Path analysis is a special class of models in structural equation modeling (SEM) where it describes causal relations among measured variables in a form of linear regression. This paper presents two estimation formulations  for confirmatory and exploratory SEM in path analysis problems where a zero pattern of the estimated path coefficient matrix explains a causality structure of the variables. In confirmatory SEM, the original nonlinear equality constraints of model parameters are relaxed to an inequality, allowing us to transform the original problem into a convex problem . A regularized estimation formulation is proposed for exploratory SEM , where the objective function is added with an l1-type penalty of the path coefficient matrix. Under a condition on problem parameters, we show that our optimal solution is low rank and provides an estimate of the path matrix of the original problem. To solve our estimation problems in a convex framework, we apply alternating direction method of multiplier (ADMM) which is shown to be suitable for a large-scale implementation. In combination with applying model selection criteria, the penalty parameter in the regularized estimation, controlling the density of nonzero entries in the path matrix, can be chosen to provide a reasonable trade-off between the model fitting and the complexity of causality structure. The performance of our approach is demonstrated in both simulated and real data sets, and with a comparison of existing methods. Real application results include learning causality among climate variables in Thailand where our findings can explain known relations among air pollutants and weather variables. The other experiment is to explore connectivities among brain regions using fMRI time series from ABIDE data sets where our results are interpreted to explain brain network differencesin autism patients .
Path analysis is a model class of structural equation modeling (SEM) , which it describes causal relations among measured variables in the form of a multiple linear regression. This paper presents two estimation formulations , one each for confirmatory and exploratory SEM , where a zero pattern of the estimated path coefficient matrix can explain a causality structure of the variables. The original nonlinear equality constraints of the model parameters were relaxed to an inequality, allowing the transformation of the original problem into a convex framework . A regularized estimation formulation was then proposed for exploratory SEM using an l1-type penalty of the path coefficient matrix. Under a condition on problem parameters, our optimal solution is low rank and provides a useful solution to the original problem. Proximal algorithms were applied to solve our convex programs in a large-scale setting. The performance of this approach was demonstrated in both simulated and real data sets, and in comparison with an existing method. When applied to two real application results ( learning causality among climate variables in Thailand and examining connectivity differences in autism patients using fMRI time series from ABIDE data sets ) the findings could explain known relationships among environmental variables and discern known and new brain connectivity differences, respectively .
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