Mobile data technologies use ``actigraphs'' to furnish information on health variables as a function of a subject's movement. The advent of wearable devices and related technologies has propelled the creation of health databases consisting of human movement data to conduct research on mobility patterns and health outcomes. Statistical methods for analyzing high-resolution actigraph data depend on the specific inferential context, but the advent of Artificial Intelligence (AI) frameworks require that the methods be congruent to transfer learning and amortization. This article devises amortized Bayesian inference for actigraph time sheets. We pursue a Bayesian approach to ensure full propagation of uncertainty and its quantification using a hierarchical dynamic linear model. We build our analysis around actigraph data from the Physical Activity through Sustainable Transport Approaches in Los Angeles (PASTA-LA) study conducted by the Fielding School of Public Health in the University of California, Los Angeles. Apart from achieving probabilistic imputation of actigraph time sheets, we are also able to statistically learn about the time-varying impact of explanatory variables on the magnitude of acceleration (MAG) for a cohort of subjects.
Figures & tables
Figure 1. Left: The data covered by each subject for each date covered in the dataset under the absolute time step setup, where time ranges from 7 am - 11 pm. Right: The same under the relative time step setup from start of trajectory to up to 20 minutes later, with trajectories going beyond 20 minutes filtered out or cut off at the 22 minute mark. The color corresponds to the MAG value for the subject at a particular date for that time. Gray cells correspond to missing data for the subject-date pair at that particular epoch.
Subject-Date
Time 1
Time 2
⋯
Time T
S1_D1
A111
A112
⋯
A11 T
S1_D2
n/a
A122
⋯
A12 T
⋮
⋮
⋮
⋮
⋮
S2_D1
n/a
n/a
⋯
A21 T
S2_D2
n/a
A212
⋯
A22 T
⋮
⋮
⋮
⋮
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Table 1. A sample actigraph timesheet with s subjects and ns dates recorded for T discrete time steps. Subjects and dates are paired (e.g. S1_D1 denotes subject 1 paired with date 1), and data is recorded at each time (e.g. A111 denotes the data recorded for subject 1 at date 1 and time 1). n/a (highlighted in grey) denotes cells where the data is missing.
Figure 2. 95% credible intervals of the estimates of β1,σ2∣y1 from BayesFlow (red) and the FFBS (cyan). The means and intervals differ slightly for all parameter estimates except σ2 , which differs considerably.
Figure 3. The estimated parameter trajectories (red) due to Algorithm 6 , estimated trajectories due to the FFBS (cyan), the FF only (green), and true trajectories (blue) of the coefficients used to generate the synthetic outcomes at each time step. The true coverage rate of the synthetic outcomes after the FFBS processes the outcome is about 97.6%. The credible intervals for the ABI output were taken from the 2.5% and 97.5% quantiles of 10,000 samples generated from the trained networks. A horizontal line is added at βt,j=0 for non-intercept elements j to visualize the statistical significance of each parameter over time.
Figure 4. The credible intervals compared with their synthetic values generated from a single run of Y_FROM_DLM from Algorithm 5 for a subset of the actigraph data not present in the training data (left) and the whole actigraph data not present (right). The coverage rates (ranging from 0 to 1) for each setting are also depicted in each plot, with FFBS in blue with cyan text and ABI in pink with red text. The FFBS intervals serve as a reference for prediction, with ABI’s visibly wider and proportionately longer than the FFBS intervals depending on the latter’s length.
Figure 5. The credible intervals compared with their synthetic values generated from Equation ( 8 ) for Actigraph Timesheet imputed data.
Figure 6. The predictive performance of the FFBS (cyan) and the trained ABI (red) on the log(MAG) . Both the coverage rates and overall rates of credible intervals are similar, showing how ABI is able to adapt to the behavior of the FFBS even after being fed new data to form a posterior distribution.
Figure 7. The estimated parameter values regressing on log(MAG) over time due to Algorithm 6 , and estimated trajectories due to the FF only (green) and the FFBS (cyan) of the coefficients at each time point. The credible intervals for the ABI output were generated using the same procedure for Figure 3 . A horizontal line is added at βt,j=0 for non-intercept elements j to visualize the statistical significance of each parameter over time.
Department of Statistics TU Dortmund University, Germany · Institute for Visualisation and Interactive Systems University of Stuttgart, Germany · Department of Cognitive Science Rensselaer Polytechnic Institute, USA
Department of Statistics and Data Science, Indian Institute of Technology Kanpur, Kanpur, India 208016 · Marshall School of Business, University of Southern California, Los Angeles, United States 90089-0809