Rapid detection of hot-spots via tensor decomposition with applications to crime rate data

In many real-world applications of monitoring multivariate spatio-temporal data that are non-stationary over time, one is often interested in detecting hot-spots with spatial sparsity and temporal consistency, instead of detecting system-wise changes as in traditional statistical process control (SPC) literature. In this paper, we propose an efficient method to detect hot-spots through tensor decomposition, and our method has three steps. First, we fit the observed data into a Smooth Sparse Decomposition Tensor (SSD-Tensor) model that serves as a dimension reduction and de-noising technique: it is an additive model decomposing the original data into: smooth but non-stationary global mean, sparse local anomalies, and random noises. Next, we estimate model parameters by the penalized framework that includes Least Absolute Shrinkage and Selection Operator (LASSO) and fused LASSO penalty. An efficient recursive optimization algorithm is developed based on Fast Iterative Shrinkage Thresholding Algorithm (FISTA). Finally, we apply a Cumulative Sum (CUSUM) Control Chart to monitor model residuals after removing global means, which helps to detect when and where hot-spots occur. To demonstrate the usefulness of our proposed SSD-Tensor method, we compare it with several other methods including scan statistics, LASSO-based, PCA-based, T2-based control chart in extensive numerical simulation studies and a real crime rate dataset.