Frechet distance-based cluster analysis for multi-dimensional functional data

Multi-dimensional functional data analysis has become a contemporary research topic in medical research as patients' various records are measured over time. We propose two clustering methods using the Frechet distance for multi-dimensional functional data. The first method extends an existing K-means type approach from one-dimensional to multi-dimensional longitudinal data. The second method enforces sparsity on functional variables while grouping observed trajectories and enables us to assess the contribution from each variable. Both methods utilize the generalized Frechet distance to measure the distance between trajectories with irregularly spaced and asynchronous measurements. We demonstrate the effectiveness of the proposed methods through a comparative study using various simulation examples. Then, we apply the sparse clustering method to multi-dimensional thyroid cancer data collected in South Korea. It produces interpretable clusters and weighs the importance of functional variables.