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2018 | OriginalPaper | Chapter

2. A Continuous-Time Approach to Intensive Longitudinal Data: What, Why, and How?

Authors : Oisín Ryan, Rebecca M. Kuiper, Ellen L. Hamaker

Published in: Continuous Time Modeling in the Behavioral and Related Sciences

Publisher: Springer International Publishing

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Abstract

The aim of this chapter is to (a) provide a broad didactical treatment of the first-order stochastic differential equation model—also known as the continuous-time (CT) first-order vector autoregressive (VAR(1)) model—and (b) argue for and illustrate the potential of this model for the study of psychological processes using intensive longitudinal data. We begin by describing what the CT-VAR(1) model is and how it relates to the more commonly used discrete-time VAR(1) model. Assuming no prior knowledge on the part of the reader, we introduce important concepts for the analysis of dynamic systems, such as stability and fixed points. In addition we examine why applied researchers should take a continuous-time approach to psychological phenomena, focusing on both the practical and conceptual benefits of this approach. Finally, we elucidate how researchers can interpret CT models, describing the direct interpretation of CT model parameters as well as tools such as impulse response functions, vector fields, and lagged parameter plots. To illustrate this methodology, we reanalyze a single-subject experience-sampling dataset with the R package ctsem; for didactical purposes, R code for this analysis is included, and the dataset itself is publicly available.

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previous chapter First- and Higher-Order Continuous Time Models for Arbitrary N Using SEM

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The variance-covariance matrix of the variables Σ is a function of both the lagged parameters and the variance-covariance matrix of the innovations, vec(Σ) = (I −Φ ⊗Φ)⁻¹ vec(Ψ), where vec(.) denotes the operation of putting the elements of an N × N matrix into an NN × 1 column matrix (Kim and Nelson 1999, p. 27).

Readers should note that there are multiple different possible ways to parameterize the CT stochastic process in integral form, and also multiple different notations used (e.g., Oravecz et al. 2011; Voelkle et al. 2012).

In general, there is no straightforward CT-VAR(1) representation of DT-VAR(1) models with real, negative eigenvalues. However it may be possible to specify more complex continuous-time models which do not exhibit positive autoregression. Notably, Fisher (2001) demonstrates how a DT-AR(1) model with negative autoregressive parameter can be modeled with the use of two continuous-time (so-called) Itô processes.

Similar functions can be used for deterministic systems (those without a random innovation part); however in these cases the term initial value is more typically used.

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Title: A Continuous-Time Approach to Intensive Longitudinal Data: What, Why, and How?
Authors: Oisín Ryan
Rebecca M. Kuiper
Ellen L. Hamaker
Publisher: Springer International Publishing
Book: Continuous Time Modeling in the Behavioral and Related Sciences
Print ISBN: 978-3-319-77218-9

Electronic ISBN: 978-3-319-77219-6

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-77219-6_2

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