or signals analysis. Through experiments and simulation calculations, the influence of this error on the traditional fatigue calculation method is discussed. The PSD approach will be discussed primarily, and time domain signals based on the rain-flow counting method will be recorded and verified. Total nine calculation model studied in this process. Finally, through a threshold, the range of skewness and kurtosis is indicated, that within this range, Gaussian signal-based calculations can be continued. By comparing the performance of different methods, a better method for signal adaptability can be obtained.

Introduction

For many mechanical components, the working load is in the form of random vibrations. In the fatigue design project, the load cycle of the structure is usually obtained by the rain flow counting method according to the conventional time domain signal, with the material property, the damage of the structure could be obtained by using the Miner's Law, and then prediction the life of the structure. But the acquisition of time domain signals relies on a large number of experimental records, which obviously sounds expensive. Later, according to the stochastic theoretical method, the power spectral density was used to characterize the random vibration characteristics, and the method of inferring the rain flow counting result of response stress was proposed. [START_REF] Wang | SERRA Life assessment of structures using random vibration testing[END_REF] [START_REF] Serra | Effets de l'environnement vibratoire sur la durée de vie d'une éprouvette en aluminum. SF2M, 35è mes Journé es de Printemps[END_REF] This theory generally assumes that the load is subject to a Gaussian distribution. The more successful method is the narrow-band approximation method proposed by Bendat in 1968. [START_REF] Crow | Measurement and Analysis of Random Data[END_REF] Later, due to the efforts of more scholars, the broadband approximation method was also proposed. At the same time, various improvement schemes were proposed to improve the accuracy of the approximate results. [START_REF] Wirsching | Fatigue under wide band random stress[END_REF][5] [START_REF] Dirlik | Application of computer in fatigue analysis[END_REF] However, since most of the actual loads are non-Gaussian distributions, it is obvious. Therefore, when the load is a non-Gaussian signal, the original Gaussian-based frequency domain damage analysis method may be used, which may cause poor deviation. Therefore, it is necessary to further discuss the influence of signal non-Gaussian on stress distribution. The research method based on the frequency domain signal discusses the overall distribution of signals in the frequency domain, and the non-Gaussian signal that satisfies this condition is not unique. This leads to the use of frequency domain method to study the distribution of non-Gaussian signals. Large deviations, when calculating damage using this distribution result, pose a significant risk to product design and life estimation. In this paper, a reference based on kurtosis judgment will be proposed to select the method of fatigue damage calculation. The rain-flow count analysis is performed on the response stress of non-Gaussian load, and the difference between the Gaussian signal and the non-Gaussian signal at the same level is obtained, and an allowable value is obtained, that is, in this range, even if it is not a Gaussian signal, the PSD method also could be used. The fatigue damage results obtained by the method are still within error tolerance. Beyond this range, the non-Gaussian signals must be considered with the special method.

Non-Gaussian signal and Kurtosis control

Generally, a signal whose probability density distribution obeys a Gaussian distribution is called a Gaussian signal and is mainly judged by the skewness and kurtosis of the signal. This indicator indicates the distribution of data within the data range. The skewness refers to the zero offsets of the centre, which is represented by S. The kurtosis can be understood as the specific gravity in the central region, denoted by K. The greater the kurtosis, the greater the accumulation of data in the centre. Usually, the Gaussian distribution has a kurtosis of 3 and skewness of 0. (2)

When the kurtosis or skewness has a condition that does not satisfy the Gaussian distribution, the signal is called a non-Gaussian distribution. At the same time, according to the index, when K>3, it is called leptokurtic, and when k<3, it is called platykurtic. According to the probability density distribution of the non-Gaussian signal, it can be found that the kurtosis reflects the distribution of acceleration in the middle region. [START_REF] Benasciutti | Fatigue life assessment in non-Gaussian random loadings[END_REF] In order to obtain an acceleration signal of a non-Gaussian distribution, it is usually obtained by Gaussian signal transformation. There are many methods used, Hermit polynomial, Gaussian mixture model, Phase selection method, Power-law model, Exponential method, Nonparametric method. In this paper, Steinwolf's phase selection method is used to modulate non-Gaussian signals with specific skewness and kurtosis. [START_REF] Steinwolf | Vibration testing of vehicle components by random excitations with increased kurtosis[END_REF]

