Atsuyuki Okabe

A kernel density estimation method for networks, its computational method and a GIS-based tool

We develop a kernel density estimation method for estimating the density of points on a network and implement the method in the GIS environment. This method could be applied to, for instance, finding ‘hot spots’ of traffic accidents, street crimes or leakages in gas and oil pipe lines. We first show that the application of the ordinary two‐dimensional kernel method to density estimation on a network produces biased estimates. Second, we formulate a ‘natural’ extension of the univariate kernel method to density estimation on a network, and prove that its estimator is biased; in particular, it overestimates the densities around nodes. Third, we formulate an unbiased discontinuous kernel function on a network. Fourth, we formulate an unbiased continuous kernel function on a network. Fifth, we develop computational methods for these kernels and derive their computational complexity; and we also develop a plug‐in tool for operating these methods in the GIS environment. Sixth, an application of the proposed methods to the density estimation of traffic accidents on streets is illustrated. Lastly, we summarize the major results and describe some suggestions for the practical use of the proposed methods.

Locational optimization problems solved through Voronoi diagrams

Abstract This paper reviews a class of continuous locational optimization problems (where an optimal location or an optimal configuration of facilities is found in a continuum on a plane or a network) that can be solved through the Voronoi diagram. Eight types of continuous locational optimization problems are formulated, and these problems are solved through the ordinary Voronoi diagram, the farthest-point Voronoi diagram, the weighted Voronoi diagram, the network Voronoi diagram, the Voronoi diagram with a convex distance function, the line Voronoi diagram, and the area Voronoi diagram.

Spatial Analysis along Networks: Statistical and Computational Methods

In the real world, there are numerous and various events that occur on and alongside networks, including the occurrence of traffic accidents on highways, the location of stores alongside roads, the incidence of crime on streets and the contamination along rivers. In order to carry out analyses of those events, the researcher needs to be familiar with a range of specific techniques. Spatial Analysis Along Networks provides a practical guide to the necessary statistical techniques and their computational implementation. Each chapter illustrates a specific technique, from Stochastic Point Processes on a Network and Network Voronoi Diagrams, to Network K-function and Point Density Estimation Methods, and the Network Huff Model. The authors also discuss and illustrate the undertaking of the statistical tests described in a Geographical Information System (GIS) environment as well as demonstrating the user-friendly free software package SANET.

Generalized network Voronoi diagrams: Concepts, computational methods, and applications

In the real world, there are many phenomena that occur on a network or alongside a network; for example, traffic accidents on highways and retail stores along streets in an urbanized area. In the literature, these phenomena are analysed under the assumption that distance is measured with Euclidean distance on a plane. This paper first examines this assumption and shows an empirical finding that Euclidean distance is significantly different from the shortest path distance in an urbanized area if the distance is less than 500 m. This implies that service areas in urbanized areas cannot be well represented by Voronoi diagrams defined on a plane with Euclidean distance, termed generalized planar Voronoi diagrams. To overcome this limitation, second, this paper formulates six types of Voronoi diagrams defined on a network, termed generalized network Voronoi diagrams, whose generators are given by points, sets of points, lines and polygons embedded in a network, and whose distances are given by inward/outward distances, and additively/multiplicatively weighted shortest path distances. Third, in comparison with the generalized planar Voronoi diagrams, the paper empirically shows that the generalized network Voronoi diagrams can more precisely represent the service areas in urbanized areas than the corresponding planar Voronoi diagrams. Fourth, because the computational methods for constructing the generalized planar Voronoi diagrams in the literature cannot be applied to constructing the generalized network Voronoi diagrams, the paper provides newly developed efficient algorithms using the ‘extended’ shortest path trees. Last, the paper develops user‐friendly tools (that are included in SANET, a toolbox for spatial analysis on a network) for executing these computational methods in a GIS environment.

The SANET Toolbox: New Methods for Network Spatial Analysis

This paper describes new methods, called network spatial methods, for analysing spatial phenomena that occur on a network or alongside a network (referred to as network spatial phenomena). First, the paper reviews network spatial phenomena discussed in the related literature. Second, the paper shows the uniform network transformation, which is used in the study of non-uniform distributions on a network, such as the densities of traffic and population. Third, the paper outlines a class of network spatial methods, including nearest neighbor distance methods, K-function methods, cell count methods, clumping methods, the Voronoi diagrams and spatial interpolation methods. Fourth, the paper shows three commonly used computational methods to facilitate network spatial analysis. Fifth, the paper describes the functions of a GIS-based software package, called SANET, that perform network spatial methods. Sixth, the paper compares network spatial methods with the corresponding planar spatial methods by applying both methods to the same data set. This comparison clearly demonstrates how different conclusions can result. The conclusion summarizes the major findings.

