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

Technical Summary: A Multi-View Contrastive Learning Framework for Spatial Embeddings in Risk Modelling

Problem Statement
Incorporating spatial information is critical for improving underwriting precision and risk management in insurance, particularly regarding climate, weather, and demographic factors. However, spatial data is often unstructured, high-dimensional, and difficult to integrate into predictive models. Traditional approaches represent location as simple coordinate pairs (latitude/longitude) or categorical variables (e.g., postal codes). These methods fail to capture real-world geographical structure; machine learning models like gradient boosting often apply linear splits on coordinates, creating artificial rectangular patterns that misalign with geography. While entity embeddings (mapping categorical variables to vectors) have been used in actuarial science, they are often inappropriate for spatial risk modelling because they treat spatial units as independent categories, failing to capture the smooth transitions of risk across space. Existing spatial embedding methods often rely on a single data modality or require access to external contextual data (e.g., census, satellite imagery) at the time of inference, limiting their utility when only coordinates are available.

Methodology
The authors propose a novel multi-view contrastive learning framework to generate spatial embeddings that combine information from multiple spatial data sources. The framework is designed to produce low-dimensional embeddings that capture both spatial structure and contextual similarity, enabling the generation of embeddings for any location using only its latitude–longitude coordinates.

1. Dataset Construction:

   • A dataset of 95,857 unique locations across Europe was constructed, sampled proportionally to population density.
   • Each location is enriched with two spatial views:
     • Google Satellite Imagery (GS): 256×256 RGB images centered on the coordinate.
     • OpenStreetMap (OSM) Features: Aggregated counts of six semantic categories (shops, tourism, food & drink, health, education, public safety) across three concentric rings of H3 hexagonal cells (37 hexagons total).

2. Model Architecture:
   The model consists of two main branches aligned via contrastive learning:

   • Spatial View Encoders (Multi-View Fusion):
     • GS Encoder: A ResNet18 architecture (pretrained on SSL4EO-S12) processes satellite images.
     • OSM Encoder: A custom convolutional neural network processes hexagonal OSM data. It utilizes a mapping strategy (Moussaid, 2023) to transform hexagonal grids into square matrices with masked corners, allowing the use of standard convolutional filters while preserving neighborhood relations.
     • Fusion: The outputs of the GS and OSM encoders are concatenated and passed through a feed-forward neural network to produce a joint spatial view embedding ($z_{MV}$).
   • Location Encoder:
     • Converts latitude–longitude coordinates ($\lambda, \phi$) into a vector ($z_{LE}$) of the same dimension as the fused view embedding.
     • Uses a positional encoder based on spherical harmonics (Rußwurm et al., 2024) to decompose spatial locations, followed by a Siren network (sinusoidal activation feed-forward network) to ensure smooth representation on a sphere.
   • Training Objective:
     • The model is trained using a contrastive loss function (similar to CLIP). The objective is to maximize the similarity between the coordinate-based embedding ($z_{LE}$) and the data-based embedding ($z_{MV}$) for the same location, while minimizing similarity for mismatched locations.
     • This process forces the coordinate encoder to internalize the contextual information contained in the satellite and OSM views.

3. Inference:
   Once trained, the location encoder can generate spatial embeddings for new locations using only their coordinates, without requiring access to the original satellite or OSM data.

Key Contributions

• Novel Architecture: The paper proposes the first architecture in actuarial literature that fuses multiple spatial views (imagery and tabular context) with a coordinate encoder via contrastive learning.
• Regional Focus: Unlike global foundation models (e.g., AlphaEarth, SatCLIP) that produce high-dimensional global representations, this framework is designed for fine-grained regional applications. It is computationally efficient enough to be retrained on consumer-grade hardware (single GPU) and can be adapted to specific regions or data sources.
• Coordinate-Only Inference: The framework solves the problem of generating rich spatial features for datasets that only contain coordinates, a common constraint in insurance portfolios.
• Open Resources: The authors provide the code on GitHub and release five pre-trained embedding models of varying dimensions (8, 16, 32, 64) trained on European data.

Experimental Results
The framework was validated through two case studies:

1. French Real Estate Pricing (Regression):

   • Setup: Predicting property prices using Generalized Linear Models (GLM), Generalized Additive Models (GAM), and Gradient Boosting Machines (GBM).
   • Performance: Replacing raw coordinates with spatial embeddings consistently improved out-of-sample predictive accuracy (lower Mean Squared Error) across all three model types. The improvement was most significant for GLMs (approx. 21% reduction in loss) and GAMs (25%), as these models struggle to capture complex spatial patterns with raw coordinates alone.
   • Interpretability: Variable importance analysis showed spatial embeddings became the most important feature. Partial dependence plots revealed that embeddings captured nuanced patterns (e.g., specific ski resorts, urban-rural gradients) that raw coordinates or standard smoothing could not.
   • Generalization: The model successfully generalized spatial effects to regions (Bas-Rhin, Haut-Rhin, Moselle) absent from the training data, capturing realistic risk patterns based on spatial characteristics rather than coordinate extrapolation.

2. Belgian Flood Claim Counts (Insurance):

   • Setup: Modeling flood claim counts per postal code using Poisson GLM, GAM, and GBM.
   • Performance: Spatial embeddings reduced the out-of-sample Poisson deviance for all model classes compared to coordinate-based inputs.
   • Risk Classification: Lorenz curve analysis demonstrated that models using embeddings achieved better risk classification (higher Gini indices), particularly in distinguishing high-risk areas (e.g., Verviers, Liège) from low-risk areas. The embeddings allowed the GLM to recover flood patterns more accurately than a GAM with bivariate coordinate smoothing.

Significance and Claims
The paper claims that this framework offers a flexible, scalable, and computationally efficient method for incorporating geographic information into actuarial pricing models. By moving beyond entity embeddings and simple spatial smoothing, the proposed spatial embeddings provide smooth, transferable, and explainable representations of risk.

The authors emphasize that these embeddings enable models to:

• Capture fine-grained local structures (topography, land use) that global models might overlook.
• Generalize to territories without outcome observations by associating predictions with spatial characteristics rather than absolute coordinates.
• Be retrained periodically to match temporal changes in the environment (e.g., using historical satellite archives).

The work positions spatial foundation models as a viable next step for actuarial science, bridging the gap between large-scale geospatial AI and practical, granular insurance risk management. The authors note that while the embeddings are powerful, their use as rating factors requires careful consideration of fairness, as they may capture socioeconomic patterns not present in traditional territorial variables.