A Framework for Exploring Urban Retail Discontinuities. 探索城市零售业不连续性的框架
2011; Wiley; Volume: 43; Issue: 2 Linguagem: Inglês
10.1111/j.1538-4632.2011.00812.x
ISSN1538-4632
Autores Tópico(s)Housing Market and Economics
ResumoA key area in the analysis of urban structural evolution is identifying discontinuities. Effective analysis could improve long-term forecasting and provide a better understanding of how to steer an urban system toward a desirable future state. We use a simple aggregate retail model to demonstrate an algorithm for identifying discontinuities in model parameter space. Explorations of retailing in both Greater London and South Yorkshire in the United Kingdom illustrate how understanding a system's potential for discontinuity can provide insights for both policy makers and retail businesses. The Harris and Wilson model, described in the section so-named, is used as a simple archetype to illustrate the new framework. This model can be developed in a straightforward way to incorporate further refinement. In "Executing the model and visualizing the results," we describe a single model run and in "Investigating discontinuities," we explain our framework for detecting and analyzing discontinuities. "Identifying discontinuities in the London retail system" shows the results of applying this methodology to the Greater London retail system, and in "Practical applications," we explore the policy applications for this technique as related to the decline of town centers in the South Yorkshire retail system. Some concluding comments are offered in "Conclusions." Un área clave en el análisis de la evolución de la estructura urbana es la identificación de discontinuidades. El uso de métodos de análisis efectivos podría mejorar los pronósticos de a largo plazo y proporcionar una mejor comprensión de cómo dirigir un sistema urbano hacia un futuro deseable. Los autores utilizan un modelo simple de venta al público agregado para demostrar la utilidad de un algoritmo que identifica discontinuidades. El conocimiento del potencial de discontinuidades en un sistema puede iluminar la acción de tomadores de decisiones de políticas urbanas y de negocios de venta. El presente artículo demuestra dicho potencial mediante el estudio de caso del comercio de venta al público en Londres y Yorkshire en el sur en el Reino Unido. Los autores utilizan el modelo de Harris y Wilson (descrito en la sección con el mismo nombre), como arquetipo para ilustrar el nuevo marco metodológico que proponen. El modelo es formulado de manera sencilla con el fin de permitir su refinamiento futuro. En la sección denominada "Ejecución del modelo y visualización de los resultados" ("Excecuting the model and visualizing the results"), se describe una ejecución del modelo. En "investigación de discontinuidades" ("investigating disconuities") se explica el marco metodológico para la detección y análisis de las discontinuidades. La sección "Identificación de discontinuidades en el sistema de ventas al público de Londres" ("Identifying dicontinuities in the London retail system") se muestran los resultados de la aplicación de esta metodología al sistema de ventas al público de Londres. En "aplicaciones prácticas" ("Practical applications") se exploran las aplicaciones de esta técnica en políticas vinculadas a la decadencia de centros urbanos en el sistema de venta al público de South Yorkshire. Para finalizar, en la sección "Conclusiones" ("Conclusions"), los autores ofrecen algunas observaciones finales. 城市结构演化的核心领域之一是识别不连续性。对该问题的有效分析可以改进长期预测,并为如何引导城市系统向未来理想状态演化提供更好的理解。我们利用一个简单的零售聚合模型展示了一个在模型参数空间中识别间断点的算法.基于对英国大伦敦区以及南约克郡零售业的探索分析说明理解系统潜在的不连续性如何为政策制定者和零售商提供洞见。如文中标题所示,Harris和Wilson的模型可作为描述新框架提供了一个简单的原型。该模型可直接构建并可包含进一步的改进。在"模型运行及结果可视化"一节中描述了单个模型的运行,在"不连续性调查"一节中,我们阐述了我们的框架如何识别和分析不连续性,在"伦敦零售系统不连续性识别"一节中给出了在大伦敦区零售系统上述方法的应用结果.在"实际应用"一节,我们探讨了该项技术在南约克郡零售系统城市中心城区衰退中的政策应用;在"结论"部分给出了一些总结性评论。 A set of problems exists that represent one of the challenges of 21st-century science: to model the evolution of complex systems. One geographical example of this set is the task of modeling the evolution of urban structure. Discontinuities are a hallmark of change in complex systems, and one of the most well-known empirical examples in urban systems is the corner shop to supermarket transition in 1950s United Kingdom (Wilson and Oulton 1983). More recently, the appearance of large out-of-town shopping centers has produced a significant loss of business for many town-center retail businesses (Baker 2006). Given the potential for climate change and energy scarcity to change urban environments in the future, a clear understanding of when and why discontinuities occur should be an important consideration in analyzing urban systems. In this article, we use the original Harris and Wilson (1978) retail model to show how the insights then available can be extended to improve our ability to detect discontinuities and to provide a platform for a major attack on the broader task of modeling urban evolution in the future. We show how recent advances in computer visualization provide a means of deepening the analysis. A wide variety of techniques is used to model discontinuities in urban evolution. Catastrophe theory (Thom 1975) provided impetus and was developed in the context of urban systems by Wilson (1981). Another approach using cellular automata (Batty 1998) and multiagent systems (Bura et al. 1996) explores the emergence of self-organizing structures from microlevel behaviors in systems existing far from equilibrium. Batty (2005) highlights the potential for discontinuities to occur in such systems, and Dendrinos and Mullally (1981) uses dynamical systems theory to demonstrate how discontinuities can occur in the evolution of city size patterns. Straussfogel (1991), applying