Trap of weights: The reuse of weights in the floating catchment area (FCA) methods to measuring accessibility

Evaluation of weights under the complexity of accessibility measurements

The complexity in modelling spatial accessibility is derived from its abundant definitions and methodological flexibility, which vary across different fields and build distinct taxonomies. For example, in transportation, accessibility is defined as the potential to reach spatially dispersed opportunities, with a focus on travel cost and transport policy (Páez et al., 2012). In health studies, accessibility of medical facilities is one of the five dimensions (accessibility, availability, accommodation, affordability, and acceptability) that describe the relationship between health facilities and patient utilization (Penchansky and Thomas, 1981). Guagliardo (2004) deconstructs the concept of health accessibility into two stages (potential and realized) and two dimensions (spatial and aspatial). The former refers to the difference between potential opportunity simulation and actual utilization, while the latter considers spatial factors and socioeconomic attributes.

The many definitions of ‘accessibility’ lead to a variety of accessibility-modelling methodologies. The initial container method takes accessibility as the number of facilities in a given unit (Talen and Anselin, 1998). The minimum distance and travel cost methods prioritise transportation in determining facility utilization (Guy, 1983). The Kernel density method estimates the relationship between facility density and population density (Yang et al., 2006). However, few models consider user behaviour in their measurements. For example, residents’ facility utilization can extend beyond their communities, and has a declining frequency as travel distance increases (Higgs, 2004). This is due to so-called ‘spatial barriers’ (Guagliardo et al., 2004; Neutens, 2015). To account for this, the gravity model introduces a distance decay function in assessing population demand and defines accessibility as the sum of facilities’ supply-demand ratios related to a given community. FCA methods are special applications of the gravity model that use catchment areas to simulate the geographical sphere of facility utilization and calculate the supply-demand ratios within catchments (Luo and Wang, 2003).

Numerous factors in GIS models affect the distributive patterns of accessibility maps. Different unit aggregations can generate divergent distributions because of the modifiable areal unit problem (MAUP) (Bell et al., 2013; Dadashpoor et al., 2016). As a long-standing statistical bias, the MAUP refers to the fact that different shapes and sizes of the unit can alter the geographic results (Hewko et al., 2002; Zhang et al., 2011). Bell et al. (2013) contrasted accessibility in neighbourhood and census tract to highlight the social characteristic of the unit aggregation. Additional confounding includes the methods of calculating distance, transport mode, catchment size, and facility capacity (Ni et al., 2019; Pan et al., 2018; Wan et al., 2012b). Apparicio et al. (2008) conducted a comprehensive comparison to examine the influence of different distance types and units aggregation. Geographic weights also play an important role in the accessibility measurements, as illustrated in the next section.

As this study aims to examine the influence of reusing weights in the last step of the FCA methods, the complexity of modelling accessibility increases the difficulty of selecting appropriate evaluation dimensions for their impact. Despite the mentioned influencers, the study chose two dimensions—model type and resources—as the evaluation dimensions for the impact of the reusing weights. Bunel and Tovar (2013) identified the model type as a key issue that can generate different empirical results in accessibility measurements. Although the methodological improvements have driven the reuse of weights, it is a conceptual transform in the FCA methods (Wan et al., 2012a). This conceptual shift should be investigated in a variety of FCA variants for its legitimacy. Moreover, the FCA methods are applied in measuring various resources’ accessibility, including health facilities, jobs, and urban parks (Delamater et al., 2019; Kawabata and Takahashi, 2005; Xie et al., 2018). The wide application of the FCA methods to different resources reflects the fact that the rationality of the methodology can transcend the resource characteristics. This means that differences in resource types do not affect the validity of the method. Thus, comparing the same method across resources allows further testing of the methodological generality. This study introduces a dimension of model type to test the legitimacy of reusing weights and a dimension of resources to examine its generality. The following section reviews how methodological improvements have driven the reuse of weights in FCA methods, as well as the potential limitations of reusing weights.

The development of FCA methods: The ‘omnipotent’ weights

Distance weights play a major role in the development of FCA methods (Langford et al., 2012). A series of methodological improvements have focused on changes in the application of weights for varied purposes (Luo and Qi, 2009; Wan et al., 2012a,b). The original 2SFCA (Luo and Wang, 2003) has two calculation stages: first, the population demand within the supplier’s catchment area is summed to calculate the supply-demand ratio for each supplier. Second, these supply-demand ratios are then summed within the catchment based on the neighbour tracts, which is the value of access (see Equation (1)).

