L Éo

Wolfgang Steiner

FACTOR-BALANCED S-ADIC LANGUAGES

A set of words, also called a language, is letter-balanced if the number of occurrences of each letter only depends on the length of the word, up to a constant. Similarly, a language is factor-balanced if the difference of the number of occurrences of any given factor in words of the same length is bounded. The most prominent example of a letter-balanced but not factor-balanced language is given by the Thue-Morse sequence. We establish connections between the two notions, in particular for languages given by substitutions and, more generally, by sequences of substitutions. We show that the two notions essentially coincide when the sequence of substitutions is proper. For the class of Thue-Morse-Sturmian languages, we give a full characterisation of factor-balancedness.

Introduction

The study of balancedness of languages goes back at least to Morse and Hedlund [START_REF] Morse | Symbolic dynamics II. Sturmian trajectories[END_REF] who proved that each block of length n in a Sturmian sequence of slope α has ⌊nα⌋ or ⌈nα⌉ occurrences of the letter that has frequency α (and thus ⌊n(1-α)⌋ or ⌈n(1-α)⌉ occurrences of the other letter). In other words, the difference between the number of occurrences of a letter in blocks of the same length is at most 1; we call this property letter-1-balanced, previously it has often been simply called balanced. More generally, a language is letterbalanced if the number of occurrences of a letter only depends on the length of a word in the language, up to an additive constant. We do not only consider the occurrence of letters but also of longer blocks, and we say that a language is factor-balanced if the number of occurrences of each block in a word of the language only depends on the length of the word, up to a constant that can depend on the block. Usually, languages coming from a symbolic dynamical system (or subshift) are considered; we use the slightly weaker property of being factorial. For infinite sequences, balancedness is equivalent to bounded symbolic discrepancy, as studied in [START_REF]Symbolic discrepancy and self-similar dynamics[END_REF]. These concepts have applications in operations research, for optimal routing and scheduling and are related to Fraenkel's conjecture; see [START_REF] Berthé | Balancedness and coboundaries in symbolic systems[END_REF] for references. Some relations between letter-balancedness and factor-balancedness have been studied in [START_REF] Adamczewski | Balances for fixed points of primitive substitutions[END_REF][START_REF] Queffélec | Substitution dynamical systems-spectral analysis[END_REF], and more recently in [START_REF] Berthé | Balancedness and coboundaries in symbolic systems[END_REF]. Here, we improve on these results and show that factor-balancedness is preserved by the application of a substitution. We consider languages (or subshifts) given by a sequence of substitutions, also called S-adic languages. When the substitutions are left or right proper, i.e., the image of each letter starts or ends with the same letter, we show that letter-balancedness (on all levels) implies factor-balancedness; this was previously known only under the assumption that the substitutions have unimodular incidence matrices [BCBD + 21]. A particular case is that of a substitutive shift with a proper substitution. Here, we cannot remove the assumption of properness, as for example the Date: November 23, 2023. This work was supported by the Agence Nationale de la Recherche through the projects CODYS (ANR-18-CE40-0007) and IZES (ANR-22-CE40-0011).

Thue-Morse shift is not factor-balanced [START_REF] Sadun | Finitely balanced sequences and plasticity of 1-dimensional tilings[END_REF][START_REF] Berthé | Balancedness and coboundaries in symbolic systems[END_REF]; we give a short proof in Section 5. For non-proper substitutions, we have to require balancedness for all factors of length 2, not only for letters, in order to get factor-balancedness.

In Section 2, we define most of the notions and give a characterisation of letter-balancedness in terms of the distance to a frequency vector. The effect of substitutions on balancedness is studied in Section 3. Section 4 contains our main results, on sequences of substitutions. Finally, we consider the balancedness of a particular class of S-adic languages in Section 5.

Balancedness

For a finite alphabet A, let A * be the set of finite words over A. A language is a subset L ⊆ A * . A word v ∈ A * is a factor of w ∈ A * if there exist p, s ∈ A * such that w = pvs; here, p is a prefix and s is a suffix of w. We denote the set of factors of w by F (w), and the set of factors of elements of L by F (L). A langage L is factorial if F (L) = L. The length of a word w ∈ A * is denoted by |w|, i.e., |w| = n if w ∈ A n . We denote the prefix (resp. suffix) of length n of a word w by pref n (w) (resp. suff n (w)). The number of occurrences of a word v in w, i.e., the number of different decompositions w = pvs, is denoted by

|w| v . A language L is called C-balanced w.r.t. v if |w| v -|w ′ | v ≤ C for all w, w ′ ∈ L with |w| = |w ′ |.
(By symmetry, this is equivalent to

|w| v -|w ′ | v ≤ C for all w, w ′ ∈ L with |w| = |w ′ |.) It is called C-balanced for length n if it is C-balanced for all v ∈ A n .
We often omit the C and say that a language is balanced for length n if it is C-balanced for length n for some C ≥ 0. Instead of "(C-)balanced for length 1", we also say letter-(C-)balanced ; other papers use the term "balanced" instead of letter-1-balanced or instead of letter-balanced, and a letterbalanced language is sometimes called "finitely balanced". A language is factor-(C-)balanced if it is (C-)balanced for all lengths n ≥ 1; in [START_REF] Sadun | Finitely balanced sequences and plasticity of 1-dimensional tilings[END_REF], a factor-balanced language is called totally finitely balanced. Note that factor-C-balancedness is a strong property that is satisfied for certain Sturmian languages [START_REF] Fagnot | Generalized balances in Sturmian words[END_REF], but we do not study this property here. We are only interested in factor-balancedness, which means that for each n there exists C n such that L is C n -balanced for length n; equivalently, for each v ∈ F (L) there exists C v such that L is C v -balanced w.r.t. v. (Note that |w| v = 0 for all w ∈ L, v / ∈ F (L).) We first show that balancedness for length n is the same as balancedness for lengths up to n.

Lemma 2.1. If a language is balanced for length n, then it is balanced for all lengths k ≤ n.

Proof. Let L ⊂ A * be C-balanced for length n, k < n. For all v ∈ A k , w ∈ A * , we have |w| v = s∈A n-k |w| vs + |suff n-1 (w)| v , thus |w| v -|w ′ | v ≤ (#A) n-k C + n -1 for w ′ ∈ A |w| .
We will also use the following characterisation of letter-balancedness in terms of distance from the line defined by a frequency vector, cf. [START_REF] Berthé | Balance properties of multi-dimensional words[END_REF][START_REF] Adamczewski | Balances for fixed points of primitive substitutions[END_REF].

Proposition 2.2. Let L ⊂ A * be an infinite letter-C-balanced factorial language. Then there exists a (frequency) vector (f a ) a∈A such that |w| a -f a |w| ≤ C for all a ∈ A, w ∈ L.

Proof. Since [0, 1] #A is compact, there exists a vector (f a ) a∈A and a sequence of words

v n ∈ L such that lim n→∞ |v n | = ∞ and lim n→∞ |v n | a /|v n | = f a for all a ∈ A. For arbitrary but fixed w ∈ L, set k n = ⌊|v n |/|w|⌋, and decompose v n = v n,1 • • • v n,kn v n,kn+1 with |v n,i | = |w| for all 1 ≤ i ≤ k n . Since v n,i ∈ L, L is letter-C-balanced, and |v n,kn+1 | < |w|, we obtain that |w| a - |v n | a |v n | |w| = |w| |v n | |v n | |w| |w| a -|v n | a < |w| |v n | |w| + kn i=1 |w| a -|v n,i | a ≤ |w| 2 |v n | + k n |w| |v n | C for all a ∈ A. Letting n → ∞, this gives that |w| a -f a |w| ≤ C. Lemma 2.3. Let L ⊂ A * and (f a ) a∈A such that |w| a -f a |w| ≤ C for all a ∈ A, w ∈ L. Then L is letter-(2C)-balanced. Proof. We have |w| a -|w ′ | a = |w| a -f a |w| + f a |w ′ | -|w ′ | a ≤ 2C for all w, w ′ ∈ L, a ∈ A, such that |w| = |w ′ |.

Substitutions

In this section, we study how the application of a substitution influences balancedness. Here, a substitution σ is a morphism from A * to B * , with the operation of concatenation, i.e., σ(vw) = σ(v)σ(w) for all v, w ∈ A * . We use the notation

σ := max a∈A |σ(a)|, σ := min a∈A |σ(a)|,
and call a substitution non-erasing if all images of letters are non-empty, i.e., σ ≥ 1. It is left (resp. right) proper when all letter images start (resp. end) with the same letter.

To show that substitutions preserve balancedness, we use the following lemma.

Lemma 3.1. Let L ⊂ A * be a letter-C-balanced factorial language and σ :

A * → B * a substitution. Then, for all w, w ′ ∈ F (σ(L)) with |w| = |w ′ |, there exist x, x ′ , z, z ′ ∈ B * , y, y ′ ∈ L, such that w = x σ(y) z, w ′ = x ′ σ(y ′ ) z ′ , |y| = |y ′ |, max{|x z|, |x ′ z ′ |} ≤ (2 + C #A) σ -2.
Proof. Since w, w ′ ∈ F (σ(L)) and L is factorial, we can write w = x σ(v) u and w

= x ′ σ(v ′ ) u ′ with v, v ′ ∈ L, u, u ′ , x, x ′ ∈ A * such that |u|, |u ′ |, |x|, |x ′ | < σ . Assume w.l.o.g. that |v| ≤ |v ′ |, let y = v, v ′ = y ′ s ′ with |y ′ | = |y|, z = u, z ′ = σ(s ′ ) u ′ . Since L is factorial, we have y, y ′ ∈ L. Since L is letter-C-balanced, we have |σ(y)| -|σ(y ′ )| = a∈A |y| a -|y ′ | a ) |σ(a)| ≤ (#A) C σ , thus |x ′ z ′ | = |w ′ | -|σ(y ′ )| ≤ |w| -|σ(y)| + (#A) C σ ≤ 2 ( σ -1) + (#A) C σ .
Therefore, w = x σ(y) z and w ′ = x ′ σ(y ′ ) z ′ satisfy all the required properties.

Proposition 3.2. Let L ⊂ A * be a factorial language and σ :

A * → B * a substitution. If L is letter-balanced, then F (σ(L)) is letter-balanced.
Proof. Suppose that L is C-letter-balanced, and let w = x σ(y) z, w ′ = x ′ σ(y ′ ) z ′ be as in Lemma 3.1. Then, for all b ∈ B,

|w| b -|w ′ | b = |xz| b -|x ′ z ′ | b + |σ(y)| b -|σ(y ′ )| b ≤ (2 + C #A) σ -2 + a∈A |y| a -|y ′ | a |σ(a)| b ≤ 2 (1 + C #A) σ -2.
To study balancedness for length n, we use the n-th higher block code of a word a 1 a 2 • • • a N ∈ A N , N ≥ 0, which is the word over the alphabet A n defined by

(a 1 a 2 • • • a N ) (n) = (a 1 a 2 • • • a n )(a 2 a 3 • • • a n+1 ) • • • (a N -n+1 a N -n+2 • • • a N ) ∈ (A n ) N -n+1 if N ≥ n, the empty word if N < n. Note that |w| v = |w (n) | v
for all v ∈ A n ; in particular, a language L is C-balanced for length n if and only if {w (n) : w ∈ L} is letter-C-balanced (over the alphabet A n ).

