An approach to telematics using mobile devices provides battery-efficient trajectory and mileage inference from inaccurate and intermittent location data. Accurate trajectories of how users or vehicles move in the physical world are formed by processing raw position estimates obtained from noisy, inaccurate, and error-prone position sensors on mobile devices, where the position data may also arrive intermittently with long time gaps. The trajectory is formed using the process of map matching, which determines the trajectory on a map that best explains the sequence of position observations.