Vincent B Éhani

LINEAR DYNAMICS OF AN OPERATOR ASSOCIATED TO THE COLLATZ MAP

Keywords: 2020 Mathematics Subject Classification. 47A16, 47A35, 30H20, 30B10, 15A18 Collatz conjecture, Hypercyclicity, Chaotic operators, Frequent Hypercyclicity, Ergodicity

In this paper, we study the dynamics of an operator T naturally associated to the so-called Collatz map, which maps an integer n ≥ 0 to n/2 if n is even and 3n+1 if n is odd. This operator T is defined on certain weighted Bergman spaces B 2 ω of analytic functions on the unit disk. Building on previous work of Neklyudov, we show that T is hypercyclic on B 2 ω , independently of whether the Collatz Conjecture holds true or not. Under some assumptions on the weight ω, we show that T is actually ergodic with respect to a Gaussian measure with full support, and thus frequently hypercyclic and chaotic.

Introduction

Our aim in this paper is to investigate from the point of view of linear dynamics the properties of an operator naturally associated to the so-called Collatz map. This map T 0 : Z + → Z + is defined as

T 0 (n) = n/2 if n is even; 3n + 1 if n is odd.
The dynamics of this simple-looking map are still very mysterious, and the famous Collatz Conjecture (called also the 3n + 1-Conjecture, or the Syracuse Conjecture) states that the orbit of any integer k ≥ 1 under the action of T 0 eventually reaches the point 1.

We will use in this paper the following modified Collatz map T : Z + → Z + , defined as

T (n) = n/2
if n is even; (3n + 1)/2 if n is odd whose dynamics are the same as those of T 0 as far as the Collatz Conjecture is concerned.

We will here give some of the etablished results about the Collatz Conjecture. We refer the reader to the survey [START_REF] Lagarias | The 3x + 1 problem: an overview, The Ultimate Challenge: The 3x + 1 Problem[END_REF] written by Lagarias for an overview of this problem.

The conjecture had been empirically verified for every k ≤ 20 × 10 9 and is now known to be true for every k ≤ 20 × 2 58 ≈ 5.764 × 10 18 ( [START_REF] Oliveira E Silva | Empirical verification of the 3x+1 and related conjectures, The Ultimate Challenge: The 3x + 1 Problem[END_REF]).

Since T (1) = 2 and T (2) = 1, (1, 2) is called the trivial cycle of T and Eliahou showed in [START_REF] Eliahou | The 3x + 1 problem: new lower bounds on nontrivial cycle lengths[END_REF] that a non-trivial cycle of T must have a length greater than 10 439 860 591.

A possible approach to the Collatz Conjecture is to study the proportion of positive integers k having a T -orbit that reaches 1. Krasikov and Lagarias proved for instance in [START_REF] Krasikov | Bounds for the 3x + 1 problem using difference inequalities[END_REF] that if X is sufficiently large, the number of integers 1 ≤ k ≤ X whose T -orbit reaches 1 is at least X 0.84 .

It is also of interest to investigate the time that it takes to a T -orbit to reach 1. Applegate and Lagarias proved in [START_REF] Applegate | Lower bounds for the total stopping time of 3x + 1 iterates[END_REF] that infinitely many positive integers k have a T -orbit that reaches 1 in at least 6.143 log(k) steps.

Moreover, Tao showed in [START_REF] Tao | Almost all orbits of the Collatz map attain almost bounded values[END_REF] that "almost all" T -orbits attain almost bounded values. That is to say, if we fix a function f : N → R such that f (k) → +∞ as k → +∞, the infinimum of the T -orbit of k is less than f (k) for almost every integer k ≥ 1, in the sense of logarithmic density.

It is possible to associate to T in a natural way bounded operators on some Hilbert spaces of analytic functions on the unit disk D, and to link the dynamics of T to those of these operators. An approach of this kind was first proposed by Berg and Meinardus in [START_REF] Berg | Functional equations connected with the Collatz problem[END_REF]. Consider the operator F defined on the space Hol(D) of holomorphic functions on the unit disk D in the following way: for every f ∈ Hol(D) with f (z) = ∞ n=0 c n z n ,

Ff (z) = ∞ n=0 c T (n) z n , z ∈ D.
Berg and Meinardus expressed in [START_REF] Berg | Functional equations connected with the Collatz problem[END_REF] the Collatz Conjecture in terms of functional equations and obtained that it is equivalent to the fact that 1 is an eigenvalue of F of multiplicity 2.

Then Neklyudov took in [START_REF] Neklyudov | Functional analysis approach to the Collatz conjecture[END_REF] a different perpective, and considered the adjoint T , for the Hardy space H 2 (D), of the operator F:

for every f ∈ H 2 (D) with f (z) = ∞ n=0 c n z n , T f (z) = ∞ n=0 c n z T (n) , z ∈ D.
This operator T is considered in [START_REF] Neklyudov | Functional analysis approach to the Collatz conjecture[END_REF] as acting on the Bergman space

B 2 = f : z → ∞ n=0 c n z n ∈ Hol(D); ∥f ∥ 2 = ∞ n=0 π |c n | 2 n + 1 < +∞ on D. Since T (0) = 0, T (1)
= 2 and T (2) = 1, the vector space H 0 = span[1, z, z 2 ] is invariant under the action of T , and T induces a bounded operator on the quotient space X = B 2 /H 0 , which we still denote by T . Considering this quotient space allows to avoid the trivial cycle (1, 2) and the fixed point 0 of the Collatz map, and simplifies the study of the dynamics of T . Neklyudov proved in [START_REF] Neklyudov | Functional analysis approach to the Collatz conjecture[END_REF] several results pertaining to the behavior of the iterates of T , in connection with the Collatz Conjecture. For instance, he showed that if T has no non-trivial cycle, then T is hypercyclic, i.e. admits a vector with dense orbit [START_REF] Neklyudov | Functional analysis approach to the Collatz conjecture[END_REF]Theorem 2.2]. He also undertook a study of the periodic points of T and showed that each hypothetical cycle of T and each hypothetical diverging T -orbit can be associated to a fixed point of T . He also constructed fixed points of T independently of any cycle or diverging orbit of T . Neklyudov asked in [START_REF] Neklyudov | Functional analysis approach to the Collatz conjecture[END_REF] about the existence of a criterion allowing to distinguish these fixed points and also whether T is chaotic (i.e. hypercyclic and admitting a dense set of periodic points).

In this work, we continue this study of operators on Hilbert spaces of analytic functions associated to the Collatz map and give an affirmative answer to this question of Neklyudov. We consider the operator T above as acting on weighted Bergman spaces of holomorphic functions on D of the form

B 2 ω = f : z → ∞ n=0 c n z n ∈ Hol(D); ∥f ∥ 2 ω = ∞ n=0 |c n | 2 ω(n) < +∞
where ω : Z + → (0, +∞) is a positive weight. Just as above, it will be more natural to study the action of T on the quotient space X ω = B 2 ω /H 0 . Let ω 0 (n) = (n + 1)/π for every n ≥ 0. Then B 2 ω 0 is the classical Bergman space B 2 . We write X ω 0 = X . Whenever the weight ω is such that T defines a bounded operator on X ω , it is of interest to study the dynamics of T . The main properties that we will consider are those of hypercyclicity (existence of a dense orbit, or, equivalently, topological transitivity), chaos, frequent hypercyclicity, ergodicity with respect to an invariant measure with full support... We give a brief overview of these notions in Section 3 of the present paper, and refer the reader to one of the works [START_REF] Bayart | Dynamics of linear operators[END_REF] and [START_REF] Grosse-Erdmann | Linear chaos[END_REF] for more on linear dynamics.

Our first main result states that under some mild assumptions on the weight ω, the operator T is hypercyclic.

Theorem 1.1. If ω is bounded from below and if ω(k2 n ) → +∞ as n → +∞ for every k ≥ 3, then T is hypercyclic. Theorem 1.1 applies in particular to the weight ω 0 , i.e. to the case where T acts on the Bergman space, and shows that the result [11, Theorem 2.2] of Neklyudov is in fact not conditional to the Collatz map having no non-trivial cycle.

Our second main result shows, under slightly stronger assumptions on ω, that T acting on X ω enjoys some strong dynamical properties: Theorem 1.2. If ω is bounded from below and if ∞ n=0 1/ω(k2 n ) < +∞ for every k ≥ 3, then T is chaotic, frequently hypercyclic and ergodic with respect to a Gaussian invariant measure with full support.

In particular, T acting on the Bergman space X is chaotic. This provides an affirmative answer to Neklyudov's question.

The paper is organized as follows. We first provide in Section 2 conditions on the weight ω ensuring that T acts as a bounded operator on X ω (Proposition 2.1); building on a description given in [START_REF] Neklyudov | Functional analysis approach to the Collatz conjecture[END_REF] of some eigenvectors of T , we show (still under some mild conditions on ω) that these eigenvectors span a dense subspace of X ω (Theorem 2.4) and that the adjoint T * of T on X ω has no eigenvalue (Theorem 2.7). After recalling in Section 3 some background in linear dynamics, we prove Theorems 1.1 and 1.2 above. The proofs of these two results rely heavily on the properties of the eigenvectorfields of T presented in Section 2, and in particular on Theorem 2.4. Section 4 collects some additional results as well as some open questions.

Boundedness of T and properties of its eigenvectors

Let ω be a positive weight on Z + . We first consider the operator T acting on the space C[ξ]/span[1, ξ, ξ 2 ], defined by

T d n=3 c n ξ n = 3≤n≤d,T (n)≥3 c n ξ T (n) for every d n=3 c n ξ n ∈ C[ξ]/span[1, ξ, ξ 2 ],
and give conditions on ω implying that T can be extended to a bounded operator on X ω .

Proposition 2.1. The operator T , defined for every f ∈ X ω with f

(z) = ∞ n=3 c n z n by T ∞ n=3 c n z n = {j≥3;T (j)≥3} c j z T (j) , z ∈ D,
is a bounded operator acting on X ω if and only if the three sequences (ω(6m)/ω(3m)) m≥1 , (ω(6m + 2)/ω(3m + 1)) m≥1 and ((ω(6m + 4) + ω(2m + 1))/ω(3m + 2)) m≥1 are bounded. In this case

∥T ∥ 2 ω = max sup m≥1 ω(6m) ω(3m) , sup m≥1 ω(6m + 2) ω(3m + 1) , sup m≥1 ω(6m + 4) + ω(2m + 1) ω(3m + 2) .
Moreover for every n ≥ 0

∥T n ∥ 2 ω = sup k≥3 T n (j)=k ω(j) ω(k) .
Taking ω = ω 0 we obtain in particular that T is bounded on X and that ∥T ∥ 2 = 8/3.

Proof. Suppose that the three sequences above are bounded, which means that the maximum

max sup m≥1 ω(6m) ω(3m) , sup m≥1 ω(6m + 2) ω(3m + 1) , sup m≥1 ω(6m + 4) + ω(2m + 1) ω(3m + 2) = sup k≥3 T (j)=k ω(j) ω(k) is finite. Let n ≥ 0 and f ∈ X ω with f (z) = ∞ k=3 c k z k . We have T n f (z) = T n (j)≥3 c j z T n (j) = ∞ k=3 T n (j)=k c j z k , z ∈ D, and
∥T n f ∥ 2 ω = ∞ k=3 1 ω(k) T n (j)=k c j 2 = ∞ k=3 1 ω(k) T n (j)=k ω(j) c j ω(j) 2 .
By the Cauchy-Schwarz's inequality

∥T n f ∥ 2 ω ≤ ∞ k=3 1 ω(k) T n (j)=k ω(j) T n (j)=k |c j | 2 ω(j) ≤   sup k≥3 T n (j)=k ω(j) ω(k)   ∥f ∥ 2 ω , which gives that T is bounded if the sequence ( T (j)=k ω(j)/ω(k)) k≥3 is bounded and that ∥T n ∥ 2 ω ≤ sup k≥3 T n (j)=k ω(j)/ω(k). Conversely, let n ≥ 0 and suppose that T is a bounded operator on X ω . Let (k p ) p≥1 such that T n (j)=kp ω(j)/ω(k p ) → sup k≥3 T n (j)=k ω(j)/ω(k) as p → +∞. Consider the func- tion f p ∈ X ω with f p (z) = T n (j)=kp ω(j)z j satisfying ∥f p ∥ 2 ω = T n (j)=kp |ω(j)| 2 /ω(j) = T n (j)=kp ω(j) and ∥T n f p ∥ 2 ω = 1 ω(k p ) T n (j)=kp ω(j) 2 =   T n (j)=kp ω(j) ω(k p )   ∥f p ∥ 2 ω . So we have ∥T n f p ∥ 2 ω /∥f p ∥ 2 ω → sup k≥3 T n (j)=k ω(j)/ω(k) as p → +∞ and it follows that sup k≥3 T n (j)=k ω(j)/ω(k) ≤ ∥T n ∥ 2
ω < +∞ for every n ≥ 0, which concludes the proof.

