Sample size and error in the determination of mode shapes by principal components analysis

Principal components analysis (PCA) is a multivariate statistical technique that transforms a data set having a large number of inter-related variables to a new set of uncorrelated variables called the principal components, determined to allow the dimensionality of the data set to be reduced while retaining as much of the variation present as possible. PCA can be applied to dynamic structural response data to identify the predominant modes of vibration of the structure. Because PCA is a statistical technique, there are errors in the computed modes due to the use of a sample of finite size. The aim of this paper is to study the effect of sample size on the accuracy with which the modes of vibration can be computed. The paper focuses predominantly on elastic response data and examines the potential influence of various parameters such as the period of the structure, the input excitation, and the spatial distribution of mass over the structure. Issues relating to errors in the modes of nonlinear structures are also discussed.