A Method for Modeling the Intrinsic Dynamics of Intraindividual Variability: Recovering the Parameters of Simulated Oscillators in Multi-Wave Panel Data

Abstract

A simple method for fitting differential equations to multi-wave panel data performs remarkably well in recovering parameters from underlying continuous models with as few as three waves of data. Two techniques for fitting models of intrinsic dynamics to intraindividual variability data are examined by testing these techniques' behavior in recovering the parameters from data generated by two simulated systems of differential equations. Each simulated data set contains 100 "subjects" each of whom are measured at only three points in time. A local linear approximation of the first and second derivatives of the subject's data accurately recovers the true parameters of each simulation. A statespace embedding technique for estimating the first and second derivatives does not recover the parameters as well. An optimum sampling interval can be estimated for this model as that interval at which multiple R 2 first nears its asymptotic value.

Latent Differential Equation Modeling with Multivariate Multi-Occasion Indicators

“…The MLDE model was able to simultaneously recover low bias estimates of the factor loadings and differential equations parameters if the total interval between the first and last observation was(a) smaller than the Nyquist limit of one half the period of the true oscillation and (b) large enough that the communalities of the indicators was greater than 0.5. The MLDE method appears to be much less sensitive to measurement interval than other methods for direct estimation of differential equations parameters from derivatives explored by the authors (Boker & Nesselroade, 2002;Boker, 2001). This means that the MLDE method does not require more than a gross estimate of the period of any hypothesized cyclicity in the process under analysis.…”

“…Oud & Jansen, 2000)) In these models individual differences in initial conditions are not confounded with individual differences in the dynamics of the behavior since time is only treated as relative to other measurements. Thus state space models have a distinct advantage when the initial conditions may be due to unknown exogenous influences: a participant in a study may have an essentially random state when he or she begins an experimental protocol, but changes in behavior during the protocol may be reliable (see Boker & Nesselroade, 2002, for a more complete discussion of this so-called phase problem).…”

“…One approach to estimating the intrinsic dynamics of a psychological process is to first estimate the derivatives of the process for each occasion of measurement and then fit either a regression model (Boker & Nesselroade, 2002), a structural model (Boker, 2001), or multilevel model (Boker & Ghisletta, 2001) to those resulting variables. This approach has advantages and disadvantages.…”

“…While LLA has the advantage of using only three occasions of measurement to estimate the derivatives, it can prove to have biased estimates of the dynamical parameters when the interval τ is not optimal (Boker & Nesselroade, 2002). In addition, measurement error can masquerade as high frequency signal.…”

Finding the attractor of anger: Bridging the gap between dynamic concepts and empirical data.

“…These equations mathematically describe the momentary relations between acceleration, velocity, and level of a dynamic variable (cf. Doucet & Sloep, 1992;Acheson, 1997), and can be applied to real data to identify the attractor governing the internal dynamics of the system (Boker, 2001;Boker & Chisletta, 2001;Boker & Nesselroade, 2002). When applied empirically the name "Empirical Differential Equations" (EDE) is used.…”