□

For instance, it follows from Proposition 2.1 that if ω has polynomial growth, then T is a bounded operator on X ω .

Eigenvectors play a very important role in linear dynamics. We will thus be interested in the eigenvalues and the eigenvectors of T , and will provide some condition on the weight ω ensuring that the eigenvectors of T span a dense subspace of X ω . Definition 2.2. For every µ ∈ C and every m ≥ 1, consider the analytic functions on D

h m (µ, •) : z → ∞ n=0 µ n z (6m+4)2 n -z (2m+1)2 n and h 0 (µ, •) : z → ∞ n=0 µ n z 2 n+2 .
These functions has been introduced by Neklyudov in [11, Theorem 2.3] and are the only eigenvectors that will be needed in order to obtain results concerning the spanning of dense subspaces.

Proposition 2.3. For every m ≥ 0 and every µ ∈ C, the vector h m (µ, •) belongs to X ω if and only if the series n≥0 |µ| 2n /ω((2m + 1)2 n ) and n≥0 |µ| 2n /ω((6m

+ 4)2 n ) converge. In this case T h m (µ, •) = µh m (µ, •).
In particular if ω is bounded from below then h m (µ, •) belongs to X ω for every m ≥ 0 and every µ ∈ D. Moreover if ω = ω 0 then h m (µ, •) belongs to X for every m ≥ 0 and every

µ ∈ C such that |µ| < √ 2. Proof. Let µ ∈ C and m ≥ 1. The function h m (µ, •) belongs to X ω if and only if ∞ n=0 |µ| 2n /ω((6m + 4)2 n ) + ∞ n=0 |µ| 2n /ω((2m + 1)2 n ) < +∞. And h 0 (µ, •) belongs to X ω if and only if the series ∞ n=0 |µ| 2n /ω(2 n+2
) < +∞. Moreover for every m ≥ 1 and every z ∈ D we have

T h m (µ, •)(z) = ∞ n=0 µ n z T ((6m+4)2 n ) -z T ((2m+1)2 n ) = ∞ n=1 µ n z (6m+4)2 n-1 -z (2m+1)2 n-1 + z 3m+2 -z 3m+2 = µh m (µ, z) and T h 0 (µ, •)(z) = ∞ n=0 µ n z T (2 n+2 ) = ∞ n=1 µ n z 2 n+1 + z 2 = µh 0 (µ, z),
which shows that the functions h m (µ, •), m ≥ 0, are eigenvectors of T associated to the eigenvalue µ as soon as they belong to X ω . □

Our aim is now to show that if the weight ω is bounded from below, the eigenvectors

h m (µ, •), m ≥ 0, µ ∈ D, span a dense subspace of X ω . Theorem 2.4. If ω is bounded from below, then span[h m (µ, •); m ≥ 0, µ ∈ D] is dense in X ω .
In particular if ω = ω 0 then {h m (µ, •); m ≥ 0, µ ∈ D} spans a dense subspace of X .

Since T admits a particular behavior on {2 i ; i ≥ 0}, we will denote by P 0 the set {2 i ; i ≥ 0} and by P 2 the set {2 i ; i ≥ 2}. The proof of Theorem 2.4 relies on the following lemma.

Lemma 2.5. Let f ∈ X ω , with f (z) = ∞ k=3 c k z k , such that for every n ≥ 0 and every m ≥ 1 c (6m+4)2 n ω((6m + 4)2 n ) = c (2m+1)2 n ω((2m + 1)2 n )
and c 2 n+2 = 0.

Then for every k ≥ 3 such that k / ∈ P 2 , there exist integer sequences (m n ) n≥1 , (p n ) n≥1 and (j n ) n≥1 depending on k such that:

(i) c k = (ω((2m 1 + 1)2 p 1 )/ω((6m n + 4)2 pn ))c (6mn+4)2 pn for every n ≥ 1;

(ii) m n ≥ 1 for every n ≥ 1;

(iii) 3m n + 2 = T jn (k) for every n ≥ 1;

(iv) j n+1 > j n if and only if 3m n + 2 / ∈ P 2 for every n ≥ 1; (v) (j n ) n≥1 and ((6m n + 4)2 pn ) n≥1 are either both strictly increasing or both stationary.

Proof. Since k ≥ 3 and k / ∈ P 2 , there exist m 1 ≥ 1 and p 1 ≥ 0 such that k = (2m 1 + 1)2 p 1 . Then

c k = c (2m 1 +1)2 p 1 = ω((2m 1 + 1)2 p 1 ) ω((6m 1 + 4)2 p 1 ) c (6m 1 +4)2 p 1 and 3m 1 + 2 = T (2m 1 + 1) = T p 1 +1 ((2m 1 + 1)2 p 1 ) = T p 1 +1 (k).
We set then j 1 = p 1 + 1. Suppose now that (m i ) 1≤i≤n , (p i ) 1≤i≤n and (j i ) 1≤i≤n have been defined so as to satisfy the following properties:

(i) c k = (ω((2m 1 + 1)2 p 1 )/ω((6m i + 4)2 p i ))c (6m i +4)2 p i for every 1 ≤ i ≤ n; (ii) m i ≥ 1 for every 1 ≤ i ≤ n; (iii) 3m i + 2 = T j i (k) for every 1 ≤ i ≤ n;
(iv) j i+1 > j i if and only if 3m i + 2 / ∈ P 2 for every 1 ≤ i ≤ n -1. Let us construct m n+1 , p n+1 and j n+1 .

If 3m n + 2 ∈ P 2 , then we set m n+1 = m n , p n+1 = p n and j n+1 = j n . Otherwise there exist m n+1 ≥ 1 and q n+1 ≥ 0 such that 3m n + 2 = (2m n+1 + 1)2 q n+1 . So (6m n + 4)2 pn = (3m n + 2)2 pn+1 = (2m n+1 + 1)2 q n+1 +pn+1 .

We set then p n+1 = q n+1 + p n + 1 > p n . Moreover

c k = ω((2m 1 + 1)2 p 1 ) ω((2m n+1 + 1)2 p n+1 ) c (2m n+1 +1)2 p n+1 = ω((2m 1 + 1)2 p 1 ) ω((2m n+1 + 1)2 p n+1 ) ω((2m n+1 + 1)2 p n+1 ) ω((6m n+1 + 4)2 p n+1 ) c (6m n+1 +4)2 p n+1 = ω((2m 1 + 1)2 p 1 ) ω((6m n+1 + 4)2 p n+1 ) c (6m n+1 +4)2 p n+1 .
Remark that

3m n+1 + 2 = T p n+1 +1 ((2m n+1 + 1)2 p n+1 ) = T p n+1 +1 ((6m n + 4)2 pn ) = T p n+1 ((3m n + 2)2 pn ).
So 3m n+1 +2 = T p n+1 -pn (3m n +2) = T p n+1 -pn+jn (k) and we set j n+1 = p n+1 -p n +j n > j n . We also have (6m n+1 + 4)2 p n+1 > (2m n+1 + 1)2 p n+1 = (6m n + 4)2 pn . Defined this way, the sequences (m n ) n≥1 , (p n ) n≥1 and (j n ) n≥1 satisfy the first four properties and it remains to prove the last one. On the one hand, if there exists n 0 ≥ 1 such that 3m n 0 + 2 ∈ P 2 , then the sequences (j n ) n≥1 and ((6m n + 4)2 pn ) n≥1 are stationary by construction. On the other hand if 3m n + 2 / ∈ P 2 for every n ≥ 1, then the sequences (j n ) n≥1 and ((6m n + 4)2 pn ) n≥1 are strictly increasing by construction, which proves that the fifth property is satisfied. □ Proof of Theorem 2.4. By Proposition 2.3, h m (µ, •) belongs to X ω for every m ≥ 0 and

every µ ∈ D since ω is bounded from below. Let f ∈ span[h m (µ, •); m ≥ 0, µ ∈ D] ⊥ in X ω with f (z) = ∞ k=3 c k z k .
Our aim is to show that f = 0. For every µ ∈ D and every m ≥ 1

φ m (µ) : = ⟨h m (µ, •), f ⟩ = ∞ n=0 c (6m+4)2 n ω((6m + 4)2 n ) - c (2m+1)2 n ω((2m + 1)2 n ) µ n = 0, φ 0 (µ) : = ⟨h 0 (µ, •), f ⟩ = ∞ n=0 c 2 n+2 ω(2 n+2 ) µ n = 0.
For every k ≥ 3 the radius of convergence of the power series n≥0 (c k2 n /ω(k2 n ))z n is greater than 1. Indeed by the Cauchy-Schwarz's inequality

∞ n=0 c k2 n ω(k2 n ) z n 2 ≤ ∞ n=0 |c k2 n | 2 ω(k2 n ) ∞ n=0 |z| 2n ω(k2 n ) < +∞
for every z ∈ D because f belongs to X ω and ω is supposed to be bounded from below. Thus the functions φ m and φ 0 are holomorphic and they identically vanish on D for every m ≥ 1. So for every m ≥ 1

c (6m+4)2 n ω((6m + 4)2 n ) = c (2m+1)2 n ω((2m + 1)2 n )
and c 2 n+2 = 0 for every n ≥ 0.

Claim 2.6. Let ω be a weight on Z + which is bounded from below, and let f ∈ X ω with

f (z) = ∞ k=3 c k z k . Suppose that for every m ≥ 1 c (6m+4)2 n ω((6m + 4)2 n ) = c (2m+1)2 n ω((2m + 1)2 n )
and c 2 n+2 = 0 for every n ≥ 0, Then c k = 0 for every k ≥ 3.

Proof. Fix k ≥ 3. If k ∈ P 2 then c k = 0. Otherwise consider the sequences (m n ) n≥1 , (p n ) n≥1 and (j n ) n≥1 associated to k given by Lemma 2.5. On the one hand suppose that {T n (k); n ≥ 0} ∩ P 0 = ∅, then in particular 3m n + 2 / ∈ P 2 for every n ≥ 1 by (iii). So by (iv), (j n ) n≥1 is strictly increasing and then so is ((6m n + 4)2 pn ) n≥1 by (v). Therefore by (i)

∥f ∥ 2 ω = ∞ j=3 |c j | 2 ω(j) ≥ ∞ n=1 c (6mn+4)2 pn 2 ω((6m n + 4)2 pn ) = ∞ n=1 ω((6m n + 4)2 pn ) ω((2m 1 + 1)2 p 1 ) 2 |c k | 2 .
Since ω is bounded from below, the previous series can converge only in the case where c k = 0. On the other hand suppose that {T n (k); n ≥ 0} ∩ P 0 ̸ = ∅, then (j n ) n≥1 is stationary. Indeed suppose that (j n ) n≥1 is not stationary, that is to say strictly increasing by (v). There exists N ≥ 1 such that T N (k) ∈ P 0 , so T n (k) ∈ P 0 for every n ≥ N and T j N (k) ∈ P 0 since j N ≥ N . Then T j N (k) = 3m N + 2 ∈ P 2 by (ii) and (iii), which implies that j N +1 = j N . This is impossible, so (j n ) n≥1 is indeed stationary. Thus there exists N ≥ 1 such that 3m N + 2 ∈ P 2 by (iv), which implies by (i) that

c k = ω((2m 1 + 1)2 p 1 ) ω((6m N + 4)2 p N ) c (6m N +4)2 p N = 0 because (6m N + 4)2 p N = (3m N + 2)2 p N +1 ∈ P 2 .
We have shown that c k = 0 for every k ≥ 3. □ Thus Claim 2.6 gives that f = 0, which concludes the proof of Theorem 2.4. □

We conclude this section by proving that the adjoint T * of T acting on X ω has no eigenvalue. We will denote by σ p (T * ) the set of its eigenvalues.

Theorem 2.7. If ω is bounded from below then σ p (T * ) = ∅.

In particular if ω = ω 0 then σ p (T * ) = ∅.

The proof proceeds along the same lines as the proof of Theorem 2.4. We first need the following Lemma, whose proof is similar to the proof of Lemma 2.5 and will be omitted here.

Lemma 2.8. Let µ ∈ C\{0} and let f ∈ X ω with f (z) = ∞ k=3 c k z k ∈ X ω .
Suppose that for every m ≥ 1 and every n ≥ 0

µ -1 c (3m+2)2 n ω((3m + 2)2 n ) = c (2m+1)2 n ω((2m + 1)2 n )
and c 2 n+2 = 0.

