Estimation of local training data point densities to support the assessment of spatial prediction uncertainty

Abstract. Machine learning is frequently used in environmental and earth sciences to produce spatial or spatio-temporal predictions of environmental variables based on limited field samples – increasingly even on a global scale and far beyond the location of available training data. Since new geographic space often goes along with new environmental properties represented by the model's predictors, and since machine learning models do not perform well in extrapolation, this raises questions regarding the applicability of the trained models at the prediction locations. Methods to assess the area of applicability of spatial prediction models have been recently suggested and applied. These are typically based on distances in the predictor space between the prediction data and the nearest reference data point to represent the similarity to the training data. However, we assume that the density of the training data in the predictor space, i.e. how well an environment is represented in a model, is highly decisive for the prediction quality and complements the consideration of distances. We therefore suggest a local training data point density (LPD) approach. The LPD is a quantitative measure that indicates, for a new prediction location, how many similar reference data points have been included in the model training. Similarity here is defined by the dissimilarity threshold introduced by Meyer and Pebesma (2021), which is the maximum distance to a nearest training data point in the predictor space as observed during cross-validation. We assess the suitability of the approach in a simulation study and illustrate how the method can be used in real-world applications. The LPD is a post hoc measure of training data density in predictor space, used to support uncertainty assessment of predictions independently of the model's structure. The simulation study indicated a positive relationship between LPD and prediction performance and highlights the value of the approach compared to the consideration of the distance to a nearest data point only. We therefore suggest the calculation of the LPD to support the assessment of prediction uncertainties.