Abstract

Bayesian hierarchical time series models are increasingly used to analyze intensive longitudinal data, yet most applications rely on restrictive parametric assumptions for both within-person trends and between-person predictors of dynamic parameters. This is limiting when substantive questions concern nonlinear developmental change or complex heterogeneity in latent dynamics rather than nuisance detrending alone. Building on recent work that introduced regression splines for time-varying effects within DSEM, we develop a semiparametric extension that allows smooth effects at both the intensive-observation (Level 1) and person (Level 2) levels. In a hierarchical Bayesian lag-1 autoregressive formulation, Level 1 trajectories are decomposed into common and individual smooth functions, and the person-specific mean, autoregressive effect, and residual variance are modeled as smooth functions of person-level covariates. Smooth terms are represented with penalized thin-plate regression splines and estimated under Bayesian regularization, allowing highly nonlinear functional forms when supported by the data while shrinking toward simpler linear specifications when appropriate.

We first formulate the model for cross-sectional intensive longitudinal designs, where person-specific dynamic parameters and person-level covariates are observed once but their relationships may be complex and nonlinear. We then illustrate the approach with simulations involving nonlinear Level 1 trends and nonlinear Level 2 covariate effects, and with an applied example examining urge-to-smoke dynamics as a function of job- and home-related stress. In that example, the model accommodates nonlinear within-person trends while simultaneously estimating smooth between-person effects on the mean, autoregressive parameter, and residual variability. The recovered Level 2 associations are largely linear, but the relation between home stress and autoregressive persistence shows a more pronounced nonlinear pattern, illustrating the value of flexible functional forms when theory offers limited guidance.

Finally, we extend the same semiparametric construction to measurement-burst designs that combine intensive observations with broader longitudinal structures, including mixed-effects and fully within-person developmental formulations. These extensions expand bayesian time series models beyond ad hoc detrending and simple linear moderation, enabling direct inference on nonlinear heterogeneity in dynamic parameters across developmental time, age, or other person-level covariates. More broadly, the proposed framework integrates intensive longitudinal processes, between-person differences, and longitudinal change within a unified Bayesian modeling framework for substantive questions in the behavioral and social sciences.