Higher-order iterated sums signatures Nikolas Tapia FG6 joint with - PowerPoint PPT Presentation

Higher-order iterated sums signatures Nikolas Tapia FG6 joint with J. Diehl (Greifswald) and K. Ebrahimi-Fard (NTNU) Weierstraß-Institut für angewandte Analysis und Stochastik April 2, 2020

Introduction Consider a time series x = ( x 0 , x 1 , . . . , x N ) ∈ � N . The goal is to extract features out of x , that are invariant to time warping . Example We measure the heartbeat in a patient’s ECG. This is modelled as y ( k ) = x h ( k ) ( j ) + ξ ( k ) j = 1 , . . . , M ; k = 1 , . . . , K , j j where M ≥ N and h ( k ) : { 1 , . . . , M } → { 1 , . . . , N } is a (unknown) surjective non-decreasing time change. 2/22 Higher-order iterated sums signatures

Some invariants Definition A functional F : � � → � is said to be invariant to standing still (or stuttering) if F ◦ τ n = F for all n ≥ 0 . Here τ n : � � → � � is the operator that acts by repeating the value at time n . 3/22 Higher-order iterated sums signatures

Some invariants � It’s not hard to see that the total increment x N − x 0 = ( x j − x j − 1 ) j � � � as well as ( x j − x j − 1 ) 2 , ( x j − x j − 1 )( x k − x k − 1 ) , ( x j − x j − 1 )( x k − x k − 1 ) j < k j j ≤ k are all features invariant to time warping. Questions 1. Are all invariants some kind of iterated sum? 2. The last three expressions are linearly dependent since summing the first two gives the third. How to store only linearly independent information? 4/22 Higher-order iterated sums signatures

Quasisymmetric functions Definition A formal series Q ∈ � � Y 1 , Y 2 , . . . � is a quasisymmetric function if for all indices i 1 < i 2 < · · · < i n , j 1 < j 2 < · · · < j n and integers α 1 , . . . , α n ≥ 1 the coefficient of the monomials ( Y i 1 ) α 1 · · · ( Y i n ) α n and ( Y j 1 ) α 1 · · · ( Y j n ) α n in Q are equal. Theorem (Diehl, Ebrahimi-Fard, T. 2019) Let F be a polynomial functional invariant to standing still and space translations. Then F is realized as a quasisymmetric function on the increments of x . This answers Question 1. 5/22 Higher-order iterated sums signatures

Monomial basis Different linear bases are known. Malvenuto and Reutenauer (1995) introduced the monomial quasisymmetric � functions ( Y i 1 ) α 1 · · · ( Y i m ) α m M ( α 1 ,..., α m ) ≔ i 1 < ··· < i m indexed by compositions of integers. Definition A composition of the integer n is a tuple ( α 1 , . . . , α m ) of positive integers such that α 1 + · · · + α m = n . We call ℓ ( α ) ≔ m the length of the composition and | α | = n its weight. The collection of all compositions of n is denoted by C( n ) . This answers Question 2. 6/22 Higher-order iterated sums signatures

Quasi-shuffle algebras The monomial quasisymmetric functions actually form a monomial basis for QSym. The product is described by contractions . Example � � Y 2 M ( 1 ) M ( 1 ) = 2 Y j Y k + j = 2 M ( 1 , 1 ) + M ( 2 ) . j < k j Example M ( 1 ) M ( 3 , 7 ) = M ( 1 , 3 , 7 ) + M ( 3 , 1 , 7 ) + M ( 3 , 7 , 1 ) + M ( 4 , 7 ) + M ( 3 , 8 ) . This is an example of a quasi-shuffle algebra . 7/22 Higher-order iterated sums signatures