Phase selection method

By fast Fourier transform, an acceleration time domain signal can be described as a superposition of harmonics in the frequency range.

x(t) = ∑ 𝐴 𝑛 cos (2𝜋𝑛∆𝑓𝑡 + 𝜑 𝑛 )

𝑁 𝑛=1

(3) The amplitudes of the harmonics are obtained

𝐴 𝑛 = √2∆𝑓𝑆(𝑛∆𝑓) (4)
Then the nth centre moment could be obtained

𝑀 𝑧 = lim 𝑇→∞ 1 𝑇 ∫ {𝑥(𝑡)} 𝑧 𝑑𝑡 𝑇 0 = 1 𝑇 ∫ {𝑥(𝑡)} 𝑧 𝑑𝑡 𝑇 0 , z > 2 (5)
After the kurtosis formula could be written as

K = 𝑀 4 𝑀 2 2 = 3 - 3 2 ( ∑ 𝐴 𝑛 4 𝑁 𝑛=1 (∑ 𝐴 𝑛 2 ) 𝑁 𝑛=1
2 ) + ( In the case of ensuring that the mean and RMS of the non-Gaussian signal are not changed, only the distribution is changed. So to make the K fitted for the experiment by modulating the specific φ. The resulting non-Gaussian distribution acceleration time series with specific K could be found. It should be noted that as the bandwidth increases,the amount of calculation becomes very large and also affecting the amount is the sampling frequency and data length. In general, the required non-Gaussian data is obtained in a combination of several methods.

Rain-flow count and damage calculation

The rain-flow counting method was proposed in the 1950s by two British engineers, M. Matsuishi and T. Endo. [START_REF] Matsuichi | Fatigue of Metals Subjected to Varying Stress[END_REF] The main function of this counting method is to simplify the measured load history into several load cycles for fatigue life estimation and fatigue test load spectrum. It is based on the two-parameter method and considers two variables of dynamic strength (magnitude) and static strength (mean). The rain flow counting method is mainly used in the engineering field, and is widely used in the calculation of fatigue life. Through the rain flow counting method, the response stress is cyclically counted, and the stress amplitude-cycle number curve is obtained, which is the p(s) curve required for the fatigue damage calculation. The ultimate goal of the frequency domain based PSD method is also to approximate the fitting through various frequency models through the frequency domain information, and finally obtain the probability density curve p(s) of the stress response. The S-N curve of the material expresses the number of life cycles of the material under different stresses. Usually expressed in the Basquin model. [START_REF] Basquin | The exponential law of endurance testing[END_REF] 𝑆 𝑚 𝑁 = 𝐶 (7) And then according Miner's Law [START_REF] Miner | Cumulative damage in fatigue[END_REF] , calculation formula of damage in unit time could be written

𝐷 = 𝑣 𝑝 𝐶 -1 𝑡 0 ∫ 𝑆 𝑚 𝑝(𝑠)𝑑𝑠 ∞ 0 (8)
Then the prediction life could be obtained by D 𝑇 = 1 𝐷 × 𝑡 0 (9)

Case study

In order to study the influence of the probability distribution of the response stress on the structural damage calculation, using the same frequency range, the 0 mean, the RMS is the same, the skewness is 0, and the different time series with the kurtosis between 2.9-10 are applied to the same double notched specimen, the effect of kurtosis on the damage calculation is obtained by comparing the damage conditions. 2 The RMS and Kurtosis of acceleration After, the response stress result was show that the specimen is not linear structure, the time series of response stress was as below. The direction opposite to the gravitational acceleration is the Z-axis forward direction, and the nodal stress at the left side of the upper surface of the model is selected as the research object. The overall research strategy is shown below. The focus is on the difference in stress cycling and damage values after rain flow counting. In general, the infinite fatigue stress of the material is selected as the threshold value of the rain flow count, that is, in this stress cycle and below, the material has an infinite life. However, in order to compare the results of the data, this threshold is set to zero.

