Principal component analysis (PCA) is a well-known tool in multivariate statistics. One significant challenge in using PCA is the choice of the number of components. In order to address this challenge, we propose an exact distribution-based method for hypothesis testing and construction of confidence intervals for signals in a noisy matrix. Assuming Gaussian noise, we use the conditional distribution of the singular values of a Wishart matrix and derive exact hypothesis tests and confidence intervals for the true signals. Our paper is based on the approach of Taylor, Loftus and Tibshirani (2013) for testing the global null: we generalize it to test for any number of principal components, and derive an integrated version with greater power. In simulation studies we find that our proposed methods compare well to existing approaches.

Network clustering is a fundamental task that discovers innate communities or groups in networks. Hence, network clustering methods such as spectral clustering and regularized spectral clustering have been applied in a wide range of realms. On top of a network structure, it is known in social network analysis that incorporates information from each vertex can be beneficial. This has led to the development of a series of attributed network clustering algorithms that utilize not only network connectivity but also vertex covariates in order to uncover latent clusters. This paper compares the performance of state‐of‐the‐art attributed network clustering approaches focused on detecting clusters of Seoul public bike stations. The data set consists of trip information over the bike station network in 2019. Spatial information about the bike stations is posed as vertex attributes. We show that certain attributed network clustering methods are well suited to detecting explainable clusters of bike rental stations. The results can help bike‐sharing operators better understand system usage and learn how to improve service quality in the existing system.