Monitoring movement of high-dimensional points is essential for environmental databases, geospatial applications, and biodiversity informatics as it reveals crucial information about data evolution, provenance detection, pattern matching etc. Despite recent research interest on processing continuous queries in the context of spatiotemporal data streams, the main focus is on managing the current location of numerous moving objects. In this paper, we turn our attention onto a historical perspective of movement and examine trajectories generated by streaming positional updates. The key challenge is how to maintain a concise, yet quite reliable summary of each object?s movement, avoiding any superfluous details and saving in processing complexity and communication cost. We propose two single-pass approximation techniques based on sampling that take advantage of the spatial locality and temporal timeliness inherent in trajectory streams. As a means of reducing substantially the scale of the datasets, we utilize heuristic prediction to distinguish which locations to preserve in the compressed trajectories. A comprehensive experimental study verifies the stability and robustness of the proposed techniques and demonstrates that intelligent compression schemes are able to act as effective load shedding operators achieving remarkable results.