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Two Filtering Methods of Forecasting Linear and Nonlinear Dynamics of Intensive Longitudinal Data

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An Erratum to this article was published on 13 April 2022

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Abstract

With the advent of new data collection technologies, intensive longitudinal data (ILD) are collected more frequently than ever. Along with the increased prevalence of ILD, more methods are being developed to analyze these data. However, relatively few methods have yet been applied for making long- or even short-term predictions from ILD in behavioral settings. Applications of forecasting methods to behavioral ILD are still scant. We first establish a general framework for modeling ILD and then extend that frame to two previously existing forecasting methods: these methods are Kalman prediction and ensemble prediction. After implementing Kalman and ensemble forecasts in free and open-source software, we apply these methods to daily drug and alcohol use data. In doing so, we create a simple, but nonlinear dynamical system model of daily drug and alcohol use and illustrate important differences between the forecasting methods. We further compare the Kalman and ensemble forecasting methods to several simpler forecasts of daily drug and alcohol use. Ensemble forecasts may be more appropriate than Kalman forecasts for nonlinear dynamical systems models, but further forecasting evaluation methods must be put into practice.

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Notes

  1. For example, many psychological time series exhibit “burn-in” or habituation periods where early observations may have different behavior than later observations. Equally weighting early and late observations would produce a worse forecast than a weighting scheme that emphasized the more recent observations (Gregson, 1983).

  2. Under ergodic conditions (i.e., Hannan, 1970, p. 201), time series are appropriate even for single-occasion data from multiple subjects.

  3. Note that to ease readability, we state this equation in its derivative form rather than its differential form. Although this lacks some rigor, we feel it increases clarity.

  4. This result necessarily follows from the linear case (e.g., Brockwell, 1995) because the linear dynamics are a special case of the nonlinear dynamics.

  5. Although the dynamical system we consider may be nonlinear, all of them are stable. Stable systems in discrete time and continuous time show no sensitive dependence on initial conditions (Arrowsmith & Place, 1990, Ch. 3). Instead, these systems have continuous dependence on initial conditions and perturbations (Hirsch et al., 2003, Ch. 17; V. I. Arnold, 1988, Ch. 3–4).

  6. Unlike in structural equation modeling, it is possible to estimate both the measurement noise and the dynamic noise for single indicator models (Hunter, 2018). Such models are identified and routinely estimated in some settings (e.g., radar tracking, Brookner, 1998)

  7. If the standard error of measurement is .707, then an 85% confidence interval would be \(\pm .707 \cdot 1.44 \approx 1.00\) assuming asymptotic normality.

  8. The naive carry-forward forecast method has no necessary forecast error, but we augment it with one by using a random walk, as is common (e.g., Hyndman & Athanasopoulos, 2018, Ch. 3).

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  1. Department of Human Development and Family Studies, Pennsylvania State University, 119 Health and Human Development Building, University Park, PA, 16802, USA

    Michael D. Hunter

  2. Department of Psychology, University of South Florida, Tampa, FL, 33620, USA

    Haya Fatimah & Marina A. Bornovalova

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Hunter, M.D., Fatimah, H. & Bornovalova, M.A. Two Filtering Methods of Forecasting Linear and Nonlinear Dynamics of Intensive Longitudinal Data. Psychometrika 87, 477–505 (2022). https://doi.org/10.1007/s11336-021-09827-5

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