Allen and Sanglier's (1981) model based on the theory of dissipative structures, outlines the difficulty of detecting bifurcations. Approaches based on fractals have a clear link to dynamical systems theory and can generate realistic urban forms (Batty and Longley 1994; Benguigui and Czamanski 2004) but do not provide a way of detecting discontinuities. We focus here on retailing as one of the main subsystems of a city. Harris and Wilson (1978) developed one of the first dynamic retail models that looked at the occurrence of discontinuous change within a system. This model was later applied at a finer scale and extended by Fotheringham and Knudsen (1986) to include locational rent, external scale economies, and agglomeration of retail outlets. Similarly, Lombardo (1986) introduced a more detailed cost function and demand-side dynamics. These ideas also have been taken forward by a number of other authors; for example, Lombardo, Petri, and Zotta (2004) connected these ideas to the agent-based modeling framework. Other explorations were offered by Phiri (1980), Clarke (1981), Rijk and Vorst (1983a, b), Oppenheim (1986), Clarke, Clarke, and Wilson (1986), Nijkamp and Reggiani (1988, 1989), Borgers, Gunsing, and Timmermans (1991), and Clarke, Langley, and Cardwell (1998). A different model, based on space-time differentials, was used by Baker (1994) to explore how trip frequency is affected by shopping center size, and it identified critical values at which large and small center behavior occurs. Part of the difficulty of understanding discontinuity in these approaches has been the complexity of the systems and models themselves. Here, we tackle this complication using modern visualization techniques to reveal the structure of a system and to allow it to be explored in detail. We use these ideas to construct a framework for identifying and analyzing discontinuities in urban models. The possibility of multiple solutions exists for nonlinear systems; solutions are dependent on initial conditions—"path dependence"; and discontinuities are present—that is, there are critical values of the parameters such as α and β, but, in fact, any exogenous parameter or variable at which the structure suddenly changes. Harris and Wilson (1978) show analytically that three cases exist; these appear in Fig. 1. The cost, KWj, is also plotted and forms a straight line. At equilibrium, the Dj(Wj) curve and the KWj line intersect. In Fig. 1a (in which α<1), the gradient of the Dj(Wj) curve is infinite at the origin, and hence there is always an intersection with the cost line that can be shown to be stable. Thus for α 1, the cost line either intersects the curve twice (excluding the origin)—as shown by the lower gradient straight line—or only at the origin—as shown by the other line. In the former case, the upper intersection is stable and a nonzero Wj is possible; in the latter case, Wj is 0. More centralized patterns have many such zeros. α=1 is a special case (Fig. 1b). The Dj curve has a finite gradient at the origin, and the possibilities of intersections generating a stable point are like the Fig. 1c case. Therefore, we should always expect a discontinuity at α=1. Three variations of the Dj(Wj) curve. This analysis illustrates the possibility of multiple solutions to the equations: in the Fig. 1c case, two solutions are possible; if many zones are present in such a state, then multiple solutions exist for the system as a whole. We need to specify a set of initial conditions—starting values for the {Wj} and the {eiPi}—and such {Wj} will almost certainly not be equilibrium values. When (and, indeed, if) the iterative process converges, equation (9) is satisfied. We iterate through the equations using a specified set of model parameters, essentially solving equation (7a). The parameter values and initial conditions determine whether a stable equilibrium can be found. The retail systems we analyze here have large numbers of retail zones, which means we cannot visualize the phase trajectory of the system as we might with a system of two or three zones. Instead, we use a three-dimensional representation of the system that visualizes the position and floor space of each retail zone, and then use animation to convey the variation of the system variables as it iterates toward equilibrium or over time. We communicate the important quantities at each iteration using size, color, and shape (the black-and-white figures in this article—and some additional figures and movie files—can be seen in color via http://sites.google.com/site/joelsresearchwiki/publications/ga. Retail zones are illustrated with three-dimensional blocks whose height and tapering indicate floor space {Wj} and growth rate {ΔWj}, respectively. Each residential zone is shown as a circle on the map with a diameter proportional to its spending power eiPi. The complete model is presented on top of a boundary shape file for our region of interest, which provides a sense of locality and helps us quickly identify the various parts of the system. There is also the option of visualizing the money flows into any retail zone ΣiSij and out of any residential zone ΣjSij during a model run. Figure 2 portrays the output for one iteration from a sample model run. To retain the potential for a retail zone to be revived from zero size, we enforce a minimum retail zone size of 1 m2. One iteration from a single model run. The differences in {Wj} configurations in a parameter space can be visualized—in the two-dimensional case—by letting α and β be the X and Y coordinates, and θ the third dimension across the grid as a surface. The algorithm is summarized as a flow chart in Fig. 3. Algorithm for generating a results grid. We use computer graphics to visualize a large amount of