Results

According the result of response stress, it could be found that the kurtosis were influence by structure. Moreover, as the acceleration kurtosis increases, the kurtosis of the response stress decays more significantly. Table 3 The RMS and Kurtosis of response stress By comparing the results of time-domain and frequency-domain methods with different kurtosis, it can be found that as the kurtosis increases, the stress distribution is more concentrated in the low-stress region, but the maximum value of the stress amplitude is significantly increased. The key to damage deviation. Moreover, there is a possibility that the maximum value of the stress amplitude is greater than the allowable stress directly causing structural failure. The DIRLIK method is selected to obtain the stress expectation based on the Gaussian distribution of the frequency domain method, which is used as a reference object and compared with the results under the time domain signal.

When 𝛾 ≤ 30%, this can accept the results of using a PSD-based Gaussian signal for non-strict Gaussian vibration signals. Beyond this range, fatigue calculation methods based on non-Gaussian vibration signals must be used. Generally, when the kurtosis is less than 3.5, the result is within an acceptable range. Of course the accuracy of this result is limited by the length of the data. The longer the data, the more obvious the distribution. Smaller samples are relatively more affected by randomness.

Conclusion

Based on the analysis results, it can be found that the data difference is affected by the material property m. At the same time, due to the structural relationship, the actual excitation kurtosis is greater than the threshold, but the response stress after attenuation can be calculated according to the Gaussian distribution. Because of the high kurtosis, the response stress exceeds the limit of material strength, and should be avoided in industrial design. For response stresses with high kurtosis, there is a possibility that the maximum value exceeds the strength limit. This should be taken seriously.

Figure 1

 1 Figure 1 Gaussian signal and probability density distribution

Figure 2

 2 Figure 2 Two notch specimen & excitation To meet the requirement, 10s time series of acceleration which allow non-Gaussian distribution were generated with different kurtosis. The bandwidth selection is based on the modal analysis result of the specimen, the second-order bending mode is 51.2 Hz, and 15-95 Hz is selected as the frequency range of the excitation signal. The sampling ratio is 1024Hz according to the band width. Frequency PSD (g 2 /Hz) (m 2 /s 4 )/Hz 15 0.25 12.03 95 0.25 12.03 RMS 4.472136g 31.02m/s 2

Figure 4 Figure 5 Figure 6 Figure 7 11 Figure 8

 4567118 Figure 4 Rain flow counting of K=3.07

  𝐴 𝑖 𝐴 𝑗 𝐴 𝑘 𝐴 𝑚 cos(𝜑 𝑖 + 𝜑 𝑗 + 𝜑 𝑘 -𝜑 𝑚 ) +

				1 2	∑ 𝑁 𝑛=1	𝐴 𝑛 2	) -2	{	3 2	∑ 𝑖+2𝑗=𝑘 𝑖≠𝑗	𝐴 𝑖 𝐴 𝑗	2 𝐴 𝑘 cos(𝜑 𝑖 + 2𝜑 𝑗 -𝜑 𝑘 ) +
	3 2	∑ 𝑖+𝑗=2𝑘 𝑖<𝑗	𝐴 𝑖 𝐴 𝑘	2 𝐴 𝑗 cos(𝜑 𝑖 + 𝜑 𝑗 -2𝜑 𝑘 ) + 3 ∑ 𝑖<𝑗,𝑘<𝑚,𝑖<𝑘 𝑖+𝑗=𝑘+𝑚	𝐴 𝑖 𝐴 𝑗 𝐴 𝑘 𝐴 𝑚 cos(𝜑 𝑖 + 𝜑 𝑗 -𝜑 𝑘 -
	𝜑 𝑚 ) + 3 ∑ 𝑖<𝑗<𝑘<𝑚 𝑖+𝑗+𝑘=𝑚						1 2	∑ 𝑖=3𝑗	𝐴 𝑖 𝐴 𝑗	3 cos(𝜑 𝑖 -3𝜑 𝑗 ) }
										(6)

Table 1

 1 Frequency domain information (g=9.81 m/s 2 ) The simulation part is finished by Abaqus. Before taken simulation, the modal analysis was used to check FE model to do verify and validation according mode frequency and stress distribution. Though the modal dynamic, the response stress of different times series could be obtained.

