Convex formulations for path analysis problems in structural equation modeling

Abstract

Structural equation modeling (SEM) is a statistical technique used for seeking a statistical causal multivariate model (called exploratory modeling) or for testing whether the model is supported by the given data (called confirmatory modeling). Path analysis is a problem in SEM analysis where its model describes causal relations among measured variables in a form of multivariable linear equations. This thesis proposes two alternative estimation formulations for solving problems of path analysis in SEM. For confirmatory SEM, our first formulation relaxes the original nonlinear equality constraints of the model parameters to an inequality, allowing us to transform the original problem into a convex problem that can be solved by many existing efficient algorithms. The second formulation is a regularized estimation proposed for exploratory SEM by adding ℓ1-type penalty of the path matrix into the cost objective of the first formulation which leads to sparse solutions. Practically, our optimal solution is useful when it has low rank which occurs under a mild condition on problem parameters. This solution can be used as an estimate of the inverse of covariance matrix from the original problem. Another contribution of this thesis is a numerical method based on ADMM algorithm that is suitable for solving the two formulations in a large-scale setting. This thesis also provides a scheme of learning a causal structure among variables by applying both proposed formulations. The best causal structure from our scheme is chosen from five model selection criterions, those are BIC, AIC, AICc, KIC and KICc. Our approach is examined with simulated and real data sets. The simulation results show that if the causal structure of true model is complex, AIC provides the better accuracy while BIC, AICc and KICc yield better performance when the causal structure of true model is simpler. An application of this scheme has been preliminarily illustrated by learning causal relations among brain regions from fMRI data, recorded from visual-hand hemifield stimuli experiments. A brain network from our findings shows strong relations among somatosensory, parietal, premotor, and motor area. In particular, the dominant pairs of strong connection are somatosensory->visual, somatosensory->parietal and somatosensory->premotor.