Then for every k ≥ 3 such that k / ∈ P 2 , there exist integer sequences (m n ) n≥1 , (p n ) n≥1 and (j n ) n≥1 such that:

(i) c k = (µ -n ω((2m 1 + 1)2 p 1 )/ω((3m n + 2)2 pn ))c (3mn+2)2 pn for every n ≥ 1;

(ii) m n ≥ 1 for every n ≥ 1;

(iii) 3m n + 2 = T jn (k) for every n ≥ 1;

(iv) j n+1 > j n if and only if 3m n + 2 / ∈ P 2 for every n ≥ 1; (v) (j n ) n≥1 and ((3m n + 2)2 pn ) n≥1 are either both strictly increasing or both stationary.

Proof of Theorem 2.7. For every f, g ∈ X ω with f (z) = ∞ n=3 c n (f )z n and with g(z) = ∞ n=3 c n (g)z n , we have

⟨T f | g⟩ = ∞ n=3 c n (T f )c n (g) ω(n) = ∞ n=3 T (j)=n c j (f )c n (g) ω(n) = j∈{k≥3;T (k)≥3} c j (f )c T (j) (g) ω(T (j)) , so ⟨T f | g⟩ = c 3 (f )c 5 (g) ω(5) + ∞ k=5 c k (f )c T (k) (g) ω(T (k)) .
Then T * is defined in the following way:

T * ∞ n=3 c n z n = ω(3) ω(5) c 5 z 3 + ∞ k=5 ω(k) ω(T (k)) c T (k) z k for every f : z → ∞ n=3 c n z n ∈ X ω . Let µ ∈ C, let f ∈ X ω with f (z) = ∞ n=3 c n z n and suppose that T * f = µf . We have    (ω(3)/ω(5))c 5 = µc 3 (ω(k)/ω(T (k)))c T (k) = µc k , k ≥ 5 0 = µc 4 , which is equivalent to    (ω(2m)/ω(m))c m = µc 2m , m ≥ 3 (ω(2m + 1)/ω(3m + 2))c 3m+2 = µc 2m+1 , m ≥ 1 0 = µc 4 .
Firstly if µ = 0 then c m = 0 for every m ≥ 3. So 0 does not belong to σ p (T * ).

We suppose now that µ ̸ = 0. Then we have T * f = µf if and only if

   c m2 n = µ -n (ω(m2 n )/ω(m))c m , m ≥ 3, n ≥ 0 µ -1 c (3m+2)2 n /ω((3m + 2)2 n ) = c (2m+1)2 n /ω((2m + 1)2 n ), m ≥ 1, n ≥ 0 c 2 n+2 = 0, n ≥ 0
because for every m ≥ 1 and every n ≥ 0

c (2m+1)2 n = µ -1 ω((2m + 1)2 n ) ω((2m + 1)2 n-1 ) c (2m+1)2 n-1 = . . . = µ -n ω((2m + 1)2 n ) ω(2m + 1) c 2m+1 , so c (2m+1)2 n = µ -n-1 ω((2m + 1)2 n ) ω(3m + 2) c 3m+2 = . . . = µ -n-1+n ω((2m + 1)2 n ) ω((3m + 2)2 n ) c (3m+2)2 n . Fix k ≥ 3. If k ∈ P 2 then c k = 0. Suppose now that k / ∈ P 2 . If |µ| ≤ 1, we can remark that ∥f ∥ 2 ω = ∞ j=3 |c j | 2 ω(j) ≥ ∞ n=0 |c k2 n | 2 ω(k2 n ) = ∞ n=0 |µ| -2n ω(k2 n ) ω(k) 2 |c k | 2 .
Since ω is bounded from below, the previous series can converge only in the case where c k = 0 and it follows that f = 0. Thus µ does not belong to σ p (T * ). If |µ| > 1, consider the sequences (m n ) n≥1 , (p n ) n≥1 and (j n ) n≥1 associated to k given by Lemma 2.8. On the one hand suppose that {T n (k); n ≥ 0} ∩ P 0 = ∅, then in particular 3m n + 2 / ∈ P 2 for every n ≥ 1 by (iii). So by (iv), (j n ) n≥1 is strictly increasing and then so is ((3m n + 2)2 pn ) n≥1 by (v). Therefore by (i)

∥f ∥ 2 ω = ∞ j=3 |c j | 2 ω(j) ≥ ∞ n=1 c (3mn+2)2 pn 2 ω((3m n + 2)2 pn ) = ∞ n=1 |µ| 2n ω((3m n + 2)2 pn ) ω((2m 1 + 1)2 p 1 ) 2 |c k | 2 .
Since ω is bounded from below, the previous series converges only in the case where c k = 0. On the other hand suppose that {T n (k); n ≥ 0} ∩ P 0 ̸ = ∅, we will follow the lines of the proof of Theorem 2.4 and claim that (j n ) n≥1 is stationary. Indeed suppose that (j n ) n≥1 is not stationary, that is to say is strictly increasing by (v). There exists N ≥ 1 such that T N (k) ∈ P 0 , so T n (k) ∈ P 0 for every n ≥ N and T j N (k) ∈ P 0 since j N ≥ N . Then T j N (k) = 3m N +2 ∈ P 2 by (ii) and (iii), which implies that j N +1 = j N . This is impossible. So (j n ) n≥1 is stationary and there exists N ≥ 1 such that 3m N + 2 ∈ P 2 by (iv). This implies by (i) that

c k = µ -N ω((2m 1 + 1)2 p 1 ) ω((3m N + 2)2 p N ) c (3m N +2)2 p N = 0 because (3m N + 2)2 p N ∈ P 2 .
We have shown that c k = 0 for every k ≥ 3. Thus f = 0 and so µ does not belong to σ p (T * ) either. □

Linear dynamics of the operator T

Our aim is now to understand the links between the dynamics of the Collatz map and those of the operator T . We begin by recalling briefly the definition of some important properties in linear dynamics (hypercyclicity, chaos, frequent hypercyclicity and ergodicity), before investigating whether the operator T acting on X ω satisfies each of these properties.

3.1. Hypercyclicity. A much studied dynamical property is called hypercyclicity. We refer the reader to the books [START_REF] Bayart | Dynamics of linear operators[END_REF] and [START_REF] Grosse-Erdmann | Linear chaos[END_REF] for an in-depth study of this notion. Let X be a separable infinite-dimensional complex Banach (or Fréchet) space. We will denote by B(X) the set of bounded (or continuous) linear operators on X. Definition 3.1 ([7, Definition 2.15]). An operator A ∈ B(X) is said to be hypercyclic if there exists x ∈ X such that its orbit {A n x; n ≥ 0} under A is dense in X. In this case x is called a hypercyclic vector for A.

Recall that a G δ -set in X is a countable intersection of open sets of X. The following theorem shows that either the hypercyclic vectors of an operator A ∈ B(X) forms a dense G δ -set in X or A has no hypercyclic vector. Theorem 3.2 (Birkhoff's transitivity theorem, [7, Theorems 1.16 and 2.19]). An operator A ∈ B(X) is hypercyclic if and only if for every non-empty open sets U and V in X, there exists n ≥ 0 such that A n (U ) ∩ V ̸ = ∅. In this case, the hypercyclic vectors for A form a dense G δ -set in X.

Important examples of hypercyclic operators can be found in the literature. On the Fréchet space Hol(C) of entire functions endowed with the seminorms defined by ∥f ∥ n,∞ = sup{|f (z)| ; |z| ≤ n} for n ≥ 1, the Birkhoff's operator T (a) : f → f (• + a) is hypercyclic if and only if a ̸ = 0. The derivation operator D acting on Hol(C) is also hypercyclic. In the Banach space setting, the simplest examples of hypercyclic operators are given by the Rolewicz's operators λB for |λ| > 1, where B is the backward shift on ℓ 2 (N) defined by

B(x n ) n≥1 = (x n+1 ) n≥1 .
The following criterion provides a pratical mean to prove the hypercyclicity of an operator.

Theorem 3.3 (Hypercyclicity Criterion, [START_REF] Grosse-Erdmann | Linear chaos[END_REF]Theorem 3.12]). Let A ∈ B(X). Suppose that there exist dense subsets X 0 and Y 0 of X, an increasing sequence (n k ) k≥1 of integers and a sequence (S n k : Y 0 → X) k≥1 of maps such that:

(i)

A n k x → 0 as k → +∞ for every x ∈ X 0 ; (ii) S n k y → 0 as k → + ∞ for every y ∈ Y 0 ; (iii) A n k S n k y → y as k → +∞ for every y ∈ Y 0 .
Then the operator A is hypercyclic.

In the case where ω = ω 0 , and provided that the map T has no non-trivial cycle, it is shown in [11, Theorem 2.2] that T satisfies the Hypercyclicity Criterion, and is thus hypercyclic. Our first theorem generalizes this result by showing that T is hypercyclic on X ω under a rather mild hypothesis on the weight ω, independently of any assumption on the existence of cycles for the Collatz map. Theorem 3.4. If ω is bounded from below and if ω(k2 n ) → +∞ as n → +∞ for every k ≥ 3, then T is hypercyclic.

In particular if ω = ω 0 then T is hypercyclic on X .

Sz k = z 2k for every k ≥ 3. Then ∥S n z k ∥ 2 ω = ∥z k2 n ∥ 2 ω = 1/ω(k2 n ) → 0 as n → +∞ and T Sz k = T z 2k = z T (2k) = z k for every k ≥ 3.
Thus this is still true for any linear combination in Y 0 . Then T satisfies the Hypercyclicity Criterion, so T is hypercyclic. □ Actually, the role played by eigenvectors in the hypercyclicity of an operator appears explicitly in the following criterion. Theorem 3.5 (Godefroy-Shapiro Criterion, [START_REF] Grosse-Erdmann | Linear chaos[END_REF]Theorem 3.1]). Let A ∈ B(X). Suppose that the subspaces

X 0 = span[ker(A -µ); |µ| < 1] and Y 0 = span[ker(A -µ); |µ| > 1]
are dense in X. Then A is hypercyclic. This criteria is satisfied as soon as there are enough complex numbers µ such that h m (µ, •) belongs to X ω for every m ≥ 0. Theorem 3.6. If ω is bounded from below and if there exists ρ > 1 such that for every k ≥ 3 the sequence (ρ n /ω(k2 n )) n≥0 is bounded, then T satisfies the Godefroy-Shapiro Criterion.

In particular if ω = ω 0 , then taking ρ = 2 gives us that T satisfies the Godefroy-Shapiro Criterion.

Proof. By Proposition 2.3, h m (µ, •) belongs to X ω for every m ≥ 0 and every µ ∈ C such that |µ| < √ ρ. By Theorem 2.4, it suffices to show that the subspace ). An operator A ∈ B(X) is said to be frequently hypercyclic if there exists a vector x ∈ X such that for every non-empty open set U in X lim inf

Y 0 = span[h m (µ, •); m ≥ 0, 1 < |µ| < √ ρ] is dense in X ω .
N →+∞ card{0 ≤ n ≤ N ; A n x ∈ U } N + 1 > 0.
In this case x is called a frequently hypercyclic vector for A.

The Hypercyclicity Criterion admits a frequently hypercyclic version, which is the following theorem. Theorem 3.8 (Frequent Hypercyclicity Criterion, [7, Theorem 9.9]). Let A ∈ B(X). Suppose that there exist a dense set X 0 in X and a map S : X 0 → X 0 such that for every x ∈ X 0 :

(1) n≥0 A n x converges unconditionally; (2) n≥0 S n x converges unconditionally; (3) ASx = x.

Then A is frequently hypercyclic.

The three hypercyclic operators presented before are actually frequently hypercyclic. The operators λB, D and T (a) are frequently hypercyclic respectively on ℓ 2 (N) and H(C) if |λ| > 1 and if a ̸ = 0.

In order to prove that T is frequently hypercyclic, we will actually show that T is ergodic with respect to a Gaussian measure of full support.

Let H be a complex Hilbert space, let B be the σ-algebra of Borel subsets of H and let m be a probability measure on (H, B).

Definition 3.9 ([2, Definition 3.9]). A transformation A : (H, B, m) → (H, B, m) in B(H) is said to preserve the measure m if m(A -1 (B)) = m(B) for every B ∈ B. A measure- preserving transformation A : (H, B, m) → (H, B, m) is said to be ergodic if for every B ∈ B, A -1 (B) = B implies that m(B) ∈ {0, 1}.
The eigenvectors of an operator play an important role in the study of its dynamics. The fact that the eigenvectors of T span a dense subspace of X ω allowed us to show that T is hypercyclic under some assumptions on the weight ω. In order to show that T is ergodic with respect to a Gaussian measure with full support, we will rely on the properties of the eigenvectors of T associated to unimodular eigenvalues. We first recall a few relevant definitions. Definition 3.10 ([2, Definition 3.1]). An operator A ∈ B(H) is said to have a perfectly spanning set of eigenvectors associated to unimodular eigenvalues if there exists a continuous probability measure σ on the unit circle T such that the eigenvectors of A associated to eigenvalues µ ∈ A span a dense subspace in H for every σ-measurable subset A of T satisfying σ(A) = 1. Definition 3.11 ([2, Definitions 3.13 and 3.14]). Let (Ω, F, P ) be a probability space. A measurable function f : (Ω, F, P ) → C is said to have complex symmetric Gaussian distribution if ℜ(f ) and ℑ(f ) have independent centered Gaussian distribution and the same variance.