	No.	1	2	3	4	5	6	7	8
	RMS(m/s 2 ) 31.44 31.32 31.27 31.31 31.10 30.84 31.54 31.44
	Kurtosis	3.01 3.50 4.01 4.53 5.02 8.04 10.24 2.85
	No.	9	10	11	12	13	14	15	16
	RMS(m/s 2 ) 31.44 31.44 31.44 31.44 31.44 31.44 31.40 31.52
	Kurtosis	2.92 2.80 3.06 2.96 3.12 2.90 3.15 3.21
		Table

Acknowledgement

This contribution has been elaborated under the China Scholarship Council


Cherif Snoun

Baptiste Bergeot

Sébastien Berger

Sébastien Berger Robust

Robust optimization of nonlinear energy sinks used for dynamic instabilities mitigation of an uncertain friction system

Keywords: Brake squeal noise, Friction-induced vibrations, Nonlinear energy Sink, Uncertainties, Robust optimization

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Dynamic instability also named friction-induced vibrations is one of the important problem that the friction system can confront. This phenomenon can appear by the generation of Limit Cycle Oscillation (LCO) induced by dry friction. They are explained, in most studies of self-excited systems, by the coupling of the tangential and normal modes [START_REF] Oden | Models and computational methods for dynamic friction phenomena[END_REF][START_REF] Fritz | Investigation of the relationship between damping and mode-coupling patterns in case of brake squeal[END_REF][START_REF] Hervé | Analysis of squeal noise and mode coupling instabilities including damping and gyroscopic effects[END_REF]. As a way to model these dynamic instabilities related to friction, the well-known two-degrees-of-freedom Hultèn's model [START_REF] Hultén | Friction phenomena related to drum brake squeal instabilities[END_REF][START_REF] Hultén | Brake squeal -a self-exciting mechanism with constant friction[END_REF] has been widely used and it is also considered in this paper.

The NES is a nonlinear spring mass damper with a strong cubic stiffness. It can adapt itself to the main system that it is attached to without being tuned to a specific frequency. Its operation is based on the concept of Targeted Energy Transfer (TET) which has become an important passive control technique for reducing or eliminating unwanted vibrations [START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: Part II -Resonance capture[END_REF][START_REF] Vakatis | Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems[END_REF].

In this paper, two ungrounded NESs are used in order to mitigate or eliminate mode coupling instability in the Hultèn's model. In [START_REF] Bergeot | Mode coupling instability mitigation in friction systems by means of nonlinear energy sinks : numerical highlighting and local stability analysis[END_REF], the authors classified the steady-state response regimes in two main regimes related to the dispersion of some uncertain parameters: the first are the mitigated regimes and the second are the unmitigated regimes. As usual in the context instability mitigation by means of NESs, the LCO amplitude profile presents a discontinuity between these regimes which makes the NES potentially very sensitive to uncertainty. For that, with a low computational cost, the aim is to maximize the performance of the NES (i.e. minimize the region parameter of unmitigated regimes), by optimizing its design parameters taking into account uncertain parameters.

NES has been extensively optimized in a deterministic context (see e.g. [START_REF] Tuan | Design criteria for optimally tuned nonlinear energy sinks-part 1: transient regime[END_REF]), but there is very little work performing optimization of NES under uncertainties in general [START_REF] Boroson | Optimization under uncertainty of parallel nonlinear energy sinks[END_REF] and for mitigation of self-excited vibrations in particular [START_REF] Pidaparthi | Stochastic optimization of nonlinear energy sinks for the mitigation of limit cycle oscillations[END_REF]. In the latter, the authors have developed a methodology of optimization under uncertainties of a NES attached to a two-degree-of-freedom airfoil. In the present work, a robust optimization tool based on the Multi-Element generalized Polynomial Chaos (ME-gPC) is developed.

The article is constructed as follows: in Section 2, the two degrees-of-freedom Hultèn's model coupled to two NESs is presented. In Section 3, the steady-state of the system as well as the discontinued profile are introduced. The optimization formulation under uncertainties is formulated in Section 4. Section 5 describes the polynomial chaos theory and the used stochastic optimization algorithm. The results of the optimization methods are presented in Section 6. Finally, conclusions are given in Section 7.

