We analyze multidimensional time series through the lens of their integrals of various moment orders, constituting their signature. The contribution of this article is to adapt the Principal Geodesic Analysis (PGA), the counterpart for manifolds of the Principal Component Analysis (PCA), to signature features which form a Lie group, by setting an appropriate connection structure. We show that, on both simulations and real data, our approach is more effective than the usual approximation which consists in projecting points of the manifold onto a tangent space and carrying a classical PCA.