Abstract
In this chapter we review continuous time series modeling and estimation by extended structural equation models (SEM) for single subjects and N > 1. First-order as well as higher-order models will be dealt with. Both will be handled by the general state space approach which reformulates higher-order models as first-order models. In addition to the basic model, the extensions of exogenous variables and traits (random intercepts) will be introduced. The connection between continuous time and discrete time for estimating the model by SEM will be made by the exact discrete model (EDM). It is by the EDM that the exact estimation procedure in this chapter differentiates from many approximate procedures found in the literature. The proposed analysis procedure will be applied to the well-known Wolfer sunspot data, an N = 1 time series that has been analyzed by several continuous time analysts in the past. The analysis will be carried out by ctsem, an R-package for continuous time modeling that interfaces to OpenMx, and the results will be compared to those reported in the previous studies.
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Notes
- 1.
In this chapter we follow the common practice to write the Wiener process by capital letter W, although it is here a vector whose size should be inferred from the context.
- 2.
The adapted version uses (1.18A)
$$\displaystyle \begin{aligned} {{\text{e}}^{{\mathbf{A}}{\Delta} t}} = \mathop {\lim }\limits_{D \to \infty } \prod_{d = 0}^{D - 1} {\Big[\Big({\mathbf{I}}} - \frac{1}{2}{\mathbf{A}}{\delta _d}{\Big)^{ - 1}}\Big({\mathbf{I}} + \frac{1}{2}{\mathbf{A}}{\delta _d}\Big)\Big]{\text{ for }}{\delta _d} = \frac{{{\Delta} t}}{D},{} \end{aligned} $$(1.18A)which converges much more rapidly than (1.18). It is based on the approximate discrete model (ADM), described by Oud and Delsing (2010), which just as the EDM goes back to Bergstrom (1984). Computation of the matrix exponential by (1.18) or (1.18A) has the advantage over the diagonalization method (Oud and Jansen 2000) that no assumptions with regard to the eigenvalues need to be made. Currently by most authors the Padé-approximation (Higham 2009) is considered the best computation method, which therefore is implemented in the most recent version of ctsem.
- 3.
A better approximation than a step function is given by a piecewise linear or polygonal approximation (Oud and Jansen 2000; Singer 1992). Then we write u(t) in (1.23) as \({\mathbf {u}}(t) = {\mathbf {u}}({t_0}) +(t - {t_0}){{\mathbf {b}}_{({t_0},t]}}\) and (1.26) becomes:
$$\displaystyle \begin{aligned} \begin{aligned} {\mathbf{x}}(t) &= {{\text{e}}^{{\mathbf{A}}(t - {t_0})}}{\mathbf{x}}({t_0}) + {{\mathbf{A}}^{ - 1}}\big[{{\text{e}}^{{\mathbf{A}}(t - {t_0})}} - {\mathbf{I}}\big]{\mathbf{Bu}}({t_0}) + \left\{ {{\mathbf{A}}^{ - 2}}\big[{{\text{e}}^{{\mathbf{A}}(t - {t_0})}} - {\mathbf{I}}\big] - {{\mathbf{A}}^{ - 1}}(t - {t_0}) \right\} {\mathbf{B}}{{\mathbf{b}}_{({t_0},t]}} \hfill \\ & \quad+ {{\mathbf{A}}^{ - 1}}\big[{{\text{e}}^{{\mathbf{A}}(t - {t_0})}} - {\mathbf{I}}\big]{\boldsymbol{\upgamma}} + \int_{{t_0}}^t {{{\text{e}}^{{\mathbf{A}}(t - s)}}} {\mathbf{G}}{\text{d}}{\mathbf{W}}(s). \hfill \\ \end{aligned}{} \end{aligned} $$(1.26A) - 4.
The programming code of the analysis is available as supplementary material at the book website http://www.springer.com/us/book/9783319772189.
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Oud, J.H.L., Voelkle, M.C., Driver, C.C. (2018). First- and Higher-Order Continuous Time Models for Arbitrary N Using SEM. In: van Montfort, K., Oud, J.H.L., Voelkle, M.C. (eds) Continuous Time Modeling in the Behavioral and Related Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-77219-6_1
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