Bayesian sparse factor regression trees

In this thesis, we focus on sparse principal component analysis (PCA) and nonlinear regression problems.We investigate several sparse PCA models and nonlinear regression techniques.We also explore the advantages of applying them sequentially and training them as an integral unit.First, we experiment with three sparse PCA models, which are optimal sparse PCA algorithms (OSPCA), Generalized Power algorithms (GP) and doubly sparse PCA algorithm (DSPCA).All the algorithms are compared using information loss and explained variance metrics, and we investigate their performance with both artificial and real data sets.OSPCA has the best control of the sparsity.GP and DSPCA both perform well on the synthetic and real data sets.The sparse factors identified by DSPCA for the real datasets are the most interpretable.Second, we report the results of experiments designed to test the performance of several nonlinear regression models (Bayesian additive regression trees (BART), random forests, neural networks, Extreme Gradient Boosting) in different scenarios with artificial and real data sets.When the number of predictors is smaller than that of data examples, no model outperforms the others consistently.However, when the data dimension increases, especially when the number of predictors exceeds that of data examples, the ensemble tree models, BART and random forest, are still able to handle the regression problem, whereas neural networks no longer provide a reasonable fit to the data because of the rapid increase in the number of model parameters and a lack of data.Finally, we investigate whether the prediction task can benefit from first applying sparse PCA to data to identify underlying sparse factor patterns and then applying the regression algorithms using the sparse representation of the data.We observe performance improvement for synthetic data.We also modified the inference algorithms of Bayesian DSPCA and BART to train these two models as an integral unit, so that prediction performance can inform the sparse PCA algorithms, guiding them to construct better representations of the data.

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