Proposition 3.3. Let L ⊂ A * be a factorial language that is balanced for length n, σ : A * → B * a substitution, and u ∈ B * a (possibly empty) word that is a prefix of σ(a)u for all a ∈ A or a suffix of uσ(a) for all a ∈ A. Then F (σ(L)) is balanced for length min w∈L∩A n-1 |σ(w)|+|u|+1. In particular,

• F (σ(L)) is balanced for length n if σ is non-erasing,

• F (σ(L)) is balanced for length n+1 if σ is left or right proper.

Proof. Let L ⊂ A * , σ : A * → B * , u ∈ B * be as in the statement of the proposition, 1 ≤ m ≤ min w∈L∩A n-1 |σ(w)|+|u|+1. Assume w.l.o.g. that u is a prefix of σ(a)u for all a ∈ A, the suffix case being symmetric. We define a substitution σ :

(A n ∩ L) * → (B m ) * by σ(a 1 a 2 • • • a n ) := σ(a 1 )pref m-1 (σ(a 2 • • • a n )u) (m) for a 1 • • • a n ∈ A n ∩ L.
(Here, σ is a substitution on the alphabet A n ∩ L, and the condition on m ensures that pref m-1 (σ(a 2 • • • a n )u) exists.) Then we have, for all w ∈ L,

(3.1) σ(w)u (m) = σ w (n) σ(suff n-1 (w))u (m) .
Let C n be such that L is C n -balanced for length n. Then, by Lemma 2.1, L is letter-C 1balanced for some C 1 ≥ 0. Let w, w ′ ∈ F (σ(L)) with |w| = |w ′ |, and write w = x σ(y) z, w ′ = x ′ σ(y ′ ) z ′ as in Lemma 3.1. Similarly to (3.1), we obtain that (wu) (m) = x pref m-1 (σ(y)u)

(m) σ y (n) σ(suff n-1 (y))zu (m) .

Using a similar decomposition for (w ′ u) (m) , we obtain for v ∈ B m that

|w| v -|w ′ | v ≤ max |xz|, |x ′ z ′ | + (n -1) σ + t∈A n ∩L |y| t -|y ′ | t σ(t) v ≤ (2 + C 1 #A) σ -2 + (n -1) σ + (#A) n C n σ .
Here, we have used that When the incidence matrix of σ is invertible, we can also infer letter-balancedness of L from that of F (σ(L)). Here, the incidence matrix of a substitution σ : A * → B * is

|w| v = |w (m) | v , that L is C n -balanced for length n, and that |σ(a 1 • • • a n )| = |σ(a 1 )| for a 1 • • • a n ∈ A n ∩ L. This proves that F (σ(L)) is balanced for length min w∈L∩A n-1 |σ(w)|+|u|+1. If σ is non-erasing, then |σ(w)| ≥ n-1 for all w ∈ A n-1 , thus F (σ(L)) is
M σ := (|σ(a)| b ) b∈B,a∈A .
Proposition 3.5. Let σ : A * → B * be a substitution with invertible incidence matrix M σ and L ⊂ A * . If F (σ(L)) is letter-balanced, then L is letter-balanced.

Proof. If L is finite, then it is trivially letter-balanced. Assume in the following that L is infinite and M σ invertible. Then σ is non-erasing and thus σ(L) infinite. If F (σ(L)) is letter-balanced, then Proposition 2.2 gives a frequency vector f = (f b ) b∈B such that

D := (|σ(w)| b -|σ(w)|f b ) b∈B : w ∈ L is a bounded set. Since (|σ(w)| b ) b∈B = M σ (|w| a ) a∈A and M σ is invertible, we have M -1 σ D = (|w| a ) a∈A -|σ(w)|M -1 σ f : w ∈ L}. Therefore, the vectors (|w| a ) a∈A , w ∈ L, have bounded distance from the line RM -1 σ f. Hence, M -1 σ f is a non-negative vector and (f ′ a ) a∈A := M -1 σ f/ M -1 σ f 1 is the frequency vector of L. Since a∈A |w| a -|w|f ′ a = |w| -|w| = 0 for all w ∈ A * , the set D ′ := (|w| a -|w|f ′ a ) a∈A : w ∈ L = (|w| a ) a∈A - |w| M -1 σ f 1 M -1 σ f : w ∈ L
lies in the hyperplane H := {(x a ) a∈A : a∈A x a = 0}. Therefore, D ′ is the projection of M -1 σ D to H along the line RM -1 σ f (which is not in H since M -1 σ f is non-negative and non-zero), thus D ′ is bounded. Hence, by Lemma 2.3, L is letter-balanced.

S-adic languages

Now, we consider sequences of substitutions σ = (σ k ) k≥0 , σ k :

A * k+1 → A * k . We set σ [k,n) := σ k • σ k+1 • • • • • σ n-1 for n ≥ k ≥ 0, where σ [k,k) is the identity map; then σ [k,n) is a substitution from A * n to A * k .
The language of σ at level k is defined by

L (k) σ := w ∈ A * k : w ∈ F (σ [k,n) (A n ))
for infinitely many n > k , and L σ := L (0) σ . (In other papers, the requirement for infinitely many n > k is replaced by "some n > k"; this can change the language only if a letter of A m does not occur in σ m .) Our definition ensures that σ is letterbalanced for infinitely many k, then L σ is factor-balanced.

(4.1) F σ [k,n) (L (n) σ ) = L (k) σ for all n ≥ k ≥ 0. A sequence of substitutions (σ k ) k≥0 is everywhere growing if lim k→∞ σ [0,k) = ∞. It is left (resp. right) proper when for each k ≥ 0 there exists n > k such that σ [k,n) is left (resp.
Proof. For σ = (σ k ) k≥0 ∈ {L, M, R} ∞ \ {L, R} ∞ , let n ≥ 1 be minimal such that σ n = M. We claim that F (σ [0,n) ({01, 10} * )) is letter-2-balanced. Indeed, we have σ [0,n) (01) = 01w
and σ [0,n) (10) = 10w for some w ∈ {0, 1} * . (This property is trivial for σ [n,n) and can be shown inductively for σ

[k,n) , 0 ≤ k < n, since σ k ∈ {L, R}.) Let v, v ′ ∈ F (σ [0,n) ({01, 10} * )) with |v| = |v ′ |. If |v| ≥ |w|+3 then we can write v = pus, v ′ = p ′ u ′ s ′ such that |u| 0 = |u ′ | 0 = |w| 0 +1, |u| 1 = |u ′ | 1 = |w| 1 +1 and ps, p ′ s ′ ∈ F (σ [0,n) ({01, 10} * )). Since |ps| = |p ′ s ′ | and |ps| 0 -|p ′ s ′ | 0 = |v| 0 -|v ′ | 0 , it is sufficient to consider |v| ≤ |w|+2. If |v| ≤ |w|+1, then v, v ′ ∈ F (σ [0,n) ({01} * ))
, and it is well known that this language is letter-1-balanced; see [START_REF] Lothaire | Algebraic combinatorics on words[END_REF]Chapter 2]. For |v| = |w|+2, we have |v| 0 = |w| 0 +1 or v ∈ {0w0, 1w1}. Therefore, F (σ [0,n) ({01, 10} * )) and thus L σ are letter-2-balanced.

Let now σ ∈ {L, R} ∞ . If σ contains infinitely many L's and R's, then L σ is Sturmian and thus letter-1-balanced; see e.g. [Lot02, Chapter 2]. Since L n (0) = 0, L n (1) = 10 n , R n (0) = 01 n , R n (1) = 1, for all n ≥ 0, the languages L L ∞ and L R ∞ are also letter-1-balanced. Finally, if

σ n = L, σ k = R for all k > n, or σ n = R, σ k = L for all k > n, then L σ = F (σ [0,n) ({01} * )), which is again letter-1-balanced.
For a characterisation of factor-balancedness of Thue-Morse-Sturmian languages, we need the following lemma in order to show that applying any composition of substitutions L, M, R to the Thue-Morse language, which is not factor-balanced, does not create a factor-balanced language. More precisely, we show for σ ∈ {L, M, R} * that σ(011) occurs only trivially in σ(w), w ∈ L M ∞ . Here, {L, M, R} * is the set of compositions of substitutions in L, M, R.

= σ [0,n) (1) p ′ , then σ [0,n) (0v ′ b ′ ) = p ′ σ [0,n) (01) s ′ for a prefix 10v ′ b ′ of L(1vb) and a strict suffix s ′ of σ [0,n) (b ′
). Now, the induction hypothesis implies that p ′ is empty, thus L(avb) = (10) k+1 , i.e., avb = 1 k+1 . Let now σ n = M. Then we have

σ [0,n) (M(avb)) = p σ [0,n) (01(10) k ) s and 111 / ∈ F (M(avb)). If p is a strict prefix of σ [0,n) (a), then σ [0,n) (av ′ b ′ ) = p σ [0,n) (011) s ′ ,
= σ [0,n) (0) p ′ , then σ [0,n) (1v ′ b ′ ) = p ′ σ [0,n) (011) s ′ , hence p ′ is empty, which contradicts that σ [0,n) (1) and σ [0,n) (0) start with different letters. Finally, let σ n = R. Then we have σ [0,n) (R(avb)) = p σ [0,n) (01 k+1 ) s and 1 k+2 / ∈ F (R(avb)).
If p is a strict prefix of σ [0,n) (a), then R(avb) = 01 k+1 , thus avb = 01 n . Otherwise, we have a = 0 and p

= σ [0,n) (0) p ′ , thus σ [0,n) (1v ′ b ′ ) = p ′ σ [0,n) (01 k+1 ) s ′ ,
σ is letter-balanced for all k ≥ 0. If σ does not end with M ∞ , then it is right proper. If σ also does not end with L ∞ or R ∞ , then it is everywhere growing, and we can apply Theorem 4.1. If σ ends with LR ∞ or RL ∞ , then we have seen in the proof of Proposition 5.1 that L σ = F (σ({01} * )) for some σ ∈ {L, M, R} * , which is factor-balanced. The cases of L L ∞ and L R ∞ are similar.

Consider now σ ending with M ∞ . We first prove that the Thue-Morse language L M ∞ is not balanced for length 2, giving more details than [Sad16, Example 3]; a more general proof can be found in [START_REF] Berthé | Balancedness and coboundaries in symbolic systems[END_REF]. To this end, define recursively words w n , w ′ n ∈ L M ∞ by w 1 = 00, M 2 (w 2n-1 ) = 0 w 2n 0, M 2 (w 2n ) = 1 w 2n+1 1,

  balanced for length n. If σ is left or right proper, then σ is non-erasing and |u| ≥ 1, thus F (σ(L)) is balanced for length n+1.Theorem 3.4. Let L ⊂ A * be a factorial language and σ :A * → B * a substitution. If L is factor-balanced, then F (σ(L)) is factor-balanced.Proof. For non-erasing σ, the theorem is a direct consequence of Proposition 3.3. If F (σ(L)) is finite, then it is also factor-balanced. If F (σ(L)) is infinite, then there exists a ∈ A such that |σ(a)| ≥ 1 and {|w| a : w ∈ L} is unbounded. If L is letter-C-balanced, then Proposition 2.2 gives some f a ≥ 0 such that |w| a ≥ f a |w| -C and thus |σ(w)| ≥ f a |w| -C for all w ∈ L. By Proposition 3.3, balancedness of L for length n implies balancedness of F (σ(L)) for length f a (n-1)-C+1. Note that f a > 0 since |w| a ≤ f a |w| + C and |w| a is unbounded. Therefore, factor-balancedness of L implies that of F (σ(L)).

  right) proper. The following theorem was proved in [BCBD + 21, Corollary 5.5] for unimodular incidence matrices, i.e., |det M σ k | = 1 for all k ≥ 0. Theorem 4.1. Let σ be a left or right proper sequence of substitutions. If L (k)

Lemma 5. 2 .