detail in the results grid, which allows us to move around and explore it. Each in the model parameter space is presented in the same way as that shown for a single model run, allowing us to identify the size of individual retail zones. This technique uses the idea of dimension stacking (LeBlanc, Ward, and Wittels 1990) to embed two geographic dimensions into one, two, or three dimensions representing the model parameter space. Feiner and Beshers (1990) introduce similar systems for displaying financial data. One future potential is to extend this methodology to visualize a larger number of dimensions recursively. We introduce the θ-surface plot in the next section. The results grid is illustrated for the Greater London retail system, and the data sources are detailed in Appendix B. We search a two-dimensional area of the (α, β) space in the range 0.1–2.0 using a step size of 0.1 for both parameters. Clarke and Wilson (1985) do an earlier version of this analysis, but contemporary graphics generate a more powerful representation. It should be emphasized that this section is intended only as a simplified illustration of the analysis techniques available. A θ-surface plot appears in Fig. 4a. Six rough "ridges" (labeled A–F in the figure) can be seen running through the grid,2 each of which potentially represents areas of discontinuous change in the system. The most prominent ridge at F depicts the previously predicted discontinuity that occurs as α=1 is crossed. The advantage of visualizing the surface and results grid in real time is that one can immediately examine the underlying grid of results at any point and look at what is actually occurring at the level of individual retail zones.3 Doing this for the ridge at E, we note that crossing a line representing a particular ratio of α and β,4 a set of structures exists on one side of the ridge with a number of substantial suburban retail centers, while on the other side, the structures are dominated by one large central retail zone. We can analyze this further by plotting two zone graphs, one on either side of the discontinuity, for one of the retail centers present on one side and not on the other. The graphs in Fig. 5 shows the Dj(Wj) curve plotted against the KWj line for the first iteration of each model run. We can see in Fig. 5a(1) that the Dj(Wj) curve does not intersect the KWj line at all, whereas Fig. 5a(2) contains two intersections—an unstable lower intersection and a stable upper intersection. The black dot on each graph shows the Wj value of the retail zone for the first iteration. (a) θ surface plot in α, β parameter space; (b) around best fit. (a) Zone graphs explaining the appearance of an edge city; (b) Rotherham either side of the discontinuity on iteration 295. We assess goodness of fit of the model by calculating an R2 value based on each in the results grid paired with its initial {Wj}. The best fit compared with the real data are at α=1.5, β=0.6, with an R2 of 0.75. To explore this pair-wise comparison in more detail, we generate a finer grid (Fig. 6b) in the parameter space defined by varying α from 1.41 to 1.60 and β over 0.51 to 0.70, using a step size of 0.01 for each. Ridges C and D from Fig. 4a run through the best fit, and another ridge that was not clearly visible in that plot is also present; we label this ridge X. The position of the best fit suggests that a change in α or β would cause a discontinuity to occur in the system. The South Yorkshire retail system. The results that appear in the grids in Fig. 4, and therefore also the discontinuities, are influenced by the initial conditions fed into the model. Starting with a different {Wj} as our initial conditions would produce a different set of discontinuities. This finding suggests there is the potential to influence the behavior of a retail system and to improve its stability through properly informed planning. Demonstrating the practical applications of these techniques is difficult without applying them to a more realistic model that includes, for example, an accurate representation of a transport network and rent data. As a way of connecting the theory to applications, we offer here a simplified case study of the relatively small retail system in South Yorkshire. Rotherham is a major town in the area featured in the news during 2009 because almost one-third of its High Street shops closed down (Addley 2009). These closures have been attributed to various factors, including the current recession and competition with two nearby out-of-town shopping centers: Meadowhall and Parkgate (Baker 2006). The local council is trying to reverse the decline with various policies, including subsidizing costs for new retailers and offering free parking after 3 pm. The British Retail Consortium (Hunt and Slater 2009) highlights that town center retailers across the United Kingdom are facing similar challenges. They argue (p. 16), "Town centres are assets which need to be managed and we need to focus on centres which are already at, or approaching, 'tipping points' rather than waiting to tackle the much more difficult task of High Streets already in decline." Here we use the aggregate retail model to explore whether our techniques for finding discontinuities can identify a "tipping point" or critical size for Rotherham. The initial conditions for this model appear in Fig. 6. To represent differences in the accessibility of each retail zone without resorting to a detailed transport network, we add a multiplier, mj, affecting travel costs cij into each retail zone j. The model parameters α, β, and {mj} are calibrated using a genetic algorithm that took the "fitness" of each solution as the R2 between the initial conditions and the . Once a best fit was found, we retained its