Abstract
Intra-individual processes are thought to continuously unfold across time. For equally spaced time intervals, the discrete-time lag-1 vector autoregressive (VAR(1)) model and the continuous-time Ornstein–Uhlenbeck (OU) model are equivalent. It is expected that by taking into account the unequal spacings of the time intervals in real data between observations will lead to an advantage for the OU in terms of predictive accuracy. In this paper, this is claim is being investigated by comparing the predictive accuracy of the OU model to that of the VAR(1) model on typical ESM data obtained in the context of affect research. It is shown that the VAR(1) model outperforms the OU model for the majority of the time series, even though time intervals in the data are unequally spaced. Accounting for measurement error does not change the result. Deleting large abrupt changes on short time intervals (that may be caused by externally driven events) does however lead to a significant improvement for the OU model. This suggests that processes in psychology may be continuously evolving, but that there are factors, like external events, which can disrupt the continuous flow.
Highlights
Background on the statistical models Thevector autoregressive (VAR)[1] model.When participants in an experience sampling method (ESM) study have, for example, rated the intensity of their affect at each measurement occasion, or have reported on specific emotions, behaviors or thought processes, several time series of measurements are obtained
The predictive accuracy does not improve for the majority of the time series when time intervals are explicitly taken into account, which is contrary to what we expected
Starting from the idea that intra-individual processes in psychology are assumed to continuously unfolding across time, and that time information should play an important role when trying to model such processes, we set out to investigate how much the predictive accuracy of discrete-time vector autoregressive (VAR) models can be improved by recasting them in a continuous-time framework
Summary
Background on the statistical models TheVAR[1] model.When participants in an ESM study have, for example, rated the intensity of their affect at each measurement occasion, or have reported on specific emotions, behaviors or thought processes, several (multivariate) time series of measurements are obtained (one time series per participant). Xdi}p denote the set of d different ratings (the variables) from a specific participant p at measurement occasion i = 0, . The VAR[1] model describes a linear dependency between consecutive measurements and is given by xi = c + Axi−1 + ǫi. The parameter c ∈ Rd denotes the intercept, A ∈ Rd×d is the transition matrix, and ǫi is a vector containing the stochastic fluctuations (innovations) at measurement occasion i. The innovations ǫi are assumed to be Gaussian distributed with mean zero and covariance ǫ. They are assumed to be uncorrelated across time. The system of Eq [1] is nothing but a system of multivariate linear regression equations, modeling each variable as a linear function of the variables at the previous measurement
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