Inclusion of data uncertainty in machine learning and its application in geodetic data science

Abstract

Data uncertainty plays an important role in the field of geodesy. We propose to include the uncertainty of data in deep neural network architectures to achieve better generalization, even in small data sets. Inspired by weighted and total least squares, we formulate the problem for both input and target uncertainties, and combine it with the Bayesian learning method. This results in a new form of the loss function in machine learning. Additionally, we consider the error propagation by using data uncertainty as features. As benchmark, we use models without the consideration of data uncertainty. The choice of the model is arbitrary. However, in this study the benchmark model is a single-layer LSTM neural network, which can represent and predict sequential data. We use input uncertainties either as auxiliary features or in a total least squares manner, and output uncertainties as weights in the L2 classical loss function. To show the efficacy of the proposed method, we use real data of Earth Orientation Parameters (EOP) and various GNSS station position time series across the globe. Prediction of EOP is important for many applications such as spacecraft navigation. For the two polar motion components, dUT1, and GNSS station position time series, we demonstrate that the least-squares-inspired method can outperform both the benchmark (by 45%, 52%, 1%, and 8%, respectively) and the feature-inspired method (65%, 55%, 77%, and 6%). In the case of the LOD time series, the feature-inspired method shows a better performance by 10% and 39% with regard to the benchmark and the least-squares-inspired method, respectively.