 2 Let σ ∈ {L, M, R} * , a, b ∈ {0, 1}, p, s, v ∈ {0, 1} * , k ≥ 1, such that σ(avb) = p σ(01 k ) s,p is a strict prefix of σ(a) and s is a strict suffix of σ(b). Then avb = 01 k (and p, s are empty) or avb = 1 k+1 (and s is empty).Proof. The statement is clearly true when σ is the identity. Forσ = σ 0 • σ 1 • • • • • σ n , σ i ∈ {L, M,R}, we prove the statement by induction on n.Let first σ n = L. Then we haveσ [0,n) (L(avb)) = p σ [0,n) (0(10) k ) s and 11 / ∈ F (L(avb)). If p is a strict prefix of σ [0,n) (a), in particular if a = 0, then σ [0,n) (av ′ b ′ ) = p σ [0,n) (01) s ′ for a prefix av ′ b ′ of L(avb) and a strict suffix s ′ of σ [0,n) (b ′ ). By the induction hypothesis and since 11 ∈ F (L(av ′ b ′ )), this implies that p is empty. Since σ [0,n) (0) starts with 0 and σ [0,n) (1) starts with 1, we obtain that L(avb) = 0(10) k and thus avb = 01 k . If a = 1 and p

  hence p is empty, and avb = 01 k . If a = 1 and p = σ [0,n) (1) p ′ , then we obtain that avb = 1 k+1 . If a = 0 and p

  w ′ 1 = 01, M 2 (w ′ 2n-1 ) = w ′ 2n 01, M 2 (w ′ 2n ) = w ′ 2n+1 10, with |w n | = |w ′ n | = 4 n +2 hence ℓ 2 (w 2n )ℓ 2 (w ′ 2n ) = n v -1 , i.e., |w 2n | 00 -|w ′ 2n | 00 = |w ′ 2n | 01 -|w 2n | 01 = |w ′ 2n | 10 -|w 2n | 10 = |w 2n | 11 -|w ′ 2n | 11 = n.To finish the proof of the theorem, we have to show thatF (σ(L M ∞ )) is not factor-balanced for all σ ∈ {L, M, R} * . Since 111 / ∈ L M ∞ ,Lemma 5.2 implies that |σ(w)| σ(011) = |w| 011 for all w ∈ L M ∞ , and we clearly have 0 ≤ |w| 11 -|w| 011 ≤ 1. Therefore, we have |σ(w)| σ(011) -|σ(w ′ )| σ(011) ≥ |w| 11 -|w ′ | 11 -1 for all w, w ′ ∈ L M ∞ . Since |w| 11 -|w ′ | 11 is unbounded for w, w ′ ∈ L M ∞ with |w| = |w ′ |, it is also unbounded when we restrict to w, w ′ with |w| 0 = |w ′ | 0 (and |w| 1 = |w| ′ 1 ). Then we have |σ(w)| = |σ(w ′ )|, thus σ(L M ∞ ) is not balanced for length |σ(011)|. We remark that, by Proposition 3.3, F (σ • L(L M ∞ )) is balanced for length |σ(0)| + 1 for any substitution σ. On the other hand, we have seen in the proof of Theorem 5.3 that F (σ • L(L M ∞ )) is not balanced for length |σ(01010)| for any substitution σ ∈ {L, M, R} * .

  hence p ′ is empty, which contradicts that σ [0,n) (1) and σ [0,n) (0) start with different letters.Theorem 5.3. Let σ = (σ k ) k≥0 ∈ {L, M, R} ∞ . Then L σ is factor-balanced if and only if σ k = M for infinitely many k.

	Proof. By Proposition 5.1, L

Proof. Assume that L (k) σ is letter-balanced for infinitely many k, which implies that it is letter-balanced for all k by (4.1) and Proposition 3.2. Since σ is left or right proper, there exist 0 = k 0 < k 1 < k 2 < • • • such that σ [k i ,k i+1 ) is left or right proper for all i ≥ 0. Therefore, by Proposition 3.3, L σ is balanced for all lengths n ≥ 1.

The following corollary is the particular case of Theorem 4.1 with constant sequence σ = (σ, σ, . . . ) for some substitution σ : A * → A * ; we write σ ∞ for (σ, σ, . . . ). The language of a substitution is L σ := L σ ∞ (and consists of those w ∈ A * that are in F (σ n (A)) infinitely often).

Corollary 4.2. Let σ : A * → A * be a substitution such that σ k is left or right proper for some k ≥ 1. If L σ is letter-balanced, then L σ is factor-balanced.

For invertible incidence matrices, letter-balancedness at level 0 implies letter-balancedness at all levels by Proposition 3.5, which gives the following corollary of Theorem 4.1.

Corollary 4.3. Let σ = (σ k ) k≥0 be a left or right proper sequence of substitutions with invertible incidence matrix M σ k for all k ≥ 0. If L σ is letter-balanced, then L σ is factor-balanced.

If σ is not proper, then we need balancedness for length 2 to infer factor-balancedness.

Theorem 4.4. Let σ be an everywhere growing sequence of substitutions such that L (k) σ is balanced for length 2 for infinitely many k. Then L σ is factor-balanced.

Proof. By Proposition 3.3, balancedness of L (k) σ for length 2 implies balancedness of L σ for length σ [0,k) +1. Since σ is everywhere growing, this implies that L σ is factor-balanced.

For primitive substitutions σ, a sufficient condition for balancedness for length n of L σ is given in [Ada03, Theorem 22], and it indicates that balancedness for length 2 and factorbalancedness are closely related; see also [START_REF] Queffélec | Substitution dynamical systems-spectral analysis[END_REF]Section 5.4.3]. We prove that balancedness for length 2 implies factor-balancedness for most substitutions.

Corollary 4.5. Let σ : A * → A * be an everywhere growing substitution. If L σ is balanced for length 2, then L σ is factor-balanced.

We remark that, in Theorem 4.1, Corollary 4.3 and Theorem 4.4, factor-balancedness holds not only for L σ but for all L (k) σ , k ≥ 0.

Thue-Morse-Sturmian languages

We conclude the paper by studying a special class of sequences of substitutions that occurs naturally in [START_REF] Steiner | Thue-Morse-Sturmian words and critical bases for ternary alphabets[END_REF][START_REF] Komornik | Unique double base expansions[END_REF]. A language L σ , σ ∈ {L, M, R} ∞ , with substitutions

in the case of σ ∈ {L, M, R} ∞ , this means that σ does not end with the constant sequence L ∞ or R ∞ . However, the following results also hold for non-primitive sequences.

Proposition 5.1. For all σ ∈ {L, M, R} ∞ , L σ is letter-2-balanced.


Yanlin Zhao

Xindong Li

Scott Cogan

Jiahui Zhao

Jianhong Yang
email: yangjianhong@me.ustb.edu.cn

Debing Yang

Jinqi Shang

Bing Sun

Lechang Yang

Jingqi Shang

Interval parameter sensitivity analysis based on interval perturbation propagation and interval similarity operator

Keywords: Sensitivity analysis, interval uncertainty, interval sensitivity operator, interval relative position operator, interval perturbation method

come

Introduction

A practical engineering model has to cope with various uncertainties existing in systems and structures. Uncertainties generally arise from the observed scattering of environmental conditions, lack of knowledge, inhomogeneity of materials, and measurement uncertainty.

Those uncertainties are typically divided into two distinct forms, i.e., aleatory uncertainty and epistemic uncertainty [START_REF] Kitahara | Nonparametric Bayesian stochastic model updating with hybrid uncertainties[END_REF]. Meanwhile, non-deterministic analysis (Wang et al. 2011) has gained wide interest, and elaborate literature is available in this field. Uncertainty analysis can be generally classified into two categories of probabilistic and non-probabilistic techniques (Ql et al. 2021;[START_REF] Singh | Randomized algorithms for probabilistic analysis of parametric uncertainties with unmanned helicopters[END_REF]. The interval model is one of the most representative non-probabilistic approaches, which quantifies the uncertainties by the bounds of datapoints. In this work, we mainly focus on the sensitivity analysis with interval uncertainties.

Sensitivity analysis (SA) has rapidly developed to quantify the impact of model parameters on outputs due to the growing complexity of mathematical models. One classical definition of sensitivity is "the study of how the uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model inputs" [START_REF] Andrea | Sensitivity analysis for importance assessment[END_REF], which is typically distinct from the uncertainty analysis. The importance of parameters is compared through ranked sensitivity indexes corresponding to each input parameter. The growth of uncertainty analysis has greatly promoted the development of sensitivity analysis.

The importance of sensitivity analysis is widely acknowledged. One object of sensitivity analysis is to identify the contributions of model inputs to the variation of the outputs, which is used as evidence for significant parameter selection before model calibration. For example, sensitivity analysis is generally distinguished between local methods [START_REF] Ha | A local sensitivity analysis for the kinetic Cucker-Smale equation with random inputs[END_REF]) and global methods [START_REF] Sobol | Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates[END_REF]). In the context of local methods, the changes in outputs are analyzed while one input parameter is changed, with the rest kept at reference values. [START_REF] Jacomel | A priori error estimates for local reliability-based sensitivity analysis with Monte Carlo Simulation[END_REF] presented a priori error estimates for local reliability-based sensitivity analysis. [START_REF] Achyut | Higher-order Taylor series expansion for uncertainty quantification with efficient local sensitivity[END_REF] proposed local sensitivity analysis by using an efficient approach called modified forward finite difference. Global sensitivity [START_REF] Cheng | Multivariate output global sensitivity analysis using multioutput support vector regression[END_REF] analysis captures the interaction effects among parameters when exploring the responses of the model by varying all inputs at the same time. Examples of global methods include the first-order sensitivity index of Sobol's method [START_REF] Liu | Gauge sensitivity analysis and optimization of the modular automotive body with different loadings[END_REF][START_REF] Sobol | Sensitivity estimates for nonlinear mathematical models[END_REF], the extended Fourier amplitude sensitivity test [START_REF] Saltelli | A Quantitative Model-Independent Method for Global Sensitivity Analysis of Model Output[END_REF], the Morris screening method [START_REF] Shin | Addressing ten questions about conceptual rainfall-runoff models with global sensitivity analyses in R[END_REF], the Multi-output support vector regression (M-SVR) [START_REF] Cheng | Multivariate output global sensitivity analysis using multioutput support vector regression[END_REF], and the distribution-based global sensitivity analysis [START_REF] Lukáš | On distribution-based global sensitivity analysis by polynomial chaos expansion[END_REF]. When structures with large-scale parameters, it leads to the expensive computational cost issue due to quantifying the effects of inputs on the output response globally.

A hybrid metamodel using the orthogonal constraints of radial basis function and sparse polynomial chaos expansions for the global sensitivity analysis of time-consuming models was developed [START_REF] Wu | Hybrid metamodel of radial basis function and polynomial chaos expansions with orthogonal constraints for global sensitivity analysis[END_REF].

