Spline interpolation on networks

Abstract

So far, spatial data analysis methods have been designed and developed mainly for manifold-like spaces such as the Euclidean plane. Recently, spatial data analysis methods for networks, called network spatial methods, have been gaining attentions since they are convenient for micro-scale analysis. In this paper, interpolation methods over networks are explored. Shiode extended the inverse distance-weighted method to networks, compared the results by the conventional and extended methods, and showed the significance of network-specific interpolation methods. This paper, however, shows that weighted average interpolation methods including the inverse distance-weighted method over networks suffer from a kind of bias. Instead of developing the way how to remove such bias from weighted average interpolation methods, this paper extends spline interpolation to networks, which is not a weighted average interpolation method, and hence free from such bias. For this purpose, we have to make it clear what C^p continuity at vertices means. In this paper, C^p continuity is defined from the hint obtained from Kirchhoff’s laws. In addition, it is shown that the parameters of the linear and cubic spline interpolation can be determined uniquely if each connected component of the network has at least one site with a data value assigned.