A novel regression method for harmonic analysis of time series

Harmonic analysis of time series is an important technique to reveal seasonal land surface dynamics using remote sensing information. However, frequency selection in the harmonic analysis is often difficult because high-frequency components are useful for delineating seasonal dynamics but sensitive to noise and gaps in time series. On the other hand, it is challenging to obtain temporally continuous satellite data with high quality because of atmospheric contamination. We developed a novel regression method named Harmonic Adaptive Penalty Operator (HAPO) for harmonic analysis of unevenly distributed time series. We introduced a new penalty function to minimize unexpected fluctuations in the model, which can substantially reduce the overfitting issue of regression in time series with temporal gaps. Specifically, the new penalty function minimizes the length of the model curve and the value range difference between the model and time series observations. We compared HAPO with three widely used regression methods (OLS: Ordinary Least Squares; LASSO: Least Absolute Shrinkage and Selection Operator; and Ridge) with different scenarios using Landsat time series data across the United States. First, we evaluated methods using Landsat surface reflectance time series within a single year. HAPO showed small and consistent monthly Root Mean Square Deviation (RMSD) values, in which most of the time RMSD values of predicted reflectance were less than 0.04. More importantly, HAPO showed consistent and less bias given varying density and irregularity of time series. Second, we evaluated methods using multi-year time series and the result suggested that HAPO was a better predictor of relatively short time series (less than4 years) with steady small RMSD values. When a longer time series (≥4 years) was used, all four methods disclosed similar RMSD values, but HAPO outperformed other three methods when there were temporal gaps. Last, we preliminarily tested how regression methods affected change detection and classification accuracy. HAPO showed the highest change detection accuracy of all tests in terms of F1 score when using the change threshold of 0.9999. In classification, HAPO produced the highest accuracy for short time series segments (one- or two-year time series). In contrast, all methods reached similar accuracy for 5-year time series. These results suggest that for areas that have large seasonal observation gaps or for time series that have less than 4 years records, HAPO can provide more consistent and accurate analytical results than other regression methods for harmonic analysis of time series.