Spatial analysis of roadside Acacia populations on a road network using the network K-function

Spatial patterning of plant distributions has long been recognised as being important in understanding underlying ecological processes. Ripley’s K-function is a frequently used method for studying the spatial pattern of mapped point data in ecology. However, application of this method to point patterns on road networks is inappropriate, as the K-function assumes an infinite homogenous environment in calculating Euclidean distances. A new technique for analysing the distribution of points on a network has been developed, called the network K-function (for univariate analysis) and network cross K-function (for bivariate analysis). To investigate its applicability for ecological data-sets, this method was applied to point location data for roadside populations of three Acacia species in a fragmented agricultural landscape of south-eastern Australia. Kernel estimations of the observed density of spatial point patterns for each species showed strong spatial heterogeneity. Combined univariate and bivariate network K-function analyses confirmed significant clustering of populations at various scales, and spatial patterns of Acacia decora suggests that roadworks activities may have a stronger controlling influence than environmental determinants on population dynamics. The network K-function method will become a useful statistical tool for the analyses of ecological data along roads, field margins, streams and other networks.

Nearest neighbourhood operations with generalized Voronoi diagrams: a review

Abstract An ordinary geographical information system has a collection of nearest neighbourhood operations, such as generating a buffer zone and searching for the nearest facility from a given location, and this collection serves as a useful tool box for spatial analysis. Computationally, these operations are undertaken through the ordinary Voronoi diagram. This paper extends this tool box by generalizing the ordinary Voronoi diagram. The tool box consists of 35 nearest neighbourhood operations based upon twelve generalized Voronoi diagrams: the order-fe Voronoi diagram, the ordered order-fc Voronoi diagram, the farthest-point Voronoi diagram, the kth-nearest-point Voronoi diagram, the weighted Voronoi diagram, the line Voronoi diagram, the area Voronoi diagram, the Manhattan Voronoi diagram, the spherical Voronoi diagram, the Voronoi diagram in a river, the polyhedral Voronoi diagram, and the network Voronoi diagram. Each operation is illustrated with examples and the literature of computational methods.

Stability of Spatial Competition for a Large Number of Firms on a Bounded Two-Dimensional Space

In this paper Hotellings model of spatial competition is extended to a two-dimensional space, and it is shown that if the number of firms is very large, the configuration of the firms is stable in the inner area of a square region; this configuration is fairly similar to a socially optimal configuration. These results are almost contrary to Eaton and Lipseys conjecture, and they are in marked contrast to the results of the one-dimensional model.

A Computational Method for Estimating the Demand of Retail Stores on a Street Network and its Implementation in GIS

This paper describes a computational method for estimating the demand of retail stores on a street network using GIS. First, the ‘network Huff model’ is formulated on a network with the shortest-path distance as an extension of the ordinary Huff model (which assumes a continuous plane with Euclidean distance). Second, using this model, a formula for estimating the demand is derived. This estimation formula is similar to that with the ordinary Huff model, but it has an advantage in that the formula exactly computes the demand on a network. Third, a practical method for computing the formula is developed. Finally, a method of implementing this computational method in a GIS environment is shown.

Uniform network transformation for points pattern analysis on a non-uniform network

In the real world, there are many kinds of phenomena that are represented by points on a network, such as traffic accidents on a street network. To analyse these phenomena, the basic point pattern methods (i.e. the nearest neighbour distance method, the quadrat method, the K-function method and the clumping method) defined on a plane (referred to as the planar basic point pattern methods) are extended to the basic point pattern methods on a network (referred to as the network basic point pattern methods). However, like the planar basic point pattern methods, the network basic point pattern methods assume a uniform network and this assumption is hard to accept when analysing actual phenomena. To overcome this limitation, this paper formulates a transformation, called the uniform network transformation, that transforms a non-uniform network into a uniform network. This transformation provides a simple procedure for analysing point patterns on non-uniform networks: first, a given non-uniform network is transformed into a uniform network; second, the network basic point pattern methods (which assume a uniform network) are applied to this transformed uniform network. No modification to the network basic point pattern methods is necessary. The paper also shows an actual application of this transformation to traffic accidents in Chosei, Japan.