A Scale‐Sensitive Test of Attraction and Repulsion Between Spatial Point Patterns

There exist a variety of tests for attraction and repulsion effects between spatial point populations, most notably those involving either nearest‐neighbor or cell‐count statistics. Diggle and Cox (1981) showed that for the nearest‐neighbor approach, a powerful test could be constructed using Kendall's rank correlation coefficient. In the present paper, this approach is extended to cell‐count statistics in a manner paralleling the K‐function approach of Lotwick and Silverman (1982). The advantage of the present test is that, unlike nearest‐neighbors, one can identify the spatial scales at which repulsion or attraction are most significant. In addition, it avoids the torus‐wrapping restrictions implicit in the Monte Carlo testing procedure of Lotwick and Silverman. Examples are developed to show that this testing procedure can in fact identify both attraction and repulsion between the same pair of point populations at different scales of analysis.     

A SIMPLIFICATION AND GENERALIZATION OF THE EXCLUSION THEOREM: A TECHNICAL NOTE

Several researchers have proven that for the integrated production-location problem on the Weberian triangle, intermediate points on the edge of the triangle can never be optimal locations. Authors of previous proofs of this result have used cumbersome trigonometric arguments. We present a much simpler algebraic proof of the result, and present it in terms of the more general n -input model, where the feasible location space is a convex polygon rather than a triangle. In addition, the result generalizes immediately to other cases, such as (1) multifacility production-location problems, (2) stochastic versions of one-facility and multifacility production-location problems, and (3) comparable pure location problems (e.g., the Weber problem).     

Accuracy of Count Data Estimated by the Point‐in‐Polygon Method

This paper analyzes the accuracy of count data estimated by the point‐in‐polygon method. A point‐in‐polygon interpolation model is proposed, based on a stochastic distribution of points and the target zone, in order to represent a variety of situations. The accuracy of estimates is numerically investigated in relation to the size of the target zone and the distribution of points, and the optimal location of representative points is discussed. The major findings of this paper are as follows: (1) though the relative accuracy of estimates generally increases monotonously with the size of the target zone, the monotoneity is often disturbed by the periodicity in the spatial configuration of source zones and the point distribution; (2) the point‐in‐polygon and the areal weighting interpolation methods have the same accuracy of estimates when points are concentrated in less than 12–15 percent area around the representative point in source zones; (3) the point‐in‐polygon method is not so robust against the locational gap between points and the representative point; (4) the optimal location of representative points is given by the spatial median of points.     

CENTER POINTS, EQUILIBRIUM POSITIONS, AND THE OBNOXIOUS LOCATION PROBLEM

ABSTRACT. Real variable analysis has een used to great benefit in a variety of classical problems in location theory. In this paper we explore basic complex variable techniques in one formulation of the obnoxious location problem. A general definition of center points is first given and used to formulate several alternate versions of the obnoxious location problem. A logarithmic transformation is then used to demonstrate some equivalences between these families of distinct location problems (defined via center points). A prototype logarithmic potential function which results from this formulation is then investigated, and it is demonstrated that the extremal solutions with this objective reside on the boundary of its domain of definition. An application using zero- and one-dimensional centers is discussed, and a generalization to the spatial obnoxious problem is also briefly examined. We define a zero-dimensional center as a critical point of the logarithmic potential function, and it is shown that these centers are equivalent to the solutions of the Complex Moment Problem.     

Location Modeling Utilizing Maximum Service Distance Criteria

The location set-covering problem (LSCP) and the maximal covering location problem (MCLP) have been the subject of considerable interest. As originally defined, both problems allowed facility placement only at nodes. This paper deals with both problems for the case when facility placement is allowed anywhere on the network. Two theorems are presented that show that when facility placement is unrestricted, for either the LSCP or MCLP at least one optimal solution exists that is composed entirely of points belonging to a finite set of points called the network intersect point set (NIPS). Optimal solution approaches to the unrestricted site LSCP and MCLP problems that utilize the NIPS and previously developed solution methodologies are presented. Example solutions show that considerable improvement in the amount of coverage or the number of facilities needed to insure total coverage can be achieved by allowing facility placement along arcs of the network. In addition, extensions to the arc-covering model and the ambulance-hospital model of ReVelle, Toregas, and Falkson are developed and solved.     

A Procedure to Generate Thiessen Polygons

An algorithm to generate Thiessen diagrams for a set of n points defined in the plane is presented. First, existing proximal polygon computation procedures are reviewed and terms are defined. The algorithm developed here uses a rectangular window within which the Thiessen diagram is defined. The computation of Thiessen polygons uses an iterative walking process whereby the processing starts at the lower left corner of the diagram and proceeds toward the right top corner. The use of a sorted point sequence and dynamical core allocation provide for efficient processing. The presentation is concluded by the discussion of an implementation of the algorithm in a FORTRAN program.     

