Modeling Individual Damped Linear Oscillator Processes with Differential Equations: Using Surrogate Data Analysis to Estimate the Smoothing Parameter

Abstract

Among the many methods available for modeling intraindividual time series, differential equation modeling has several advantages that make it promising for applications to psychological data. One interesting differential equation model is that of the damped linear oscillator (DLO), which can be used to model variables that have a tendency to fluctuate around some typical, or equilibrium, value. Methods available for fitting the damped linear oscillator model using differential equation modeling can yield biased parameter estimates when applied to univariate time series. The degree of this bias depends on a smoothing–like parameter, which balances the need for increasing smoothing to minimize error variance but not smoothing so much as to obscure change of interest. This article explores a technique that uses surrogate data analysis to select such a parameter, thereby producing approximately unbiased parameter estimates. Furthermore the smoothing parameter, which is usually researcher-selected, is produced in an automated manner so as to reduce the experience required by researchers to apply these methods. Focus is placed on the damped linear model; however, similar issues are expected with other differential equation models and other techniques in which parameter estimates depend on a smoothing parameter. An example using affect data from the Notre Dame Longitudinal Study of Aging (2004) is presented, which contrasts the use of a single smoothing parameter for all individuals versus use of a smoothing parameter for each individual.