A New Method of Adaptive Zoning for Spatial Interaction Models

Spatial interaction models commonly use discrete zones to represent locations. The computational requirements of the models normally arise with the square of the number of zones or worse. For computationally intensive models, such as land use–transport interaction models and activity‐based models for city regions, this dependency of zone size is a long‐standing problem that has not disappeared even with increasing computation speed in PCs—it still forces modelers to compromise on the spatial resolution and extent of model coverage as well as on the rigor and depth of model‐based analysis. This article introduces a new type of discrete zone system, with the objective of reducing the time for estimating and applying spatial interaction models while maintaining their accuracy. The premise of the new system is that the appropriate size of destination zones depends on the distance to their origin zone: at short distances, spatial accuracy is important and destination zones must be small; at longer distances, knowing the precise location becomes less important and zones can be larger. The new method defines a specific zone map for every origin zone; each origin zone becomes the focus of its own map, surrounded by small zones nearby and large zones farther away. We present the theoretical formulation of the new method and test it with a model of commuting in England. The results of the new method are equivalent to those of the conventional model, despite reducing the number of zone pairs by 96% and the computation time by 70%. Los modelos de interacción espacial suelen utilizar zonas discretas para representar áreas o puntos de interés. Los requisitos computacionales de estos modelos normalmente aumentan a razón del número de zonas elevadas al cuadrado o más. Para modelos computacionalmente intensivos como los modelos de interacción entre uso de suelo y transporte y los modelos basados en actividades para ciudades‐región, el impacto del tamaño de la zona es un problema persistente no superado aun. Esta limitación persiste a pesar de los grandes avances en la velocidad de procesamiento en computadoras, pues obliga a los modeladores a hacer concesiones entre la resolución espacial y la extensión que abarca el modelo, así como en el rigor y profundidad del análisis. En este artículo se presenta un nuevo tipo de sistema de zonas discretas que: a) tienen como objetivo reducir el tiempo de estimación de la aplicación de modelos de interacción espacial; y b) al mismo tiempo mantienen su nivel de precisión. La premisa que gobierna este nuevo sistema es que el tamaño apropiado de las zonas de destino depende de la distancia a su zona de origen: a distancias cortas, la precisión espacial es importante y las zonas de destino deben ser pequeño; a distancias mas largas, conocer la ubicación precisa es progresivamente menos importante y las zonas pueden ser mayores. El nuevo método define un mapa específico de zonas para cada zona de origen; cada zona de origen se convierte en el foco de su propio mapa, rodeada de zonas cercanas pequeñas y zonas grandes a mayor distancia. El estudio presenta la formulación teórica del nuevo método y su demostración vía un modelo de desplazamientos residencia‐trabajo en Inglaterra. Los resultados del nuevo método son equivalentes a las del modelo convencional, a pesar de reducir del número de pares de zonas en un 96% y el tiempo de cálculo en un 70%. 空间相互作用模型通常采用离散区域代表区位。模型的计算量往往与区域数量呈平方甚至更高阶增长。对于可计算的精细模型,如土地利用‐交通相互作用模型和基于行为的城市区域模型,区域尺度的依赖性是长期存在的问题,即使计算机的计算速度增加,该问题仍无法消除。因此,建模者需在模型空间分辨率和覆盖范围以及模型分析的严谨性和深度上做出权衡。本文介绍了一种新型的离散分区系统,目的在于减少空间相互作用模型估算和计算时间,同时维持其精度。新系统的前提是目标区域的适当尺度取决于与初始区域的距离：在短距离范围内,空间精确性是重要的,且目标区域必须是小的;在更远距离上,位置精度的重要性降低,目标区域可以变大。该方法为每个初始区域制定了具体的尺度地图。每个初始区域成为其自身地图的中心,被近邻的小区域和更远距离的大区域所包围。本文给出了新方法的理论公式,并以英格兰地区的通勤模型进行检验。结果显示,尽管区域对的数量减少了96%,计算时间缩短了70%,但新方法的计算结果等效于常规模型。     

Dynamic Models of Agricultural Location in a Spatial Interaction Framework

Recent research demonstrates that spatial interaction models may also be made to function as location models by the addition of appropriate hypotheses about structural adjustment. An appealing feature of the approach is that dynamics are explicitly incorporated. In this paper, the attempt is made to recast a problem from classical economic-geographic theory—that of agricultural location—within a spatial interaction framework. It is shown that a wide variety of models is potentially available, and the properties of a range of these models are selectively explored.     

Using Simple Genetic Algorithms to Calibrate Spatial Interaction Models

The paper investigates the use of two genetic algorithms in an attempt to obtain globally optimal parameter estimates for a mix of simple and complex spatial interaction models. The genetic algorithms work well and are strongly advocated as a more robust approach particularly for use with the more complex multiparameter models where the differences in both performance and parameter values are judged to be significant.     