parameters and reduced the Wj value for Rotherham from its 2004 size of 105,800 m2 down to 10,580 m2, in 100 steps, to produce a one-dimensional results grid. Figure 7 portrays the range of initial values of Rotherham against the resulting size of Rotherham at equilibrium, along with its three nearest competitors. We can see a discontinuity at approximately 56,000 m2, at which Rotherham jumps from being nonzero to zero. The zone graphs for Rotherham on either side of the discontinuity are identical for the initial iteration; however, by iteration 295 we can see major differences. The zone plot in Fig. 5b(1) for shows the point at which the Dj(Wj) curve stops intersecting the KWj line at any point except the origin, pushing Rotherham to zero, whereas the plot in Fig. 5b(2) for shows a stable intersection at about 40,000 m2. This analysis may offer a methodology to identify the point at which "High Streets""fail," either in a recession or in competition with out-of-town centers. The critical size at which Rotherham becomes unsustainable. We demonstrate a relatively simple and widely applicable software package, together with its theoretical underpinnings, for identifying and analyzing discontinuities in a simple urban retail model using semirealistic data. The visualization capability is an important part of the system that allows discontinuities to be explored in detail. The technique has the potential to provide useful insights for decision makers in both government and retail businesses in order to better understand the impact of planned changes to a system. For example, when commissioning new out-of-town shopping centers, one might look at the conditions required to create a reasonable balance of demand at both "High Street" and out-of-town shopping centers. Many avenues of further research could be pursued, including the following: The cost function, KWj, is a retail zone production function. More realistic functions should be explored along with alternative rent functions. At the expense of considerable computing power, the assumption needed to construct the zonal graphs, that all {Wk}, k≠j are fixed, should be able to be relaxed. In constructing the Dj(Wj) curve for each Wj point, the model could be rerun to equilibrium, maintaining the condition of constant total floor space. Once the zonal graphs can be produced with greater accuracy (following the research implied by the previous point), their properties can be explored: as a set for the whole system, thus enabling the exploration of system-wide discontinuities; and individually, because what the simulations have revealed is that in the instances where two intersections exist, the position of the unstable intersection relative to the origin can be explored, which has implications for the response of the system to different initial conditions. For example, if a particular is nearer to the origin than the unstable point, then the system is more likely to "jump" to the "stable" zero at equilibrium; and vice versa. In this way, path dependence may be able to be altered. Estimating plausible time lines for the exogenous variables should be possible, allowing reproduction of the history of the evolution of retail systems. Explorations of alternatives would then lead us to search for discontinuities generated by changes in any of {ei, Pi, cij, or K}. The method can be extended to other urban models. An obvious starting point for illustrative purposes would be the Lowry (1964) model. This extension should illustrate discontinuities resulting from the interdependence of the submodels of a comprehensive model. More detail can be added to any of the submodels. A residential location model disaggregated by social groups, for example, could be used to seek discontinuities that generate gentrification of city centers. We argue that related systems can be modeled using this kind of methodology (Wilson 2008). Possible applications exist in network analysis because this model system can be seen as a network generator. The real world is far from deterministic (Allen 1997). Formulation of a version of this analysis with a stochastic element would be useful. If we set the rent value K equal to the total spending power in a system divided by the total floor space in the system, we can show that the model maintains the same overall floor space W at equilibrium as it has at the start of a model run. Because we are using a self-normalizing model, we must also state that growth is coming from market capture from one zone over another. The second multiplies the value of ɛ by the Wj value of each retail zone, effectively reducing the rate at which small retail zones reach equilibrium. In some ways, this specification produces more realistic results; however, the time taken to reach equilibrium is so long that it makes it impractical to use this specification to explore equilibrium structures. Simulation experiments demonstrate that both equations produce the same equilibrium structures. The residential zone data were derived from a combination of Census Area Statistics (CAS) ward-level population data from the U.K. 2001 census and Consolidated Analysis Centers Inc. (CACI) Paycheck, which provide income data at postcode level. For each ward, we found the average income level from all the postcode areas inside that ward and then multiplied this quantity by the resident population for the ward to give a spending power eiPi. The ward center points were used as the location of each residential zone. The retail zone data are from 2004 and come from the Town Centres Project (available online at http://www.planningstatistics.org.uk). The discontinuities in the Greater London results grid are described in the following table:
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