Using bundle embeddings to predict daily cortisol levels in human subjects.

BACKGROUND: 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. METHODS: 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. RESULTS: 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. CONCLUSIONS: 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.

Using State Space Methods to Reveal Dynamical Associations Between Cortisol and Depression.

Despite extensive research, the link between etiological factors and depression remains poorly understood. This may in part be due to a focus on strictly linear definitions of causality, derived at the group level. However, etiological relations in depression are likely to be dynamical, nonlinear and potentially unquantifiable with traditional statistics. Therefore the aim of this study was to evaluate the use of the convergent cross-mapping (CCM) method in investigating possible nonlinear relationships between supposed etiological factors and depressive symptomatology. Time series data from six healthy individuals were used to model the relationship between 24-h urinary free cortisol and negative affect using CCM and dewdrop embeddings. CCM is a nonlinear measure of causality, based on state space reconstruction with lagged coordinate embeddings. The results showed that nonlinear dynamical relationships between cortisol and negative affect may be present within participants, as demonstrated by a positive cross-map convergence from negative affect to cortisol. However, analyses also showed that noise and influential points had considerable impact on the results. Convergent crossmapping can be used to reveal possible nonlinear dynamical relationships between etiological factors and psychopathology that may remain undetected with traditional linear causality measures.