A Fourier Fitting Method for Floating Vehicle Trajectory Lines

Abstract

With the advancement of spatial positioning technology, trajectory data have been growing rapidly. Trajectory data record the spatiotemporal information and behavioral characteristics of moving objects, and in-depth analysis can provide decision support for urban transportation. This paper explores effective methods for trajectory data representation, with a focus on the study of data fitting methods. Data fitting can extract key information and reveal underlying patterns, and the use of fitting methods can significantly improve the efficiency and accuracy of spatiotemporal trajectory data analysis, offering new perspectives and methodological support for related research fields. This paper integrates road network data to enhance trajectory data, treating trajectory data as a dynamic signal that changes over time. Through Fourier transformation, the data are converted from the time domain to the frequency domain, and trajectory points are fitted in the frequency spectrum domain, transforming discrete trajectory points into time-continuous linear elements. By referencing the minimum visually discernible distance and velocity precision requirements at a specific scale, thresholds for positional and velocity errors are set. The similarity between the Fourier-fitted trajectory and the original trajectory is measured in both spatial and temporal dimensions. By calculating the number of expansion terms of the Fourier series at a specific spatiotemporal scale, a functional relationship between the number of expansion terms, duration, and distance is fitted within the set threshold range (R² = 0.8424). This enables the Fourier series representation of any trajectory data under specific positional and velocity error thresholds. The errors in position and velocity obtained using this expression are significantly lower than the theoretical errors. The experimental results demonstrate that the Fourier fitting method exhibits strong generality and precision, effectively approximating the original trajectory, and provides a robust mathematical foundation for the quantification and detailed analysis of trajectory data.