Geostatistical semi-supervised learning for spatial prediction

Geoscientists are increasingly tasked with spatially predicting a target variable in the presence of auxiliary information using supervised machine learning algorithms. Typically, the target variable is observed at a few sampling locations due to the relatively time-consuming and costly process of obtaining measurements. In contrast, auxiliary variables are often exhaustively observed within the region under study through the increasing development of remote sensing platforms and sensor networks. Supervised machine learning methods do not fully leverage this large amount of auxiliary spatial data. Indeed, in these methods, the training dataset includes only labeled data locations (where both target and auxiliary variables were measured). At the same time, unlabeled data locations (where auxiliary variables were measured but not the target variable) are not considered during the model training phase. Consequently, only a limited amount of auxiliary spatial data is utilized during the model training stage. As an alternative to supervised learning, semi-supervised learning, which learns from labeled as well as unlabeled data, can be used to address this problem. However, conventional semi-supervised learning techniques do not account for the specificities of spatial data. This paper introduces a spatial semi-supervised learning framework where geostatistics and machine learning are combined to harness a large amount of unlabeled spatial data in combination with typically a smaller set of labeled spatial data. The main idea consists of leveraging the target variable's spatial autocorrelation to generate pseudo labels at unlabeled data points that are geographically close to labeled data points. This is achieved through geostatistical conditional simulation, where an ensemble of pseudo labels is generated to account for the uncertainty in the pseudo labeling process. The observed labels are augmented by this ensemble of pseudo labels to create an ensemble of pseudo training datasets. A supervised machine learning model is then trained on each pseudo training dataset, followed by an aggregation of trained models. The proposed geostatistical semi-supervised learning method is applied to synthetic and real-world spatial datasets. Its predictive performance is compared with some classical supervised and semi-supervised machine learning methods. It appears that it can effectively leverage a large amount of unlabeled spatial data to improve the target variable's spatial prediction.