$A_i^{2S}$ represents the access at tract i based on the 2SFCA method, C_i is the catchment centred at tract i, R_j is the supply-demand ratio of supplier j which falls in the catchment centred at tract i, S_j is the supply capacity of supplier j, C_j is the catchment centred at supplier j, and D_k is the population demand of tract k, which falls in the catchment centred at supplier j.

The E2SFCA method enhances the stimulation of population demand in 2SFCA (Luo and Qi, 2009). It applies a distance decay function in the population demand to change it from a dichotomous calculation to a continuous variable. Equation (2b) is the distance decay function in the form of a power function. The distance weight and population demand decline as the distance increases.

$A_i^{E2}$ represents the access calculated by the E2SFCA method, $R_j^{E2}$ is the supply-demand ratio of supplier j which falls in the catchment centred at tract i calculated by the E2SFCA method, w_kj is the distance weight between tract k and supplier j, d_kj is the distance between tract k and supplier j, β is the decay coefficient, and the other variables are the same as in equation (1).

Furthermore, the M2SFCA method argues that not only population demand decreases with distance, but also service effectiveness of suppliers (Delamater, 2013). In M2SFCA, the effectiveness is calculated using distance weights plus the supply capacity for each pair of tracts and suppliers (see Equation (3a)). $A_i^{M2}$ represents the access at tract i based on the M2SFCA method, w_ij is the distance weight between tract i and supplier j. Equation (3b) is the distance decay function, d_ij is the distance between tract i and supplier j, and the other variables are the same as in Equation (2a) and (2b).

Unlike M2SFCA, the 3SFCA method tries to solve the exaggeration of population demand in the E2SFCA (Wan et al., 2012b). As in Equation (2a), the population of tract k is calculated multiple times in accordance with the number of suppliers within tract k’s catchment. The solution is to introduce the supplier weights into population demand. For certain tract k, its supplier weight of supplier j ($w_{kj}^S$) equals its distance weights w_kj divided by the sum of the distance weights of all suppliers within its catchment. In Equation (4a), $A_i^{3S}$ represents the access at tract i based on the 3SFCA method, w_kj is the distance weight between tract k and supplier j. Equation (4b) is the calculation of supplier weight $w_{kj}^S$ between tract k and supplier j, where supplier m is the supplier’s fall in the catchment of tract k, and w_km is the distance weights between tract k and supplier m.

Figure 1 shows the development of four FCA methods, as well as their differences. For various intents, the adaptation of geographic weights has continued throughout the FCA improvement process. Based on the original 2SFCA, the E2SFCA introduces distance weights in population demand to model spatial barriers in resource utilization, which results in declining population demand as the distance between the supplier and the consumer increases. Further to this, the M2SFCA has placed distance weights on the supply capacity to model distance barriers in resource effectiveness. The 3SFCA improves on the E2SFCA’s population demand calculation by factoring in the supplier weight. For a distance between a census unit and a supplier, its supplier weight equals its distance weight divided by the sum of the distance weights of all suppliers that are located in the catchment area of the unit. The supplier weight distributes population demand in accordance with spatial barriers and simulates the competition among suppliers. Especially in the M2SFCA and the 3SFCA, the reuse of weights is the main method to achieve their improvement goals.

Figure 1. The development of floating catchment area (FCA) methods, where E2SFCA is the enhanced two-step FCA, M2SFCA is the modified two-step FCA, 3SFCA is the three-step FCA, and 2SFCA is the flow-based method.

The adaptation of geographic weights has continued throughout the improvements.

