Isotropic covariance functions on graphs and their edges
The authors study a generalization of a linear network called a graph with Euclidean edges, i.e., a graph where each edge is associated to an abstract set in bijective correspondence with a line segment of \(\mathbb R\). The aim of the manuscript is constructing isotropic parametric families of covariance functions over graphs with Euclidean edges. These covariance functions are defined on the vertices and edge points of graphs and are isotropic in the sense that they depend only on a specific distance. Two distances, the geodesic one and a new resistance metric, are considered. The resistance metric is discussed in detail in comparison with the geodesic metric. It is shown that numerous known isotropic covariance models, for example, the power exponential, Matérn, generalized Cauchy and Dagum classes, can be used as covariance functions for any graph with Euclidean edges with respect to the resistance metric. At the same time, they are covariance functions with respect to the geodesic metric only in special cases.The paper also establishes constraints, depending on the graph structure, for features of any radial profile which generates an isotropic covariance function with respect to both metrics. Proofs of theoretical results use Hilbert space and embedding methods.The results can be used to construct isotropic Gaussian random fields on graphs and to develop their statistical inference via likelihood methods.
parametric classes of covariance functions
restricted covariance function properties
scientific article