Dynamic relations in sampled processes

I will start my talk by a short description of a historical connection between my university, Shanghai Jiao Tong University (SJTU), and Collegio Romano. There is an interesting chain from Clavius, who led the Gregorian calendar reform, via Matteo Ricci and Xu Guangqi, the first modern scientist of China, to the founding of the two elite universities on Shanghai, Fudan and SJTU. The technical part of my talk is based on joint work with Tryphon Georgiou. Linear dynamical relations that may exist in continuous-time, or at some natural sampling rate, are not directly discernable at reduced observational sampling rates. Indeed, at reduced rates, matrix-valued spectral densities of vector-valued time series have maximal rank and thereby cannot be used to ascertain potential dynamic relations between their entries. This hitherto undeclared source of inaccuracies appears to plague off-the-shelf identification techniques seeking remedy in hypothetical observational noise. In this paper we explain the exact relation between stochastic models at different sampling rates and show how to construct stochastic models at the finest time scale that data allows. We then point out that the correct number of dynamical dependences can only be ascertained by considering stochastic models at this finest time scale, which in general is faster than the observational sampling rate.