Similarly, geographic weights are the solution to the high sensitivity of the distance decay coefficient, which is a common problem in the methods described above (Xing et al., 2018). The value of the distance decay coefficient β (see Equation (2b),(3b), and (4b)) can significantly change the accessibility distribution, even from a monocentric to a decentralised structure. This raises two questions: how should a suitable value for the decay coefficient be determined, and to what extent should it influence the accessibility distribution? For the first question, the normal arbitrary value of β ranges from 1.0 to 2.2 (Luo and Wang, 2003; Luo and Qi, 2009). One practical and rationale way to determine the coefficient values for the different facilities is to use questionnaires to simulate the spatial barriers in facility utilization (Giles-Cortia and Donovan, 2002). For the second question, Wan et al. (2012a) argue that accessibility distribution should remain stable regardless of the coefficient value. This is because the decay coefficient describes how utilization intention and service effectiveness declines with increasing distance. For example, when β equals 1, the distance weight for a census tract 5 km from the hospital equals 0.2, while when β is 2, the weight equals 0.04 (see Equation (2b)). Both values can be appropriate according to divergent individual preferences. However, it is difficult to deal with the uncertainty around the individual use of facilities in the place-based FCA methods. Their solution is to retain the distance-decaying nature, but to remove the influence of β by taking accessibility as the sum of weighted supply-demand ratios, i.e. the access ratio. The distribution of access ratio is relatively stable, with similar patterns under different values of the distance decay coefficient. Table 1 shows the comparison of access (A_i) and access ratio ($A_i^R$) of the three methods. The difference between ‘access’ and ‘access ratio’ is whether to multiply the suppliers’ supply-demand ratios (R_j) by distance weights (w_ij) in the final step. The differences between E2SFCA, M2SFCA, and 3SFCA is the way to calculate the suppliers' supply-demand ratios (R_j). The E2SFCA calculates R_j as supplier capacity divided by weighted population. The M2SFCA argues that the supplier capacity also decreases while distance increases and multiplies supplier capacity by the distance weight. The 3SFCA reveals the double-counted population demand and multiplies population demand by the distance weight and the supplier weight.

Defining accessibility as the sum of weighted ratios has been widely applied in various FCA methods due to its relatively simple implementation (Fransen et al., 2015). In the measurements of access ratios ($A_i^R$), there is limited consideration of the decay coefficient value or the stability of the accessibility distribution. However, this weighted ratio has not been examined for its impact on the FCA variants. There is a potential misidentification of the ‘omnipotence’ of weights in FCA methods. For example, reused weights in population demand, which is the population multiplied by the supplier weight and the distance weight in the 3SFCA, are not equivalent to the row normalisation of the previous population (see Figure 2). The next section discusses the possible side effects of reusing weights in FCA methods.

Figure 2.

Comparison of accessibility values as the sum of supply-demand ratios (A_i) and the sum of weighted supply-demand ratios (A^R_i). A^R_i enlarges the resource acquisition of grids closer to supplier’s location.

Side effects of reusing weights

Although researchers have discovered several limitations of FCA methods, the reuse of weights has not gained sufficient attention. Among those identified limitations (i.e. catchment sizes, overestimation of population, and the MAUP), the application of weights focuses on the value of the distance decay coefficient and the function type of distance decay (McGrail, 2012; Bunel and Tovar, 2013). Evidently, the impacts of distance weights have been simplified to only consider the differences between decay coefficient β values and common forms of distance decay, including the exponential function, the inverse-power function, and the Gaussian function (Kwan, 1998; Neutens, 2015).

However, the manipulation of weights can also have a substantial impact on modelling accessibility. Delamater (2013) has discovered one side effect of reusing weights in the 3SFCA. He modelled a simple dynamic topology that gradually moved one far-away unit closer to a facility-adjacent area. During this process, the 2SFCA method calculates the accessibility for this unit as monotonically increasing, while the 3SFCA method calculates the accessibility as first decreasing and then increasing. This is because reusing weights in population demand, which considers supplier weights and distance weights simultaneously, generates a non-monotonically varying distance decay function in the 3SFCA. Although, the non-monotonic variation in the 3SFCA may have a limited impact in complex topologies.

The use of weighted ratios can produce additional problematic results. Figure 2 illustrates one side effect of defining accessibility as the sum of weighted ratios. The grey quadrilaterals are census tracts and the red pentagrams are supplier locations. To simplify the calculation, this example applies the E2SFCA method and a piecewise function as the distance decay function. The weights of zone 1, zone 2, and zone 3 equal 1, 0.75, and 0.5, respectively. The first step calculates the supply-demand ratio of each supplier (S1 and S2), which is the weighted population of tracts divided by the suppliers’ capacity. The second step aggregates the supply-demand ratio of suppliers within the catchment of each grid centroid. Detailed calculations are provided below:

R_Sn: the supply-demand ratio of supplier Sn.

A_Gn: the access of grid Gn (accessibility as the sum of supple-demand ratios).

$A_{Gn}^R$: the access ratio of grid Gn (accessibility as the sum of weighted ratios).

