Most of the tools of analysis of time series (autocorrelation, various forms of Fourier transform etc) rely on the invariance of the model under the time shifts, and do not survive nonlinear reparameterizations of the timeline. But what if we require our descriptors to be reparameterization invariant?

In higher dimensions, an essentially complete set of descriptors is given by the iterated path integrals, a (filtered) collection of functionals of trajectories with fascinating combinatorial properties. In this talk I will focus on second order invariants and show how they can be useful to recover lead-lag structure between components of a high-dimensional time series, with applications in neuroscience and economics.