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 letter 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 dependencies can only be ascertained by considering stochastic models at this finest time scale, which in general is faster than the observational sampling rate.
Highlights
S UPPOSE that we seek to identify linear dynamical relations that may exist between the components of a continuous-time process
Dependencies that may exist in continuous-time, or at some other fine “natural” sampling rate1, are obfuscated by the process of sampling or sub-sampling. This is reflected in the fact that while the nullity of the spectral density of Dythneaormigiincal prreolcaestsicooninscidiens wsitah mthepnluemdberporfolicneeasr sreelastions, the density of the sampled process has TgryepnheornicTa. lGlyeormgioaux, iFmelalolwr, aIEnEkE. aEnvdiAdnednetrlsyL,inwdqhueisnt, tLhifee Foebllsoewr,vIEaEtiEonal sampling rate is sufficiently fast, the density of the sampled process is close to a singular one with the correct nullity. In such cases, hypothesizing bstract—Linear dynamical relations that may exist in nuous-time, or at some natural sampling rate, are not tly discernable at reduced observational sampling rates. ed, at reduced rates, matricial spectral densities of vectorial series have maximal rank and thereby cannot be used to tain potential dynamic relations between their entries
Statistical reasoning was sought to mathematize the search for exact linear algebraic relations giving rise to methods of principle component analysis, factor analysis and so on, see e.g., [3], [4]
Summary
S UPPOSE that we seek to identify linear dynamical relations that may exist between the components of a continuous-time process. The preponderance of techniques in the literature implicitly assume that the observed time series inherits any dynamical dependencies and typically seek to identify relations for the discrete-time process at the observation sampling rate. This assumption cannot be made in general. AEnvdiAdnednetrlsyL,inwdqhueisnt, tLhifee Foebllsoewr,vIEaEtiEonal sampling rate is sufficiently fast, the density of the sampled process is close to a singular one with the correct nullity In such cases, hypothesizing bstract—Linear dynamical relations that may exist in nuous-time, or at some natural sampling rate, are not tly discernable at reduced observational sampling rates.
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