As shown in Figure 2, the access (A_i) of Grid 1 is greater than that of Grid 2, while their access ratios ($A_i^R$) are reversed. Grid 1 has a reduced access ratio because of its location within two suppliers’ Z2 catchments, while Grid 2 lies within only one supplier’s Z1 catchment. Defining accessibility as the sum of weighted ratios may exaggerate a community’s resource consumption when it lies in close proximity to suppliers. In this instance, it remains uncertain which census tract contains higher service capacities. Still, this example reveals that access ratios exaggerate the accessibility of population units with closer proximities to suppliers and, therefore, greater geographic weights.

The exaggeration of the access ratio in units with closer proximities to amenities should be examined in the context of complicated topologies. There remains the possibility that this exaggeration is either imperative for the precise description of resource distribution, or irrelevant in the context of complex geography. Moreover, how this distortion of the access ratio relates to the modelling approach and the type of resource in question needs to be further investigated. Access ratios produced by various applications of distance weights in the FCA methods could alter its impact on accessibility results. Furthermore, due to their diverse topologies, varied resources with distinct differences in user behaviours appropriate to the size and attributes of their catchments may also influence the effect of weighted ratios. As mentioned above, there are two levels of weighting applications. The differential applications of geographical weights in supply and demand simulations shape FCA variants, while the access ratio helps quantify accessibility. Accordingly, this study focuses on the impact of the access ratio and attempts to build a systematic evaluation by comparing A_i and $A_i^R$, with consideration of the aforementioned influencing factors.

Overview

As a global centre for finance, innovation, and transportation, Shanghai is one of the most populated metropolises in the People’s Republic of China. Its rapid urban expansion has caused a gap in distributing various resources between urban and rural areas (Xiao et al., 2017). Accessibility measurements can detect the spatial equity of various resources in the city’s urban development. Moreover, taking Shanghai as a case study provides an empirical analysis of accessibility models in complex topology and a large city. This study chose Shanghai’s administrative land region (6340.5 km²) as the study area.

Data source

The original data was purchased from Urban Data Party (UDP) via its data service (please see Underlying data). The datasets provided by UDP are accessed from AMAP through web crawler technology. The Urban Data Party membership provides cell phone signalling data, road network data, and the points of primary schools, companies, and hospitals in Shanghai in 2017. This study used the cell phone signalling data to generate the Shanghai population grid. UDP has subsequently deleted the 2017 cell phone signalling data, and instead cooperated with WorldPop et al. (2017) to provide a broader map of population density that is publicly available.

As an alternative to the restricted UDP datasets, OpenStreetMap provides publicly available POIs and road network data of Shanghai, while WorldPop provides the population density of Shanghai (please see Underlying data for more information).

Study design

This study examines the role of weighted ratios in accessibility measurements from two dimensions. The study is based on the contrast between access (A_i) and access ratios ($A_i^R$) , as shown in Table 1. Their difference lies in whether to multiply the suppliers’ supply-demand ratios by the distance weights in the last step of the FCA methods. The first dimension is the different FCA metrics. The E2SFCA, M2SFCA, and 3SFCA models were chosen because of their prevalence and multiple weight applications with various aims. Methodological comparisons assess whether the impacts of access ratios change across different models. Therefore, the purpose of this study is to examine the relationship between the type of FCA model and the side effects of reusing geographic weights.

The second dimension is the impact of access ratios for different types of resources, including primary schools, jobs opportunities, and major hospitals. These empirical comparisons evaluate how access ratios perform in realistic and complex topology.

Table 2 shows the detailed characteristics of the three objects, including their catchment areas and spatial distributions. Primary schools are distributed dispersedly in accordance with neighbourhood locations, as pupils have limited capabilities for long-distance travel, and will therefore usually attend the school closest to their residence. In contrast, jobs and major hospitals are distributed in a centralized way, as workers and patients can typically endure longer distances to obtain income and medical services. Furthermore, healthcare-related behaviours are more tolerant to increased distance than work-related commuting behaviours because obtaining healthcare is typically a matter of necessity and occurs less frequently. Even with the same distance radius, the catchments of hospitals involve more flexible behaviours than those of job opportunities. Therefore, primary schools have a catchment radius of three kilometres for their limited-service areas, while job opportunities and major hospitals both have a 20 kilometre radius.