A probability measure m on (H, B) is said to be a Gaussian measure if the function y → ⟨y, x⟩ has complex symmetric Gaussian distribution for every x ∈ H.

A measure m on (H, B) is said to have a full support if m(U ) > 0 for every non-empty open set U ∈ B. Theorem 3.12 ([2, Theorem 3.22]). If an operator A ∈ B(H) admits a perfectly spanning set of eigenvectors associated to unimodular eigenvalues, then there exists a Gaussian invariant measure with full support m on H such that A : (H, B, m) → (H, B, m) is ergodic.

We are now ready to prove the ergodicity of T . Theorem 3.13. If ω is bounded from below and if ∞ n=0 1/ω(k2 n ) < +∞ for every k ≥ 3, then T is ergodic with respect to a Gaussian invariant measure with full support on X ω . This is in particular the case when ω = ω 0 .

Proof. By Theorem 3.12, it suffices to prove that T admits a perfectly spanning set of eigenvectors associated to unimodular eigenvalues. Since ∞ n=0 1/ω(k2 n ) < +∞ for every k ≥ 3, the function h m (µ, •) belongs to X ω for every m ≥ 0 and every µ ∈ C such that |µ| ≤ 1 by Proposition 2.3. We will prove that for every Borel set A ⊂ T such that σ(A) = 1, where σ is the Lebesgue measure on the unit circle, the subspace span[h m (µ, •); m ≥ 0, µ ∈ A] is dense in X ω .

would have 2a(r + 1) ≥ 3 n (2b + 2) > 3 n (2b + 1), and on the other hand we would have 2a(r + 1)/3 ≥ 3 n (2b + 2)/3 ≥ 3 n ((2b -1)/3 + 1). Then this is true for P The operator T can actually be seen as acting on a space of sequences (c n ) n≥3 instead of on a space of holomorphic functions f : z → ∞ n=3 c n z n . Question 4.4. What would remain from these results if we considered T as acting on the space of complex sequences C N ?

Proof.

  It suffices to prove that T satisfies the Hypercyclicity Criterion (Theorem 3.3). Consider the subspace X 0 = span[h m (µ, •); µ ∈ D, m ≥ 0], which is dense in X ω by Theorem 2.4. We have T n h m (µ, •) = µ n h m (µ, •) → 0 as n → +∞ for every µ ∈ D and every m ≥ 0. Thus this is true for any linear combination in X 0 . Consider now the dense subspace Y 0 = span[z k ; k ≥ 3] of X ω and the map S : Y 0 → Y 0 defined by

3 . 2 .

 32 This can be proved by following the steps of the proof of Theorem 2.4 and considering the holomorphic functions φ m and φ 0 on {µ ∈ C; |µ| < √ ρ} instead of D. □ Frequent hypercyclicity and ergodicity. The notion of frequent hypercyclicity is a reinforcement of that of hypercyclicity. It quantifies the frequency with which the orbit of a vector visits a non-empty open set. We refer the reader to[START_REF] Bayart | Frequently hypercyclic operators[END_REF] for more on this notion. Definition 3.7 ([7,Definition 9.2]

Question 4 . 3 .

 43 satisfies it. So for every aξ + b ∈ P If lim inf n→+∞ ω(n) = 0, could T be bounded, hypercyclic or ergodic ? Could T * have any eigenvalue ?

This work was supported in part by the project FRONT of the French National Research Agency (grant ANR-17-CE40-0021) and by the Labex CEMPI (ANR-11-LABX-0007-01).

I am grateful to Sophie Grivaux for our discussions and her helpful proofreadings.

Claim 3.14. Let ω be a weight on Z + which is bounded from below and such that ∞ n=0 1/ω(k2 n ) < +∞ for every k ≥ 3. Then the subspace span[h m (µ, •); m ≥ 0, µ ∈ D] is dense in X ω for every dense subset D of T.

Our aim is to prove that f = 0. For every µ ∈ D and every m ≥ 1

For every k ≥ 3 the power series ∞ n=0 (c k2 n /ω(k2 n ))z n is uniformly convergent on the closed disk D. Indeed for every n ≥ 0 and every

and by the Cauchy-Schwarz's inequality

since f belongs to X ω and by assumption on ω. Then φ m and φ 0 are holomorphic on D, continuous on D and they vanish on D for every m ≥ 1. Since D is dense in the unit circle T, the functions φ m and φ 0 vanish on T by continuity. By the maximum modulus principle, φ m and φ 0 identically vanish on D for every m ≥ 1 and

and c 2 n+2 = 0 for every n ≥ 0.

It follows then from Claim 2.6 that f = 0, which gives that span[h m (µ,

Thus since a Borel set A ⊂ T satisfying σ(A) = 1 is dense in T, Claim 3.14 gives that T admits a perfectly spanning set of eigenvectors associated to unimodular eigenvalues, which concludes the proof.

□

We now will link the ergodicity of an operator to its frequent hypercyclicity. Let A ∈ B(H) be ergodic with respect to an invariant measure with full support. First it follows from the Birkhoff's transitivity theorem that A is hypercyclic. Then it follows from Birkhoff's pointwise ergodic theorem that: Theorem 3.15 ([3, Proposition 6.23]). If an operator A ∈ B(H) is ergodic with respect to a probability measure m on H with full support, then A is frequently hypercyclic. Moreover the frequently hypercyclic vectors of A form a set of full measure for m.

We deduce from Theorems 3.13 and 3.15 the following result:

In particular if ω = ω 0 , T is frequently hypercyclic on X .

We finish this section by proving that, under the assumptions of Theorem 3.13, T is chaotic. Definition 3.17 ([7, Definition 2.29]). An operator A ∈ B(X) is said to be chaotic if A is hypercyclic and has a dense set of periodic points. Theorem 3.18. If ω is bounded from below and if ∞ n=0 1/ω(k2 n ) < +∞ for every k ≥ 3, then T is chaotic on X ω .

In particular if ω = ω 0 , then T is chaotic on X .

Proof. Since ∞ n=0 1/ω(k2 n ) < +∞ for every k ≥ 3, the function h m (µ, •) belongs to X ω for every m ≥ 0 and every µ ∈ C such that |µ| ≤ 1 by Proposition 2.3. By Theorem 3.4, T is hypercyclic and it suffices to show that set of its perdiodic points Per(T ) is dense in X ω . It follows from [START_REF] Grosse-Erdmann | Linear chaos[END_REF]Proposition 2.33

and its density is given by Claim 3.14. □

Open questions

We finish this paper by presenting a few open questions connected to the results we have presented.

Question 4.1. In the case where ω = ω 0 , it follows from Propositions 2.1 and 2.3 that the spectrum σ(T ) satisfies

What is the exact value of the spectral radius ρ(T ) of T ? Is it true that ρ(T ) = √ 2 ?

In order to answer this question, one could perhaps build on the following proposition, which provides an explicit expression for the norm of the iterates T n of T : Proposition 4.2. If ω = ω 0 , then for every n ≥ 1 Proof. According to Proposition 2.1

If α, β, γ, δ > 0, one can remark that a sequence ((αm+β)/(γm+δ)) m≥0 is strictly increasing or constant if and only if αδ ≥ βγ. We will prove by recursion that a(r + 1) ≥ 3 n (b + 1) for every aξ + b ∈ P


Mame Diarra Fall

QUANTIFYING UNCERTAINTY IN KNEE OSTEOARTHRITIS DIAGNOSIS

Keywords: Knee OsteoArthritis, Deep-learning, Uncertainty, X-ray images, classification

Knee OsteoArthritis (OA) is one of the most common causes of physical disability in the world, causing a large personal and socio-economic burden. Visual assessment of OA still suffers from subjectivity. Deep learning (DL), and in particular convolutional neural networks (CNN), has recently led to remarkable improvements in knee OA detection. However, traditional deep learning-based knee OA classification algorithms lack the ability to quantify decision uncertainty. This is a key point in the medical field where, due to the high cost of labelling, we are faced with a lack of sufficient data to train a learning model. We propose here an alternative approach based on the the concept of Evidential Deep Learning (EDL). Unlike Bayesian neural networks which indirectly infer prediction uncertainty through uncertainties in the network weights, EDL approaches explicitly model this uncertainty using the theory of subjective logic. Experimental results on the Osteoarthritis (OAI) database demonstrate the potential of the proposed approach.

INTRODUCTION

Knee OsteoArthritis is a degenerative disease characterized by deterioration and damage of the articular cartilage, joint edges, and reactive hyperplasia of the subchondral bone [START_REF] Loeser | Osteoarthritis: a disease of the joint as an organ[END_REF]. Multiple factors, including age, weight, stress, trauma, etc. may contribute to its occurrence [START_REF] Litwic | Epidemiology and burden of osteoarthritis[END_REF]. The disease is associated with stiffness, swelling, and pain. Knee OA is recognized as the main cause of reduced mobility in the elderly, and is now recognized as an independent risk factor for increased mortality. Since no treatment can prevent the degenerative structural changes responsible for the progression of knee osteoarthritis, early detection is essential so that timely behavioural therapies, such as weight loss, can be implemented to delay the onset and progression of knee OA [START_REF] Wluka | Tackling obesity in knee osteoarthritis[END_REF].

The Kellgren and Lawrence (KL) grading system [START_REF] Kellgren | Radiological assessment of osteo-arthrosis[END_REF] defines knee OA severity in five grades, from 0 (normal) to 4 (severe), according to the existence and severity of symptoms such as osteophytes and joint space narrowing (see Fig. 1). This criterion is however semi-quantitative, and suffers from subjectivity and ambiguity, which makes early OA diagnosis It is therefore necessary to develop automated methods to facilitate the knee OA diagnosis. Several deep learning (DL) have been proposed to this aim. Most of them are based on convolutional neural networks (CNN). The latter have become state-of-the-art technology for image classification tasks due to their ability to capture spatial information in images. One can refer to [START_REF] Nasser | Discriminative regularized autoencoder for early detection of knee osteoarthritis: Data from the osteoarthritis initiative[END_REF] for a brief review of current CNN-based methods for knee OA. Nevertheless, one major drawback of CNN models is their dependency on large amounts of labelled data. In the field of medical imaging the small quantity of valid medical datasets, due to the high cost of labelling, still poses a major challenge for training the learning model. In this context it is crucial to quantify the uncertainty associated with the model prediction.

In classical CNN for classification, the softmax function is used to predict class assignment probabilities. A softmax output is however often misinterpreted as an indication of model confidence. Nevertheless a model can be uncertain in its predictions even with a high softmax output for a particular class [START_REF] Gal | Dropout as a bayesian approximation: Representing model uncertainty in deep learning[END_REF]. Bayesian neural networks can address this issue by putting a prior distribution on the neural network parameters and inferring their posterior distribution using approximations such as Variational Bayes [START_REF] Gal | Dropout as a bayesian approximation: Representing model uncertainty in deep learning[END_REF]. The posterior predictive distribution is approximated using Monte-Carlo sampling methods. These models are however computationally demanding since each data point has to pass through the network at several times, and also require in addition considerable modifications to existing baselines.

Recently, a new class of models based on the concept of evidential deep-learning (EDL) has been proposed [START_REF] Sensoy | Evidential deep learning to quantify classification uncertainty[END_REF], [START_REF] Malinin | Predictive uncertainty estimation via prior networks[END_REF]. They allow to compute uncertainty in a single forward pass by parameterizing distributions on distributions, and involve minimal modifications to the architecture of standard neural networks.

Is this paper, we develop an EDL-based method for knee OA diagnosis. The potential of the proposed approach is eval-uated for binary and multi-class classification tasks. To the best of our knowledge, this is the first work based on deep neural networks that allows to quantify uncertainty in the diagnosis of knee OA.

The rest of the paper is organized as follows. In Section 2 the EDL theory is briefly reviewed before presenting our model. Experimental setups are described in Section 3. Results are presented in Section 4. We conclude in Section 5 and point some future research directions.