MEDIAN,MEAN, AND OPTIMUM AS FACILITY LOCATIONS*

ABSTRACT The mean and the median are both measures of centrality. In one dimension, the median minimizes the average absolute distance from a facility to a set of customers {x_i}, and the mean minimizes the average squared distance. In two dimensions, the median minimizes the average rectangular distance, and the mean minimizes the average squared distance. This paper investigates the “location penalty” when a nonoptimal location is substituted for the optimal location. In one dimension, the average absolute distance at the mean is never more than twice the average absolute distance at the median. Surprisingly, this happens when the median and mean are close together. In two dimensions, the ratio of the average Euclidean distance at the median to the average Euclidean distance at the optimum is never more than . However, this upper bound depends on an unlikely scenario with just two customers. With three equal-sized customers, this ratio is never more than 1.12. However, if the triangle formed by the customers is rotated relative to the rectangular grid, the ratio never exceeds 1.028.     

A Topological Representation for Taking Cities as a Coherent Whole

A city is a whole, as are all cities in a country. Within a whole, individual cities possess different degrees of wholeness, defined by Christopher Alexander as a life‐giving order or simply a living structure. To characterize the wholeness and in particular to advocate for wholeness as an effective design principle, this article develops a geographic representation that views cities as a whole. This geographic representation is topology‐oriented, so fundamentally differs from existing geometry‐based geographic representations. With the topological representation, all cities are abstracted as individual points and put into different hierarchical levels, according to their sizes and based on head/tail breaks—a classification and visualization tool for data with a heavy tailed distribution. These points of different hierarchical levels are respectively used to create Thiessen polygons. Based on polygon–polygon relationships, we set up a complex network. In this network, small polygons point to adjacent large polygons at the same hierarchical level and contained polygons point to containing polygons across two consecutive hierarchical levels. We computed the degrees of wholeness for individual cities, and subsequently found that the degrees of wholeness possess both properties of differentiation and adaptation. To demonstrate, we developed four case studies of all China and U.K. natural cities, as well as Beijing and London natural cities, using massive amounts of street nodes and Tweet locations. The topological representation and the kind of topological analysis in general can be applied to any design or pattern, such as carpets, Baroque architecture and artifacts, and fractals in order to assess their beauty, echoing the introductory quote from Christopher Alexander.     

HIGH-TECHNOLOGY JOBS AND WORKER MOBILITY*

ABSTRACT. The premise of this paper is that one can discover much about the location determinants of high-technology industry by examining the location (migration) decision of workers with high-technology occupations. A model of the high-technology worker migration decision is developed that permits estimation of worker response to both personal and area characteristics, the latter chosen to mirror location factors often cited by high-technology firms as important for both the attraction and retention of their specialized work force. Logit estimates of this model between 1975 and 1980 are presented. The paper is concluded with a discussion of policy implications.     

MILL PRICING AND SPATIAL PRICE DISCRIMINATION MONOPOLY PERFORMANCE AND LOCATION WITH SPATIAL RETAIL MARKETS*

ABSTRACT. Most of the monopoly spatial price discrimination literature explicitly assumes uniform population density over space. It also implicitly assumes that firms (plants) are spatially isolated from each other with production and retail points that coincide in location. While departures from these assumptions have been explored separately in the literature, it remains to examine performance and location when these assumptions are relaxed simultaneously. What emerges in this paper is a model where density functions approximate a pair of cities isolated from other cities. Each city has its own retail market, while the location of a single production or wholesale point is determined by characteristics of the two markets. Comparisons of mill pricing and spatial price discrimination found in the spatial monopoly literature can be interpreted as special cases of the more general framework provided here.     

A FIJIAN PEASANTRY: GALALA AND VILLAGERS

Village societies in rural Fiji have often been studied in terms of a dichotomy between traditional village communalism and independent galala individualism. This was especially so in the 1960s, at a time when government efforts to promote individualist Fijian agriculture led to the creation of a number of resettlement schemes based on the peasant model. Social scientists hailed the galala movement as a fundamental and progressive process of change in Fijian society. However, recent evidence and the demise of regulations which institutionalised galala point to the present lack of distinction between galala and villagers. Indeed, there is considerable confusion within some villages over how galala should be defined and it seems to be much more a matter of spatial location of residence than any attitudinal or economic break with communalism.     

乡村吸引力研究进展与展望

乡村吸引力是表征乡村地域系统整体功能的新指标,是乡村地理学研究的新领域.提升乡村吸引力是应对乡村衰退、实现乡村振兴的必由之路,开展乡村吸引力理论与实证研究意义重大.基于相关成果的逻辑关联性,从乡村吸引力综合研究、分类研究及相关内容研究三个方面进行文献梳理,并进行简要述评.总体上看,以乡村吸引力为主题的综合性研究尚处于起...     

A Minimal Representation for Dead-Reckoning Navigation: Updating the Homing Vector

A number of organisms are capable of traveling directly back to the origin of locomotion after following a circuitous outbound path, using dead-reckoning navigation. We describe a minimal representation that could be used for such navigation. It involves keeping track of only the polar coordinates of the origin of locomotion with respect to the observer's current location. An exact solution is provided first, followed by a simplified solution, then a solution involving linearization of trigonometric functions. The paper also examines performance attainable with these solutions and briefly considers ways in which the model might be modified to account for human navigation without reference to landmarks.     