To be more specific, there are some important nuances in the tested formulae of each object in the three methods. First of all, it is crucial to realize that every tiny parameter change can lead to significant accessibility distributions, including the different sizes of catchments, different distance decay equations and different values of the distance decay coefficient β. However, due to different facilities usage habits, it is impossible to keep all parameters the same crossing different objects. This study tries to maintain the same parameter according to the object type. Equation (5a) and (5b) are the tested formulae of ‘access’ and ‘access ratio’ in the E2SFCA method. To avoid irrelevant influences, this study uses the Gaussian function as the distance decay equation in all three FCA methods, as Equation (5c) shows.

$A_i^{E2}$ and $A_i^{R\left(E2\right)}$ respectively represent the ‘access’ and ‘access ratio’ calculated by the E2SFCA method, $R_j^{E2}$ is the supply-demand ratio of supplier j which falls in the catchment centred at tract i calculated by the E2SFCA method, S_j is the supplier capacity, D_k is the population demand of tract k, $w_{\mathit{kj}}$ is the distance weight between tract k and supplier j, d_kj is the distance between tract k and supplier j, β is the decay coefficient, d_c is the catchment radius and d₀ is the inner critical radius of the catchment, which means the tract population can be fully counted if the tract is close enough to the supplier, i.e. the distance between tract and supplier is less than d₀. For jobs and major hospitals, d₀ equals 2 km and their catchment radius d_c is 20 km. Their distances are calculated as the network distance in the driving mode. While pupils usually have low tolerance to distance, d₀ for primary school is 500 m and d_c equals 3 km. To ensure the Gaussian equation has the same decay dimension as the power function in Equation (2b) and provides a slow changing trend at the same time (Delamater et al., 2019), β equals 110 in measurements of job opportunities and major hospitals, which is 185 in measurements of primary schools.

Equation (6a) and (6b) are the ‘access’ and ‘access ratio’ calculated by the M2SFCA method. The distance decay function and coefficient remain the same as in Equation (5c).

$A_i^{M2}$ and $A_i^{R\left(M2\right)}$ are the ‘access’ and ‘access ratio’ calculated by the M2SFCA method, $R_j^{M2}$ is the supply-demand ratio of supplier j which falls in the catchment centred at tract i calculated by the M2SFCA method, and the other variables are the same as in Equation (5a) and (5b).

Analogously, Equation (7a) and (7b) are the ‘access’ and ‘access ratio’ under the 3SFCA method. Besides the same distance decay function (Equation (5c)), Equation (7c) is the calculation of supplier weights, which is also the main difference between R_j under the M2SFCA and 3SFCA methods.

$A_i^{3S}$ and $A_i^{R\left(3S\right)}$ are the ‘access’ and ‘access ratio’ calculated by the 3SFCA method, $R_j^{3S}$ is the supply-demand ratio of supplier j which falls in the catchment centred at tract i calculated by the 3SFCA method, $w_{\mathit{kj}}^S$ is the supplier weight between tract k and supplier j, where supplier m is the supplier's fall in the catchment of tract k, and w_km is the distance weights between tract k and supplier m, and the other variables are the same as in Equation (5a) and (5b).

Data analysis

There are three stages of data analysis: data pre-processing, generation of the distance matrices and the calculation stage. The first pre-processing stage provides the basic datasets for the distance matrices. The distance matrices refer to the network distance between each population grid and each POI. The calculation stage computes the values of access and access ratios for each grid.

The first stage processes the population grid and the POI datasets in the software Quantum Geographic Information System (QGIS 3.16.6). This study first generated the population density from the 2017 cell phone signalling data for the population grid. It then used the default Zonal Statistics plug-in in QGIS to transfer the population density into a 250*250 m population grid (28,037 units) in conjunction with the Statistical Yearbook data (Shanghai Statistics Bureau, 2018). For the POIs downloaded from UDP (or OpenStreetMap), it is necessary to delete wrong and duplicate points in the QGIS. Specifically, the number of companies (over two hundred thousands points) is too large to generate distance metrics in QGIS. Therefore, the job opportunities are estimated in the 250*250 m grid by multiplying the company size and the number of companies. Major hospitals are those whose official level is greater than Grade II. All service capacities are calculated as the number of objects. For datasets downloaded from OpenStreetMap and WorldPop, it is crucial to ensure they have the same coordinate systems and projections.

The second stage is to form the distance matrices between population grids and POIs datasets. The QNEAT3 plug-in in QGIS generated the distance matrices between the population grid’s geometric centre points and the POIs based on the road networks. In this study, the distance is computed in the car-drive mode.

