Bayesian Generative Modeling of Complex Dynamical Systems

This dissertation presents a Bayesian generative modeling approach for complex dynamical systems for emotion-interaction patterns within multivariate data collected in social psychology studies. While dynamical models have been used by social psychologists to study complex psychological and behavior patterns in recent years, most of these studies have been limited by using regression methods to fit the model parameters from noisy observations. These regression methods mostly rely on the estimates of the derivatives from the noisy observation, thus easily result in overfitting and fail to predict future outcomes. A Bayesian generative model solves the problem by integrating the prior knowledge of where the data comes from with the observed data through posterior distributions. It allows the development of theoretical ideas and mathematical models to be independent of the inference concerns. Besides, Bayesian generative statistical modeling allows evaluation of the model based on its predictive power instead of the model residual error reduction in regression methods to prevent overfitting in social psychology data analysis. In the proposed Bayesian generative modeling approach, this dissertation uses the State Space Model (SSM) to model the dynamics of emotion interactions. Specifically, it tests the approach in a class of psychological models aimed at explaining the emotional dynamics of interacting couples in committed relationships. The latent states of the SSM are composed of continuous real numbers that represent the level of the true emotional states of both partners. One can obtain the latent states at all subsequent time points by evolving a differential equation (typically a coupled linear oscillator (CLO)) forward in time with some known initial state at the starting time. The multivariate observed states include self-reported emotional experiences and physiological measurements of both partners during the interactions. To test whether well-being factors, such as body weight, can help to predict emotion-interaction patterns, we construct functions that determine the prior distributions of the CLO parameters of individual couples based on existing emotion theories. Besides, we allow a single latent state to generate multivariate observations and learn the group-shared coefficients that specify the relationship between the latent states and the multivariate observations. Furthermore, we model the nonlinearity of the emotional interaction by allowing smooth changes (drift) in the model parameters. By restricting the stochasticity to the parameter level, the proposed approach models the dynamics in longer periods of social interactions assuming that the interaction dynamics slowly and smoothly vary over time. The proposed approach achieves this by applying Gaussian Process (GP) priors with smooth covariance functions to the CLO parameters. Also, we propose to model the emotion regulation patterns as clusters of the dynamical parameters. To infer the parameters of the proposed Bayesian generative model from noisy experimental data, we develop a Gibbs sampler to learn the parameters of the patterns using a set of training couples. To evaluate the fitted model, we develop a multi-level cross-validation procedure for learning the group-shared parameters and distributions from training data and testing the learned models on held-out testing data. During testing, we use the learned shared model parameters to fit the individual CLO parameters to the first 80% of the time points of the testing data by Monte Carlo sampling and then predict the states of the last 20% of the time points. By evaluating models with cross-validation, one can estimate whether complex models are overfitted to noisy observations and fail to generalize to unseen data. I test our approach on both synthetic data that was generated by the generative model and real data that was collected in multiple social psychology experiments. The proposed approach has the potential to model other complex behavior since the generative model is not restricted to the forms of the underlying dynamics