Generalized iterated-sums signatures (Q6169180)

Unlike iterated-integrals and iterated-sum signatures, Kiraly and Oberhauser's generalization of signatures (introduced in a specific context) are not compatible with the shuffle product, nor the quasishuffle product. In order to recover a character property for them, the quasishuffle product is deformed by coalgebra isomorphisms, and especially a family of isomorphism indexed by formal diffeomorphisms. These products are shown to be compatible with a generalized iterated-sum signature defined with the same formal diffeomorphism. In a second part, inspired by machine learning applications, it is shown that polynomial transformations of the time series increments induce transformation of iterated-sum signatures dually corresponding to a quasishuffle and Hopf algebra endomorphism, whereas polynomial transformations of the time series itself induce transformation of iterated-sum signatures dually corresponding to a quasishuffle, but not Hopf algebra, endomorphism.