Gaussian Processes (GPs) are powerful non-parametric Bayesian models for regression of scalar fields, formulated under the assumption that measurement locations are perfectly known and the corresponding field measurements have Gaussian noise. However, many real-world scalar field mapping applications rely on sensor-equipped mobile robots to collect field measurements, where imperfect localization introduces state uncertainty. Such discrepancies between the estimated and true measurement locations degrade GP mean and covariance estimates. To address this challenge, we propose a method for updating the GP models when improved estimates become available. Leveraging the differentiability of the kernel function, a second-order correction algorithm is developed using the precomputed Jacobians and Hessians of the GP mean and covariance functions for real-time refinement based on measurement location discrepancy data. Simulation results demonstrate improved prediction accuracy and computational efficiency compared to full model retraining.

翻译：高斯过程（GPs）是用于标量场回归的强大非参数贝叶斯模型，其构建基于测量位置完全已知且对应场测量值具有高斯噪声的假设。然而，许多现实世界的标量场映射应用依赖于配备传感器的移动机器人来采集场测量数据，其中不完善的定位会引入状态不确定性。这种估计测量位置与真实测量位置之间的差异会降低GP均值和协方差的估计精度。为应对这一挑战，我们提出了一种在获得改进估计时更新GP模型的方法。利用核函数的可微性，基于预计算的GP均值与协方差函数的雅可比矩阵和海森矩阵，开发了一种二阶校正算法，用于根据测量位置偏差数据进行实时优化。仿真结果表明，与完整模型重新训练相比，该方法在预测精度和计算效率上均有提升。