From physical linear systems to discrete-time series. A guide for analysis of the sampled experimental data

Modeling physical data with linear discrete-time series, namely, the autoregressive fractionally integrated moving average (ARFIMA) model, is a technique that has attracted attention in recent years. However, this model is used mainly as a statistical tool only, with weak emphasis on the physical background of the model. The main reason for this lack of attention is that the ARFIMA model describes discrete-time measurements, whereas physical models are formulated using continuous-time parameters. In order to eliminate this discrepancy, we show that time series of this type can be regarded as sampled trajectories of the coordinates governed by a system of linear stochastic differential equations with constant coefficients. The observed correspondence provides formulas linking ARFIMA parameters and the coefficients of the underlying physical stochastic system, thus providing a bridge between continuous-time linear dynamical systems and ARFIMA models.