Sensitivity analysis has been implemented in various areas, such as model Verification and Validation (V&V) [START_REF] Eamon | Integrated reliability and sizing optimization of a large composite structure[END_REF][START_REF] Ehre | A framework for global reliability sensitivity analysis in the presence of multi-uncertainty[END_REF][START_REF] Papaioannou | Variance-based reliability sensitivity analysis and the FORM α -factors[END_REF][START_REF] Suzana | Review of finite element model updating methods for structural applications[END_REF], structural optimization design [START_REF] Eamon | Integrated reliability and sizing optimization of a large composite structure[END_REF][START_REF] Liu | Gauge sensitivity analysis and optimization of the modular automotive body with different loadings[END_REF], structural reliability analysis [START_REF] Ehre | A framework for global reliability sensitivity analysis in the presence of multi-uncertainty[END_REF][START_REF] Papaioannou | Variance-based reliability sensitivity analysis and the FORM α -factors[END_REF], mechanical property analysis of laminated plates (Longfei et al. 2012), and robust design in aerospace engineering [START_REF] Dasari | Predictive modelling to support sensitivity analysis for robust design in aerospace engineering[END_REF]. However, most variance-based or momentindependent sensitivity analyses [START_REF] Zhang | A new framework of variance based global sensitivity analysis for models with correlated inputs[END_REF][START_REF] Zhou | Moment independent sensitivity analysis with correlations[END_REF] involve evaluating partial derivatives of probabilistic model outputs at the nominal values of the input parameters, which should combine the sampling-based probabilistic methods and mathematical models to quantify uncertainties. In the case of a complex model with massive numbers of input variables, the sensitivity analysis from the probabilistic view is highly time-consuming.

In the context of non-probabilistic uncertainty quantification, the input parameters and output responses are both non-probabilistic. Traditional variance-based sensitivity analysis is inapplicable to the interval model due to the lack of probabilistic information on inputs and outputs. Recently, a prediction on the static response of structures with uncertain-but-bounded parameters based on the adjoint sensitivity analysis was developed, where the sensitivity analysis is implemented without considering the interval characteristics of uncertain parameters [START_REF] Luo | Prediction on the static response of structures with large-scale uncertain-but-bounded parameters based on the adjoint sensitivity analysis[END_REF]. It is necessary to consider interval uncertainties when sensitivity and uncertainty analysis is performed. Up to now, however, most sensitivity analysis methods and sensitivity coefficients are mainly established for probabilistic parameter selection, such as the Sobol sensitivity index [START_REF] Sobol | Sensitivity estimates for nonlinear mathematical models[END_REF], the total-effect index [START_REF] Homma | Importance measures in global sensitivity analysis of nonlinear models[END_REF], the Morris sensitivity index [START_REF] Morris | Factorial Sampling Plans for Preliminary Computational Experiments[END_REF]),and FAST sensitivity index [START_REF] Mcrae | Global sensitivity analysis-a computational implementation of the Fourier Amplitude Sensitivity Test (FAST)[END_REF].

Sensitivity analysis is developed to provide information for the reliability-based design. [START_REF] Xiao | Reliability sensitivity analysis for structural systems in interval probability form[END_REF] proposed a reliability sensitivity analysis method for the model with both epistemic and aleatory uncertainties using P-boxed. [START_REF] Bi | The Bhattacharyya distance: Enriching the P-box in stochastic sensitivity analysis[END_REF] developed a stochastic sensitivity analysis with a novel sensitivity index based on the Bhattacharyya distance. Those research efforts have been made on sensitivity analysis when both hybrid epistemic and aleatory uncertainties. However, with the limitation of samples, the stochastic characteristics of parameters cannot be precisely determined. Besides, those mentioned probabilistic sensitivity coefficients for the stochastic models or models with hybrid uncertainties are not applicable to the model with purely non-probabilistic uncertainties. Hence, it is necessary to extend the sensitivity analysis to a wider application with only interval uncertainties. A novel sensitive coefficient based on the geometric interval quantification method is presented to quantify the parameter sensitivity in this paper.

The interval sensitivity analysis process relies on the accurate propagation of uncertainties in the form of intervals. However, the interval arithmetic operations are difficult to implement directly in uncertainty propagation. Interval analysis is introduced to estimate the interval outputs according to the interval inputs, which is named interval uncertainty propagation.

Interval analysis is the basis of interval sensitivity, which predicts the interval output for estimating the sensitivity indices. Monte-Carlo simulation [START_REF] Callens | MULTILEVEL QUASI-MONTE CARLO FOR INTERVAL ANALYSIS[END_REF]) is one of the typical uncertainty propagation methods which has been introduced into Sobol's sensitivity analysis. Interval analysis typically requires a global optimization procedure with Monte Carlo simulation to determine the interval bounds on the output side of a computational model. However, in the context of complex models in practice, massive Monte Carlo simulation brings excessive computation, sharply increasing the computation cost of sensitivity analysis. Some efficient interval propagation methods, such as advanced interval analysis [START_REF] Fujita | An efficient methodology for robustness evaluation by advanced interval analysis using updated second-order Taylor series expansion[END_REF], multivariate interval quantification approach based on the concept of the convex hull [START_REF] Faes | A multivariate interval approach for inverse uncertainty quantification with limited experimental data[END_REF][START_REF] Faes | Identification and quantification of multivariate interval uncertainty in finite element models[END_REF], Infor-gap uncertainty quantification models [START_REF] Ben-Haim | Uncertainty, probability and information-gaps[END_REF], and some interval surrogate models [START_REF] Fang | An interval model updating strategy using interval response surface models[END_REF][START_REF] Khodaparast | Interval model updating with irreducible uncertainty using the Kriging predictor[END_REF] are rapidly developed to reduce the computation cost. The interval perturbation methodology [START_REF] Li | Adaptive sub-interval perturbation-based computational strategy for epistemic uncertainty in structural dynamics with evidence theory[END_REF][START_REF] Wang | An interval perturbation method for exterior acoustic field prediction with uncertain-but-bounded parameters[END_REF], a representative interval propagation method, has some advantages over the Monte Carlo simulation, including lower computation cost because of calculating only by the information of a single point that allows the consideration of the complexity of structure. Therefore, this work introduces the interval perturbation propagation method to effectively estimate the output interval according to interval inputs.

A parameter sensitivity analysis method with a novel interval-based sensitivity metric is developed by introducing the interval propagation methodology in this work. The interval similarity operator (ISO) is employed as a sensitivity metric to measure the discrepancy between two interval model outputs corresponding to initial and changed interval parameters, respectively. This metric is developed for interval uncertainty quantification as it is computed only based on the extreme bounds of the interval without their inner data points. The interval perturbation method is adopted to estimate the bounds of model outputs to improve the computation effectiveness. The feasibility and accuracy of the proposed method is verified by two typical academic cases. This work is organized as follows. Section 2 presents an overview of sensitivity analysis with interval uncertainties. Section 3 presents how to calculate the proposed interval-based sensitivity index. Section 4 illustrates the comprehensive framework of sensitivity analysis with interval uncertainties. Section 5 gives two study cases, i.e., an academic case and a more complex satellite case, to investigate the proposed method.

Background of sensitivity analysis for interval parameters

A finite element model or other complex black box models can be expressed as 𝑭ሺ•ሻ as follows

𝒇 = 𝑭ሺ𝜽ሻ (1)
where 𝒇 is the model output. 𝑭 represents a propagation function of the model system. 𝜽 is the model parameters, where 𝜽 = ሼ𝜃 𝑖 ሽ, 𝑖 = 1,2,3, … , 𝑛 , and 𝑛 is the number of model parameters.

The sensitivity index S=ሼ𝑆 𝑖 ሽ of model parameter 𝜽 is simply expressed as:

𝐒 = ∆𝒇 ∆𝜽 = 𝑭ሺ𝜽+∆𝜽ሻ-𝑭ሺ𝜽ሻ ∆𝜽 (2)
where ∆𝜽 = ሼ∆𝜃 𝑖 ሽ represents the variation of parameter 𝜽 . ∆𝒇 = ൛∆𝑓 𝑗 ൟ represents the variation of 𝒇,where j=1, 2, …, m, m represents the dimensions of model output features.

In the context of a model with interval parameters 𝜽 𝑰 , Eq. ( 1) can be given as

𝒇 𝑰 = 𝑭൫𝜽 𝑰 ൯ (3)
where 𝜽 𝑰 = ൛𝜃 𝑖 𝐼 ൟ represents the interval parameters of model 𝑭, and 𝒇 𝑰 = ൛𝑓 𝑗 𝐼 ൟ are interval uncertainties. Then, the sensitivity index for interval parameters is accordingly converted as:

𝐒 𝐼𝑛𝑡 = 𝝏𝒇 𝑰 𝜕∆𝜽 𝐼 = 𝐹൫𝜽 𝑪 +∆𝜽 𝑰 ൯-𝐹൫𝜽 𝑪 +∆𝜽 𝑰 ൯ ∆𝜽 𝐼 -∆𝜽 𝑰 (4)
where ∆𝜽 𝑰 = ൛∆𝜃 𝑖 𝐼 ൟ means the changed interval radius. It can be seen that ∆𝜽 𝐼 and ∆𝜽 𝑰 is independent of the interval midpoints 𝜽 𝑪 . We mainly focus on identifying the contributions of the slight change in model interval inputs to the variation of uncertainty degree of the outputs in this work.

For an interval variable 𝜽 𝑰 , it can be determined by the extreme bounds 𝜽 and 𝜽 or by the inter center 𝜽 𝐶 and the interval radius ∆𝜽, as follows:

𝜽 𝑰 = ሾ𝜽 𝑐 -∆𝜽, 𝜽 𝑐 + ∆𝜽ሿ = ൣ𝜽, 𝜽൧ (5)
The fundamental operations of arithmetic for intervals (Moore 1996) are interpreted by two interval variables 𝜃 1 𝐼 and 𝜃 2 𝐼 , which is given as

ە ۖ ۔ ۖ ۓ 𝜃 1 𝐼 + 𝜃 2 𝐼 = ൣ𝜃 1 𝑙 + 𝜃 2 𝑙 , 𝜃 1 𝑢 + 𝜃 2 𝑢 ൧ 𝜃 1 𝐼 -𝜃 2 𝐼 = ൣ𝜃 1 𝑙 -𝜃 2 𝑢 , 𝜃 1 𝑢 -𝜃 2 𝑙 ൧ 𝜃 1 𝐼 ×𝜃 2 𝐼 =ൣ𝑚𝑖𝑛൛𝜃 1 𝑙 𝜃 2 𝑙 ,𝜃 1 𝑙 𝜃 2 𝑢 ,𝜃 1 𝑢 𝜃 2 𝑙 ,𝜃 1 𝑢 𝜃 2 𝑢 ൟ,𝑚𝑎𝑥൛𝜃 1 𝑙 𝜃 2 𝑙 ,𝜃 1 𝑙 𝜃 2 𝑢 ,𝜃 1 𝑢 𝜃 2 𝑙 ,𝜃 1 𝑢 𝜃 2 𝑢 ൟ൧ 𝜃 1 𝐼 𝜃 2 𝐼 =𝜃 1 𝐼 ×ቈ 1 𝜃 2 𝑢 , 1 𝜃 2 𝑙 (6)
This work mainly focuses on the problem of local sensitivity analysis, which ignores the relationship between input parameters. The sensitivity analysis index 𝐒 𝐼𝑛𝑡 is to quantify the impact of the change of interval parameters from 𝜽 𝑰 into 𝜽 𝑰 + ∆𝜽 𝑰 on the model uncertainty output 𝒇 𝑰 . Interval sensitivity analysis is simply illustrated in Fig. 1. We can find that for the sensitive interval parameters, once there is some variation in their boundaries, both the interval center and the interval radius of outputs significantly change. However, on the contrary, even if the insensitive interval parameters encounter large perturbations, it may cause a small impact on model outputs. Fig. 1 Diagrammatic sketch for sensitivity analysis of interval parameters.