• γ • γ c 1 c 2 k 2 , k N L 2 k 1 , k N L 1 x 2 x 1 m NES 1 NES 2 (a) NES 1 c h k h , k N L h h 1 (b) NES 2 c h k h , k N L h h 2 (c)

The mechanical system

The mechanical system considered in this work is composed by the two degrees-of-freedom (DOF) Hultèn's model [START_REF] Hultén | Friction phenomena related to drum brake squeal instabilities[END_REF][START_REF] Hultén | Brake squeal -a self-exciting mechanism with constant friction[END_REF], which represents the primary system, coupled to two identical NESs with masses m h , damping coefficients c h and cubic stiffnesses k NL h . The NESs are attached on the primary system in an ungrounded configuration as shown in Fig. 1.

The equations which describe the mechanical system are given by :

d 2 x 1 dt 2 + η 1 ω 1 dx 1 dt + ω 2 1 x 1 -γω 2 2 x 2 + ϕ 1 x 3 1 -γϕ 2 x 3 2 + µω 1 dx 1 dt - dh 1 dt + ξ h (x 1 -h 1 ) + ϕ h (x 1 -h 1 ) 3 = 0 (1a) ε d 2 h 1 dt 2 + µω 1 dh 1 dt - dx 1 dt + ξ h (h 1 -x 1 ) + ϕ h (h 1 -x 1 ) 3 = 0 (1b) d 2 x 2 dt 2 + η 2 ω 2 dx 2 dt + ω 2 2 x 2 + γω 2 1 x 1 + γϕ 1 x 3 1 + ϕ 2 x 3 2 + µω 1 dx 2 dt - dh 2 dt + ξ h (x 2 -h 2 ) + ϕ h (x 2 -h 2 ) 3 = 0 (1c) ε d 2 h 2 dt 2 + µω 1 dh 2 dt - dx 2 dt + ξ h (h 2 -x 2 ) + ϕ h (h 2 -x 2 ) 3 = 0, (1d)
where h 1 (t) and h 2 (t) (respectively x 1 (t) and x 2 (t)) represent the NESs displacements (respectively the displacements of the primary system), 3 Preliminary results

η i = c i / √ mk i , ω i = k i /m, ϕ i = k NL i /m (with i = 1, 2), ε = m h /m assuming 0.01 < ε < 0.1 , ξ h = k h /m, µ = c h / √ mk 1 and ϕ h = k NL h /m.

Vibratory levels and possible steady-state regimes

A example of the numerical simulation of the system without NESs is plotted in Fig. 2 for two different values of the friction coefficient: γ = 0.3 and γ = 0.4. The other parameters are

ω 1 = 2π100, ω 2 = 2π85, η 1 = 0.02, η 2 = 0.06, ϕ 1 = 10 5 , ϕ 2 = 0, ε = 0.05, ξ h = 0.001, µ = 0.02, ϕ h = 1.4 • 10 5 .
(

Moreover, the initial conditions are small perturbation of the trivial equilibrium position: x 1 (0) = x 2 (0) = 0, ẋ1 (0) = ẋ2 (0) = 10. When γ = 0.4, Limit Cycle Oscillations (LCO) are observed. We focus our analysis on the capacity of the NESs to suppress or mitigate these LCOs.

Four main types of steady-state regimes can be generated when two NESs is attached on the: complete suppression of the instability, mitigation through Periodic Response (PR), mitigation through Strongly Modulated Response (SMR) or no mitigation.

An illustration of these regimes is shown in Fig. 3 which plots the displacements x 1 (t) with respect to times with and without the NESs. Hereafter, mitigated regimes referred to complete suppression, PR and SMR.

The objective function

We define the amplitude A wNES 1 of the variables x 1 of the coupled system Eq. ( 1) and within a steady-state regime as

A wNES 1 = max x SSR 1 (t) -min x SSR 1 (t) 2 , (3)
where x SSR 1 (t) is the times series of the variables x 1 obtained from the numerical integration of the coupled system Eq. ( 1) within the steady-state regime 1 .

The amplitudes A wNES 1 and A woNES 1 (i.e. the amplitude of the system without NESs) are plotted as a function of the friction coefficient γ in Fig. 4 for the set of parameters Eq. ( 2).

