Detection of collective and point anomalies at the presence of trend and seasonality

Abstract: Detecting anomalies in time series data is a challenging task with broad relevance in many applications. Existing methods work effectively only under idealized conditions, typically focusing on point anomalies or assuming a constant baseline. Our approach overcomes these limitations by detecting both collective and point anomalies, while allowing for polynomial trends and seasonal patterns. We establish statistical theory demonstrating that our method accurately decomposes the time series into anomaly, trend, seasonality, and a remainder component. We further show that it estimates the number of anomalies consistently and their locations with minimal error. Simulation studies confirm its strong detection performance with finite samples, and an application to energy price data illustrates its practical utility. An R package is available on request.