Isotropic covariance functions on graphs and their edges

Research output: Book/ReportReportResearch

Abstract

We develop parametric classes of covariance functions on linear networks and
their extension to graphs with Euclidean edges, i.e., graphs with edges viewed
as line segments or more general sets with a coordinate system allowing us to
consider points on the graph which are vertices or points on an edge. Our covariance
functions are defined on the vertices and edge points of these graphs
and are isotropic in the sense that they depend only on the geodesic distance
or on a new metric called the resistance metric (which extends the classical
resistance metric developed in electrical network theory on the vertices of a
graph to the continuum of edge points). We discuss the advantages of using
the resistance metric in comparison with the geodesic metric as well as
the restrictions these metrics impose on the investigated covariance functions.
In particular, many of the commonly used isotropic covariance functions in
the spatial statistics literature (the power exponential, Matérn, generalized
Cauchy, and Dagum classes) are shown to be valid with respect to the resistance
metric for any graph with Euclidean edges, whilst they are only valid
with respect to the geodesic metric in more special cases.
Original languageEnglish
PublisherCSGB, Institut for Matematik, Aarhus Universitet
Number of pages29
Publication statusPublished - 2017
SeriesCSGB Research Report
Number10

Fingerprint

Covariance Function
Metric
Graph in graph theory
Geodesic
Euclidean
Spatial Statistics
Electrical Networks
Line segment
Continuum
Valid
Restriction

Keywords

  • geodesic metric
  • linear network
  • parametric classes of covariance functions
  • reproducing kernel Hilbert space
  • resistance metric
  • restricted covariance function properties

Cite this

Anderes, E., Møller, J., & Rasmussen, J. G. (2017). Isotropic covariance functions on graphs and their edges. CSGB, Institut for Matematik, Aarhus Universitet. CSGB Research Report, No. 10
Anderes, E. ; Møller, Jesper ; Rasmussen, Jakob Gulddahl. / Isotropic covariance functions on graphs and their edges. CSGB, Institut for Matematik, Aarhus Universitet, 2017. 29 p. (CSGB Research Report; No. 10).
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Anderes, E, Møller, J & Rasmussen, JG 2017, Isotropic covariance functions on graphs and their edges. CSGB Research Report, no. 10, CSGB, Institut for Matematik, Aarhus Universitet.

Isotropic covariance functions on graphs and their edges. / Anderes, E. ; Møller, Jesper; Rasmussen, Jakob Gulddahl.

CSGB, Institut for Matematik, Aarhus Universitet, 2017. 29 p. (CSGB Research Report; No. 10).

Research output: Book/ReportReportResearch

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N2 - We develop parametric classes of covariance functions on linear networks andtheir extension to graphs with Euclidean edges, i.e., graphs with edges viewedas line segments or more general sets with a coordinate system allowing us toconsider points on the graph which are vertices or points on an edge. Our covariancefunctions are defined on the vertices and edge points of these graphsand are isotropic in the sense that they depend only on the geodesic distanceor on a new metric called the resistance metric (which extends the classicalresistance metric developed in electrical network theory on the vertices of agraph to the continuum of edge points). We discuss the advantages of usingthe resistance metric in comparison with the geodesic metric as well asthe restrictions these metrics impose on the investigated covariance functions.In particular, many of the commonly used isotropic covariance functions inthe spatial statistics literature (the power exponential, Matérn, generalizedCauchy, and Dagum classes) are shown to be valid with respect to the resistancemetric for any graph with Euclidean edges, whilst they are only validwith respect to the geodesic metric in more special cases.

AB - We develop parametric classes of covariance functions on linear networks andtheir extension to graphs with Euclidean edges, i.e., graphs with edges viewedas line segments or more general sets with a coordinate system allowing us toconsider points on the graph which are vertices or points on an edge. Our covariancefunctions are defined on the vertices and edge points of these graphsand are isotropic in the sense that they depend only on the geodesic distanceor on a new metric called the resistance metric (which extends the classicalresistance metric developed in electrical network theory on the vertices of agraph to the continuum of edge points). We discuss the advantages of usingthe resistance metric in comparison with the geodesic metric as well asthe restrictions these metrics impose on the investigated covariance functions.In particular, many of the commonly used isotropic covariance functions inthe spatial statistics literature (the power exponential, Matérn, generalizedCauchy, and Dagum classes) are shown to be valid with respect to the resistancemetric for any graph with Euclidean edges, whilst they are only validwith respect to the geodesic metric in more special cases.

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KW - reproducing kernel Hilbert space

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KW - restricted covariance function properties

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Anderes E, Møller J, Rasmussen JG. Isotropic covariance functions on graphs and their edges. CSGB, Institut for Matematik, Aarhus Universitet, 2017. 29 p. (CSGB Research Report; No. 10).