# Understanding the Modifiable Areal Unit Problem and Identifying Appropriate Spatial Units while Studying the Influence of the Built Environment on the Traffic System State

Authors: Tian Li; Mengmeng Zhang; Haobin Jiang; Peng Jing
Journal: Journal of Advanced Transportation
 (2022)

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## Abstract

Spatially aggregated data are prone to the effects of the modifiable areal unit problem (MAUP), which applies to built environments and traffic data. Although various studies have been carried out to explore the impact of built environment factors on traffic systems, few have considered MAUPs, which may result in statistical inconsistency. The purpose of this study is to assess the effects of MAUPs on statistical variables and geographically weighted regression results when evaluating the influence of the built environment on the traffic system state. Fifty sets of spatial configurations were created using the different aggregation criteria. The variance inflation factor and spatial autocorrelation of the variables, as well as the R2 and root mean squared error of the GWR model, were used to assess the MAUP effect. The results show that the index variation is more dependent on the scale of the spatial unit than on zoning type. In the case study presented, based on the available dataset, the optimal spatial unit size for analyzing the influence of the built environment on Jinan’s traffic system was 900m × 900 m.


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## Body

## 1. Introduction

Although increasing attention has been paid to the MAUP, it is still neglected in research on built environments (BEs) and traffic system states (TSSs). Some authors discuss BE study areas at the city [8, 9], community [10], or block [11] levels, but these studies generally use scale as a category rather than as a basis for MAUP discussion. Recent studies have investigated the effect of MAUP on the impact of BE on travel behaviors. Zhong et al. [12] investigated the impact of BE on travel behavior on both traffic community and buffer zone scales and found that at a neighborhood scale, the BE cannot explain all resident types’ travel behavior effectively. Hong et al. [13] studied the effect of BEs on the vehicle travel distance at a transport analysis zone (TAZ) scale and at a 1 km buffer zone scale and found that the travel distance is more sensitive to the BE variables at the TAZ scale. However, the spatial scale classifications of the above study are too simple, and the SU scale and zoning type design were not carried out; therefore, the corresponding MAUP effect analyses were insufficient, while the conclusions are only applicable to the TAZ and buffer zones. In addition, the above study mainly evaluated the changes in the model parameters while ignoring the MAUP effect on the variables. In addition, the above research focuses on the relationship between BE and travel behavior, and the latter will eventually affect TSS. The influence of the MAUP on the relationship between the BE and TSS has received little attention. Thus, it is necessary to compare the influence of BE on TSS from multiple spatial scales and types to determine the effect of the MAUP. In addition, the wide availability and utilization of transportation and geographic big data provide an opportunity to study the MAUP within the context of the BE-TSS relationship from the perspective of variable variation and model parameter fluctuation.In this study, we examined how variables and model results change when using SUs of various scales and zoning types, as well as the optimal SU choice.


## 2. The Built Environment

BE refers to various buildings and urban spaces that are different from the natural environment, especially those that can be changed through policies and human behavior. BE contains multidimensional elements. In the field of urban geography and urban planning, Cervero et al. [14] put forward the “3 Ds” element, referring to density, diversity, and design, which is widely applied. Ewing et al. [15] extended the “3 Ds” to “5 Ds,” adding “destination accessibility” and “distance to transit”. In this paper, we use the “5 Ds” dimensions of BE. The main indicators and calculation methods for each dimension are listed in the Table 1.Table 1
Main indicators of the 5Ds.
VariablesMain indicators of previous studiesIndicator variables of this paperCalculation methodDensityPopulation density, employment density, building densityPopulation densityPopulation per square kilometerDiversityLand-use mix, job-housing imbalanceLand-use mix [16]HHIi=∑j=1nxij/xi2Where HHIi is the herfindahl-hirschman index of grid i, xi is the total POI in grid i and xij is the total POI of category j in grid iDesignIntersection density, neighborhood, road densityIntersection densityNumber of intersections per gridDestination accessibilityAccessibility to jobs, distance to CBDDistance to CBDAi=∑j=1n1dijWhere Ai is the destination accessibility of grid i, dij is the distance between grid i and CBDjDistance to transitBus stop density, distance to transit stopBus stop densityNumber of bus stops per grid


