### Abstract

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 language | English |
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Publisher | CSGB, Institut for Matematik, Aarhus Universitet |
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Number of pages | 29 |

Publication status | Published - 2017 |

Series | CSGB Research Report |
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Number | 10 |

### Fingerprint

### Keywords

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

### Cite this

*Isotropic covariance functions on graphs and their edges*. CSGB, Institut for Matematik, Aarhus Universitet. CSGB Research Report, No. 10

}

*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.

Research output: Book/Report › Report › Research

TY - RPRT

T1 - Isotropic covariance functions on graphs and their edges

AU - Anderes, E.

AU - Møller, Jesper

AU - Rasmussen, Jakob Gulddahl

PY - 2017

Y1 - 2017

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.

KW - geodesic metric

KW - linear network

KW - parametric classes of covariance functions

KW - reproducing kernel Hilbert space

KW - resistance metric

KW - restricted covariance function properties

M3 - Report

BT - Isotropic covariance functions on graphs and their edges

PB - CSGB, Institut for Matematik, Aarhus Universitet

ER -