A multi-view contrastive learning framework for spatial embeddings in risk modelling

TL;DR

This work tackles the problem of integrating rich spatial information into risk-modelling tasks by learning low-dimensional spatial embeddings from multiple data sources. It introduces a multi-view contrastive learning framework that fuses Google Satellite imagery and OpenStreetMap features with coordinate encodings based on spherical harmonics, enabling embeddings to be generated for any location from its latitude-longitude pair. The authors build a European dataset of 95,857 locations and demonstrate that replacing raw coordinates with the learned spatial embeddings improves predictive performance across GLMs, GAMs, and GBMs in a French real estate pricing case study, while also providing interpretable spatial effects and transferability to unseen regions. The approach is lightweight, adaptable to additional spatial data sources, and suitable for actuarial deployments, suggesting meaningful implications for risk pricing, reserving, and climate-related risk analytics using spatial foundation-model style representations.

Abstract

Incorporating spatial information, particularly those influenced by climate, weather, and demographic factors, is crucial for improving underwriting precision and enhancing risk management in insurance. However, spatial data are often unstructured, high-dimensional, and difficult to integrate into predictive models. Embedding methods are needed to convert spatial data into meaningful representations for modelling tasks. We propose a novel multi-view contrastive learning framework for generating spatial embeddings that combine information from multiple spatial data sources. To train the model, we construct a spatial dataset that merges satellite imagery and OpenStreetMap features across Europe. The framework aligns these spatial views with coordinate-based encodings, producing low-dimensional embeddings that capture both spatial structure and contextual similarity. Once trained, the model generates embeddings directly from latitude-longitude pairs, enabling any dataset with coordinates to be enriched with meaningful spatial features without requiring access to the original spatial inputs. In a case study on French real estate prices, we compare models trained on raw coordinates against those using our spatial embeddings as inputs. The embeddings consistently improve predictive accuracy across generalised linear, additive, and boosting models, while providing interpretable spatial effects and demonstrating transferability to unseen regions.

Figures (18)

• Figure 1: Conceptual illustration of entity versus spatial embeddings for a selection of postal codes in France. Each postal code is embedded into a one-dimensional latent space, illustrated by colour.
• Figure 2: Architecture of our multi-view spatial embedding model. The green block represents the embedding of latitude and longitude coordinates. Each blue block embeds a different spatial view, and the orange block fuses these different spatial embeddings. The model is trained with a contrastive learning objective to learn spatial embeddings by contrasting the coordinate embedding with the information learnt from the different spatial views. Image inspired by klemmer2023satclip.
• Figure 3:
• Figure 4: Illustration of one data point in the dataset $\mathcal{D}$, with coordinates $(\lambda,\phi) = (4.70591^{\circ}, 50.88173^{\circ})$ in Leuven, Belgium. Panel (a) shows the location of the coordinate pair. Panel (b) displays the associated Google Satellite image, where the matrix $V^{\mathrm{GS}}\in\mathbb{R}^{256\times256\times3}$ stores the RGB pixel values. Panel (c) displays the corresponding OpenStreetMap features, where the matrix $V^{\mathrm{OSM}}\in\mathbb{N}^{37\times6}$ contains the counts of six categories of tags for each of the $37$ surrounding hexagons.
• Figure 5: Illustration of the encoder used for the Google satellite imagery spatial view. We use a ResNet18 architecture He2016-ab, pretrained following Wang2023. The input image (green block) of dimension $256\times256\times3$ is processed by convolutional layers (blue blocks) that extract spatial patterns. These patterns are then pooled and flattened (yellow blocks) before passing through a fully connected network (rightmost blue block). The final embedding is a vector of dimension $d_{\mathrm{GS}}$, as shown in the red block. Blue blocks contain the trainable parameters of the model.