## 3. The Traffic System State

The TSS refers to the running and changing states of complex traffic systems. In a narrow sense, TSS is usually expressed through parameters such as the urban road traffic service level, vehicle speed, and road congestion index, with the focus being an accurate representation of traffic efficiency. The generalized TSS includes the state of traffic travel, safety, and comfort, and is a comprehensive representation of the multidimensional state of the traffic system (shown in Table2). The TSS studied in this study was in a generalized state.Table 2
Definition and calculation method of TSS indicators.
VariablesDefinitionCalculation methodTraffic efficiencyRoad speedTTIi=vf/1/N∑wNvwiWhere TTIi is the travel time index of grid ivf is the free-flow speed, vwi is the actual speed of track point w in grid iTraffic safetyNumber of traffic accidentsTotal number of incidents per yearTraffic comfortComfort level while drivingThe sum of uncomfortable track points, as defined in the literature [17]


## 4. Methodology

### 4.1. Design of SU Scale and Type

The MAUP effect can be subdivided into scale and zoning effects [18] The scale effect refers to the fact that when the size of the SU changes owing to aggregation, the statistical results of the same dataset will be different (shown in Figure 1). The zoning effect occurs when a change in the spatial element boundaries for the same unit area results in different statistical results (shown in Figure 2).Figure 1
Scale effect.Figure 2
Zoning effect.Commonly used SUs in the study of BE and TSS interactions include the TAZ [19, 20], buffer zone [21], and grid [22]. Considering the need for multiscale design of the SU scale and types of this study, we note that once the traditional TAZ division has been completed, it is difficult to change. In addition, TAZs are too large and, therefore, unsuitable for fine-scale research. Buffers are more focused on “point/line” research and are not suitable for the whole region’s “surface” research. Grids can be varied freely in scale and type and are more conducive to data clustering analysis at different scales. Considering the strong operability and wide application of square grids, a square grid was used as the basic SU in this study.The selection of the SU scale should consider the road length and block area enclosed by the road in the urban built-up area. On February 6, 2016, the Central Committee and the State Council issued “several opinions on further strengthening the management of urban planning, construction,” which introduced the concept of “narrow roads and dense networks,” and proposed that road density should reach 8 km/km2. Based on this initiative, the resulting road spacing was between 100 m and 200 m. Therefore, in this study, the minimum SU scale selected was 100 m. For the selection of the maximum scale of SU, we referred to the literature on BE and transportation [23–27], where 400 m, 500 m, and 1000 m were used as maximum SU scales. Eventually, the 1000 m SU scale was chosen as the maximum, with 10 subscales ranging from 100 m to 1000 m at 100 m intervals.To explore the role of SU type in the influence of BE on TSS, five zoning methods with grid aspect ratios of 1 : 1, 1 : 2, 2 : 1, 1 : 3, and 3 : 1 were adopted (shown in Figure3). Finally, we designed 50 SU types (10 scales multiplied by 5 zoning types).Figure 3
Diagram of basic SU type design: (a) 100m × 100 m (1 : 1). (b) 100 m × 200 m (1 : 2). (c) 200 m × 100 m (2 : 1).
(a). (b)(c)


### 4.2. Two Layer MAUP Effect Analysis

#### 4.2.1. Analysis of MAUP Effect on Variables

Spatial autocorrelation analysis can reflect whether the pattern reflected by spatial elements is aggregated, discrete, or random. Spatial autocorrelation analysis is often used as a partition reference in spatial data research. Marcos et al. [6] used the spatial autocorrelation of traffic accident data as the basis for dividing research units. In this study, spatial autocorrelation was also used to test the MAUP effect of variables. The metric of spatial autocorrelation is Moran’s I and is calculated as follows:(1)I=n∑i=1n∑j=1nWij×∑i=1n∑j=1nWijxi−x¯xj−x¯∑i=1nxi−x¯2,where i is the SU index, xi and xj are the attribute values of SUs i and j, x¯ is the mean value of all SU values, and Wij is the spatial weight matrix. I > 0 indicates that the attribute values of all SUs have a positive spatial correlation; that is, as the spatial distribution locations are aggregated, their correlation becomes more significant. I = 0 represents random regional distribution with no spatial correlation. I < 0 indicates that the attribute values of all regions have a negative correlation in space; that is, with the dispersion of spatial distribution positions, their correlation becomes increasingly significant.In a linear regression model, multicollinearity refers to a large deviation between the linear regression result and the true value owing to a high linear correlation between variables. The scale of SU and the type of zoning also have an important influence on variable multicollinearity. The variance inflation factor (VIF) is commonly used in multivariate multicollinearity tests, with high values indicating possible collinearity between the explanatory variables. The VIF is calculated asVIF=1/1−Rj2, where Rj2 is the degree of linear correlation between the variable xj and the other variables.


