Modeling Noisy Data with Differential Equations Using Observed and Expected Matrices

Abstract

intraindividual variability, differential equation model(s)(ing), time series, damped linear oscillator, analytic solution(s),

Adaptive Equilibrium Regulation: Modeling Individual Dynamics on Multiple Timescales

“…Boker et al (2010) demonstrated that increasing the embedding dimension results in lower estimates of frequency, while increasing the order of derivatives to be estimated partly mitigates the bias. Deboeck and Boker (2010) provide an analysis of parameter bias in Local Linear Approximation and a model-based method of correction using time-embedding windows of multiple widths per model fit. With OpenMx, the command mxSE() can be used obtain standard error estimates for the corrected parameter estimates.…”

“…Real behavioral or psychological data are likely to exhibit non-systematic random influence from unmeasured sources, the effects of which would be modeled as additional, normally distributed variance of the short-term second derivative. IOs have been demonstrated to bias the estimates of damping to zero or positive values, misrepresenting data as undamped or even amplifying (Deboeck & Boker, 2010). In modeling affect data for the current study, no method of detecting IOs was used.…”

“…Algorithms have been developed to correct for outliers (Ih, Tiao, & Chen, 1988). Their application has yet to be shown with LDEs and data structured in a time-embedded format, though estimate biases due to such random processes have been demonstrated (Deboeck & Boker, 2010). For this reason, IOs were not included in the simulation process.…”

Dynamics of Change and Change in Dynamics

“…One side–effect of this is that the ordering of the rows of a time delay embedded matrix does not affect an analysis since the time–dependent information has been captured within each row. This fact carries with it the great advantage that so–called phase resetting phenomena do not have large impacts on the parameter estimates of models fit from time delay embedded matrices (Deboeck & Boker, 2010). A phase reset might occur when a substantial external event creates an abrupt change in the target variable value.…”

A tutorial on ordinary differential equations in behavioral science: What does physics teach us?

“…Social sciences are no exception, and researchers realized certain advantages of the continuous-time approach compared with the discrete one (Ryan et al, 2018). Since the beginning of the 1950s (Arminger, 1986; Simon, 1952), ordinary differential equations (ODE) have been used to capture the dynamics of a wide range of psychosocial variables such as affects (Deboeck et al, 2008; Deboeck & Bergeman, 2013; Montpetit et al, 2010; Steele & Ferrer, 2011), emotions (Chow et al, 2005; Pettersson et al, 2013; Reed et al, 2015), behavior (Hu & Huang, 2018), memory (Gasimova et al, 2014), craving (Boker & Graham, 1998; Timms et al, 2013; Trail et al, 2014), sexual desire or emotions in dyads (Bringmann et al, 2018; Farr et al, 2014; Randall et al, 2021), mood (Oud et al, 2018), body position (Butner et al, 2005; McKee & Neale, 2019), distance between persons (Deboeck & Boker, 2010), interactions in social groups (Simon, 1952), even complex interaction between nationalism, ethnocentrism, and authoritarianism (Angraini et al, 2014).…”