Contextual anomaly and outlier detection is about to finding the peculiar phenomenons
which don’t conform the normal pattern. Anomaly detection has vast applications
in many areas such as fraud detection for credit cards, disease outbreak detection,
sensor networks and environmental science. We address the anomaly detection
problem in high dimensional spatio-temporal data. This study is motivated by finding
non-conforming patterns in environmental data sets such as NDVI and CO2 flux
linkage. In the first part, we propose two class of detection algorithms based on the
low rank and sparse Fourier representation of data. The simulation studies show
that the suggested algorithms can efficiently detect the abnormal patterns. We also
state the theoretical conditions under which the sparse Fourier anomaly detection
scheme attains the minimax consistency rate. The second part is devoted to study
the fused lasso signal approximator (FLSA) as a change point detection algorithm.
We present some upper bounds on `1 and `2 norm of estimation error of convex and
non-convex FLSA under possible model mismatch.