Qiu Xiaomeng, Wang Fengwei, Zhang Qiuxi, Tao Guoqiang, Zhou Shijian
Gandong University, Fuzhou, People's Republic of China.
State Key Laboratory of Marine Geology, Tongji University, Shanghai, People's Republic of China.
Sci Rep. 2024 Aug 20;14(1):19268. doi: 10.1038/s41598-024-70421-7.
Due to various unavoidable reasons or gross error elimination, missing data inevitably exist in global navigation satellite system (GNSS) position time series, which may result in many analysis methods not being applicable. Typically, interpolating the missing data is a crucial preprocessing step before analyzing the time series. The conventional methods for filling missing data do not consider the influence of adjacent stations. In this work, an improved Gaussian process (GP) approach is developed to fill the missing data of GNSS time series, in which the time series of adjacent stations are applied to construct impact factors, together with a comparison of the conventional GP and the commonly used cubic spline methods. For the simulation experiments, the root mean square error (RMSE), mean absolute error (MAE) and correlation coefficient (R) are adopted to evaluate the performance of the improved GP. The results show that the filled missing data of the improved GP are closer to the true values than those of the conventional GP and cubic spline methods, regardless of the missing percentages ranging from 5 to 30%, with an interval of 5%. Specifically, the mean relative RMSE and MAE improvements for the improved GP with respect to the conventional GP are 21.2%, 21.3% and 8.3% and 12.7%, 16.2% and 11.01% for the North (N), East (E) and Up (U) components, respectively. In the real experiment, eight GNSS stations are analyzed using improved GP, together with conventional GP and a cubic spline. The results indicate that the first three principal components (PCs) of the improved GP can perverse 98.3%, 99.8% and 77.0% of the total variance for the N, E and U components, respectively. This value is obviously higher than those of the conventional GP and cubic spline. Therefore, we can conclude that the improved GP can better fill in the missing data in GNSS position time series than the conventional GP and cubic spline because of the impacts of adjacent stations.
由于各种不可避免的原因或粗大误差剔除，全球导航卫星系统（GNSS）位置时间序列中不可避免地存在缺失数据，这可能导致许多分析方法无法适用。通常，在分析时间序列之前，对缺失数据进行插值是一个关键的预处理步骤。传统的填充缺失数据的方法没有考虑相邻站点的影响。在这项工作中，开发了一种改进的高斯过程（GP）方法来填充GNSS时间序列的缺失数据，其中应用相邻站点的时间序列来构建影响因子，并将传统GP与常用的三次样条方法进行比较。对于模拟实验，采用均方根误差（RMSE）、平均绝对误差（MAE）和相关系数（R）来评估改进GP的性能。结果表明，无论缺失百分比在5%至30%之间（间隔为5%），改进GP填充的缺失数据都比传统GP和三次样条方法更接近真实值。具体而言，改进GP相对于传统GP在北（N）、东（E）和上（U）分量上的平均相对RMSE和MAE改进分别为21.2%、21.3%和8.3%，以及12.7%、16.2%和11.01%。在实际实验中，使用改进GP以及传统GP和三次样条对8个GNSS站点进行了分析。结果表明，改进GP的前三个主成分（PC）分别可以保留N、E和U分量总方差的98.3%、99.8%和77.0%。该值明显高于传统GP和三次样条的值。因此，我们可以得出结论，由于相邻站点的影响，改进GP在填充GNSS位置时间序列中的缺失数据方面比传统GP和三次样条表现更好。
