Distribution-Free Stochastic Analysis and Robust Multilevel Vector Field Anomaly Detection

Massive vector field datasets are common in multi-spectral optical and radar sensors, among many other emerging areas of application. We develop a novel stochastic functional (data) analysis approach for detecting anomalies based on the covariance structure of nominal stochastic behavior across a domain. An optimal vector field Karhunen-Loeve expansion is applied to such random field data. A series of multilevel orthogonal functional subspaces is constructed from the geometry of the domain, adapted from the KL expansion. Detection is achieved by examining the projection of the random field on the multilevel basis. A critical feature of this approach is that reliable hypothesis tests are formed, which do not require prior assumptions on probability distributions of the data. The method is applied to the important problem of degradation in the Amazon forest. Due to the complexity and high dimensionality of satellite imagery, it is not feasible to assume known distributions, nor to estimate them. In addition to providing reliable hypothesis tests, our approach shows the advantage of using multiple bands of data in a vectorized complex, leading to better anomaly detection. Furthermore, using simulated data, our approach is capable of detecting subtle anomalies that are impossible to detect with PCA-based methods.