Interval perturbation FE method

In the context of a model containing interval uncertainties, Fig. 2 Interval uncertainties can be rewritten as

𝜽 𝑰 = ൛𝜃 𝑖 𝐼 ൟ = ሼ𝜃 𝐶 + ∆𝜃 𝐼 ሽ 𝑖 , 𝑖 = 1,2, … 𝑁 (7)
where 𝜃 𝑖 𝐼 = 𝜃 𝑖 𝑐 + ∆𝜃 𝑖 𝐼 = 𝜃 𝑖 𝑐 + ∆𝜃 𝑖 • 𝜀 𝑖 𝐼 , and

𝜀 𝑖 𝐼 =[-1,1].
According to the perturbation method, the interval output 𝐹൫𝜽 𝑰 ൯ can be given in a series mode as

𝑭൫𝜽 𝑰 ൯ = 𝑭൫𝜀 𝑖 𝐼 ൯ = 𝑓 0 + 𝜀 𝑖 𝐼 𝑓 1 + ሺ𝜀 𝑖 𝐼 ሻ 2 𝑓 1 +…+ሺ𝜀 𝑖 𝐼 ሻ 𝑚 𝑓 𝑚 , 𝑖 = 1,2, … 𝑁 (8)
where 𝑚 is the truncation order of the series. Based on the Taylor series expansion expanded at the middle point of the interval vector 𝜽 𝑰 , Eq. ( 8) can be transformed as

𝑭൫𝜽 𝑰 ൯ = 𝑭൫𝜽 𝑪 ൯ + σ 𝜕𝑭ሺ𝜽ሻ 𝜕𝜃 𝑝 1 𝑁 𝑝 1 =1 ฬ 𝜃 𝑖 𝐼 =𝜃 𝐶 ,𝑖≠𝑝 1 • ∆𝜃 𝑝 1 + 1 2 σ σ 𝜕𝑭 2 ሺ𝜽ሻ 𝜕𝜃 𝑝 1 𝜕𝜃 𝑝 2 𝑁 𝑝 2 =1 𝑁 𝑝 1 =1 ฬ 𝜃 𝑖 𝐼 =𝜃 𝐶 ,𝑖≠𝑝 1 ,𝑝 2 • ∆𝜃 𝑝 1 ∆𝜃 𝑝 2 + ⋯ + 1 𝑚 σ … 𝑁 𝑝 1 =1 σ 𝜕𝑭 𝑚 ሺ𝜽ሻ 𝜕𝜃 𝑝 1 …𝜕𝜃 𝑝 𝑚 𝑁 𝑝 𝑚 =1 ฬ 𝜃 𝑖 𝐼 =𝜃 𝐶 ,𝑖≠𝑝 1 ,…,𝑝 𝑚 • ∆𝜃 𝑝 1 … ∆𝜃 𝑝 𝑚 + 𝑅 ሺ9ሻ Since the values of ∆𝜃 𝑝 1 • ∆𝜃 𝑝 2 , … , ∆𝜃 𝑝 1 • … • ∆𝜃 𝑝 𝑚 in
more than second-order form is very small, Eq.( 9) can be approximated to a first-order form based on the interval algorithm, which is given as

𝑭൫𝜽 𝑰 ൯ ≅ 𝑭 ൫𝜽 𝑰 ൯ = 𝑭൫𝜽 𝑪 ൯ + σ 𝜕𝑭ሺ𝜽ሻ 𝜕𝜃 𝑗 𝑁 𝑗=1 ฬ 𝜃 𝑖 𝐼 =𝜃 𝐶 ,𝑖≠𝑗 • ∆𝜃 𝑗 , 𝑗 = 1,2, … , 𝑁 (10)
Then, we can obtain the following equations that

𝑭 ൫𝜽 𝑰 ൯ = σ ൫𝜃 𝑗 𝑐 + ∆𝜃 𝑗 𝐼 ൯𝐹 𝑖 𝑛 𝑗=1 = 𝑭൫𝜽 𝑪 ൯ + σ 𝐹 𝑖 𝑛 𝑗=1 • ∆𝜃 𝑗 𝐼 (11)
where

𝐹 𝑖 = σ 𝜕𝑭ሺ𝜽ሻ 𝜕𝜃 𝑗 𝑁 𝑗=1 ฬ 𝜃 𝑖 𝐼 =𝜃 𝐶 ,𝑖≠𝑗
.

From Eq.( 12), we can find that once we have the uncertain part of σ 𝐹 𝑖 𝑛 𝑗=1

• ∆𝜃 𝑗 𝐼 , we can calculate the approximate bounds of the model output 𝒇 𝑰 = 𝑭൫𝜽 𝑰 ൯. The differential method is introduced to calculate the lower and upper bounds of the model output 𝒇 𝑰 , which is shown as

ە ۖ ۔ ۖ ۓ 𝒇 = 𝐹ሺ𝜽 𝒄 ሻ + σ 𝑭ቀ𝜃 𝑗 𝑐 +𝛿𝜃 𝑗 ቁ-𝑭ቀ𝜃 𝑗 𝑐 ቁ 𝛿𝜃 𝑗 𝑁 𝑗=1 𝛥𝜃 𝑗 𝒇 = 𝐹ሺ𝜽 𝒄 ሻ -σ 𝑭ቀ𝜃 𝑗 𝑐 +𝛿𝜃 𝑗 ቁ-𝑭ቀ𝜃 𝑗 𝑐 ቁ 𝛿𝜃 𝑗 𝑁 𝑗=1 𝛥𝜃 𝑗 (12)
where 𝛿𝜃 𝑗 is the minor variable of the interval variable 𝜃 𝑗 .

4 Interval sensitivity analysis with Interval Similarity Operator

Interval Similarity Operator

In the context of local sensitivity analysis, for example, when the parameters are changed from 𝜽 𝑰 = ሼ𝜃 1 𝐼 , 𝜃 2 𝐼 , … , 𝜃 𝑛 𝐼 ሽ to 𝜽 𝑰 = ൛𝜃 1 𝐼 , 𝜃 2 𝐼 , … , 𝜃 𝑛 𝐼 ൟ , namely 𝜃 1 𝐼 becomes 𝜃 1 𝐼 , the outputs are accordingly changed from the initial value of 𝒇 𝑰 = ൛𝑓 𝑗 𝐼 ൟ to the perturbed value of 𝒇 𝑰 = ൛𝑓 መ 𝑗 𝐼 ൟ.

It should be noted that although only one input parameter interval changes, all the output intervals are changed simultaneously. [START_REF] Zhao | The sub-interval similarity: A general uncertainty quantification metric for both stochastic and interval model updating[END_REF] proposed an uncertainty quantification metric of interval similarity operator (ISO) to address the issue of structural model updating.

In this work, this metric is introduced to propose a novel sensitivity index to quantify the discrepancy between one-dimensional intervals 𝒇 𝑰 and 𝒇 𝑰 , reflecting the sensitivity of each input interval parameter.

Firstly, the Interval Relative Position Operator (IRPO) is utilized to measure the difference between two intervals based on the mathematical rule of interval length 𝐿ሺ•ሻ, defined as follows: The IRPO is calculated according to different overlap cases in Fig. 3, and its calculation rules are given as

𝐿ሺ𝑓 𝐼 ሻ = 𝑓 -𝑓 (
𝐼𝑅𝑃𝑂൫𝑓 𝐼 , 𝑓 መ 𝐼 ൯ = ە ۖ ۖ ۖ ۔ ۖ ۖ ۖ ۓ ቀ𝑓-𝑓 መ ቁ 𝑚𝑎𝑥൛𝐿ሺ𝑓 𝐼 ሻ,𝐿൫𝑓 መ𝐼 ൯ൟ 𝐶𝑎𝑠𝑒 1,2 ቀ𝑓-𝑓 ቁ 𝑚𝑎𝑥൛𝐿ሺ𝑓 𝐼 ሻ,𝐿൫𝑓 መ𝐼 ൯ൟ 𝐶𝑎𝑠𝑒 3 ቀ𝑓 መ -𝑓 መ ቁ 𝑚𝑎𝑥൛𝐿ሺ𝑓 𝐼 ሻ,𝐿൫𝑓 መ𝐼 ൯ൟ 𝐶𝑎𝑠𝑒 4 ቀ𝑓 መ -𝑓 ቁ 𝑚𝑎𝑥൛𝐿ሺ𝑓 𝐼 ሻ,𝐿൫𝑓 መ𝐼 ൯ൟ 𝐶𝑎𝑠𝑒 5,6 (16)
where m𝑎𝑥൛𝐿ሺ𝑓 𝐼 ሻ, 𝐿൫𝑓 መ 𝐼 ൯ൟ represents the maximum interval length between 𝑓 𝐼 and 𝑓 መ 𝐼 .

For the cases 1 and 6, 𝑓 𝐼 and 𝑓 መ 𝐼 have an overlapping space, and then the IRPO is negative.

When the length of intervals 𝑓 𝐼 and 𝑓 መ 𝐼 are infinite, the denominator of IRPO tends to zero.

In cases 2-5, there is an overlap between 𝑓 𝐼 and 𝑓 መ 𝐼 ,and the value of the IRPO is clearly positive and restrained to the range of (0,1). If both the position and the length of 𝑓 𝐼 is consistent with that of 𝑓 መ 𝐼 , 𝐼𝑅𝑃𝑂 achieves its maximum value of 1. Hence the range of 𝐼𝑅𝑃𝑂 is

𝐼𝑅𝑃𝑂൫𝑓 𝐼 , 𝑓 መ 𝐼 ൯ ∈ ሺ-∞, 1ሿ (17)
Next, impose that the IRPO has a high gradient as the value moves close to one; we develop an Interval Sensitivity Operator based on the IRPO to quantify the similarity between two interval vectors concerning their geometric position and shape. The fundamental calculation rule of ISO is given by

𝐼𝑆𝑂൫𝒇 𝑰 , 𝒇 𝑰 ൯ = 𝑚𝑒𝑎𝑛 ቆ1 - 1 1+𝑒𝑥𝑝ሼ-𝐼𝑅𝑃𝑂ቀ𝑓 𝑗 𝐼 ,𝑓 መ 𝑗 𝐼 ቁቅ ቇ , 𝑗 = 1,2, … , 𝑚 (18)
where 𝑚𝑒𝑎𝑛ሺ•ሻ represents the mean value of ሺ•ሻ.

From Eq. ( 18), we can find that the value of ISO is limited within ሺ0, +∞ሻ. A high gradient of Eq. ( 18) reflects the similarity between 𝒇 𝑰 and 𝒇 𝑰 . On the contrary, when 𝐼𝑆𝑂 moves to the infinite positive, this means 𝒇 𝑰 is significantly different from 𝒇 𝑰 . In the context of sensitivity analysis, this implies that variations in parameter 𝜃 𝑖 𝐼 have a strong influence on the outputs 𝒇 𝑰 .

