Latent Differential Equation Modeling with Multivariate Multi-Occasion Indicators

Abstract

A number of models for estimating the coefficients of dynamical systems have been proposed. One is the exact discrete model, another is the latent differences model or proportional change model, and a third is a continuous time manifest variable differential structural model. These models differ from standard growth curve modeling in that they intend to illuminate processes generating individual trajectories over time rather than just estimate an aggregate best trajectory. The current work proposes a novel approach to the modeling of multivariate change by first creating a lagged covariance matrix. Then rather than using factor loadings to estimate average growth characteristics as in other growth curve modeling related methods, confirmatory factor loadings are fixed across the time dimension and freed across the variables dimension, in such a way as to force the factor scores to estimate instantaneous derivatives. Then, the factor covariances are structured as a regression that allows the estimation of parameters of differential equation models of dynamical systems theories proposed to account for the data. Results of a simulation will be presented illustrating strengths and weaknesses of the latent differential equation model.

Regularized Structural Equation Modeling

“…To provide the background on general SEM and make clear what matrices are available for regularization, we briefly detail Reticular Action Model RAM;McArdle & McDonald, 1984) notation as it provides a 1:1 correspondence between the graphical and matrix specifications (Boker, Neale, & Rausch, 2004), as well as only requiring three matrices to specify the full SEM. Additionally, the current implementation of RegSEM only uses the RAM matrices, making it necessary to understand the notation.…”

Modeling Noisy Data with Differential Equations Using Observed and Expected Matrices

“…This is accomplished by using localized vectors (e.g., x t−2 to x t+2 ); these localized vectors are the same conceptually as the rows of a time delay embedded matrix, which is one approach for recreating a state space (Takens, 1981). Embedding matrices have been used previously in psychology (e.g., Boker, Neale & Rausch, 2004), but such studies have initially estimated the derivatives of a time series; this can result in correlations between derivative estimates, which in turn can bias model parameter estimates.…”

“…Modeling these relationships, rather than the observed trajectories, provide several advantages including: expression of complex trajectories using linear equations with few parameters, parameters with meaningful interpretations, parameter estimates that do not depend on the selection of initial time (t = 0), and vastly differing trajectories for differing individuals described by a common dynamical system. Parameter estimation of a differential equation model is often approached by either (1) estimating the parameters directly from a time series using a model (e.g., Ramsay & Silverman, 2005), or (2) estimating the derivatives of a time series which are then used to estimate the model parameters (e.g., Boker, Neale & Rausch, 2004). The first approach works well when variables that change over time are not expected to violate the dynamics of the system described by a set of differential equation models; examples are abundant in the physical sciences, where one would not expect a physical object to display a discontinuous change in position.…”

Drawing Conclusions from Cross-Lagged Relationships: Re-Considering the Role of the Time-Interval

“…To overcome both of the problems posed by the time-interval dependency issue, the use of a Continuous-Time (CT) Modeling approach has been suggested repeatedly in the literature (e.g., Boker, 2002;Chow et al, 2005;Oravecz, Tuerlinckx, & Vandekerckhove, 2009;Oud & Delsing, 2010a). In particular, both Voelkle, Oud, Davidov, and Schmidt (2012) and Boker, Neale, and Rausch (2004) have demonstrated how these models can be estimated in SEM-based software.…”