Graph Neural Networks for Spatial Data Analysis: Theoretical Results and Applications

by Marianne Abemgnigni Njifon
Doctoral thesis
Date of Examination:2025-03-25
Date of issue:2025-12-02
Advisor:Prof. Dr. Dominic Schuhmacher
Referee:Prof. Dr. Dominic Schuhmacher
Referee:Prof. Dr. Stephan Huckemann
Referee:Prof. Dr. Gerlind Plonka-Hoch
Referee:Prof. Dr. Anja Sturm
Referee:Prof. Dr. David Russell Luke
Referee:Prof. Dr. Fabian Sinz
Persistent Address: http://dx.doi.org/10.53846/goediss-11668

 

 

Files in this item

Name:Marianne_corrected_thesis.pdf
Size:2.04Mb
Format:PDF

The following license files are associated with this item:

Abstract

English

Overall, this thesis explores the advantages and challenges of spatial data prediction tasks with Graph Neural Networks (GNN). Kriging, the best linear unbiased predictor presents a great number of advantages over other methods. However, besides the scalability issues that Kriging presents and for which many solution attempts are proposed in the literature, one problem remains that in practice, the optimality conditions of Kriging are often not met. GNNs offer the flexibility of directly embedding the target variables of neighbors in predictions. Unfortunately, limitations remain in the design of adjacency matrices for effectively capturing the dependencies. Moreover, most of the available GNNs for spatial prediction still require a large amount of data. This is the context in which our contributions subscribe. Performance comparisons of our first proposed method SPONGE present the relevance of the architecture we developed, both in terms of prediction accuracies but also in terms of ”automation” in the hyperparameter selection process of adjacency matrices. The mathematical formalism we introduced allows to unify the various methodologies for spatial prediction tasks into one framework. We specifically highlight the great performance of the Sponge 1-hop (the smallest in parameters) in Kriging unfriendly world including in small datasets regime, which prove to be challenging for standard neural networks in general and GNNs in particular. A further step is taken to derive an exact formula of the joint distribution between the true unobserved value and the output of a shallow ReLU neural network with one hidden layer, with Gaussian correlated input. Although this shallow neural network (shallow NN) is in design different from GNNs, this generalisation to nonlinear predictors is a first major step for ReLU networks analysis. We additionally compute an exact derivation of the mutual information (MI) as well as the MSE of the shallow NN predictor for getting unobserved gaussian correlated input. This result is original in the sense that standard methods for mutual information analysis are approximate methods. Finally, our contribution in tumor heterogeneity predictions are threefold: A methodology to obtain graph datasets from an artificially simulated tumor model, a proposition of GNN-based methodology that yielded exciting results on the artificial test set, an analysis of our proposed spatial- and birth/death markers-based features to explain the observed accuracy. Validation on real world datasets are in perspective.
Keywords: Graph Neural Networks; Spatial data; Kriging; Mutual Information; Tumor Heterogeneity; MSE; SPONGE; BGNN; Shallow networks
 
Statistik