According to Eq. ( 4), the proposed interval sensitivity index 𝑺 based on ISO can be expressed as Eqs. ( 19)-( 20). This index vector consists of a series of sensitive index variables ሼ𝑆 𝑖 , 𝑖 = 1, … , 𝑛ሽ, and each variable 𝑆 𝑖 quantifies the sensitivity of parameter 𝜃 𝑖 𝐼 .

𝑺 = ሼ𝑆 𝑖 ሽ, 𝑖 = 1, … , 𝑛 (19)
𝑆 𝑖 = ∆𝒇 𝑰 ∆𝜃 𝐼 = 𝐼𝑆𝑂൫𝒇 𝑰 ,𝒇 𝑰 ൯ ∆𝜃 𝑖 𝐼 (20)
We can rank the sensitivity index 𝑆 𝑖 in descending order to select the sensitive parameters.

It should be noted that this proposed interval sensitivity index is calculated based on the boundaries of model outputs without any inner data points, which is especially appropriate for the model with pure interval uncertainties. Meanwhile, when there are stochastic uncertainties or hybrid stochastic and interval uncertainties in the model since the geometric position and range of model outputs can be measured through ISO, this index can be adopted for models with complicated uncertainties as well.

Framework of sensitivity analysis with interval uncertainties

This work belongs to the One-at-a-Time method (OAAT), so the problem of coupling between model parameters is not considered here. Besides, in contrast to the sensitivity analysis with stochastic uncertainties, this interval sensitivity analysis is performed with no hypothesis of probabilistic distributions. The objective is to quantify the variation of interval outputs 𝒇 𝑰 → 𝒇 𝑰 caused by the change in the interval parameters 𝜃 𝑖 𝐼 → 𝜃 𝑖 𝐼 , such that the sequence of 𝑆 𝑖 represents the sensitivity of inputs 𝜽 𝑰 .

The framework of the proposed sensitivity analysis method is illustrated in Fig. 4, which consists of three parts: (1) the major body, (2) the interval uncertainty propagation, and (3) the sensitivity index calculation. As mentioned above, the interval perturbation method is introduced to estimate model output intervals effectively. Then, the sensitivity index 𝑆 𝑖 corresponding to each parameter 𝜃 𝑖 𝐼 is accordingly calculated to measure the discrepancy between the initial interval output 𝒇 𝑰 and the perturbated output 𝒇 𝑰 . The major body contains a pre-determined initial and perturbed range of parameters, namely 𝜃 𝑖 𝐼 and 𝜃 𝑖 𝐼 . Finally, we rank the sensitivity index 𝑆 𝑖 in descending order. The detailed steps are illustrated as follows:

Step 1: The framework starts from a pre-determined range of the model parameters 𝜽 𝑰 = ሼ𝜃 1 𝐼 , … , 𝜃 𝑛 𝐼 ሽ with its initial interval center𝜽 𝑪 = ሼ𝜃 𝑖 𝑐 ሽ and corresponding interval radius ∆𝜽 𝑰 = ൛∆𝜃 𝑖 𝐼 ൟ. The initial interval 𝜃 𝐼 represents the gross knowledge from engineering judgments.

Step 2: Select an interval parameter component 𝜃 𝑖 𝐼 and change its interval radius proportionally to model the small change of 𝜃 𝑖 𝐼 . Keep other interval parameter components unchanged. We finally obtain the interval perturbed parameter 𝜽 𝑰 = ൛𝜃 1 𝐼 , … , 𝜃 𝑖 𝐼 , … , 𝜃 𝑛 𝐼 ൟ.

Step 3: Estimate the initial model output interval 𝒇 𝑰 = ሼ𝑓 1 𝐼 , … , 𝑓 𝑚 𝐼 ሽ and the perturbed output interval 𝒇 𝑰 = ൛𝑓 መ 1 𝐼 , … , 𝑓 መ 𝑚 𝐼 ൟ through the interval perturbation method, which concerns the initial interval parameter 𝜽 𝑰 = ሼ𝜃 1 𝐼 , … , 𝜃 𝑛 𝐼 ሽ and the perturbed interval parameter 𝜽 𝑰 = ൛𝜃 1 𝐼 , 𝜃 2 𝐼 , … , 𝜃 𝑛 𝐼 ൟ, respectively.

Step 4: Calculate the IRPO and the ISO of 𝒇 𝑰 and 𝒇 𝑰 , and calculate the sensitivity value 𝑆 መ 1 corresponding to 𝜃 1 𝐼 .

Step 5: Repeat Step 2-Step 4 n times to obtain a series of sensitivity indexes 𝑆 መ 𝑖 , 𝑖 = 1,2, … , 𝑛 corresponding to interval parameters ሼ𝜃 1 𝐼 , … , 𝜃 𝑛 𝐼 ሽ.

Step 6: Sequence the value of 𝑆 መ 𝑖 , 𝑖 = 1,2, … , 𝑛 in descending order to present the sensitivity of interval parameters ሼ𝜃 1 𝐼 , … , 𝜃 𝑛 𝐼 ሽ.

Fig. 4 Flowchart of sensitivity analysis for interval parameters.

Case studies

6.1 Case 1: Ishigami function

Problem description

A tutorial case study of the Ishigami function is presented in this section, which originated from Ref. [START_REF] Ishigami | An importance quantification technique in uncertainty analysis for computer models[END_REF]) and is analyzed by Ref. [START_REF] Bi | The Bhattacharyya distance: Enriching the P-box in stochastic sensitivity analysis[END_REF]). The Ishigami function is a general example of sensitivity analysis, which is given as follows:

𝑦ሺ𝑃ሻ = sinሺ𝑝 1 ሻ + 𝑎 sinሺ𝑝 2 ሻ 2 + 𝑏𝑝 3 4 sinሺ𝑝 1 ሻ (21)
where 𝑃 = ሼ𝑝 1 ,𝑝 2 ,𝑝 3 ሽ is the input parameters; 𝑦 is the output feature; a and b are constant coefficients with pre-determined values as Ref. [START_REF] Marrel | Calculations of Sobol indices for the Gaussian process metamodel[END_REF]), a=7 and b=0.1.

Since [START_REF] Bi | The Bhattacharyya distance: Enriching the P-box in stochastic sensitivity analysis[END_REF] mainly focus on the stochastic sensitivity analysis with both aleatory and epistemic uncertainties, the uncertainties 𝑝 1-3 in Ref. [START_REF] Bi | The Bhattacharyya distance: Enriching the P-box in stochastic sensitivity analysis[END_REF]) assumed that 𝑝 1

and 𝑝 2 are prescribed to follow the uniform distribution, and 𝑝 3 follows the Gaussian distribution. However, this uncertainty characteristic is inappropriate for the application of the proposed method because the determined distribution types of 𝑝 1-3 belong to probabilistic uncertainties without any interval uncertainties.

In this case, 𝑝 1-3 are defined as interval parameters, and according to Eq. ( 4), we assume that the lengths of interval radius ∆𝑷 are changed 50% as presented in Table 1. For example, the initial length of ∆𝑝 1 is 0.8, which is changed to 0.4 in the perturbed interval 𝑝Ƹ 1 𝐼 , namely the interval of 𝑝 1 𝐼 is changed from [-2.4,-0.8] to [-2,-1.2]. Since this work is mainly about local sensitivity analysis, the other variables 𝑝 2 𝐼 and 𝑝 3 𝐼 of 𝑷 𝑰 are unchanged. By comparing the variation of the outputs caused by the initial and perturbed intervals of parameters, the sensitivity of each input can be identified.

Table 1 The initial and perturbed parameter intervals. Since this work belongs to the problem of local sensitivity analysis, this interval sensitivity analysis aims to quantify the importance of each input according to how many intervals of uncertainty space of all outputs can be changed when the interval length of this initial input is proportionally increased, where the changed inputs are called perturbed inputs here. For this case, the perturbed interval parameters 𝑝 1-3 are presented in Table 1. The objective of proportionally changing the interval radius ∆𝑝 𝐼 = {∆𝑝 1 𝐼 , ∆𝑝 2 𝐼 , ∆𝑝 3 𝐼 } to generate perturbed inputs with a radius of ∆𝑝 𝐼 = {∆𝑝 1 𝐼 , ∆𝑝 2 𝐼 , ∆𝑝 3 𝐼 } is illustrated in Figure 5, where the interval centers are completely fixed. The most intuitive manner is to measure how much the output interval space is changed to reflect the sensitivity of parameters. Hence the following operator to calculate the interval outputs according to 𝑝 𝐼 and 𝑝Ƹ 𝐼 . As the Ishigami function is simple, 5000 Monte Carlo simulations are conducted to estimate the initial interval outputs 𝑦 𝐼 and perturbed output 𝑦 ො 𝐼 .

Parameters

Fig. 5 Initial and perturbed input intervals.

Table 2 presents the initial output interval 𝑦 𝐼 calculated based on the initial parameter intervals 𝑝 1 𝐼 , 𝑝 2 𝐼 , and 𝑝 3 𝐼 . In the context of local sensitivity analysis, there are three perturbed interval output spaces corresponding to variations sequentially occurring in 𝑝 1 𝐼 , 𝑝 2 𝐼 , and 𝑝 3 𝐼 .

The three perturbed interval outputs are given in Table 2. The output variability significantly reflects the degree of dispersion in the output uncertainty space, which is investigated in Figure 6. For example, the perturbed procedure is executed for an interval of 𝑝Ƹ 1 𝐼 meanwhile, keeping the interval bounds of 𝑝 2 𝐼 , and 𝑝 3 𝐼 . Accordingly, the perturbed output 𝑦 ො 𝑝1 𝐼 is available. From Figure 6, it can be seen that the perturbation in 𝑝 3 𝐼 results in the most obvious changes in output, i.e. 𝑦 𝐼 to 𝑦 ො 𝑝3 𝐼 , implying the 𝑝 3 𝐼 is sensitive to the outputs 𝑦 𝐼 . ( [START_REF] Bi | The Bhattacharyya distance: Enriching the P-box in stochastic sensitivity analysis[END_REF]) can be adopted in this case, whose results are similar to those calculated by the proposed indices as shown in Figure 7.

Parameters with hybrid probabilistic and interval uncertainties

To assess the effectiveness of the proposed sensitivity analysis method, a published work on this Ishigami function, namely Ref. [START_REF] Bi | The Bhattacharyya distance: Enriching the P-box in stochastic sensitivity analysis[END_REF]) is utilized herein as a reference to compare with the results of the current work. In Ref. [START_REF] Bi | The Bhattacharyya distance: Enriching the P-box in stochastic sensitivity analysis[END_REF], the parameters of the Ishigami function contain both hybrid probabilistic and interval uncertainties, which are expressed by the P-box technique. More complex uncertainty characteristics of the parameters are assigned as presented in Table 4, and correspondingly, the outputs 𝑦 𝐼𝑅 are with hybrid stochastic and interval uncertainties. The P-box of an imprecise uniform distribution can be easily determined, i.e., the P-box of 𝑝 1 is enveloped by the CDFs of 𝑝 1 ~Uሺ-4, 2ሻ and 𝑝 1 ~Uሺ-3, 3ሻ as shown in Figure 9. The uncertainty of 𝑝 1 is controlled by the coefficients of 𝑎 1 and 𝑏 1 in 错误!未 找到引用源。, and the P-boxes of 𝑝 2-3 can be determined by the uncertain coefficients of 𝑎 2 , 𝑏 2 , 𝜇 3 , and 𝜎 3 . A two-level procedure is proposed in Ref. [START_REF] Bi | The Bhattacharyya distance: Enriching the P-box in stochastic sensitivity analysis[END_REF]) to calculate the P-boxes of outputs according to the input P-boxes. The Bhattacharyya distance is utilized to measure the discrepancy between the bounded CDF of the output P-box. For comparison, the proposed ISO is adopted to replace the Bhattacharyya distance for sensitivity analysis Fig. 9 The P-box of 𝑝 1-3 in Ref. [START_REF] Bi | The Bhattacharyya distance: Enriching the P-box in stochastic sensitivity analysis[END_REF].

