This article proposes a novel data-driven adaptive robust optimization (ARO) framework based on principal component analysis (PCA). By performing PCA on uncertainty data, the correlations among uncertain parameters are effectively captured, and principal components are identified. Uncertainty data are then projected onto each principal component, and distributional information is extracted from the projected uncertainty data using kernel density estimation. To explicitly account for asymmetric uncertainties, we introduce forward and backward deviations into uncertainty sets. The proposed data-driven ARO approach enjoys a less conservative solution compared with conventional robust optimization methods. A numerical example and an application in process network planning are presented to demonstrate the effectiveness of the proposed approach. Some promising extensions are also made within the proposed framework. Specifically, we investigate a data-driven uncertainty set in a low-dimensional subspace, and derive a theoretical bound on the performance gap between ARO solutions due to the dimension reduction of uncertainties.