Statistical Methods for Evaluating the Geometrical Hierarchy of a Network

The objective of this paper is to present statistical methods for evaluating the geometrical hierarchy of a network in comparison with a random hierarchy network. First, the random hierarchy network is explicitly formulated by four stochastic processes in which line segments or lines, which represent roads, are randomly placed and their placement is independent of their ranks. Second, under these stochastic processes, this paper derives the expected value and variance of i) the number of intersections between roads of different ranks; and ii) the distance from an arbitrary point to the nearest highest-ranked road through two kinds of routes. Using these expected values, the intersection index and the detour index are proposed, and it is shown that these indicators are useful for evaluating the geometrical hierarchy of a network. As calculation of these indices is laborious, an efficient computational method is developed, and its application to an actual example is described.     

A Computational Method for Market Area Analysis on a Network

This paper shows a computational method for market area analysis assuming that stores and consumers are distributed over a network and the distance between two points on the network is given by the route distance. First, we consider five basic questions often raised in market area analysis, and show a general method, called the network transformation method, that gives an intuitive way of looking at computational methods for solving these questions. Second, assuming that consumers follow the Huff model, we consider four questions concerning market area delineation and market potential often discussed in market area analysis in practice. We show that the network transformation method is also useful to develop computational methods for solving these questions. One of the notable results is that market area delineation of the Huff model (which is analytically difficult to obtain on a plane) can be exactly obtained on a network.     

The Modifiable Areal Unit Problem,in a Regression Model Whose Independent Variable Is a Distance from a Predetermined Point

This paper deals with the modifiable areal unit problem in the context of a regression model where a dependent variable is an attribute value (say, income) of an atomic data unit (say, a household) and an independent variable is a distance from a predetermined point (say, a central business district) to the atomic data unit (a disaggregated model). We apply this disaggregated model to spatially aggregated data in which the dependent variable is the average income over a spatial unit and the independent variable is the average distance from each household in a spatial unit to the predetermined point (an aggregated model). First, estimating the slope coefficient by the least squares method, we prove that the variance of the estimator for the slope coefficient in the aggregated model is larger than that in the disaggregated model. Second, focusing on variations in the variance of the estimator for the slope coefficient in the aggregated model with respect to the number of zones, we obtain the number of zones in which the variance is close to that in the disaggregated model. Third, we obtain the zoning system that has the minimum variance for a fixed number of zones. We also calculate the maximum variance in order to examine the range of the variance.     

The K‐Function Method on a Network and Its Computational Implementation

This paper proposes two statistical methods, called the network K‐function method and the network cross K‐function method, for analyzing the distribution of points on a network. First, by extending the ordinary K‐function method defined on a homogeneous infinite plane with the Euclidean distance, the paper formulates the K‐function method and the cross K‐function method on a finite irregular network with the shortest‐path distance. Second, the paper shows advantages of the network K‐function methods, such as that the network K‐function methods can deal with spatial point processes on a street network in a small district, and that they can exactly take the boundary effect into account. Third, the paper develops the computational implementation of the network K‐functions, and shows that the computational order of the K‐function method is O(n²_Q log n_Q) and that of the network cross K‐function is O(n_Q log U3Q), where n_Q is the number of nodes of a network.     

The Multi Nearest Neighbor Distance Method for Analyzing the Compound Effect of Infrastructural Elements on the Distribution of Activity Points

We propose a statistical method to analyze the compound effect of infrastructural elements (such as stations, arterial streets, and lakes) on the distribution of activity points (such as retail stores) over a region. First, we formulate a function that explicitly shows the compound effect of infrastructural elements. Second, we show an efficient computational method for estimating this compound function from data. Third, we develop multivariate statistical methods for testing several hypotheses about these compound effects. Last, we examine the compound effect of arterial streets and subway stations on the distribution of “high-class” apartment buildings in Sumida-Kohto, Tokyo.     

A Statistical Method for Analyzing the Spatial Relationship between the Distribution of Activity Points and the Distribution of Activity Continuously Distributed over a Region

This paper shows a statistical method for analyzing the spatial relationship between the distributions of two different kinds of activity in a region. One kind of activity is discretely distributed as points in a region (such as the distribution of retail stores), and the other kind of activity is continuously distributed over the region (such as the distribution of population). First, three models representing the relationship between the above two distributions are formulated. Second, statistical methods for fitting these models to data are developed and the measures of fitness are proposed. Third, using these measures, the relationship between the distributions of thirty-seven kinds of retail stores and the distribution of population is examined in a suburb of Osaka in Japan.     

Local Indicator of Spatial Agglomeration between Newly Opened Outlets and Existing Competitors on a Street Network

Distance from competitors is a key factor in retail site selection and profitability. To understand the locational tendency that each newly opened outlet locates close to or far from existing competitors in a target area, a specific method is needed. Hence, this study aims first to develop a statistical method to discover the local spatial associations between newly opened and existing point-like outlets on a street network. We achieve this objective by extending the network local cross K function. The second objective is to evaluate the practicality of the proposed method by applying it to restaurants in a trendy district in Tokyo. Specifically, this study focuses on answering two questions: first, whether each newly opened restaurant is closely located to existing ones or not and, second, whether each existing restaurant attracts newly opened restaurants or not. The results show that the method is useful for revealing the location tendencies of retail outlets toward competitors.     

Statistical Analysis of the Distribution of Points on a Network

This paper shows four statistical methods that examine the distribution of points on a network (such as the distribution of retail stores along streets). The first statistical method is an extension of the nearest-neighbor distance method (the Clark-Evans statistic) defined on a plane to the method defined on a network. The second statistical method examines the effect of categorical attribute values of links (say, types of streets) on the distribution of activity points on a network. The third statistical method examines the effect of infrastructural elements (such as railway stations) on the distribution of activity points on a network. The fourth statistical method examines the compound effect of multiple kinds of infrastructural elements (say, railway stations and big parks) on the distribution of activity points on a network. These methods are discussed with empirical examples.