Feature Selection under Multicollinearity & Causal Inference on Time Series

Abstract

In this work, we study and extend algorithms for Sparse Regression and Causal Inference problems. Both the problems are fundamental in the area of Data Science. The goal of regression problem is to nd out the \best" relationship between an output variable and input variables, given samples of the input and output values. We consider sparse regression under a high-dimensional linear model with strongly correlated variables, situations which cannot be handled well using many existing model selection algorithms. We study the performance of the popular feature selection algorithms such as LASSO, Elastic Net, BoLasso, Clustered Lasso as well as Projected Gradient Descent algorithms under this setting in terms of their running time, stability and consistency in recovering the true support. We also propose a new feature selection algorithm, BoPGD, which cluster the features rst based on their sample correlation and do subsequent sparse estimation using a bootstrapped variant of the projected gradient descent method with projection on the non-convex L0 ball. We attempt to characterize the efficiency and consistency of our algorithm by performing a host of experiments on both synthetic and real world datasets. Discovering causal relationships, beyond mere correlation, is widely recognized as a fundamental problem. The Causal Inference problems use observations to infer the underlying causal structure of the data generating process. The input to these problems is either a multivariate time series or i.i.d sequences and the output is a Feature Causal Graph where the nodes correspond to the variables and edges capture the direction of causality. For high dimensional datasets, determining the causal relationships becomes a challenging task because of the curse of dimensionality. Graphical modeling of temporal data based on the concept of \Granger Causality" has gained much attention in this context. The blend of Granger methods along with model selection techniques, such as LASSO, enables efficient discovery of a \sparse" sub-set of causal variables in high dimensional settings. However, these temporal causal methods use an input parameter, L, the maximum time lag. This parameter is the maximum gap in time between the occurrence of the output phenomenon and the causal input stimulus. How-ever, in many situations of interest, the maximum time lag is not known, and indeed, finding the range of causal e ects is an important problem. In this work, we propose and evaluate a data-driven and computationally efficient method for Granger causality inference in the Vector Auto Regressive (VAR) model without foreknowledge of the maximum time lag. We present two algorithms Lasso Granger++ and Group Lasso Granger++ which not only constructs the hypothesis feature causal graph, but also simultaneously estimates a value of maxlag (L) for each variable by balancing the trade-o between \goodness of t" and \model complexity".