Using bundle embeddings to predict daily cortisol levels in human subjects

Roelof B. Toonen, Klaas J. Wardenaar, Elisabeth H. Bos, Sonja L. van Ockenburg, Peter de Jonge

Many biological variables sampled from human subjects show a diurnal pattern, which poses special demands on the techniques used to analyze such data. Furthermore, most biological variables belong to nonlinear dynamical systems, which may make linear statistical techniques less suitable to analyze their dynamics. The current study investigates the usefulness of two analysis techniques based on nonlinear lagged vector embeddings: sequentially weighted global linear maps (SMAP), and bundle embeddings. Time series of urinary cortisol were collected in 10 participants, in the morning ('night' measurement) and the evening ('day' measurement), resulting in 126 consecutive measurements. These time series were used to create lagged vector embeddings, which were split into 'night' and 'day' bundle embeddings. In addition, embeddings were created based on time series that were corrected for the average time-of-day (TOD) values. SMAP was used to predict future values of cortisol in these embeddings. Global (linear) and local (non-linear) predictions were compared for each embedding. Bootstrapping was used to obtain confidence intervals for the model parameters and the prediction error. The best cortisol predictions were found for the night bundle embeddings, followed by the full embeddings and the time-of-day corrected embeddings. The poorest predictions were found for the day bundle embeddings. The night bundle embeddings, the full embeddings and the TOD-corrected embeddings all showed low dimensions, indicating the absence of dynamical processes spanning more than one day. The dimensions of the day bundles were higher, indicating the presence of processes spanning more than one day, or a higher amount of noise. In the full embeddings, local models gave the best predictions, whereas in the bundles the best predictions were obtained from global models, indicating potential nonlinearity in the former but not the latter. Using a bundling approach on time series of cortisol may reveal differences between the predictions of night and day cortisol that are difficult to find with conventional time-series methods. Combination of this approach with SMAP may especially be useful when analyzing time-series data with periodic components.