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

## TL;DR

This paper establishes a theoretical connection between discrete-time ARFIMA models and continuous-time linear stochastic differential equations, enabling better physical interpretation of sampled experimental data.

## Contribution

It introduces formulas linking ARFIMA parameters to the coefficients of underlying physical stochastic systems, bridging discrete-time models with continuous-time physical models.

## Key findings

- ARFIMA models can be viewed as sampled trajectories of continuous-time systems
- Formulas relate ARFIMA parameters to physical system coefficients
- Provides a framework for physically interpretable time series analysis

## Abstract

Modelling physical data with linear discrete time series, namely Fractionally Integrated Autoregressive Moving Average (ARFIMA), is a technique which achieved attention in recent years. However, these models are used mainly as a statistical tool only, with weak emphasis on physical background of the model. The main reason for this lack of attention is that ARFIMA model describes discrete-time measurements, whereas physical models are formulated using continuous-time parameter. In order to remove this discrepancy we show that time series of this type can be regarded as sampled trajectories of the coordinates governed by 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.