Dynamic relations in sampled processes

TL;DR

This paper investigates how sampling rates affect the detection of linear dynamical relations in stochastic processes, revealing that lower sampling rates can obscure true dependencies and proposing methods to model at the finest possible time scale.

Contribution

It establishes the exact relationship between models at different sampling rates and introduces a way to construct models at the finest time scale for accurate dependency detection.

Findings

Spectral densities at reduced sampling rates have maximal rank, hiding true relations.

Correct dynamical dependences are only identifiable at the finest time scale.

Proposes methods to construct stochastic models at the finest possible sampling rate.

Abstract

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, matricial spectral densities of vectorial 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.