Spatial Characteristics and Comparisons of Interaction and Median Clustering Models

Cluster analysis has been pursued from a number of directions for identifying interesting relationships and patterns in spatial information. A major emphasis is currently on the development and refinement of optimization‐based clustering models for the purpose of exploring spatially referenced data. Within this context, two basic methods exist for identifying clusters that are most similar. An interesting feature of these two approaches is that one method approximates the relationships inherent in the other method. This is significant given that the approximation approach is invariably utilized for cluster detection in spatial and aspatial analysis. A number of spatial applications are investigated which highlight the differences in clusters produced by each model. This is an important contribution because the differences are in fact quite significant, yet these contrasts are not widely known or acknowledged.     

Analysis of Dynamic Spatial Interaction Models by Means of Optimal Control

This paper deals with the design of general classes of dynamic spatial interaction models. On the basis of a general (well-behaved) multiperiod objective function and of a dynamic model representing the evolution of a spatial interaction system, an optimal control model is constructed. Particular attention is given to the equilibrium and stability conditions. It turns out that it is possible to identify steady-state solutions for a dynamic spatial interaction model. Furthermore, it can also be demonstrated that the entropy model is a specific case of this spatial interaction system. A simple illustration for urban dynamics is given as well.     

Prediction of Bike-sharing Trip Counts: Comparing Parametric Spatial Regression Models to a Geographically Weighted XGBoost Algorithm

Regression models are commonly applied in the analysis of transportation data. This research aims at broadening the range of methods used for this task by modeling the spatial distribution of bike-sharing trips in Cologne, Germany, applying both parametric regression models and a modified machine learning approach while incorporating measures to account for spatial autocorrelation. Independent variables included in the models consist of land use types, elements of the transport system and sociodemographic characteristics. Out of several regression models with different underlying distributions, a Tweedie generalized additive model is chosen by its values for AIC, RMSE, and sMAPE to be compared to an XGBoost model. To consider spatial relationships, spatial splines are included in the Tweedie model, while the estimations of the XGBoost model are modified using a geographically weighted regression. Both methods entail certain advantages: while XGBoost leads to far better values regarding RMSE and sMAPE and therefore to a better model fit, the Tweedie model allows an easier interpretation of the influence of the independent variables including spatial effects.     

Learning in Single Hidden-Layer Feedforward Network Models: Backpropagation in a Spatial Interaction Modeling Context

Learning in neural networks has attracted considerable interest in recent years. Our focus is on learning in single hidden-layer feedforward networks which is posed as a search in the network parameter space for a network that minimizes an additive error function of statistically independent examples. We review first the class of single hidden-layer feedforward networks and characterize the learning process in such networks from a statistical point of view. Then we describe the backpropagation procedure, the leading case of gradient descent learning algorithms for the class of networks considered here, as well as an efficient heuristic modification. Finally, we analyze the applicability of these learning methods to the problem of predicting interregional telecommunication flows. Particular emphasis is laid on the engineering judgment, first, in choosing appropriate values for the tunable parameters, second, on the decision whether to train the network by epoch or by pattern (random approximation), and, third, on the overfitting problem. In addition, the analysis shows that the neural network model whether using either epoch-based or pattern-based stochastic approximation outperforms the classical regression approach to modeling telecommunication flows.     

Explorations and Syntheses of Linear Programming And Spatial Interaction Models of Residential Location

Various “linear programming” models of residential location are explored in some detail. It is then shown that entropy maximizing “sub-optimal” versions of these can be developed, which may be more realistic at typical levels of resolution used for analysis but can still be given some of the interpretations of the programming models. Further, any programming model can be seen as a limiting case of some entropy maximizing model in which some of the parameters tend to infinity. Some empirical tests of both kinds of models are presented, and the limiting relationships are discussed.     

Building an Automated Modeling System to Explore a Universe of Spatial Interaction Models

A methodology is described for harnessing the power of supercomputer hardware to build an automated modeling system. The problems of applying this new approach to mathematical modeling in geography are discussed by reference to the design of better performing spatial interaction models. It is also possible that the results may provide clues about new theories and knowledge.     

Note on the Measurement of Spatial Imbalance

This note presents a new measure of spatial imbalance that uses the center of gravity (or the mean center) of a distribution and the geometric center (or the centroid) of a region to define a vector indicating the relative position of the distribution in the region. The spatial imbalance of a distribution in a region is defined as the distance and directional deviation of the center of gravity of the distribution from the geometric center of the region. The convex hull of a region is introduced as the reference for maximum spatial imbalance of the distribution in the region. The spatial imbalance is measured in terms of (1) the relative distance of the center of gravity between the geometric center and the convex hull; and (2) the direction from the geometric center to the center of gravity. This new measure is tested by quantifying the westward movement of the contiguous U.S. population between 1790 and 1950. It describes the well‐known observation that the U.S. population was moving westward across the contiguous United States and toward reduced spatial imbalance.