The figure highlights a jump (or discontinuity) in the amplitude profile A wNES

1

. This discontinuity corresponds, when γ increases, to the transition from SMR to no suppression regimes and separates mitigated regimes and unmitigated regimes.The value of γ at the jump is called mitigation limit (with respect to γ) and denoted γ ml . Then the Propensity of the system to undergo a Harmless Steady-State Regime (PHSSR) of the oscillation by the NESs is estimated. We compute a set of S total samples of the uncertain parameters within the considered uncertain space and following their distribution law. Then, the PHSSR is defined as follows

PHSSR = S HSSR S total × 100 (4)
where S HSSR is the number of samples within the region of uncertain parameters space in which the LCO of the system is mitigated or the system is stable.

The PHSSR represents the objective function of the study.

Optimization Formulation under uncertainties

The sensitivity of the friction coefficient is such that the steady state of the mechanical system is discontinuous and presents a jump. This jump induces areas in which the efficiency of the NES is either high or low. In order to obtain a robust design of NES which is insensitive to the dispersion of the uncertain parameters, a stochastic optimization problem is considred as follow: .

• x u represent the uncertain parameters of the primary system.

Optimization Algorithm

In order to solve an optimization problem under uncertainties, the determination of the LCO of the mechanical system need to use a stochastic method to take into account the uncertain parameters. In this work, the Multi-Element generalized Polynomial Chaos is used as an alternative approach to reduce the computational cost of the traditional methods (Monte Carlo or deterministic simulations).

Polynomial Chaos theory

generalized Polynomial Chaos

The gPC theory [START_REF] Roger | Stochastic finite elements: A spectral approach[END_REF][START_REF] Cameron | The orthogonal development of non-linear functionals in series of fourierhermite functionals[END_REF] allows to express a random process X(ξ ) called also the Quantity of Interest (QoI) with a truncated orthogonal polynomial function series such as

X(ξ ) ≈ N p ∑ j=0 x j φ j (ξ ), (6)
where:

• x j are the gPC coefficients of the stochastic process X(ξ ),

• φ j (ξ ) are orthogonal polynomial functions,

• N p = (p+r)! p!r! -1 where p is the order of the gPC and r is the number of the uncertain parameters, • ξ (ξ 1 , ..., ξ r ) is a vector of r independent random variables within[-1, 1] r .

Multi-Element Generalized Polynomial Chaos (ME-gPC)

The ME-gPC consists to split ξ into a collection of m non-intersecting elements and to approximate the stochastic process X(ξ ) using the gPC in each element [START_REF] Wan | An adaptive multi-element generalized polynomial chaos method for stochastic differential equations[END_REF]. It is given by

X(ξ ) ≈ m ∑ k=1 X k ( ξ k )J k , (7)
where

• X k ( ξ k ) = N p ∑ j=1
xk, j φ j ( ξ k ) where ξ k is an uniform random variables corresponding to the k th element,

• J k is the element size.

We denote by σ 2 p,k the variance of the QoI estimated in the k th element directly from the gPC coefficients as:

σ 2 p,k = 1 2 r N p ∑ j=1 x2 k, j φ 2 j . ( 8
)

Proposed algorithm

The ME-gPC metamodel would not be efficient in its initial configuration due to the presence of discontinuities. In fact, the goal is not to obtain an accurate representation of the response of the system, but to be able to locate the discontinuity. Figure 5 shows the different steps of the ME-gPC based method to detect the jump. The method consists to split the stochastic parameter space into two equal non-intersecting elements and to evaluate in each element the amplitude of displacement of the system. Then, the verification of the presence or not of discontinuity is addressed according to the comparison of the variance σ 2 p,k to a chosen threshold θ 1 . The process stops either if the element size reaches a threshold

J kmin = θ 2 J k0 , (9)
where J k0 represents the size of the initial element or if the number N of numerical simulations required with the ME-gPC based method is smaller than the typical number N of simulations required with the reference method. The mitigation limit is, therefore approximated by the upper bound of the last found element and then the robust evaluation of the PHSSR is calculated.

Results

In this work, the friction coefficient γ is considred as the uncertain parameter (x u ) and the damping rate of the NES µ as the design variables (x d ).

