Clustering of generalized gamma distributions by using information geometry: An application to medical imaging
Résumé
Probability density functions can be treated as functional data and may be represented as points of a statistical manifold using Information Geometry. Within this frame, densities are endowed with a Riemannian manifold structure, the metric being generally given by the Fisher information. The purpose is to present some new results about the generalized gamma manifold and how information geometry improved the performance of the classification of Alzheimer's disease population. In the medical field, a growing number of quantitative image analysis techniques have been developed, including analysis of histograms, which is widely used to quantify the diffuse pathological changes of some neu-rological diseases. For using the entire information included in the data, the underlying probability density functions themselves should be rather used as a biomarker of the whole brain. Some information geometric properties of the generalized gamma family are investigated, especially when restricted to the gamma submanifold, that is particularly relevant in the Alzeihmers disease context. The Fisher information and results in the case of the generalized gamma manifold will be first detailed. Next, a clustering technique has been successfully extended by using a geodesic distance of which an approximation is computed numerically with a two steps algorithm.