Quasi-shuffle algebras (cont.) Definition (Gaines 1994; Hoffman 2000) Let A be an alphabet having a semigroup structure [−−] : A × A → A . On the tensor algebra T ( A ) define the quasi-shuffle product ∗ recursively by e ∗ u ≔ u ≕ u ∗ e and ua ∗ vb ≔ ( u ∗ vb ) a + ( ua ∗ v ) b + ( u ∗ v )[ ab ] for u , v ∈ T ( A ) and a , b ∈ A . Example Take A = ( � + , + ) . Then 1 ∗ 1 = 2 · 11 + 2 and 1 ∗ 37 = 137 + 317 + 371 + 47 + 38 . Theorem (Hoffman, 2000) Let δ : T ( A ) → T ( A ) ⊗ T ( A ) be the deconcatenation coproduct. Then, ( T ( A ) , ∗ , δ , | · |) is a graded, connected, commutative and non-cocommutative Hopf algebra. 8/22 Higher-order iterated sums signatures

Iterated-sums signature Notation We set A = { 1 , . . . , d } and A is the free commutative semigroup over A . For a = [ i 1 · · · i ℓ ] = [ 1 k 1 · · · d k d ] ∈ A , j = ∆ x i 1 j · · · ∆ x i ℓ j ) k 1 · · · ( ∆ x d let ∆ x a j = ( ∆ x 1 j ) k d . Definition (Diehl, Ebrahimi-Fard, T. 2019) For a 1 , . . . , a p ∈ A , � m � ISS ( x ) n , j − 1 , a 1 · · · a p − 1 � ∆ x a p � ISS ( x ) n , m , a 1 · · · a p � ≔ j . j = n +1 Example � � N ( ∆ x 1 j ) 2 , ∆ x 1 j ∆ x 1 k ∆ x 2 � ISS ( x ) 0 , N , [ 11 ]� = � ISS ( x ) 0 , N , 1 [ 12 ]� = k . j =0 1 ≤ j < k ≤ N 9/22 Higher-order iterated sums signatures

Iterated-sums signature (cont.) We have the factorization � � � � � � � � N � � � � ∆ x a 1 j 1 ∆ x a 2 ∆ x a ISS ( x ) 0 , N = ε + a 1 a 2 + · · · a + j j 2 � � � � � � � � � � a ∈ A j =1 a 1 , a 2 ∈ A j 1 < j 2 � � � ∆ x a ∆ x a ∆ x a = ε + 1 a ε + 2 a · · · ε + N a � � a ∈ A a ∈ A a ∈ A � � − − − − → ∆ x a = ε + j a 1 ≤ j ≤ N a ∈ A Compare with � � � � � j e i 1 i 2 + · · · � − − − − → − − − − → � � j e i + 1 ∆ x i 1 j ∆ x i 2 ∆ x i S ( X ) 0 , 1 = exp ⊗ ( ∆ x j ) = ε + + · · · 2 � � 1 ≤ j ≤ N 1 ≤ j ≤ N i ∈ A i 1 , i 2 ∈ A 10/22 Higher-order iterated sums signatures

Quasi-shuffle algebras (cont.) Proposition (Diehl, Ebrahimi-Fard, T. 2019) The Poincaré–Hilbert series of T ( A ) is � ( 1 − t ) d t n dim T ( A ) n = H ( t ) ≔ 2 ( 1 − t ) d − 1 n ≥ 0 t 2 + d ( 13 d 2 + 9 d + 2 ) = 1 + dt + d ( 3 d + 1 ) t 3 + O ( t 4 ) 2 6 � Compare with 1 1 − td = 1 + dt + d 2 t 2 + d 3 t 3 + O ( t 4 ) . t n dim T ( A ) n = n ≥ 0 11/22 Higher-order iterated sums signatures

Iterated-sums signature (cont.) Theorem (Diehl, Ebrahimi-Fard, T. 2019) For each n ≤ m , ISS ( x ) n , m is a quasi-shuffle character, i.e. � ISS ( x ) n , m , u ∗ v � = � ISS ( x ) n , m , u �� ISS ( x ) n , m , v � for all u , v ∈ T ( A ) . Theorem (Chen’s property; Diehl, Ebrahimi-Fard, T. 2019) For all n ≤ p ≤ m we have ISS ( x ) n , p ⊗ ISS ( x ) p , m = ISS ( x ) n , m Remark In this case Chow’s theorem fails! � � 2 � � N N ∆ x 1 ∆ x 1 ([ 11 ] − 1 log ⊗ ISS ( x ) 0 , N = j 1 + · · · + 2 11 ) + · · · j j =1 j =1 12/22 Higher-order iterated sums signatures

