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.

翻译：大规模向量场数据集常见于多光谱光学和雷达传感器等众多新兴应用领域。我们提出了一种新颖的随机函数（数据）分析方法，通过分析域内标称随机行为的协方差结构来检测异常。将最优向量场Karhunen-Loève展开应用于此类随机场数据。基于域几何结构，从KL展开出发构建一系列多层级正交函数子空间。通过考察随机场在多层级基上的投影实现检测。该方法的关键特征在于构建可靠的假设检验，无需对数据概率分布做先验假设。我们将该方法应用于亚马孙森林退化这一重要问题。由于卫星影像的复杂性和高维度性，既无法假设已知分布，也无法对其进行估计。除了提供可靠的假设检验外，我们的方法还展示了在向量化复合体中使用多波段数据的优势，从而实现了更优的异常检测。此外，基于仿真数据，我们的方法能够检测到基于PCA方法无法检测的细微异常。