Modern technologies have accelerated the pace at which data is collected, leading to new frontiers in quantitative research. For example, there is a proliferation of wearable devices and sensors that continuously capture large, multi-modal longitudinal data. In order to utilize these complex temporally correlated datasets, we return to a first principles motivation of statistical modeling: exploiting similarities between ambient statistical objects to obtain powerful computational and inference strategies. Useful analyses for understanding temporal data include clustering and pattern recognition, modeling non-stationary systems, and nonparametric models. For these modeling objectives, we employ several techniques: latent variable and factor analysis, robust modeling of mixed hidden Markov models, and auxiliary data augmentation for Bayesian computation of nonparametric effects. We apply these methods to (1) cluster Pennsylvania counties by their time series of COVID-19 cases, (2) model active-rest cycles using passively collected smartphone data, and (3) model nonparametric effects with applications to modeling concussion recovery as a nonparametric function of time and distributional regression of activity profiles.