Based on the distance matrices, the third stage calculates the spatial accessibilities to three types of facilities within three FCA models. Python (3.7.0) is used to calculate the accessibility of each population grid in three existing models (2SFCA, M2SFCA, and 3SFCA). Using the custom codes available in Extended data (Zhang, 2021) can realise the calculations of access (A_i) and access ratios ($A_i^R$) in the three models. An alternative is to reference the codes of access (A_i) in 2SFCA and 3SFCA models at the spatial_access package. This package requires strict forms of original data and generates limited accessibility results.

Weighted ratios: Pattern stability beyond model types

From a methodological perspective, weighted ratios produce stable accessibility patterns regardless of the FCA model type. Compared to the non-weighted maps (Figure 3), weighted ratios (Figure 4) result in similar global spatial structures of accessibility distribution. The weighted ratios produce stable patterns not only under different values of the distance decay coefficient (Wan et al., 2012a), but also under various FCA methods. Particularly for resources with a low distance tolerance, such as primary schools, weighted ratios produce maximum accessibility to the same locations. As the tolerance of user behaviour increases, the nuances between the three methods become more pronounced. The 3SFCA creates more local differences in global structure between job accessibility and health accessibility compared to E2SFCA and M2SFCA. This implies that user behaviours in resource utilization and the aggregation of original POIs can affect the stability of the weighted ratios.

One possible explanation for this pattern stability is the side effect of weighted ratios shown in Figure 2. Weighted ratios overestimate the accessibility to resources of communities with high distance weights and a close proximity to suppliers, as higher access ratios are observed in peripheral areas rather than city centres (see Figure 4). Units in peripheral areas tend to have fewer suppliers in their catchments, resulting in high supplier weights. For those units with only one facility and close proximity to it, their access ratios ($A_i^R$) are almost the same as their access values (A_i). Conversely, units in the city centre with multiple suppliers have access ratios lower than their access values because of their relative small distance weights multiplied several times. Therefore, access ratios of units with fewer suppliers in the periphery are relatively enlarged.

However, there remains a common point between weighted and non-weighted ratios. Both results show that the accessibility distribution differs depending on the type of amenity. It is logical that accessibility structures for primary schools, jobs, and main hospitals are dissimilar, as each object has its own distinguishing distribution. This suggests that the accessibility maps have an evidence-based foundation in addition to methodological influences.

Although weighted ratios eliminate the effect of the decay coefficient values, the validity of model stability across model types remains contentious. In Figure 4, the nuances between three FCA methods exist in partial districts and at the local level. Those subtle differences make it difficult to verify the intended characteristics of FCA methods and further lead to the dispute over the necessity of improvements. On the contrary, Figure 3 visualises the different model improvements in results. The improvement of the M2SFCA, that is, the distance decay of service effectiveness, converts the uniform value of accessibility into a gradual value (especially A3 and B3 in Figure 3). The enhancement of the 3SFCA, which models demand division among suppliers, reduces the overestimation of the population in the facility cluster and produces a more concentrated distribution (see B1, C1, B2 and C2 in Figure 3). On the other hand, it could also be considered a methodological advantage of weighted ratios to create stability across FCA methods. To determine which calculation, using either non-weighted or weighted ratios, is rational, one possible approach is to apply the evidence-based foundation and introduce empirical judgements.

An epistemological trap

Although their results vary, both calculations of access and access ratios are supported by methodological rubrics. Despite these methodological rationalisations, the variability of results leads to diametrically opposed empirical judgements. These contrasting discernments could make it easier to choose a more rational method, based on experiences. Therefore, the evidence-based foundation of accessibility measures may also provide empirical criteria for the choice between A_i and $A_i^R$, by linking the mapped results with common sense.

While there are varying degrees of variability between access and access ratios for different resources, they show a common characteristic among the resources. Maps of access ratios ($A_i^R$ in Figure 4) share similar structures with those of the access calculated by the M2FCA (A_i B-maps in Figure 3). This is due to the reuse of weights in the M2SFCA, as described in the formulas shown in Table 1. Furthermore, the reused weights divide accessibility distribution into two kinds of judgements. In the case of primary schools, the access ratios in Figure 4 indicate that the central city contains a shortage of primary education resources, while the north-eastern island (Chongming Island) has the greatest abundance of those resources. However, the 3SFCA method in Figure 3 concludes the opposite: that the city centre is the most resource-rich, while Chongming Island lacks basic education facilities. Similarly opposed conclusions are the polycentric and monocentric structures of accessibility to job opportunities and major hospitals generated from A_i and $A_i^R$.