Hoffman’s isomorphism Definition (Hoffman, 2000) Let a 1 , . . . , a n ∈ A . Given I = ( i 1 , . . . , i p ) ∈ C( n ) define I [ a 1 . . . a n ] = [ a 1 · · · a i 1 ][ a i 1 +1 · · · a i 1 + i 2 ] · · · [ a i 1 + ··· + i p − 1 · · · a n ] ∈ T ( A ) Theorem (Hoffman, 2000) The linear map Φ H : ( T ( A ) , ✁ , δ ) → ( T ( A ) , ∗ , δ ) defined by � 1 Φ H ( a 1 · · · a n ) ≔ i 1 ! · · · i p ! I [ a 1 · · · a n ] I ∈C( n ) is an isomorphism of Hopf algebras. � Its inverse is given by (− 1 ) n − p Φ − 1 H ( a 1 · · · a n ) = I [ a 1 · · · a n ] . i 1 · · · i p I ∈C( n ) 13/22 Higher-order iterated sums signatures

Hoffman’s isomorphism (cont.) Theorem (Diehl, Ebrahimi-Fard, T. 2019) Let x be a time series and consider the (infinite dimensional) path ( X a : a ∈ A ) where, for a = [ 1 k 1 · · · d k d ] ∈ A the path X a is the piecewise linear interpolation of the path � � n n j ) k 1 · · · ( ∆ x d ∆ x a ( ∆ x 1 j ) k d . n ↦→ j = j =1 j =1 Then � S ( X ) 0 , N , u � = � ISS ( x ) 0 , N , Φ H ( u )� for all u ∈ T ( A ) . 14/22 Higher-order iterated sums signatures

Higher-order iterated sums signature Definition (Diehl, Ebrahimi-Fard, T. 2020+) �� � ⊗ r  Let 1 ≤ p ≤ ∞ ,    � �     p − − − − → 1 ISS ( p ) ( x ) n , m ≔ ∆ x a   ε + j a   . r !   n < j ≤ m r =1 a ∈ A Remark If 1 < p < ∞ , ISS ( p ) ( x ) is not a character for neither ∗ nor ✁ . Indeed, e.g. p = 2 , � � ISS ( 2 ) ( x ) n , m , i � = ∆ x i j , � � j k 2 + 1 k 1 ∆ x j k ∆ x j � ISS ( 2 ) ( x ) n , m , ij � = ∆ x i ∆ x i k = � ISS ( x ) n , m , ij + 1 2 [ ij ]� . 2 k 1 < k 2 k So, � ISS ( 2 ) ( x ) n , m , ij + j i � = � ISS ( 2 ) n , m , i �� ISS ( 2 ) n , m , j � . 15/22 Higher-order iterated sums signatures

Higher-order iterated sums signature (cont.) Remark (cont.) � � k 3 + 1 ∆ x i 1 k 1 ∆ x i 2 k 2 ∆ x i 3 ( ∆ x i 1 k 1 ∆ i 2 k 1 ∆ x i 3 k 2 + ∆ x i 1 k 1 ∆ x i 2 k 2 ∆ x i 3 � ISS ( 2 ) ( x ) n , m , i 1 i 2 i 3 � = k 2 ) 2 k 1 < k 2 < k 3 k 1 < k 2 = � ISS ( x ) n , m , i 1 i 2 i 3 + 1 2 [ i 1 i 2 ] i 3 + 1 2 i 1 [ i 2 i 3 ]� . So, � ISS ( 2 ) ( x ) n , m , i 1 �� ISS ( 2 ) ( x ) n , m , i 2 i 3 � = � ISS ( 2 ) ( x ) n , m , i 1 i 2 i 3 + i 2 i 1 i 3 + i 2 i 3 i 1 − [ i 1 i 2 ] i 3 � . Theorem (Diehl, Ebrahimi-Fard, T. 2020+) � We have 1 � ISS ( p ) ( x ) n , m , a 1 · · · a ℓ � = i 1 ! · · · i k ! � ISS ( x ) n , m , I [ a 1 · · · a ℓ ]� . I ∈C p ( ℓ ) 16/22 Higher-order iterated sums signatures