## 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.

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 |

## Keywords

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