Statistical Inference in PCA for Hyperspectral Images

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 or anomaly detection, where the precision of further steps in the algorithm may depend on the reliable knowledge of the estimated eigenvalues and eigenvectors. The issue is also relevant in the context of small sample sizes occurring in local types of detectors (such as RX) and in investigations of individual components and small multi-material clusters within a hyperspectral image. 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, eigenvectors, and intrinsic dimensionality in six images of various sizes. In a broader context, we show an example of the correct statistical inference (construction of confidence intervals and bias-adjusted estimates) that can be implemented in other imaging applications.