#### 4.2.2. Analysis of MAUP Effect on Model Parameters

In this section, the variation rules of model parameters under different SU scales and types are studied to explore the influence of the MAUP on the effects of BE on the TSS models. Since both BE and TSS data are spatial, the impact of BE on TSS may vary depending on SU, and the traditional ordinary least squares global linear regression model is not capable of explaining spatial heterogeneity. The geographically weighted regression (GWR) model fully considers the spatial characteristics of the research object, and the local regression function can be adjusted according to the spatial variation of the sample. Therefore, in this study, the GWR model was used as a unified model. The mathematical expression corresponding to the GWR model is shown as follows [28]:(2)yi=β0i+∑j=1pβijxij+εi,where yi is the ith dependent variable, xij is the value of the jth variable at the ith grid, βij is the jth regression parameter of grid i, and p is the number of independent variables.GWR models of BE, traffic efficiency, traffic safety, and traffic comfort were constructed and are denoted as GWR-TE, GWR-TS, and GWR-TC, respectively. To evaluate the effect of the GWR model on different SUs, the root mean square error (RMSE) of the model was used, where smaller RMSE values corresponded to a better fit of the GWR model to the observed data, as well asR2, which is the determination coefficient of the GWR.


### 4.3. Multiobjective Selection Model of SU

#### 4.3.1. Model Construction

In the study of the influence of BE on TSS, it should be noted that the aggregation values of variables, such as population density, land use diversity, and intersection density, are affected by the MAUP. Simultaneously, the performance of the model is also affected by MAUP, as the parameter estimates and performance indicators of different SUs are different. Therefore, multiobjective functions with both variable -and model result-related objectives should be designed. The goal of a multiobjective selection model is to select an appropriate SU that minimizes the influence of the MAUP on the variables and models. We set four goals as:Goal 1: The spatial autocorrelation of the variables is minimal to achieve homogeneity in the region and randomness in the interval. The degree of spatial autocorrelation was measured by Moran’s indexI, with smaller values corresponding to less spatial autocorrelation.Goal 2: The collinearity degree between variables is minimal to ensure that the variables are independent. The degree of collinearity was measured using VIF, with smaller values corresponding to less collinearity.Goal 3: The model-fitting effect is the best. The fitting effect of the GWR model was measured usingR2, with larger values considered more desirable.Goal 4: The model-fitting performance, measured using the RMSE in the case of the GWR model, is optimal. Smaller RMSEs correspond to a better fit.A multiobjective function is now established according to the above objectives.The decision variables are the SU type (z).The objective function is:minIzminVzmaxRzminRSSzThe constraints are as follows:z=100m∗100m,100m∗200m,200m∗100m,…,3000m∗1000mIz, Vz, Rz, and RSSz are the mean values of Moran’s I, VIF, R2, and RMSE, respectively, for SU type z.In this study, multiple objective functions were combined into a unified function via a weighted grouping method.(3)minFz=w1Iz+w2Vz−w3Rz+w4RSSz,where w1,w2,w3,w4 are the weights of Iz, Vz, Rz, and RSSz, respectively.


#### 4.3.2. Model Solution

To determine the weights while ensuring that (i) they include both subjective and objective factors, (ii) the uncertainty of subjective factors is reduced, and (iii) any discrepancy between the objective weight and the actual degree of importance is minimized, we chose a hybrid weighting method combining the analytic hierarchy process (AHP) and the entropy weight method.The calculation process of AHP is mainly divided into three steps:(i)
The comparison matrixB=aijn×n is established, where aij is a value indicating the importance of factor i relative to factor j and n is the number of scoring objects.(ii)
The largest eigenvector of matrix U was found. The maximum eigenvalue of the comparison matrixλmax is then obtained, followed by the corresponding eigenvector w using the equation Bw=λmaxw′.(iii)
A consistency test was conducted. If the random consistency ratioCR<0.1, it can be concluded that the consistency of the comparison matrix is strong, where CR=CI/RI, with CI=/λmax−n/n−1, and RI are obtained from Table 3.The entropy weighting method determines the index weights according to their degree of difference. The main calculation process is as follows.(i)
The variables are normalized and forwarded.(ii)
The information entropy of thej-th index is calculated according to the equation as:(4)Ej=−1lnn∑i=1mxij′lnxij′,wherexij′ is the value of the j-th index of the i-th sample after normalization and forward transformation.(iii)
The difference coefficientgj of the j-th index is calculated as the following type.(5)gj=1−Ej.(iv)
The weight indexwj″ is calculated as the following type.(6)wj′=gj∑j=1ngj.Table 3
Determination of matrix RI value.
n2345678910RI00.520.91.121.241.321.411.451.49Then, the combined weight is calculated as the following type:(7)wj=0.7wj′+0.3wj′.


