Current Topics in Acoustical Research Vol. 3, (2003) Nonlinear prediction of time series obtained from an experimental pendulum Eduardo G. F. Pinto 1 and Marcelo A. Savi 2 1 Instituto Militar de Engenharia, Department of Mechanical Engineering and Materials, 22.290.270 - Rio de Janeiro - RJ – Brazil 2 Universidade Federal do Rio de Janeiro, COPPE / EE – Department of Mechanical Engineering, 21.945.970 – Rio de Janeiro – RJ – Brazil, Cx. Postal 68.503, E-Mail: savi@serv.com.ufrj.br ABSTRACT Time series is a sequence of observations of one or a few time variable of a dynamical system. Linear analysis assumes that the intrinsic dynamics of the system is related to the fact that small causes lead to small effects. On the other hand, nonlinear data set may be related to irregular data with purely deterministic inputs. Nonlinear time series analysis is of special interest of several areas. Time series prediction is an area of this general topic that has the objective of estimating future values from a known time series, called past, without any knowledge of the governing equations of phenomena. This article considers the analysis of some prediction techniques applied to time series obtained from an experimental nonlinear pendulum. Noise suppression is not contemplated and all signals are analyzed without filtering. Periodic and chaotic signals are analyzed employing three different predictors: simple nonlinear, polynomial and radial basis functions. The influence of state space reconstruction is exploited showing that it is an important task to be taking into account in prediction problems. 1. Introduction. The most direct link between chaos theory and the real world is the analysis of time series from real systems in terms of nonlinear dynamics. Time series is a sequence of observations of one or a few time variable of the system [1]. Usually, it is related to a nonlinear dynamical system which experimental analysis furnishes a scalar sequence of measurements. Linear methods interpret regular structure in a data set meaning that the intrinsic dynamics of the system is related to the fact that small causes lead to small effects. Nonlinear, chaotic systems, however, can produce irregular data with purely deterministic inputs [1]. Chaotic behavior presents sensitive dependence on initial conditions and long-term unpredictability. Nonlinear analysis involves different tools from linear ones. State space reconstruction and the determination of dynamical invariants are some of these tools. Lyapunov exponents and attractor dimension are some examples of dynamical invariants that could be used to identify chaotic behavior. The signs of the Lyapunov exponents provide a qualitative picture of the system’s dynamics and any system containing at least one positive exponent presents chaotic behavior. The attractor dimension, on the other hand, counts the effective number of degrees of freedom in a dynamical system and its strangeness is related to situations where a noninteger number describes the system dimension. An area related to time series analysis is the prediction which objective is estimating future values from a known time series, called past, N n S n ,..., 1 , = . Therefore, it is necessary to estimate future time series, , employing some prediction technique that results in an estimated series: . p N N N S S S + + + , . . . , , 2 1 p N N N P P P + + + , . . . , , 2 1 Figure 1 shows a schematic plot related to the prediction problem. From a known time series, called past, some predictor evaluates future values of time series, prediction. These estimated values can be compared with future values associated with the original series in order