A Journey to Derivatives: From Historical Foundations to Spatial Omics

Abstract

Spatial data pervades numerous fields, ranging from geospatial research to genomics and image analysis, typically within a 2-dimensional plane. Gaussian Processes (GPs) have become foundational in studying these datasets. However, the conventional approach often centers solely on the function. This talk revisits the historical essence of calculus, championed by Newton and Leibniz, and extends it to the novel territory of Gaussian Processes. We commence with the derivative and curvature processes derived from Gaussian Processes, known as the ‘derivative process’ and the ‘curvature process’, and their pivotal role in spatial omics data—a burgeoning domain in genomics. Transitioning to non-Euclidean domains, intrinsic to studies of surfaces such as the brain, heart, and teeth, we outline the recent advancements of GPs in these manifold domains. Through a rigorous definition of the associated derivative and curvature processes, we provide a blueprint of the conditions required for their existence and derive their joint distributions for comprehensive statistical inference. This talk, thus, traces the journey of the derivative concept, bridging its historical significance with modern applications in spatial data analysis.