Table 4 Uncertainty characteristics of 𝑝 1-3 in Ref. [START_REF] Bi | The Bhattacharyya distance: Enriching the P-box in stochastic sensitivity analysis[END_REF]).

Parameters

Ref. [START_REF] Ishigami | An importance quantification technique in uncertainty analysis for computer models[END_REF] Parameter probabilistic distribution Uncertain coefficient

𝑝 1 𝑝 1 ~𝑈ሺ𝑎 1 𝐼 , 𝑏 1 ሻ 𝑎 1 𝐼 ∈ ሾ-4.0, -3.0ሿ; 𝑏 1 𝐼 ∈ ሾ2.0,3.0ሿ 𝑝 2 𝑝 2 ~𝑈ሺ𝑎 2 𝐼 , 𝑏 2 ሻ 𝑎 2 𝐼 ∈ ሾ-3.0, -1.0ሿ; 𝑏 2 𝐼 ∈ ሾ3.0,5.0ሿ 𝑝 3 𝑝 3 ~𝑈ሺ𝜇 3 𝐼 , 𝜎 3 2 ሻ 𝜇 3 𝐼 ∈ ሾ0.0,1.0ሿ; 𝜎 3 𝐼 ∈ ሾξ5, ξ2ሿ
To calculate the sensitivity of the parameter 𝑝 1 , ten levels of interval 𝑎 1 and 𝑏 1 are investigated by assigning ten equidistant values within the intervals. The full factorial design results in 100 configurations of 𝑎 1 and 𝑏 1 , and correspondingly 100 perturbated P-boxes of 𝑝 1 are obtained. Then 100 groups of perturbated P-boxes of outputs are simulated according to 100 groups of an interval variable ሼ𝑎 1 𝐼 , 𝑏 1 𝐼 ሽ 𝑗 ,j=1,..,100, respectively. To compare the uncertain space to give an explicit sensitivity ranking of 𝑝 1-3 , the metrics of Bhattacharyya distance and of the 100 perturbated P-boxes according to 𝑝 1-3 in the form of CDFs, and Figure 11 gives specific data points of perturbated outputs. The sensitivity rank of parameters is evaluated based on the difference between the perturbed and original outputs concerning 𝑝 1-3 . Figure 10 illustrates the sensitivity rank rule of Ref. [START_REF] Bi | The Bhattacharyya distance: Enriching the P-box in stochastic sensitivity analysis[END_REF], where the reduced uncertainty space of output is measured from the view of CDF. Figure 11 investigates the variation between the original and perturbed output uncertainty space by applying the interval concept. It is difficult to rank the parameter sensitivity with manual observation, but it is easy to directly tell the sensitivity rank through an interval observation mode, as shown in Figs 10-11. Since those uncertain data points are quantified by different uncertainty quantification tools, the results of sensitivity rank are inconsistent, as shown in Table 5 and Fig. 12. From the point of view of the interval concept, the uncertain output space of 𝑝 3 is changed most obviously compared with the original one, where the subjective judgment consists of the objective calculation results calculated by ISO. The ranks calculated by ISO differ from that of Ref. [START_REF] Bi | The Bhattacharyya distance: Enriching the P-box in stochastic sensitivity analysis[END_REF]. This is because the interval is mainly affected by the extreme data points, while the CDF is calculated according to the dispersion of the datapoints. The sensitivity rank computed by interval uncertainty quantification of ISO is credible. This is because we mainly focus on the low-probability tail risk, which occurs when the model outputs are extremely large or small in engineering practice. It can be found that the change in data from the probabilistic point of view cannot imply a significant change in data from the interval point of view.

1 𝑆 መ 𝑝 3 𝑆 መ 𝑝 2 2 𝑆 መ 𝑝 1 𝑆 መ 𝑝 1 3 𝑆 መ 𝑝 2 𝑆 መ 𝑝 3

Case 2: Satellite FE model

This case of a satellite model is derived from Ref. [START_REF] Zhang | A frequency response model updating method based on unidirectional convolutional neural network[END_REF], which is utilized for model calibration analysis. Here, this model is analyzed to demonstrate the performance of the ISO metric within the proposed sensitivity analysis. The FE model of the satellite is presented in Figure 13, and this model consists of the upper platform, the shear platform, the central panels, and the lower platform.

Fig. 13 Finite element model of the satellite.

Interval uncertainties propagation

In this satellite FE model, the Elastic modulus of the FE model 𝜃 1 is 7.0×10 10 pa, the density 𝜃 2 is 2.7×10 3 kg•m -3 , and the thickness of the lower platform 𝜃 4 is 1mm. Some parameters of this FE model are assumed to be interval as given in Table 6. The first two eigenfrequency intervals 𝑓 1 𝐼 and 𝑓 2 𝐼 are regarded as the model output features. 7. It can be found that the bounds calculated by the perturbated method are consistent with that of MC simulation, demonstrating the accuracy of the interval perturbation method. Besides, the calculation time of MC simulations is 85960s, while that of the interval perturbation method is about 240s, as shown in Fig. 15. This illustrates the effectiveness of interval perturbation method when coping with the problem of interval propagation. From Figs. 17-18, we can find that the variations between the initial and perturbed outputs space of 𝜃 5 𝐼 are the largest, which reveals the impact of the model input interval parameter 𝜃 5 𝐼 , namely the thickness of the shear platform on the model output intervals 𝑓 1 𝐼 and 𝑓 2 𝐼 is significant. On the contrary, 𝜃 6 𝐼 is the least important parameter. The last column of Table 9 presents the ranking results of the current work, namely 𝜃 5 > 𝜃 3 > 𝜃 6 , according to the proposed sensitivity index, which is visualized in Fig. 19. The sensitivity ranks imply that the interval uncertainty of shear platform thickness 𝜃 5 significantly impacts the primary model natural frequencies of the satellite model. The proposed sensitivity analysis is performed for six parameters 𝜃 1-6 𝐼 , and the initial and perturbed intervals of parameters are presented in Table 10. We can find that as the variation proportion ratio of the interval radius increases from 10%

to 60%, the sensitivity ranks of parameters experience slight changes. Of notable significance is parameter 𝜃 4 𝐼 , which represents the thickness of the lower platform, as it consistently maintains the first order rank in influencing output features, regardless of the variation of the parameter interval radius. Conversely, uncertainties surrounding parameter 𝜃 6 𝐼 , which denotes the thickness of the upper platform, have minimal impact on output uncertainty and rank towards the end of the sensitivity order. Henc, it is recommended to pay more attention to the thickness of the lower platform, and it may be more beneficial to focus on adjusting and optimizing parameter 𝜃 4 𝐼 in order to achieve desired output results.

Conclusion

An exhaustive interval sensitivity analysis method based on the interval perturbation method and interval similarity operator is developed. In this interval sensitivity analysis framework, the ISO metric is adopted to quantify the discrepancy between two intervals based on the interval geometric position and the interval bounds without requiring inner interval samples.

This metric can rank different sensitivity analysis frameworks, e.g. interval sensitivity analysis and sensitivity analysis with hybrid stochastic and interval uncertainties, which is illustrated by

  depicts the relationships among interval propagation, sensitivity analysis, and model updating, also called model calibration. The interval uncertainty propagation approach is investigated to determine the bounds of model outputs. The relative significance of interval parameters is assessed using sensitivity analysis. The model updating for sensitive inputs is applied to improve the accuracy of the simulation model. Interval propagation is regarded as a critical operator for both the sensitivity analysis and the model updating. The interval perturbation method (Zhao et al. 2018; Zhao et al. 2020), a typical interval propagation method, is investigated in this work.

Fig. 2

 2 Fig.2The relationship between the sensitivity analysis, the interval uncertainty propagation, and the model calibration.

  15) Two intervals 𝑓 𝐼 = ሾ𝑓, 𝑓ሿ and 𝑓 መ 𝐼 = ሾ𝑓 መ , 𝑓 መ ሿ are utilized to explain the proposed Interval Similarity Operator. Six typical positional relationships between intervals 𝑓 𝐼 and 𝑓 መ 𝐼 are presented in Figure 3.

Fig. 3

 3 Fig. 3 Six interval relative positions.

Fig. 6

 6 Fig. 6 Comparison between the initial and perturbated outputs intervals. The IRPO, ISO, and comprehensive sensitivity index of 𝑝 1 𝐼 are evaluated according to Section 3. The same strategy is performed for 𝑝 2 𝐼 , and 𝑝 3 𝐼 , and their corresponding sensitivity indices are presented in Table 3. According to the determined input parameters, 5000 times Monte Carlo samples with the assumption of parameters following uniform distribution are performed to propagate the uncertainty from the inputs to the outputs, yielding 5000 data points bounded by the output intervals. Hence the sensitivity indices of Bhattacharyya distance in Ref.

Fig. 7

 7 Fig. 7 Sensitivity analysis of 𝑝 1 𝐼 , 𝑝 2 𝐼 , and 𝑝 3 𝐼 The sensitivity ranking is 𝑆 መ 𝑝 3 > 𝑆 መ 𝑝 1 > 𝑆 መ 𝑝 2 , and 𝑆 መ 𝑝 1 and 𝑆 መ 𝑝 2 are significantly smaller than 𝑆 መ 𝑝 3 , indicating that 𝑆 መ 𝑝 3 is more sensitive than 𝑆 መ 𝑝 1 and 𝑆 መ 𝑝 2 from the point of view of interval uncertainty. Meanwhile, 𝐼𝑆𝑂 𝑝1 and 𝐼𝑆𝑂 𝑝2 are close to 0, implying a high geometric similarity between intervals 𝑦 𝐼 and 𝑦 ො 𝐼 . It should be noted that the Bhattacharyya distance is calculated based on the distribution function of data, implying it is unable to deal with interval uncertainties. The proposed ISO is calculated only by the bounds of output intervals, which is reliable in engineering. Besides, the calculation time of the ISO is much less than that of the Bhattacharyya distance, which is proved by Fig. 8.

Fig. 10

 10 Fig. 10 Reduced output stochastic space when epistemic uncertainties of input parameters are reduced compared with the original P-box, which is calculated according to Ref. (Bi et al. 2019).