METHODS

Theory of Evidence and uncertainty

Let x ∈ R d be an input image (in vectorized form) and f (•|θ) : R d → R K a NN with parameters θ, where K ≥ 2 stands for the number of classes. We denote a = f (x|θ) ∈ R K the raw output vector of the NN corresponding to the input x. The output is ŷ = Φ(a). A schematic representation of such a network is depicted in Fig. 2. In classical NN for classification, Φ is given by the softmax function. The training dataset consists of D = {(x i , y i ), i = 1, . . . , N D } where y i is the ground-truth class associated to x i , in the form of a one-hot encoded vector. The idea behind EDL is to train the network to give an opinion on a classification problem [START_REF] Sensoy | Evidential deep learning to quantify classification uncertainty[END_REF]. This originates from the Dempster-Shafer Theory of Evidence (DST), a generalization of the Bayesian theory to subjective probabilities. Using the DST theory, we assign a belief mass b k ≥ 0 to each class k and consider u ≥ 0 an overall uncertainty mass such that

u + K k=1 b k = 1.
A belief mass can be computed using the evidence. If we let e k ≥ 0 be the evidence derived for the k-th class, then b k = e k S and u = K S , where S = K k=1 (e k + 1). The uncertainty is inversely proportional to the total evidence; and if there is no evidence, the belief for each class is zero and the uncertainty u is one.

Using subjective logic, we can formalize the DST as a Dirichlet distribution, quantifying belief masses and uncertainty [START_REF] Jsang | Subjective Logic: A formalism for reasoning under uncertainty[END_REF].

Dirichlet Modeling

The Dirichlet distribution is a distribution over probability mass functions. A random vector P = (P 1 , . . . , P K ) is distributed according to the Dirichlet distribution with parameter α = (α 1 , . . . , α K ) if its probability density function is

f (p|α) = 1 β(α) K k=1 p α k -1 k , where β(α) = K k=1 Γ(α k ) Γ(S α ) is the K-dimensional multinomial beta function, S α = K k=1 α k is the Dirichlet strength, with α k > 0 ∀k.
Since classical CNN using sotfmax only provide a point estimate of the class probabilities for a given sample x, one solution is to use a Dirichlet distribution to model the probability distribution of these class probabilities. To this end, one should define a relationship between the NN output a and the Dirichlet parameters in the form α = Φ(a), where Φ must comply with the strictly positive constraint of α. We consider Φ(a) = ReLU(a) + 1 which satisfies both conditions. To any input data x i we associate the random variable

P i = (P i1 , ..., P iK ) ∼ Dir(p i |α i ), (1)
where

α i = Φ(a i ) = ReLU(f (x i |θ)) + 1.
The class prediction ŷi is given by the mean of the Dirichlet distribution

Dir(p i |α i ), that is, ŷi = αi Sα i = αi1 Sα i , . . . , α iK Sα i .
We used a similar idea in [START_REF] Simonnet | Proportion inference using deep neural networks. applications to x-ray diffraction and hyperspectral imaging[END_REF] to estimate proportions in a mixture for X-Ray diffraction and hyperspectral imaging.

The problem considered in this paper is different, since it is a classification problem for which we want to estimate the uncertainty at the same time.

Loss function

To train the neural network, we must define a loss function using the Dirichlet model [START_REF] Loeser | Osteoarthritis: a disease of the joint as an organ[END_REF]. As in [START_REF] Simonnet | Proportion inference using deep neural networks. applications to x-ray diffraction and hyperspectral imaging[END_REF], we consider minimizing the mean square error. That is for any input x i , minimize the expectation of the squares of the errors between y i and P i ,

L SE i (θ) = E( y i -P i 2 ) = y i -ŷi 2 + Var(P i ), (2)
where Var(P i ) = K j=1 Var(P ij ). This loss aims to achieve the joint goal of minimizing the prediction error and the variance of the Dirichlet distribution output by the NN.

In addition, we also need to ensure that the NN has the expected behaviour under foreign data inputs, by reinforcing for example the flatness of the Dirichlet distribution. We consider a regularization term in the loss function, consisting of the KL divergence with respect to the uniform Dirichlet distribution [START_REF] Sensoy | Evidential deep learning to quantify classification uncertainty[END_REF]. The overall loss considered is then,

L i (θ) = L SE i (θ) + λ t KL [Dir(p i | αi )||Dir(p i |1)] , (3)
where Dir(p i |1) refers to the uniform Dirichlet distribution, αi = y i + (1 -y i ) α i (with referring to the elementwise product) are the Dirichlet parameters after removing the non-misleading evidence from predicted parameters α i for sample i, λ t = min(1, t/k) is an annealing coefficient, t is the index of the current training epoch and k is a fixed number (often set to k = 10).

The KL divergence in the loss (3) exhibits a closed-form,

KL [Dir(p i | αi )||Dir(p i |1)] = log      Γ K k=1 αik Γ (K) K k=1 Γ (α ik )      + K k=1 (α ik -1) ψ (α ik ) -ψ K k=1 αik ,
where ψ(•) is the digamma function.

EXPERIMENTAL SETUP

We evaluate our method on the publicly available Osteoarthritis Initiative (OAI) data set. This cohort recruited 4796 participants with age ranging from 45 to 79. As the OAI is a multi-center study, the physical resolution and dimension of the knee X-ray images collected from the baseline cohort are not homogeneous. Pre-processing is required to ensure that all images have the same size. We use the 4130 X-ray images with 8260 knee joints from [START_REF] Chen | Fully automatic knee osteoarthritis severity grading using deep neural networks with a novel ordinal loss[END_REF]. The data distribution is represented in Fig. 3 below.

Fig. 3. Dataset distribution

Dataset preparation

The dataset of X-ray images of knee joints is not suitable in terms of clarity and localization as input for the DL models. It is therefore necessary to perform a data pre-processing step, during which the images are transformed to clearly capture the joint area where OA information is likely to exist. This involves first cropping the image to the desired region of the knee, so that all undesirable regions are excluded. To achieve this, we cropped the images by 60 pixels top and bottom, resulting in images of size 224 × 104 × 3. The second step was to perform histogram equalization to improve contrast and ensure good visibility of the desired areas.

We randomly assigned images to the training, validation, and test sets with respectively the following percentages 64%, 16% and 20%. All data were normalized, and we also used a data augmentation scheme in the training phase using a horizontal random flip with a probability of 0.5.

Training phase

We conduct the experiments with the ResNet-101 architecture [START_REF] He | Deep residual learning for image recognition[END_REF]. The standard softmax layer is replaced by a ReLU whose output is used as an evidence vector for the Dirichlet distribution. The network is trained from scratch for 200 epochs in experiments 1 (described below), and 500 in experiments 2 and 3. We choose Adam as the optimizer, with an initial learning rate of 10 -3 and a weight decay of 10 -4 to avoid overfitting. We also choose a learning rate schedule, decreasing the learning rate by 10% every 20 epochs.

Performance metrics

In our experiments, quantitative evaluation was performed using four different metrics:

1. Accuracy: percentage of predictions that match exactly the ground-truth.

2. Precision: fraction of true positives among predicted positives.

3. Recall: fraction of the total number of true positives retrieved.

4. F1-score: harmonic mean of precision and recall.

We now present the results on different test sets derived from the original data.

NUMERICAL RESULTS

In this section, the performance of the proposed method is assessed on binary and multi-class classification tasks.

Binary classification

First, the model has been trained to detect two classes. We consider two classification scenarios .

• Experiment 1. The aim of this experiment is to detect knee OA at an early stage. We therefore classify normal patients (KL0) versus those with mild OA (KL2). The test database includes 1086 images.

• Experiment 2. This aims at detecting the presence/absence of knee OA. We create a binary dataset by combining classes KL0 and KL1 to represent negative diagnosis of KOA (denoted as "Normal"); KL2, KL3, and KL4 are combined to represent positive diagnosis ("Abnormal"). There are 1656 images in the test database.

Multi-class classification

• Experiment 3. The aim of this third experiment is to determine the severity of KOA. We therefore remove classes KL0 and KL1 and consider classes KL2, KL3, and KL4. The number of images in the test database is 721.

Table 1 summarizes the results of the three classification experiments on the test set. Note that the table can be misleading for our approach, since totally uncertain predictions (i.e. u = 1) are also considered failures when computing overall accuracy. Despite this, the performance results are good, and quite similar for Experiments 1 and 2. This is promising, as experiment 1 is challenging due to the similarity between KL0 and KL2 images. This also shows the model's ability for early detection. The results of experiment 3 are slightly less good. Confusion matrices (not shown here) indicate large misclassifications of knee joints categorized as KL3 (moderate) and predicted as KL2 (minimal). These images show minimal variations in terms of joint space width and osteophytes formation, making them challenging to distinguish. Moreover, due to the very unbalanced data distribution (see Fig. 3), we have much less data for these classes, making the classification task even more difficult.

Uncertainty quantification

The strength of our approach lies in its ability to quantify uncertainty. We first examine the average uncertainty for right and wrong predictions. The results are presented in Table 2 for the three experiments. Our model associates higher uncertainty to erroneous predictions, in all the three data sets. Uncertainty is also higher for experiment It may be useful to look at the images and the results obtained when the classification results matches the groundtruth. Fig. 4 shows that the proposed method in this case produces high probability and low uncertainty, indicating a confident classification. Finally, we examine the results obtained in the case of misclassification. Fig. 5 displays some of these results. In the case of a severe misclassification in Experiment 3, for example with a KL4 (severe) grade predicted as KL2 (minimal), the model produces a high uncertainty (u = 0.57).

CONCLUSION AND PERSPECTIVES

In this paper, we have proposed a new classification method for knee OA diagnosis. These preliminary results are promising since classification task can be challenging particularly when dealing with complex data such as X-ray images used for the knee OA diagnosis. Being able to quantify uncertainty is a key point of the proposed method There are, however, some limitations. The dataset was relatively small and highly unbalanced. Another limitation was the lack of clean classes due to the high similarity of Xray images particularly in early stages of knee OA. Future work will be devoted to explore other kind of data augmentation [START_REF] Wang | Key-exchange convolutional auto-encoder for data augmentation in early knee osteoarthritis classification[END_REF], and combining texture and shape information [START_REF] Nasser | A discriminative shape-texture convolutional neural network for early diagnosis of knee osteoarthritis from x-ray images[END_REF].

Fig. 1 .

 1 Fig. 1. Radiological grades of OA according to the KL scale very challenging.It is therefore necessary to develop automated methods to facilitate the knee OA diagnosis. Several deep learning (DL) have been proposed to this aim. Most of them are based on convolutional neural networks (CNN). The latter have become state-of-the-art technology for image classification tasks due to their ability to capture spatial information in images. One can refer to[START_REF] Nasser | Discriminative regularized autoencoder for early detection of knee osteoarthritis: Data from the osteoarthritis initiative[END_REF] for a brief review of current CNN-based methods for knee OA. Nevertheless, one major drawback of CNN models is their dependency on large amounts of labelled data. In the field of medical imaging the small quantity of valid medical datasets, due to the high cost of labelling, still poses a major challenge for training the learning model. In this context it is crucial to quantify the uncertainty associated with the model prediction.In classical CNN for classification, the softmax function is used to predict class assignment probabilities. A softmax output is however often misinterpreted as an indication of model confidence. Nevertheless a model can be uncertain in its predictions even with a high softmax output for a particular class[START_REF] Gal | Dropout as a bayesian approximation: Representing model uncertainty in deep learning[END_REF]. Bayesian neural networks can address this issue by putting a prior distribution on the neural network parameters and inferring their posterior distribution using approximations such as Variational Bayes[START_REF] Gal | Dropout as a bayesian approximation: Representing model uncertainty in deep learning[END_REF]. The posterior predictive distribution is approximated using Monte-Carlo sampling methods. These models are however computationally demanding since each data point has to pass through the network at several times, and also require in addition considerable modifications to existing baselines.Recently, a new class of models based on the concept of evidential deep-learning (EDL) has been proposed[START_REF] Sensoy | Evidential deep learning to quantify classification uncertainty[END_REF],[START_REF] Malinin | Predictive uncertainty estimation via prior networks[END_REF]. They allow to compute uncertainty in a single forward pass by parameterizing distributions on distributions, and involve minimal modifications to the architecture of standard neural networks.Is this paper, we develop an EDL-based method for knee OA diagnosis. The potential of the proposed approach is eval-

Fig. 2 .

 2 Fig. 2. Schematic representation of the NN.

Fig. 4 .

 4 Fig. 4. Results with correctly classified data.

Fig. 5 .

 5 Fig. 5. Results with misclassified data

Table 1 .

 1 Model performance on the three experiments.

	Experiment Accuracy F1score Precision Recall
	1	0.72	0.71	0.72	0.70
	2	0.73	0.73	0.73	0.73
	3	0.66	0.64	0.62	0.66

Table 2 .

 2 3, for which the classification results are less good. Average uncertainty for right and wrong predictions.

