educationI&M in Kurt Barbé, Karen Pien, Camille Raets, and Koen Putman Tutorial Paper: Autoregressive Integrated Moving Average Time Series Analysis for Measurement Professionals D ynamical measurements are abundantly available when using instruments to collect data. Such dynamical measurements are time series which hold quantities with time-dependent values. The main focus of analyzing time series is either to obtain insight in the data generating process or to render predictions about the future given the past measurements. In this tutorial we give a comprehensive introduction to time series analysis wherein we cover both objectives. We concentrate on the so called ARIMA-framework or Autoregressive Integrated Moving Average framework. Nonetheless we also discuss some generalizations towards nonlinear time series for the interested reader. This introduction requires readers with a profound understanding of general linear models and the basics of dynamical systems. Continuous Time Dynamical Systems Versus Discrete Time Models A dynamical system describes an observed phenomenon as a function of time. As a result, the data generation is regulated in continuous time. Modeling dynamical systems can be performed in continuous-time as well as discrete-time models. It depends on the assumption considered about the realization of the system which viewpoint is the most relevant. Continuous-time models require the measurements or acquired observations to be band-limited whereas discrete-time systems require the signals to be well approximated by a piece-wise constant function. Discrete time models are furthermore easier to handle as they do not require the direct modeling of differential operators. In this paragraph we June 2023 revisit continuous time systems and discrete time mathematical models. Toy Example: Horizontally Driven Pendulum Consider a damped pendulum with a mass attached to a rod whose pivot point can be moved along a frictionless beam (Fig. 1.) The pivot point's position along the horizontal beam as a function of time is given by x(t). The pendulum's rod is of length l with an attached mass m. In Fig. 1, the force vectors are drawn such that by virtue of Newton's second law of motion the angle y(t) as a function of time satisfies the following second order nonlinear differential equation: 2 ml yt l y t mg yt F t y t dt dd () dt2 x F () sin( ( )) ()cos( ()) and gravitational force F (1) where μ, g are the damping and gravitational constants corresponding to the friction force f F g respectively such that Fx(t) is the horizontal force applied to the pivot point along the horizontal beam and m is the force applied to the mass m. It is common practice to linearize the differential equation (1) for small angles leading to: 2 ml y t dt2 dd () (2) l y t mgy t F t dt () x () ( ) Various methods exist to discretize the differential equation (2) where most methods are used for solving the equation numerically. However, for modeling one aims at providing a discrete time equation such that the impulse response corresponds. This method is known as the impulse invariance transform [1]. In order to use the impulse invariant transformation, we compute the impulse response function of the differential equation: Fig. 1. Horizontally driven pendulum: Schematic Newtonian mechanics. IEEE Instrumentation & Measurement Magazine 1094-6969/23/$25.00©2023IEEE 27