Fig. 12

 12 Fig. 12 Sensitivity analysis comparison with respect to the metrics of ISO and Bhattacharyya distance.

Fig. 14

 14 Fig. 14 Interval of 𝑓 1 𝐼 and 𝑓 2 𝐼 calculated by MCS and the interval perturbation method.

Fig. 16

 16 Fig. 16 Comparison between initial and perturbed input intervals.

Fig. 17

 17 Fig. 17 Initial and perturbated output intervals of 𝑓 1 𝐼

Fig. 19

 19 Fig. 19 Sensitivity order of 𝜃 3 𝐼 , 𝜃 5 𝐼 , and 𝜃 6 𝐼

Fig. 21

 21 Fig. 21 The variation of 𝑓 1 𝐼 to 𝑓 8 𝐼 results from the changes of 𝜃 4 𝐼

Fig. 23

 23 Fig. 23 Sensitivity ranks of 𝜃 1 𝐼 -𝜃 6 𝐼 correpongding to different variation proportion ratio of interval radius.

  the tutorial case of the Ishigami function. The interval perturbation method is introduced for interval uncertainty propagation, which has the advantage of not requiring abundant FE simulation to estimate precise extreme bounds of interval outputs. It is a significant benefit for sensitivity analysis in the presence of practical engineering structures. The feasibility and effectiveness of this proposed interval sensitivity analysis algorithm are verified by two numerical examples of the classical Ishigami function and the satellite example. Further development for interval analysis includes the applications for nonlinearity systems, the consideration of the inner relationship between multi-dimensional outputs, and the hybrid stochastic and interval uncertainties propagation.

Table 2

 2 Initial and perturbed input intervals.

	Initial parameters	Initial output 𝑦 𝐼	Perturbed parameter orders	Perturbed output 𝑦 ො 𝐼
			𝑝Ƹ 1 𝐼 , 𝑝 2 𝐼 , 𝑝 3 𝐼	𝑦 ො 𝑝1 𝐼 =[-113.07,-27.78]
	𝑝 1 𝐼 , 𝑝 2 𝐼 , 𝑝 3 𝐼	𝑦 𝐼 =[-111.55,-21.52]	𝑝 1 𝐼 , 𝑝Ƹ 2 𝐼 , 𝑝 3 𝐼	𝑦 ො 𝑝2 𝐼 =[-111.82,-22.04]
			𝑝 1 𝐼 , 𝑝 2 𝐼 , 𝑝Ƹ 3 𝐼	𝑦 ො 𝑝3 𝐼 =[-85.30,-29.65]

Table 3

 3 Sensitivity analysis of 𝑝 1-3 .

	Sensitivity index	𝑆 መ 𝑝 1	𝑆 መ 𝑝 2	𝑆 መ 𝑝 3	Sensitivity rank
	Bhattacharyya distance Ref. (Bi et al. 2019)	0.02525	0.0055	0.228	𝑆 መ 𝑝 3 > 𝑆 መ 𝑝 1 > 𝑆 መ 𝑝 2
	ISO	0.03475	0.003	0.20325	𝑆 መ 𝑝 3 > 𝑆 መ 𝑝 1 > 𝑆 መ 𝑝 2

Table 5

 5 Uncertainty characteristics of 𝑝 1-3 .

	Rank	Results according to different indices 𝑆 መ 𝐼𝑆𝑂 𝑆 መ

𝐵𝐷 in Ref.

[START_REF] Bi | The Bhattacharyya distance: Enriching the P-box in stochastic sensitivity analysis[END_REF]

Table 6

 6 Satellite parameter table

		Parameters	Interval centers Interval radius
	𝜃 3 𝐼	The thickness of the central panel	2 (mm)	0.2 (mm)
	𝜃 5 𝐼	The thickness of the shear platform	3 (mm)	0.2 (mm)
	𝜃 6 𝐼 The thickness of the upper platform	2 (mm)	0.2 (mm)
	Since it is time-consuming to calculate the FE model of the satellite structure, the interval
	perturbation method is adopted to estimate the interval outputs efficiently. 10000 Monte Carlo
	simulations are conducted to estimate 𝑓 1 𝐼 and 𝑓 2 𝐼 . The results of the two methods are presented
	in Fig. 14 and Table

Table 7

 7 Interval perturbation method Fig. 15 Comparison of calculation time between MCS and the interval perturbation method.Figure 16 intuitively presents the changed interval inputs, andTable 8 lists the initial parameter interval 𝜃 𝐼 and the perturbed parameter interval 𝜃 𝐼 .

	Parameters	MC simulation	Interval perturbation method	Relative error
	𝑓 1 𝐼	[ 18.33,20.94]	[18.29, 21.04]	[ 0.21%, 0.47%]
	𝑓 2 𝐼	[ 18.78, 21.6]	[18.74, 21.7]	[0.21%, 0.46% ]

𝐼 }, where the interval centers are completely fixed.

Table 8

 8 The initial and perturbed intervals of 𝜃 3 𝐼 , 𝜃 5 𝐼 and 𝜃 6

	𝐼

Table 9

 9

		Sensitivity analysis for 𝜃 3 𝐼 , 𝜃 5 𝐼 and 𝜃 6 𝐼
	Parameters	Perturbated 𝑓 መ 𝐼	Initial 𝑓 𝐼	𝑆 መ 𝜃 𝑖	Sensitivity rank
	𝜃 3 𝐼 , 𝜃 5 𝐼 , 𝜃 6 𝐼	𝑓 መ 1 𝐼 = ሾ18.50,20.83ሿ 𝑓 መ 2 𝐼 = ሾ18.97,21.47ሿ		3.004
	𝜃 3 𝐼 , 𝜃 5 𝐼 , 𝜃 6 𝐼	𝑓 መ 1 𝐼 = ሾ18.71,20.62ሿ 𝑓 መ 2 𝐼 = ሾ19.19,21.26ሿ	𝑓 1 𝐼 =ሾ18.29,21.04ሿ 𝑓 2 𝐼 =ሾ18.74,21.70ሿ	3.326	𝑆 መ 𝜃 5 > 𝑆 መ 𝜃 3 > 𝑆 መ 𝜃 6
	𝜃 3 𝐼 , 𝜃 5 𝐼 , 𝜃 6 𝐼	𝑓 መ 1 𝐼 = ሾ18.35,20.98ሿ 𝑓 መ 2 𝐼 = ሾ18.81,21.64ሿ		2.772

Table 10

 10

		The initial and perturbated intervals of 𝜃 1 𝐼 to 𝜃 6 𝐼
	Parameters	𝜃 𝐼	𝜃 𝐼	Variation proportion ratio of interval radius
	𝜃 1 𝐼	[6.8,7.2]	[6.9,7.1]	50%
	𝜃 2 𝐼	[2.5,2.9]	[2.6,2.8]	50%
	𝜃 3 𝐼	[1.8,2.2]	[1.9,2.1]	50%
	𝜃 4 𝐼	[0.8,0.2]	[0.9,1.1]	50%
	𝜃 5 𝐼	[2.8,3.2]	[2.9,3.1]	50%
	𝜃 6 𝐼	[1.8,2.2]	[1.9,2.1]	50%
	We calculate the sensitivity indexes with respect to each parameter, and finally rank them as
	given in Table 11 and Fig 20. 𝜃 4 𝐼 , the lower platform of the satellite model, is the most sensitive
	parameter among those six interval parameters, and 𝜃 6 𝐼 , the thickness of the upper
	platform is the most not sensitive parameter. Besides, it should be noted that the rank of
	𝜃 3 𝐼 , 𝜃 5 𝐼 , and 𝜃 6

𝐼 in this case is consistent with that in Section 6.2.2, while with the different sensitivity index value. This is because the dimensions of outputs are increased from 3 to 8. For multiple outputs, the sensitivity of parameters is not changed in this Satellite FE model, which illustrates the stability of the proposed sensitivity analysis method.

Table 11

 11 Sensitivity analysis for 𝜃 1-6 𝐼 . 𝑆 መ 𝜃 4 > 𝑆 መ 𝜃 2 > 𝑆 መ 𝜃 5 > 𝑆 መ 𝜃 3 > 𝑆 መ 𝜃 1 > 𝑆 መ 𝜃 6 𝑓 8 𝐼 and 𝑓 መ 8 𝐼 cased by the changes of 𝜃 4 𝐼 are apparent, and the value of ISO between 𝑓 8 𝐼 and 𝑓 መ 8 𝐼 is 0.3568. Meanwhile, the most stable parameter of 𝜃 6 𝐼 , the changes in 𝜃 6 𝐼 lead to slight changes in outputs 𝑓 1 𝐼 to 𝑓 8 𝐼 as shown in Fig. 22. The proposed method not only can order the sensitivity of interval parameters but also can measure the changes of each interval output features.

	Parameters 𝑆 መ 𝜃 𝑖 with ISO	Sensitivity rank
	𝜃 1 𝐼	2.844
	𝜃 2 𝐼	3.033
	𝜃 3 𝐼	2.880
	𝜃 4 𝐼	3.206
	𝜃 5 𝐼	2.951
	𝜃 6 𝐼	2.836

𝐼

Since the proposed method is a local sensitivity analysis method, we can determine the sensitivity of each output feature of 𝑓 1 𝐼 to 𝑓 8 𝐼 to each parameter based on the sensitivity index with ISO. For example, Fig.

21

presents the initial and perturbed intervals of 𝑓 1 𝐼 to 𝑓 8 𝐼 concerning the most sensitive parameter 𝜃 4 𝐼 . We can observe that the variations between the

Table 12

 12 TSensitivity ranks according to different variation proportion ratio of interval radius.𝜃 4 > 𝑆 መ 𝜃 2 > 𝑆 መ 𝜃 5 > 𝑆 መ 𝜃 6 > 𝑆 መ 𝜃 3 > 𝑆 መ 𝜃 1 20% 𝑆 መ 𝜃 4 > 𝑆 መ 𝜃 5 > 𝑆 መ 𝜃 2 > 𝑆 መ 𝜃 1 > 𝑆 መ 𝜃 3 > 𝑆 መ 𝜃 6 30% 𝑆 መ 𝜃 4 > 𝑆 መ 𝜃 2 > 𝑆 መ 𝜃 5 > 𝑆 መ 𝜃 3 > 𝑆 መ 𝜃 1 > 𝑆 መ 𝜃 6 40% 𝑆 መ 𝜃 4 > 𝑆 መ 𝜃 5 > 𝑆 መ 𝜃 2 > 𝑆 መ 𝜃 6 > 𝑆 መ 𝜃 3 > 𝑆 መ 𝜃 1 50% 𝑆 መ 𝜃 4 > 𝑆 መ 𝜃 2 > 𝑆 መ 𝜃 5 > 𝑆 መ 𝜃 3 > 𝑆 መ 𝜃 1 > 𝑆 መ 𝜃 6 60% 𝑆 መ 𝜃 4 > 𝑆 መ 𝜃 5 > 𝑆 መ 𝜃 2 > 𝑆 መ 𝜃 3 > 𝑆 መ 𝜃 1 > 𝑆 መ 𝜃 6

	Variation proportion ratio of interval radius	Sensitivity ranks of 𝜃 1 𝐼 -𝜃 6 𝐼
	10%	𝑆 መ

Acknowledgments: The authors gratefully acknowledge the support of the Postdoctoral Research Foundation of Shunde Innovation School, University of Science and Technology Beijing (2021BH012), the Fundamental Technical Project (JSZL2020203B001), the International Communication Foundation of the University of Science and Technology Beijing (QNXM20220028), the National Natural Science Foundation of China (52005032 and 72271025), and the Guangdong Basic and Applied Basic Research Foundation(2022A1515110276).

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Replication of results:

All modeling parameters are given in the case, and the corresponding finite element model can be obtained according to the case modeling. The presented results are produced using our in-house code surrogate-based optimization and sensitivity analysis. The code and data for producing the presented results will be made available by request. The relevant codes for the algorithms could be available on request by emailing the first author. The authors wish to withhold the source code for commercialization purposes. This includes the finite strain elastoplastic analysis code implementing the finite strain elastoplastic analysis and