	Experiment Right prediction Wrong prediction
	1	0.14	0.23
	2	0.18	0.34
	3	0.43	0.59

Acknowledgement

This work was supported by the French National Agency of Research (ANR) through the MIMOSA project (ANR-20-CE45-0013) and the BACKUP project (ANR-23-CE40-0018-01).


Montana Sergio

Michiorri Andrea

Long-Term Dynamic Thermal Ratings of Underground Cables Integrating Soil Dynamics and Climate Projections

Keywords: Underground Cables, Climate change, Dynamic Thermal Rating, Transmission

Underground Cables capacity is influenced by thermal properties, particularly the interplay between dynamics of the physical properties and the surrounding soil's heat dissipation capabilities. This paper introduces a new approach to quantifying the long-term effect of climate conditions and soil dynamics on underground cable capacity that leverages machine learning techniques to estimate soil temperature and moisture in the depth range of 0.8 to 1.2 meters. The research extends its focus to assess the impact of climate change using various representative concentration pathways across 52 European stations. The main strength of this model lies in its ability to offer detailed estimations by capturing the relationships among different types of soils. The results show that the quasi Dynamic Thermal Rating methodology produces an average increase in ratings of 7.1%. In addition, it has the potential to further widen the disparity between winter and summer seasons, with an increase of up to 8.1%. Regarding climate impacts, there are moderate reductions over the medium-term emissions horizon, with an average reduction of 0.27% to 0.63%.

Introduction

The rising adoption of Underground Cables (UGC), driven by expanding renewable generation plants [START_REF] Sugihara | Fundamental analysis of electrothermal coordination of underground cables toward high-penetration renewable generation[END_REF], poses a challenge for a warmer, energy-demanding future [START_REF] Van Ruijven | Amplification of future energy demand growth due to climate change[END_REF]. Meteorological factors and climate changes significantly affect UGC capacity, mainly reliant on the surrounding soil's effectiveness in dissipating joule losses. This heat transfer mechanism, impacted by moisture and soil thermal resistivity, holds the potential for up to 50% capacity reduction [START_REF] Brakelmann | Ampacity reduction factors for cables crossing thermally unfavorable regions[END_REF], which cannot be captured using the traditional approach of energy utilities for capacity limits. In this context, large-scale soil factor measurements play a crucial. However, only historical high-resolution models combining data with observations have emerged in the last decade [START_REF] Sungmin | Global soil moisture data derived through machine learning trained with in-situ measurements[END_REF][START_REF] Sabater | ERA5-Land hourly data from 1950 to present[END_REF], with a gap in long-term projections for burial depths between 1 and 5 meters. This underscores the significance of our research in understanding the complexities of climate-induced changes and their implications for the transmission sector over the long term.

To cope with the effects of temperature rise and optimize cable current carrying capacity, Dynamic Thermal Rating (DTR) [START_REF] Wang | Dynamic thermal analysis for underground cables under continuously fluctuant load considering time-varying van wormer coefficient[END_REF][START_REF] Diaz-Aguiló | Introducing mutual heating effects in the ladder-type soil model for the dynamic thermal rating of underground cables[END_REF][START_REF] Olsen | Modelling of dynamic transmission cable temperature considering soil-specific heat, thermal resistivity, and precipitation[END_REF][START_REF] Wang | Dynamic thermal analysis of high-voltage power cable insulation for cable dynamic thermal rating[END_REF][10][START_REF] Anders | Transient ratings of buried power cables. i. historical perspective and mathematical model[END_REF][START_REF] Karimi | Dynamic thermal rating of transmission lines: A review[END_REF][START_REF] Morozovska | Dynamic rating with applications to renewable energy[END_REF] is actively proposed as a technology that aims to identify the real-time current carrying capacity to optimize asset utilization, reduce congestion and improve efficiency as a function of environmental conditions such as ambient and soil temperature. It is crucial to emphasize that the primary objective, as presented in [START_REF] Wang | Dynamic line rating systems for transmission lines: Topical report[END_REF], positions DTR as a strategic tool for system operators. It aids in accurately and reliably determining current rating limits, thereby alleviating constraints based on thermal considerations.

As the temperature is the primary variable in the DTR, weather or mechanical characteristics can be determined from two methods: i) Real-time measurement equipment, known as direct. Usually performed nowadays for brand new circuits with an embedded optical fiber as described in [START_REF] Cherukupalli | Types of Power Cables and Cable with Integrated Fibers[END_REF], offering excellent accuracy and precision, but with barriers such as capital costs and non-applicability to already installed networks. ii) Thermal mathematical model based on information from meteorological conditions, with a non-steady state thermal model using historical data or numerical weather predictions [START_REF] Douglass | Increased power flow through transmission circuits: Overhead line case studies and quasidynamic rating[END_REF]. Deployed as a method that increases capacity compared to static thermal power but reduces the risk of overloading.

When this model is applied, meteorological factors and climatic changes have a discernible impact on the DTR ratio [START_REF] Olsen | Modelling of dynamic transmission cable temperature considering soil-specific heat, thermal resistivity, and precipitation[END_REF], as established by well-known international standards such as IEEE Std 1863 [START_REF]IEEE Guide for Thermal Resistivity Measurements of Soils and Backfill Materials[END_REF] and IEC 60287 [START_REF] Commission | Electric cables -Calculation of the current rating -Part 1-1: Current rating equations (100 % load factor) and calculation of losses -General[END_REF]. This impact is vital in defining the parameters used for rating estimation, described by energy conservation and heat transfer mechanisms [19,[START_REF]Handbook of applied hydrology[END_REF] of the cable's physical properties and the surrounding soil's heat dissipation capabilities. Notably, the heat balance highlights the challenge of low heat dissipation in UGC exposed directly to the environment (direct soil buried). Understanding and quantifying these factors is critical to assessing UGC systems' thermal performance and overall efficiency.

A crucial parameter influencing the rating within the soil's thermal properties is the soil thermal resistivity ( o Cm/W), highly correlated with the moisture content [START_REF]IEEE Guide for Thermal Resistivity Measurements of Soils and Backfill Materials[END_REF][START_REF] Commission | Electric cables -Calculation of the current rating -Part 1-1: Current rating equations (100 % load factor) and calculation of losses -General[END_REF][START_REF] Dorison | Ampacity calculations for deeply installed cables[END_REF]. This effect becomes particularly significant when its impact is translated into temperature. In [START_REF] Brakelmann | Ampacity reduction factors for cables crossing thermally unfavorable regions[END_REF], a 50% reduction in the rating is observed when the soil thermal resistivity is three times greater and the ambient temperature increases by 5 o C. Reinforcing its high dependence and correlation between temperature and humidity of the surrounding soil, among other factors, such as laying depth and physical properties [START_REF] Cosby | A statistical exploration of the relationships of soil moisture characteristics to the physical properties of soils[END_REF].

The utilization of in-situ sensors for measuring soil moisture and temperature at various spatial scales, as discussed in [START_REF] Dorigo | The International Soil Moisture Network: a data hosting facility for global in situ soil moisture measurements[END_REF][START_REF] Tavakol | The soil moisture data bank: The ground-based, model-based, and satellite-based soil moisture data[END_REF], presents an opportunity to comprehend the interrelationship and devise a solution to estimate the intricate interaction in exchange processes between land-atmosphere interactions and soil. Consequently, considerable research efforts have been directed towards this objective over the past few decades. Noteworthy contributions include satellite observations [START_REF] Dorigo | Esa cci soil moisture for improved earth system understanding: State-of-the art and future directions[END_REF], microwave remote sensing, and physically-based models [START_REF] Balsamo | A revised hydrology for the ecmwf model: Verification from field site to terrestrial water storage and impact in the integrated forecast system[END_REF], providing spatiotemporal estimations. However, these methods may experience deviations in performance due to complex topography and transient climate conditions [START_REF] Sungmin | Global soil moisture data derived through machine learning trained with in-situ measurements[END_REF][START_REF] Dorigo | Esa cci soil moisture for improved earth system understanding: State-of-the art and future directions[END_REF][START_REF] Dutra | Robustness of process-based versus data-driven modeling in changing climatic conditions[END_REF]. On the other hand, Machine-Learning models (ML) employing remote sensing data have been applied in various research studies [START_REF] Sungmin | Global soil moisture data derived through machine learning trained with in-situ measurements[END_REF][START_REF] Hengl | Soilgrids250m: Global gridded soil information based on machine learning[END_REF][START_REF] Elsaadani | Assessment of a spatiotemporal deep learning approach for soil moisture prediction and filling the gaps in between soil moisture observations[END_REF][START_REF] Taheri | A review of machine learning approaches to soil temperature estimation[END_REF][START_REF] Delbari | Modeling daily soil temperature over diverse climate conditions in Iran-a comparison of multiple linear regression and support vector regression techniques[END_REF][START_REF] Rani | Chapter 6 -machine learning for soil moisture assessment[END_REF] with a broad spectrum of data integration. These models, primarily influenced by climatic conditions and soil taxonomy, including texture and soil type, present a novel alternative that challenges conventional processes. By "learning" the intricate interdependencies between meteorological variables and soil dynamics, they redefine the approach to understanding and modeling these intricate connections.

A limited number of strategies have been developed, incorporating ML in DTR for UGC [START_REF] Huang | Use of day-ahead load forecasting for predicted cable rating[END_REF][START_REF] Huang | Dynamic cable ratings for smarter grids[END_REF]. As outlined in [START_REF] Bracale | Day-ahead and intraday forecasts of the dynamic line rating for buried cables[END_REF], an approach that involves the development of a day-ahead DTR forecast relies on a Support Vector Regression (SVR) method for soil temperature. However, the physical-statistical model for predicting thermal conductivity, applying a regression model for soil-specific heat, exhibits reduced simulation accuracy across diverse soil types, limiting its general applicability. To analyze the thermal conductivity and cover a wide range of soil texture, bulk density, temperature, and moisture, in the last decades, several estimations models have been proposed [START_REF] Campbell | Soil Physics with BASIC: Transport Models for Soil-Plant Systems[END_REF][START_REF] Devries | Thermal Properties of Soils[END_REF][START_REF] He | A modified normalized model for predicting effective soil thermal conductivity[END_REF][START_REF] Xiong | A new model to predict soil thermal conductivity[END_REF][START_REF] Lu | An Empirical Model for Estimating Soil Thermal Conductivity from Texture, Water Content, and Bulk Density[END_REF][START_REF] Johansen | Thermal conductivity of soils[END_REF]. In [START_REF] Ren | A new empirical model for the estimation of soil thermal conductivity[END_REF], a new empirical model for estimating soil thermal conductivity was developed. For this, a comprehensive evaluation of several approaches was performed and is influenced by many factors such as organic matter, soil texture, and particle composition. Resulting in a performance of R 2 of 0.98, increased simulation accuracy for ten soil types.

As temperature and soil moisture conditions become increasingly common variables in planning and managing the capacity of UGC [START_REF] Olsen | Modelling of dynamic transmission cable temperature considering soil-specific heat, thermal resistivity, and precipitation[END_REF], establishing projections related to the energy transition is a critical step in quantifying and understanding the impacts associated with climate change. To tackle the issues outlined above, the contributions of this paper are: a) Introduce a probabilistic DTR method aiming to exploit transmission capacity efficiently and minimize the risk of overloading as part of long-term power system planning c) Assess trends and possible impacts of climate change on UGC in the long term horizon.

The paper is organized as follows: Section 2 illustrates the machine learning process, thermal model, and methodology. Results are described in Section 3, and conclusions are drawn in Section 4.

Methodology

Overview

We develop a model to quantify the long-term effect of climate conditions on UGC capacity, summarized in the block diagram Fig. 1. First, supervised ML models are trained using historical in-situ measurements [START_REF] Dorigo | The international soil moisture network: serving earth system science for over a decade[END_REF] and meteorological information [START_REF] Sabater | ERA5-Land hourly data from 1950 to present[END_REF][START_REF] Hiederer | Mapping Soil Properties for Europe -Spatial Representation of Soil Database Attributes[END_REF] in module 2.2. Secondly, meteorological time series data from both historical [START_REF] Dorigo | The international soil moisture network: serving earth system science for over a decade[END_REF][START_REF]Climate and energy indicators for Europe from 2005 to 2100 derived from climate projections[END_REF] and machine learning projections are collected in module 2.3. These data are fed to components' thermal models to calculate the time series for the transmission capacity in modules 2.5,2.4. Subsequently, in the final module 2.6, a probabilistic approach integrated into DTR is proposed [START_REF] Salas | Weather-Based Quasi Dynamic Thermal Ratings for Power Transformers[END_REF][START_REF] Hadiwidjaja | Quasi-Dynamic Line Rating spatial and temporal analysis for network planning[END_REF] by fitting a probability distribution to the lower tail of the simulations for each month/hour combination and selecting an accepted level of risk tolerance to thermal overload (0.1% in this case).