Reference study

In this section, the reference is evaluated with the deterministic model [START_REF] Oden | Models and computational methods for dynamic friction phenomena[END_REF]. In the first case, 50 deterministic samples of the damping rate of the NES (µ ∈ [0, 0.04]) are used. For each sample of µ, 1 000 samples of the friction coefficient (γ ∈ [0, 0.4]) are used in order to determine the mitigation limit γ m l. The number of simulations is therefore equal to 50 000 (50 × 1 000). Fig. 6(a) shows the variation of the mitigation limit as a function of µ. Because of the presence of a plateau, we realize that the number of samples of γ is not sufficient to determine the optimal design value of µ. For that, 10 added deterministic samples of the damping rate of the NES (µ ∈ [0.02, 0.03]) with now 10 000 samples of the friction coefficient γ to determine the mitigation limit. The total number of simulations required is now equal to 150 000 (50 000 + 10 × 10 000). Fig. 6(b) shows the new variation of the mitigation limit as a function of µ. In this case, the maximum (i.e. optimal) value of γ ml is 0.2004 which gives a maximum value of PHSSR equals to 50.1%. The corresponding optimal design value for the damping rate is µ Ref opt = 0.024.

Optimization using the ME-gPC based method

In this section, the algorithm presented in Fig. 5 is applied to estimate A wNES 1 (the considered QoI) and then the mitigation limit and the PHSSR. The gPC order is p = 1 and the threshold θ 1 = 2.10 -3 %. Fig. 7 shows an example, with a given value of µ, of the algorithm described in Section 5.2. As described above, the mitigation limit is approximated by the upper bound of the last found element. Now, 50 deterministic samples of the damping rate of the NES (µ ∈ [0, 0.04]) are used. The samples of the friction coefficient γ are built using the Latin Hypercube Samples (LHS) method. Fig. 8(a) shows the variation of the mitigation limit as a function of µ with θ 2 = 1%. In this case, the number of simulations is equal to 6547. The optimal value of µ is in the interval I opt,1 = [0.016, 0.032]. In order to reduce the size of I opt the threshold θ 2 and the mitigation limit is determined within I opt,1 . Fig. 8(b) shows the variation of the mitigation limit as a function of µ with θ 2 = 0.1%. The added number of simulations is equal to 2766 and the optimal value of µ is in the interval I opt,2 = [0.019, 0.026]. Fig. 8(c) shows the variation of the mitigation limit as a function of µ with θ 2 = 0.05% used within I opt,2 . In this case, the added number of simulations is equal to 1395 and the optimal value of µ is now in the interval I opt,2 = [0.0208, 0.0248] = 0.0228 ± 8%. This interval has a midpoint equal to 0.0228 which gives a relative error equal to 5% compared to the reference. The total number of simulations is equal to 10 708 which represents 14 times less than the reference.

Conclusion

In this paper, the optimization of two NESs attached to the two degrees-of-freedom (DOF) Hultèn's model is presented. It accounts the discontinuity presented in the steady-state profiles using a stochastic optimization method. The results show the efficiency of the ME-gPC based method to detect the discontinuity and therefore the so-called mitigation limit. This allows us to provide an optimal design of the NESs with a low computational cost compare to that the deterministic optimization reference method.

Figure 1 :

 1 Figure 1: (a) Mechanical system with NESs. (b) Zoom on the NES 1 . (c) Zoom on the NES 2 .

Figure 2 :

 2 Figure 2: Numerical integration in the direction x 1 of the Hult'en's model without NESs for two values of the friction coefficient: (a) Times series; (b) Representation in the phase plane.

Figure 3 :

 3 Figure 3: Comparison between time serie x 1 (t) resulting from the numerical integration of the braking system with and without NESs. (a) Complete suppression, γ = 0.16; (b) Mitigation: PR, γ = 0.18; (c) Mitigation: SMR, γ = 0.2; (d) No suppression, γ = 0.22. The set of parameters Eq. (2) is used.

Figure 4 :

 4 Figure 4: Amplitudes A wNES 1

Figure 5 :Figure 6 :

 56 Figure 5: Algorithm of the ME-gPC based method to identify the mitigation limit.

Figure 7 :

 7 Figure 7: Last step of the ME-gPC based method to identify of the mitigation limit (γ ml ).

Figure 8 :

 8 Figure 8: Variation of the mitigation limit as a function of the damping ratio of the NES using the ME-gPC based method. (a) θ 2 = 1%; (b) θ 2 = 0.1%; (c) θ 2 = 0.05%.

The calculation was performed on 4 seconds in order to be sure that the steady-state has been reached.