## 4.1. Design of SU Scale and Type

The MAUP effect can be subdivided into scale and zoning effects [18] The scale effect refers to the fact that when the size of the SU changes owing to aggregation, the statistical results of the same dataset will be different (shown in Figure 1). The zoning effect occurs when a change in the spatial element boundaries for the same unit area results in different statistical results (shown in Figure 2).Figure 1
Scale effect.Figure 2
Zoning effect.Commonly used SUs in the study of BE and TSS interactions include the TAZ [19, 20], buffer zone [21], and grid [22]. Considering the need for multiscale design of the SU scale and types of this study, we note that once the traditional TAZ division has been completed, it is difficult to change. In addition, TAZs are too large and, therefore, unsuitable for fine-scale research. Buffers are more focused on “point/line” research and are not suitable for the whole region’s “surface” research. Grids can be varied freely in scale and type and are more conducive to data clustering analysis at different scales. Considering the strong operability and wide application of square grids, a square grid was used as the basic SU in this study.The selection of the SU scale should consider the road length and block area enclosed by the road in the urban built-up area. On February 6, 2016, the Central Committee and the State Council issued “several opinions on further strengthening the management of urban planning, construction,” which introduced the concept of “narrow roads and dense networks,” and proposed that road density should reach 8 km/km2. Based on this initiative, the resulting road spacing was between 100 m and 200 m. Therefore, in this study, the minimum SU scale selected was 100 m. For the selection of the maximum scale of SU, we referred to the literature on BE and transportation [23–27], where 400 m, 500 m, and 1000 m were used as maximum SU scales. Eventually, the 1000 m SU scale was chosen as the maximum, with 10 subscales ranging from 100 m to 1000 m at 100 m intervals.To explore the role of SU type in the influence of BE on TSS, five zoning methods with grid aspect ratios of 1 : 1, 1 : 2, 2 : 1, 1 : 3, and 3 : 1 were adopted (shown in Figure3). Finally, we designed 50 SU types (10 scales multiplied by 5 zoning types).Figure 3
Diagram of basic SU type design: (a) 100m × 100 m (1 : 1). (b) 100 m × 200 m (1 : 2). (c) 200 m × 100 m (2 : 1).
(a). (b)(c)


## 4.2. Two Layer MAUP Effect Analysis

### 4.2.1. Analysis of MAUP Effect on Variables

Spatial autocorrelation analysis can reflect whether the pattern reflected by spatial elements is aggregated, discrete, or random. Spatial autocorrelation analysis is often used as a partition reference in spatial data research. Marcos et al. [6] used the spatial autocorrelation of traffic accident data as the basis for dividing research units. In this study, spatial autocorrelation was also used to test the MAUP effect of variables. The metric of spatial autocorrelation is Moran’s I and is calculated as follows:(1)I=n∑i=1n∑j=1nWij×∑i=1n∑j=1nWijxi−x¯xj−x¯∑i=1nxi−x¯2,where i is the SU index, xi and xj are the attribute values of SUs i and j, x¯ is the mean value of all SU values, and Wij is the spatial weight matrix. I > 0 indicates that the attribute values of all SUs have a positive spatial correlation; that is, as the spatial distribution locations are aggregated, their correlation becomes more significant. I = 0 represents random regional distribution with no spatial correlation. I < 0 indicates that the attribute values of all regions have a negative correlation in space; that is, with the dispersion of spatial distribution positions, their correlation becomes increasingly significant.In a linear regression model, multicollinearity refers to a large deviation between the linear regression result and the true value owing to a high linear correlation between variables. The scale of SU and the type of zoning also have an important influence on variable multicollinearity. The variance inflation factor (VIF) is commonly used in multivariate multicollinearity tests, with high values indicating possible collinearity between the explanatory variables. The VIF is calculated asVIF=1/1−Rj2, where Rj2 is the degree of linear correlation between the variable xj and the other variables.


