A Smooth Reconstruction of Covariance Kernels from Fragmented Functional Data

Estimating the covariance operator from incomplete observations is a known problem in functional data analysis. While existing methods provide consistent estimates of the operator when functional samples are partially observed on varying subintervals of their domain, they fail in settings where missing values follow a persistent pattern across all samples, and entire portions of the domain remain systematically unobserved. We propose a nonparametric covariance reconstruction method designed specifically for this setting. Our approach estimates the covariance kernel through the sample covariance matrix and extends matrix completion techniques by introducing a Laplacian regularization which enforces smoothness in the solution. Our method effectively mitigates instability and roughness issues associated with purely low-rank approaches, while preserving accuracy in the reconstruction. The proposed methodology is showcased on multi-temporal interferometric images of the Phlegraean Fields, Italy, where covariance reconstruction is key for reliable displacement estimation and natural hazard monitoring.