Aggregation Decomposition and Aggregation Guidelines for a Class of Minimax and Covering Location Models

Facility location problems often involve movement between facilities to be located and customers/demand points, with distances between the two being important. For problems with many customers, demand point aggregation may be needed to obtain a computationally tractable model. Aggregation causes error, which should be kept small. We consider a class of minimax location models for which the aggregation may be viewed as a second‐order location problem, and use error bounds as aggregation error measures. We provide easily computed approximate “square root” formulas to assist in the aggregation process. The formulas establish that the law of diminishing returns applies when doing aggregation. Our approach can also facilitate aggregation decomposition for location problems involving multiple “separate” communities.     

Distance in Spatial Analysis: Measurement,Bias, and Alternatives

Distance is an important and basic concept in geography. Many theories, methods, and applications involve distance explicitly or implicitly. While measuring the distance between two locations is a straightforward task, many geographical processes involve areal units, where the distance measurement can be complicated. This research investigates distance measurement between a location (point) and an area (polygon). We find that traditional polygon-to-point distance measurements, which involve abstracting a polygon into a central or representative point, could be problematic and may lead to biased estimates in regression analysis. To solve this issue, we propose a new polygon-to-point distance metric along with two algorithms to compute the new distance metric. Simulation analysis shows the effectiveness of the new distance metric in providing unbiased estimates in linear regression.     

LOCATION,THEORY OF PRODUCTION,AND VARIABLE TRANSPORTATION RATES*

ABSTRACT. The question of what conditions insure independence between location and output decisions is investigated within the context of triangle space. Assuming that transportation rates vary with quantity and distance, it is shown that this independence is insured if the production function generates a linear expansion path and the following are all constant: transportation rate elasticities with respect to distance and quantity, and the price elasticity of demand for inputs.     

Redundancy as an Index of Change in Point Pattern Analysis

Point pattern analysis based on concepts from information theory can go beyond existing techniques. Direct measurement of spatial form is achieved when Thiessen polygons are constructed around the points; in this scheme, each point's proportion of total area may be treated like a probability. Three information-theoretic indices are available for analysis of a distribution of such probabilities. Entropy is density dependent. Redundancy, defined as the difference between observed and maximum entropy, seems to avoid this problem when the number of individuals in a pattern exceeds twenty. Comparisons of prior and posterior redundancy provide an indication of change in overall pattern form. An information gain expectation reflects changes for each individual in a pattern. Here, point-area redundancy parameters are determined for Poisson-generated patterns, using a gamma distribution of polygon areas and a computer-generated set. An application to an urban crime problem illustrates the use of these parameters in the analysis of pattern change.     

TRANSPORTATION RATES AND LOCATION OF THE FIRM: A COMPARATIVE STATIC ANALYSIS*

ABSTRACT This paper examines the role of the structure of transportation rates in the Weber-Moses triangle model. It shows that, in Zeigler's analysis, the appearance of the price elasticities of demand for inputs actually vanish by a simple application of the envelope theorem. Applying comparative statics analysis, we show when transportation rates are a function of quantity and distance, the assumptions that the production function is homogeneous of degree one and the transportation rates elasticities with respect to quantity and distance are constant are not sufficient to insure that the optimum location is independent of the level of output. This result is significantly different from that obtained by either Miller and Jensen or Zeigler.     

On the Finite Optimality Set of the Vector Assignment p‐Median Problem

The vector assignment p‐median problem (VAPMP) is one of the first discrete location problems to account for the service of a demand by multiple facilities, and has been used to model a variety of location problems in addressing issues such as system vulnerability and reliability. Specifically, it involves the location of a fixed number of facilities when the assumption is that each demand point is served a certain fraction of the time by its closest facility, a certain fraction of the time by its second closest facility, and so on. The assignment vector represents the fraction of the time a facility of a given closeness order serves a specific demand point. Weaver and Church showed that when the fractions of assignment to closer facilities are greater than more distant facilities, an optimal all‐node solution always exists. However, the general form of the VAPMP does not have this property. Hooker and Garfinkel provided a counterexample of this property for the nonmonotonic VAPMP. However, they do not conjecture as to what a finite set may be in general. The question of whether there exists a finite set of locations that contains an optimal solution has remained open to conjecture. In this article, we prove that a finite optimality set for the VAPMP consisting of “equidistant points” does exist. We also show a stronger result when the underlying network is a tree graph.     

On Covering Problems and the Central Facilities Location Problem

A number of variations of facilities location problems have appeared in the research literature in the past decade. Among these are problems involving the location of multiple new facilities in a discrete solution space, with the new facilities located relative to a set of existing facilities having known locations. In this paper a number of discrete solution space location problems are treated. Specifically, the covering problem and the central facilities location problem are shown to be related. The covering problem involves the location of the minimum number of new facilities among a finite number of sites such that all existing facilities (customers) are covered by at least one new facility. The central facilities location problem consists of the location of a given number of new facilities among a finite number of sites such that the sum of the weighted distances between existing facilities and new facilities is minimized. Computational experience in using the same heuristic solution procedure to solve both problems is provided and compared with other existing solution procedures.