The difficulty in judging the validity of opposing results lies in the ratio attribute of accessibility. The provision of services and the needs of the population are reflected in visual and definitive indicators of their quantity. Each facility points has certain indicator of its service capacity, such as the number of physicians in a hospital. Each residential unit obtains an accurate number for its population. However, the relationship between supply and demand is indirect, uncertain, and dynamic. The maximum supply-demand ratio could occur in the suburbs or the city centre based on empiricism.

One possible solution is to apply Kantian ‘a priori knowledge’, which combines empiricism and rationalism. First, key locations are contrasted to provide actual and intuitive measurements. Field research on the key locations provide empirical evidences for the resource distribution. Second, the overall judgements of resource distributions in previous studies are discussed to choose the most accurate global structure of accessibility. These previous studies provide the ‘a priori knowledge’ of resource distributions based on actual utilizations.

Fieldwork may provide insight into the comparison of the accessibility values of key points and thus inform model recommendations. In accessibility to primary schools, the catchment of School A at Chongming Island (see (a1) in Figure 4) has one of the highest access ratios $A_i^R$, although the island is famous for its agricultural products and natural landscapes. School B is one of the medium values also located on the island. Field research reveals that School A serves only its collective farm, which is a socialist legacy with limited productivity and a decreasing population. School B, on the other hand, is located in a more densely populated area and provides a higher level of educational resources for the children in the county. It is counter-intuitive that the catchment area of School A is more accessible than School B.

Furthermore, the urban-rural dichotomy regarding quality of life caused by the Chinese dual system has been a long-standing issue (Chan and Wei, 2019; Ma et al., 2020). In the dichotomy, population and land management systems are both separate and unequal between urban and rural areas. The socialist governance system leads to the concentration of public resources in the urban areas, while rural areas lack the same resources. Xiao et al. (2017) identified that high access to urban parks occurred mainly in downtown Shanghai. Zhao and Cao (2020) investigated 81 million trips that used transit smart cards to identify the gradual decrease of job accessibility from the city centre towards the outer suburbs. In the case of hospitals, the urban-rural dichotomy of medical resources has already been identified, revealing a lack of medical resources in suburbs (Meng et al., 2015). In 2009, the Shanghai Municipal Government launched the ‘5+3+1’ project to promote the construction of high-quality hospitals in the periphery. Using this ‘a priori knowledge’, it seems impossible that the most abundant resources are located in the agricultural island in all three categories. Therefore, the 3SFCA model with non-weighted ratios might be the most appropriate method to reflect the accessibility distribution in Shanghai.

In summary, the reuse of weights, including the M2SFCA method (Figure 3 b-maps) and defining accessibility as the sum of weighted ratios (Figure 4), can generate an epistemological trap. The outcomes of access ratios can produce pattern stability incongruous with model features and resource types, when compared to the access results. Moreover, the side effect of reusing weights, which exaggerates accessibility for units with closer proximities and fewer suppliers, can produce counter-intuitive results. The reuse of weights may lead to methodologically plausible, but common-sense-defying distributions of accessibility. It is therefore necessary to combine empiricism and rationalism in the evaluation of accessibility methods; similarly, the use of geographic weights needs to be cautious and epistemologically consistent.

Underlying data

The underlying data consists of the cell phone signalling data, road networks, point of interest (POI) data (primary schools, companies, and hospitals) in Shanghai in 2017, which was purchased from Urban Data Party (UDP) via its data service. These datasets cannot be made publicly available because they are under copyright to Urban Data Party. The following URLs are only available with Urban Data Party member registration:

As an alternative to the restricted UDP datasets (points 1-3), OpenStreetMap provides publicly available POI and road network data for Shanghai which is representative of the analysed datasets: https://www.openstreetmap.org/relation/913067. This data can be downloaded directly on the website via the Overpass API. The QGIS plug-in QuickOSM can also download the OpenStreetMap data and convert it to shapefiles, see tutorials here.

Extended data

Zenodo: Python codes for accessibility and access ratios in the E2SFCA, M2SFCA, 3SFCA methods. http://doi.org/10.5281/zenodo.4890880 (Zhang, 2021).

This project contains the following extended data:

Data are available under the terms of the Apache License 2.0.