On the reliability of PCA for complex hyperspectral data

Principal Component Analysis (PCA) is a popular tool for initial investigation of hyperspectral image data. There are many ways in which the estimated eigenvalues and eigenvectors of the covariance matrix are used. Further steps in the analysis or model building for hyperspectral images are often dependent on those estimated quantities. It is therefore important to know how precisely the eigenvalues and eigenvectors are estimated, and how the precision depends on the sampling scheme, the sample size, and the covariance structure of the data. This issue is especially relevant for applications such as difficult target detection, where the precision of further steps in the algorithm may depend on the reliable knowledge of the estimated eigenvalues and eigenvectors. The sampling properties of eigenvalues and eigenvectors are known to some extent in statistical literature (mostly in the form of asymptotic results for large sample sizes). Unfortunately, those results usually do not apply in the context of hyperspectral images. In this paper, we investigate the sampling properties of eigenvalues and eigenvectors under three scenarios. The first two scenarios consider the type of sampling traditionally used in statistics, and the third scenario considers the variability due to image noise, which is more appropriate for hyperspectral imaging applications. For all three scenarios, we show the precision associated with the estimated eigenvalues and eigenvectors.