Machine Learning Model

This module aims to target soil moisture ( ψ) and temperature ( Θs ) data at 0.25 • , with a daily resolution in a depth range between 0.8 and 1.2 meters based on hourly-recorded input data. The pre-selection of the features Figure 1: Block scheme of the approach proposed for calculating quasi Dynamic Thermal Ratings in UGC.

was performed under a combination of time series of meteorological data, soil properties, and observations, mainly based on bibliographic analysis [START_REF] Sungmin | Global soil moisture data derived through machine learning trained with in-situ measurements[END_REF][START_REF] Elsaadani | Assessment of a spatiotemporal deep learning approach for soil moisture prediction and filling the gaps in between soil moisture observations[END_REF][START_REF] Taheri | A review of machine learning approaches to soil temperature estimation[END_REF][START_REF] Rani | Chapter 6 -machine learning for soil moisture assessment[END_REF]. To ensure the spatiality of the system throughout the European region, we rely on the open sources described in the section 2.3.

Due to the problem's complexity, we implemented the following preprocessing and numerical transforms: a) Anomaly extraction. b) Eliminating poor-quality data [START_REF] Dorigo | Global automated quality control of in situ soil moisture data from the international soil moisture network[END_REF] c) Excluding stations with less than three months of annual information d) Daily interpolation e) Encoding cyclical Day of the Year (t c ) f) Minimum and maximum scaling [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF] f) Feature selection conducted to achieve the highest prediction accuracy. g) Finally, to ensure broad operability, each analyzed soil is encoded (S code ) by soil texture classification according to [START_REF]Soil Taxonomy: A Basic System of Soil Classification for Making and Interpreting Soil Surveys[END_REF] and texture represented in Table 1.

In the initial phase, an evaluation of various machine-learning techniques was carried out in order to select the most appropriate for our method. This assessment focused on key error measurement indicators such as Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE), with particular emphasis on the computation time of the predictions. Evaluation metrics are given with the following equations:

RMSE q = 1 H H i=1 (y q,i -ŷq,i ) 2 (1a) MAE q = 1 H H i=1 |y q,i -ŷq,i | (1b)
In the equations, ŷi denotes the estimate of humidity and temperature, while y i represents the observation for the same variables from ISMN stations. The subscript H signifies the projected horizon of the estimation. As described in the previous section, a reduction in UGC ratings is associated with lower humidity and higher temperature. Thus, our analysis focuses on the heightened risk of overloading, the metrics described in equations , are evaluated the lower 20th quantile for humidity and the 80th quantile for temperature, and denoted as q.

0-20 0-20 ≥ 60 1
The importance of the calculation time is due to the expected regional application of the continentalscale model, highlighting the need for accuracy and efficiency. The techniques evaluated include Artificial Neural Network (ANN) [START_REF] Lecun | Deep learning[END_REF], Regressions Tree (RT), K-Nearest Neighbours (k-NN), Extreme Gradient Boosting (XgBoost) [START_REF] Chen | XGBoost: A scalable tree boosting system[END_REF], Random Forest (RF) [START_REF] Breiman | Random forests[END_REF], Long short-term memory (LSTM), and Lasso linear regression [START_REF] Tibshirani | Regression Shrinkage and Selection Via the Lasso[END_REF].

Considering the soil-inertia nature in the output variables as detailed in [START_REF] Sungmin | Global soil moisture data derived through machine learning trained with in-situ measurements[END_REF], this model is supervised-trained with observations lagged by five days. This lag was determined through time series analysis using a partial auto-correlation function (PACF) [START_REF] Seabold | statsmodels: Econometric and statistical modeling with python[END_REF]. The dataset is divided by stations and soil types to train and test the models. The training set is then randomly divided into 70%, with 10% reserved for tuning the model hyperparameters using techniques from [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF][START_REF] Akiba | Optuna: A next-generation hyperparameter optimization framework[END_REF]. The remaining 30% serves as a validation set to test the model. To avoid overfitting, N-fold cross-validation is performed by applying the metrics described in (1) by dividing the test dataset into five folds, ensuring all soil types are present in each fold, thus guaranteeing homogeneity of selection.

Datasets

Quantifying the impact on UGC capacity is conducted by processing data derived from a combination of soil properties and meteorological variables of the interest region, as outlined in Table 2.

Considering 100year climatologists, ECMWF ERA5 and Copernicus climate change service (C3S) datasets provide the historical reanalysis (a) [START_REF] Sabater | ERA5-Land hourly data from 1950 to present[END_REF] and climate projections with a Representative Concentration Pathway (RCP) of 2.6, 4.5, and 8.5 (b) [START_REF]Climate and energy indicators for Europe from 2005 to 2100 derived from climate projections[END_REF]. The high-grid resolution of 0.25 • datasets provides a time series from January 01, 1970, to December 31, 2072. However, climatic projections are linearly interpolated with a time resolution of one day to match the frequency of the In-Situ data.

Observing temporal variations in soil is crucial for estimating soil moisture and temperature behavior and assessing sensitivity to climate change and soil taxonomic properties. This mandates the availability of soil moisture datasets that exhibit superior quality, extended duration, continuity, and consistency. In this study, in situ soil moisture (d) , taxonomy (c,d) , and temperature measurements (d) sourced from the International Soil Moisture Network [START_REF] Dorigo | The international soil moisture network: serving earth system science for over a decade[END_REF] and [START_REF] Hiederer | Mapping Soil Properties for Europe -Spatial Representation of Soil Database Attributes[END_REF] are employed. The dataset involves data from 52 networks, spanning the years 2006 to 2022 and covering depths from 0.05 to 120 m.

Steady-State Thermal model with Moisture Migration

In line with standard procedures for quantifying the ampacity of buried UGC, a thermo-electric model from [START_REF] Commission | Electric cables -Calculation of the current rating -Part 1-1: Current rating equations (100 % load factor) and calculation of losses -General[END_REF] is employed. According to this standard, the capacity, with the influence of a dry area formation, is calculated per each day (i) as follows: The above equation focuses on the critical temperature of the boundary between the wet and dry zones ∆θ x and the ratio between the thermal resistivities of the dry and wet zones of the back-fill soil v i . With high dependence on variable factors such as ambient temperature, moisture, and precipitation, the other parameters such as λ 1-2 , C, R, T 1-3 , W d obey the cable construction. These parameters can be determined and calculated according to [START_REF] Brakelmann | Ampacity reduction factors for cables crossing thermally unfavorable regions[END_REF][START_REF] Commission | Electric cables -Calculation of the current rating -Part 1-1: Current rating equations (100 % load factor) and calculation of losses -General[END_REF], guide and suggestions.

B = W d 1 2 T 1 + n (T 2 + T 3i + v i T 4i ) (2a) C = nR i (1 + λ 1 + λ 2 ) (T 2 + v i T 4i ) (2b)
I i = ∆Θ i -B + (v -1)∆Θ x R i T 1 + nR i (1 + λ 1 )T 2 + C ∀i = 1..H (2c)
Where the thermal resistance T 4 of the surrounding medium for a single core cable, lay horizontally, is defined as

T 4i = ρ si 2π ln u + u 2 -1 + ln 1 + 2L S 1 2 (3)
Where ρ si is the thermal resistivity of soil, S 1 spacial axis distance between cables, u is the ratio of the distance from the surface to the cable axis L, and the external diameter of the cable D e . Hence, it is imperative to calculate ρ si , and the analysis must also incorporate considerations based on the cable environment. In conclusion, the relationship between the buried UGC's capacity and the input parameters can be described as a function of Eq. ( 4). This relationship holds true for uniform soils, where thermal properties remain constant at each coordinate.

I DT Ri = I(Θ i , ρ si , ψ i , constant parameters) (4)

Dynamic Soil Model

Modeling soil moisture and temperature dynamics is an area that has been developed over the last century. Differential equations involving consideration of various factors have been presented, as described in the previous section. A one-dimensional representation of heat flow in a vertical direction (x) is outlined in [START_REF] Sabater | ERA5-Land hourly data from 1950 to present[END_REF], in which the thermal diffusivity α s is characterized as a direct relationship between the thermal conductivity (λ s ) and the inverse of the soil density and specific heat capacity. Our approach will implement the dynamics model addressing the physical properties and processes within soils using machine learning, as described in Section 2. However, a mathematical model correlating soil hydraulic-physical properties with thermal conductivity (λ s ) needs to be established to determine the thermal resistivity of each analyzed soil.

∂Θ ∂t = α s ∂ 2 Θ ∂x 2 (5)
To address this, the estimation of soil thermal conductivity, which is the inverse of ρ s , is conducted following the interrelation proposed in [START_REF] Johansen | Thermal conductivity of soils[END_REF][START_REF] Ren | A new empirical model for the estimation of soil thermal conductivity[END_REF]. This involves unsaturated soils and incorporates parameters such as λ dry , λ sat , and the Kersten coefficient K e .

λ si = (λ sati -λ dry i )K ei + λ dryi ∀i = 1..H (6a) K ei = exp (α -ψ -β i ) (6b)
Here, α and β represent the shape factors of the volumetric water content (ψ) and thermal conductivity curve, respectively. These factors exhibit linear relationships with particle size distribution (S % ) and organic matter content (S org % ), as defined below:

α = a 1 S Sand % + a 2 S Silt % + a 3 S org % + a 4 (7a) β = b 1 S Clay % + b 2 S org % + b 3 (7b)
Where, the weighting factors of the physical parameters of the model are as follows At this point, the thermal conductivity is calculated for each soil using the features described in Table 2. This, in turn, allows us to estimate the v ratio and the thermal resistivity of the soil ρ s per day.

Quasi Dynamic Thermal Ratings

Finally, a quasi-dynamic Thermal Ratings (qDTR) is applied to calculate ratings for a long-term horizon, as proposed in [START_REF] Salas | Weather-Based Quasi Dynamic Thermal Ratings for Power Transformers[END_REF][START_REF] Hadiwidjaja | Quasi-Dynamic Line Rating spatial and temporal analysis for network planning[END_REF]. After collecting the time series and parameters described in Sections 2.3 and 2.5, the DTR is calculated using (2),(3). These time series are fitted with a probability distribution described in [START_REF] Clauset | Power-law distributions in empirical data[END_REF], in the lower tail of the simulations for each month/day combination. An accepted level of tolerance to the risk of thermal overload (0.1% in this case) is then selected.

Figure 2: The conceptual illustrations of this proposed method display the DTR calculated throughout a year (2000-2022) for a specific coordinate, represented in blue. The STR H is depicted in black, the monthly qDTR H in green, and the qDTR Hs for the winter and summer seasons in indigo and orange, respectively This method is conceptually illustrated in Fig. 2. First, the thermal model calculates the maximum capacity-DTR over a 1-year horizon. In contrast plotted as a dash black line, the Static Thermal Rating (STR H ) applies a fixed value as a constrained capacity limit throughout the year, under-utilizing the UGC and potentially overloading, regardless of changing conditions, with a 0.1% exceedance probability. The qDTR H is then calculated with the same risk of exceeding over the lower tail values of the DTR across a 50-year time horizon for a monthly/yearly approach, as shown in the central figure. Finally, the qDTR Hs is calculated for winter and summer with the same industry seasonal approach.

Results

This study applies qDTR to estimate the maximum allowable currents of UGC using historical (from 1970 to 2022) and future (from 2023 to 2070) weather projections for Europe. The HV cable used in this paper is described in Table 4 obtained from [START_REF]High and extra high voltage cables[END_REF] and complemented from [START_REF] Commission | Electric cables -Calculation of the current rating -Part 1-1: Current rating equations (100 % load factor) and calculation of losses -General[END_REF], with a manufacturer reference A2X(F)K2Y-800RM.

mm

Firstly, to apply qDTR, the data described in 2.3 needs retrieval. For projecting humidity and temperature based on each representative concentration pathway, a machine learning model with a daily step was trained, validated, and selected as outlined in 2.2.

