Uncertainty quantification for principal component regression

Abstract: Principal component regression is an effective dimension reduction method for regression problems. To apply it in practice, one typically starts by selecting the number of principal components k, then estimates the corresponding regression parameters using say maximum likelihood, and finally obtains predictions with the fitted results. The success of this approach highly depends on the choice of k, and very often, due to the noisy nature of the data, it could be risky to just use one single value of k. Using the generalized fiducial inference framework, this paper develops a method for constructing a probability function on k, which provides an uncertainty measure on its value. In addition, this paper also constructs novel confidence intervals for the regression parameters and prediction intervals for future observations. The proposed methodology is backed up by theoretical results and is tested by simulation experiments and compared with other methods using real data. To the best of our knowledge, this is the first time that a full treatment for uncertainty quantification is formally considered for principal component regression.