The performance of each model was compared with the ERA-5 data, which were used as a benchmark. Results not shown in this paper indicate that techniques such as KNN and LST, with deviations of +0.05m 3 m -3 over the observations, increase the computational cost. On the other hand XGBoost [START_REF] Chen | XGBoost: A scalable tree boosting system[END_REF], as illustrated in [START_REF] Olsen | Modelling of dynamic transmission cable temperature considering soil-specific heat, thermal resistivity, and precipitation[END_REF], and Lasso [START_REF] Tibshirani | Regression Shrinkage and Selection Via the Lasso[END_REF] described in [START_REF] Wang | Dynamic thermal analysis of high-voltage power cable insulation for cable dynamic thermal rating[END_REF], exhibited the most favorable performance for humidity and temperature, in the techniques evaluation process. This was evidenced by achieving the lowest error values among all models, outlined in Table . 5 and6

. ψi = K k=1 f k (x i ) (8a)
x i = [[w s , Θ a , S r , t s ] i-5:i-1 , S % , S code1-5 ] (8b)
Where x i is the flattened feature vector for the i-th observation

Θi = x i • βL (9a) βL = arg min β    1 2H H i=1 (Θ i -x i • β) 2 + λ p j=1 |β j |    (9b)
x i = [[Θ a , S r , t s ] i-5:i-1 , S % , S code1-5 ] (9c)
β is the coefficient vector associated with the features, βL is the estimated parameter vector using LASSO and λ is the regularization parameter. The results are derived from diverse station selected through cross-validation, as detailed in Table 7. This table provides information on the various soil codes used. For a graphical representation of the performance across different soil codes, refer to in Fig. 3. At first glance, the XGBoost Moisture model demonstrated lower average error values than the ERA5 model. It is essential to note that MAE has the same units, m 3 m -3 . With marginally superior performance, the Lasso temperature models denote the robustness of ERA5, establishing themselves as a dependable choice for historical references.

From here, we quantify the climate impact of transmission network capacity for underground cables. The qDTRs are fitted with a probability distribution commonly employed to characterize natural phenomena [START_REF] Clauset | Power-law distributions in empirical data[END_REF]. Specifically, the power law distribution is applied to the lowest 2% of the calculated ratings for each month/day combination. Across all month/hour combinations, considering an exceedance probability of 0.1%, the results show that the coefficient of determination (r 2 ) exceeds 0.9. This procedure is applied to the three projected RCP scenarios, revealing a moderate but noticeable effect of temperature increase according to climate projections, as shown in Fig. 4. Preliminary results suggest that colder temperatures, increased precipitation, and reduced irradiance levels contribute to increased capacity during winter. For instance, in the qDTR Hw scenario, the increase may be as high as 8.1%. Additional analysis of this chart reveals a decrease -albeit small -of 0.56% in ∆ % (qDTR RCP4.5 , qDTR H ) for this specific location. Conversely, a temporal shift in the usual annual temperature cycle is noticeable, with lower capacity observed around September. This phenomenon is attributed to the thermal memory and inertia inherent in buried cabling, causing a time delay in temperature exchange between different layers. The same inertia translates into low daily variability, suggesting that these devices can be effectively analyzed at lower frequencies. This is exemplified by the dashed line, which indicates a monthly (qDTR H month ) analysis approach. Finally, the same procedure is applied on a country scale, and the results are summarized in Table 8 and geographically represented in Fig. 6. The UGC location retrieved from [START_REF] Brown | PyPSA-Eur: An open optimisation model of the european electricity system[END_REF] and described as average, maximum decrease for the qDTRs calculated for the ten lines. It is possible to observe an average reduction in ratings in the order of 0.16%, 0.27%, and 0.33%, respectively, for the 2.6, 4.5, and 8.5 RCP. With maximum peaks in the high emission scenario of 1.38 %

The research investigates the methodology employed in [START_REF] Montana | The climate change impact on power grid transmission capacity[END_REF], while also offering a broader perspective on additional components.

Conclusions

Through an assessment conducted at a grid resolution, this analysis investigates the impact of climate on transmission network capacity using established thermal models. Employing the Quasi-Dynamic Thermal Rating method to estimate maximum capacity with a low-risk exceedance probability, Our findings highlight moderate reductions in the mid-term emission horizon for underground cables, showing sensitivity to potential climate changes, with a mean reduction from 0.27% to 0.63%. It reflects that the variation caused by the meteorological conditions in the lower risk scenario (0.1%) is reduced at depth buried cables. Conversely, the In addition, this paper estimated soil moisture and soil temperature at a depth of 1 meter. The results, obtained through 5-Fold validation, demonstrate mean absolute errors of 0.08 (± 0.02) m 3 m -3 for soil moisture and 2.51 (± 0.17) ( o K) for soil temperature. This highlights a better accuracy than the benchmark model and gives more strength to the present study. As part of a future improvement proposal for this module, we suggest integrating additional soil layers, potentially up to 5, and extending the extrapolation to depths of 5 meters. Furthermore, expanding the scope of station inclusion to a global scale is recommended to ensure comprehensive coverage of all soil types.

The primary advantage of the qDTR methodology is its ability to yield higher average ratings throughout most of the year compared to the annual industry rating, resulting in an average increase of 7%. Moreover, it has the potential to further widen the disparity between winter and summer seasons, with an increase of up to 8.1%.

Figure 3 :Figure 5 :

 35 Figure 3: Daily time series of estimated soil moisture with (a) XGBoost at Soil 5 and (b) LASSO (Green) at Soil 2 compared to ERA data (RED) and Test Values (Black dash)

Figure 4 :

 4 Figure 4: Yearly representation of qDTR week/month for historical qDTR H values and projections qDTR RCP2.6, 4.5, 8.5 in node located in South Spain, in-dash blue the qDTR H month as a reference

Figure 6 :

 6 Figure 6: Comparative Analysis: Assessing qDTR variations among historical qDTR H , STR H , and RCP projections qDTR RCP2.6, 4.5, 8.5 at ten French UGC (b) Regional Analysis: Examining variations in UGL assessments between historical data and RCP 8.5 projections across France

Table 1 :

 1 REVISED USDA SOIL TEXTURE CLASSIFICATION, INDICATING SOIL CODES USED IN THIS STUDY

	Texture Class	Sand %	Silt %	Clay %	Code
	Sand (S)	> 85	0-15	0-15	5
	Loamy Sand (LS)	70-85	0-20	0-15	5
	Sandy Loam (SL)	43-70	7-20	0-27	4
	Loam (L)	20-40	40-60	7-27	4
	Silt Loam (SiL)	0-20	≥ 70	0-27	4
	Silt (Si)	0-10	80-90	0-10	4
	Sandy Clay Loam (SCL)	20-35	28-50	27-40	3
	Clay Loam (CL)	20-35	28-50	27-40	3
	Silty Clay Loam (SCLa)	0-20	50-70	27-40	3
	Sandy Clay (SC)	0-20	0-28	40-60	2
	Silty Clay (SCla)	0-10	50-70	40-60	2
	Clay (C)

Table 2 :

 2 DATASETS AND PARAMETERS

	Variable	Units Process Source
		Meteorological
	Temp. Air at 2	• C	2.2	a,b	Dynamic
	m (Θa)
	Total precipita-	mm	2.2	a,b	Dynamic
	tion (Pt)
	Net surface solar	Jm -2 2.2	a,b	Dynamic
	radiation (Sr)
	u -v -wind at 10	ms -1 2.2	a,b	Dynamic
	m
		Soil Proprieties
	Silt (S silt % )	%	2.5,2.2	c	Static
	Sand (S sand % )	%	2.5,2.2	c	Static
	Clay (S clay % )	%	2.5,2.2	c	Static
	Organic (Sorg % )	%	2.5	c	Static
	Texture Compo-	-	2.2	c	Static
	sition (Stext)
	Bulk (S bulk )	kgm -3 2.5,2.2	c	Static
		Soil Measurements
	Temperature (Θ) • C	2.2,2.6	d	Dynamic
	Moisture (ψ)	%	2.2,2.5	d	Dynamic

Table 3 :

 3 WEIGHTING FACTORS FOR THE EMPIRICAL MODEL OF THE SOIL THERMAL CONDUCTIVITY

	a 1	a 2	a 3	a 4	b 1	b 2	b 3
	0.493 0.86 0.014 0.778 0.736 0.006 0.222

Table 4 :

 4 UNDERGROUND CABLE PARAMETERS

	Parameter	Default Value	Unit/Symbol
	Name	Cable A2X(F)	-
	Conductor Material	Aluminium	-
	Insulation Type	XLPE	-
	Conductor type	Segmental	-
	L	1000	mm
	ρ 20	2.826 × 10 -8	Ω • m
	α 20	4.03 × 10 -3	1/K
	Θ	90	°C
	tan (δ)	1 × 10 -3	-
	U 0	72500	V
	ε	2.5	-
	Frequency	50	Hz
	S	400	mm
	C	0.295 × 10 -9	F/m
	R 0	0.0367 × 10 -3	Ω/m 2
	dc	33	mm
	De

Table 5 :

 5 MACHINE LEARNING PERFORMANCE -TEMPERATURE AT 80TH PERCENTILE

					Model
		CV	0	1	2	3	4	ERA
		0	2.23	2.03	2.29	2.15	2.39	2.72
	RMSE80	1 2 3	2.69 2.92 2.02	2.43 4.81 1.85	2.73 3.19 2.25	2.63 3.16 1.90	2.83 3.37 2.14	3.64 2.76 2.90
		4	2.89	3.48	2.60	3.11	2.70	3.56
		Avg. 2.71 3.16 2.83 2.74 2.95 3.10
		0	2.23	2.03	2.29	2.15	2.39	2.72
	MAE80	1 2	2.56 1.71	4.61 1.55	2.85 1.95	2.84 1.61	3.06 1.87	2.18 2.32
		3	2.55	3.14	2.26	2.77	2.38	2.58
		4	2.64	2.96	3.03	2.56	3.41	2.21
		Avg. 2.34 2.86 2.48 2.39 2.62 2.40

Table 6 :

 6 MACHINE LEARNING PERFORMANCE -MOISTURE AT 20TH PERCENTILE

					Model
		CV	0	1	2	3	4	ERA
		0	0.29	0.07	0.13	0.08	0.14	0.22
	RMSE20	1 2 3	0.05 0.05 0.05	0.21 0.03 0.06	0.08 0.10 0.06	0.09 0.04 0.16	0.08 0.04 0.05	0.15 0.09 0.07
		4	0.13	0.08	0.11	0.05	0.19	0.24
		Avg. 0.11 0.09 0.10 0.08 0.10 0.16
		0	0.29	0.06	0.11	0.07	0.13	0.22
	MAE20	1 2	0.04 0.04	0.21 0.03	0.08 0.11	0.08 0.03	0.07 0.04	0.14 0.08
		3	0.04	0.05	0.05	0.17	0.04	0.06
		4	0.12	0.07	0.10	0.04	0.19	0.24
		Avg. 0.11 0.08 0.09 0.08 0.09 0.15

Table 7 :

 7 FIVE-FOLD CROSS VALIDATION FOR MOISTURE AND TEST

		Description		CV Fold
	ID	Station	Soil	1 2 3 4 5 Test
			Code
	1	SOD140	3
	2	Borda Coll	4
	3	Hupsel	5
	4	Kehrigk	5
	5	Hoal-08	3
	6	Ribera de Sio	4
	7	ITCSM-15b	5
	8	ITCSM-05d	5
	9	ITCSM-06b	3
	10	Camidels Nerets	4
	11	Kuusamo	5
	12	ITCSM-10c	5
	13	Marsh Bubnow	3
	14	San Pietro Capo	4
	15	Voulton	4
	16	Abrahams	5
	17	ITCSM-17c	5
	18	SOD071	3
	19	Falkenberg	4
	20	PoriII	4
	21	ITCSM-14c	5
	22	ITCSM-05c	5
	23	Clot de les Peres	1
		II
	24	Clot de les Peres	1
		I
	26	Suizy	1
	27	Serra de	1
		CostaAmpla
	28	ITCSM-03b	5
	29	Kvarstadseter	5
	30	Pessonada	4
	31	La Cultiada	4
	32	Hoal-07	3

Table 8 :

 8 qDTR differences over a fifty-year period for qDTR RCP2.6, 4.5, 8.5 and qDTR H datasets for 10 UGC in France

	Line	Hist	∆ % RCP8.5	∆ % RCP4.5	∆ % RCP2.6
	ID	(A)	Mean Max Mean Max Mean Max
	1257 1073 -0.44 -1.19 -0.31 -0.88 -0.13 -0.79
	8566 1012 -0.42 -0.48 -0.24 -0.09 -0.21 -0.01
	5676 1110 -0.36 -0.69 -0.21 -0.59 -0.14 -0.29
	8554	941	-0.33 -0.77 -0.42 -0.98 -0.21 -0.37
	5808 1002 -0.33 -0.53 -0.39 -0.99 -0.22 -0.23
	5792 1006 -0.30 -1.14 -0.13 -0.41 -0.15	0.00
	2271	947	-0.30 -1.38 -0.19 -0.91 -0.08 -0.70
	8178 1046 -0.27 -1.20 -0.27 -0.60 -0.19 -0.91
	8544 1094 -0.27 -0.53 -0.25 -0.43 -0.07 -0.27
	5730 1085 -0.23 -0.55 -0.30 -0.40 -0.20 -0.50
	Average	-0.33 -0.85 -0.27 -0.63 -0.16 -0.41