### 4.2.2. Analysis of MAUP Effect on Model Parameters

In this section, the variation rules of model parameters under different SU scales and types are studied to explore the influence of the MAUP on the effects of BE on the TSS models. Since both BE and TSS data are spatial, the impact of BE on TSS may vary depending on SU, and the traditional ordinary least squares global linear regression model is not capable of explaining spatial heterogeneity. The geographically weighted regression (GWR) model fully considers the spatial characteristics of the research object, and the local regression function can be adjusted according to the spatial variation of the sample. Therefore, in this study, the GWR model was used as a unified model. The mathematical expression corresponding to the GWR model is shown as follows [28]:(2)yi=β0i+∑j=1pβijxij+εi,where yi is the ith dependent variable, xij is the value of the jth variable at the ith grid, βij is the jth regression parameter of grid i, and p is the number of independent variables.GWR models of BE, traffic efficiency, traffic safety, and traffic comfort were constructed and are denoted as GWR-TE, GWR-TS, and GWR-TC, respectively. To evaluate the effect of the GWR model on different SUs, the root mean square error (RMSE) of the model was used, where smaller RMSE values corresponded to a better fit of the GWR model to the observed data, as well asR2, which is the determination coefficient of the GWR.


## 4.2.1. Analysis of MAUP Effect on Variables

Spatial autocorrelation analysis can reflect whether the pattern reflected by spatial elements is aggregated, discrete, or random. Spatial autocorrelation analysis is often used as a partition reference in spatial data research. Marcos et al. [6] used the spatial autocorrelation of traffic accident data as the basis for dividing research units. In this study, spatial autocorrelation was also used to test the MAUP effect of variables. The metric of spatial autocorrelation is Moran’s I and is calculated as follows:(1)I=n∑i=1n∑j=1nWij×∑i=1n∑j=1nWijxi−x¯xj−x¯∑i=1nxi−x¯2,where i is the SU index, xi and xj are the attribute values of SUs i and j, x¯ is the mean value of all SU values, and Wij is the spatial weight matrix. I > 0 indicates that the attribute values of all SUs have a positive spatial correlation; that is, as the spatial distribution locations are aggregated, their correlation becomes more significant. I = 0 represents random regional distribution with no spatial correlation. I < 0 indicates that the attribute values of all regions have a negative correlation in space; that is, with the dispersion of spatial distribution positions, their correlation becomes increasingly significant.In a linear regression model, multicollinearity refers to a large deviation between the linear regression result and the true value owing to a high linear correlation between variables. The scale of SU and the type of zoning also have an important influence on variable multicollinearity. The variance inflation factor (VIF) is commonly used in multivariate multicollinearity tests, with high values indicating possible collinearity between the explanatory variables. The VIF is calculated asVIF=1/1−Rj2, where Rj2 is the degree of linear correlation between the variable xj and the other variables.


## 4.2.2. Analysis of MAUP Effect on Model Parameters

In this section, the variation rules of model parameters under different SU scales and types are studied to explore the influence of the MAUP on the effects of BE on the TSS models. Since both BE and TSS data are spatial, the impact of BE on TSS may vary depending on SU, and the traditional ordinary least squares global linear regression model is not capable of explaining spatial heterogeneity. The geographically weighted regression (GWR) model fully considers the spatial characteristics of the research object, and the local regression function can be adjusted according to the spatial variation of the sample. Therefore, in this study, the GWR model was used as a unified model. The mathematical expression corresponding to the GWR model is shown as follows [28]:(2)yi=β0i+∑j=1pβijxij+εi,where yi is the ith dependent variable, xij is the value of the jth variable at the ith grid, βij is the jth regression parameter of grid i, and p is the number of independent variables.GWR models of BE, traffic efficiency, traffic safety, and traffic comfort were constructed and are denoted as GWR-TE, GWR-TS, and GWR-TC, respectively. To evaluate the effect of the GWR model on different SUs, the root mean square error (RMSE) of the model was used, where smaller RMSE values corresponded to a better fit of the GWR model to the observed data, as well asR2, which is the determination coefficient of the GWR.


## 4.3. Multiobjective Selection Model of SU

### 4.3.1. Model Construction

In the study of the influence of BE on TSS, it should be noted that the aggregation values of variables, such as population density, land use diversity, and intersection density, are affected by the MAUP. Simultaneously, the performance of the model is also affected by MAUP, as the parameter estimates and performance indicators of different SUs are different. Therefore, multiobjective functions with both variable -and model result-related objectives should be designed. The goal of a multiobjective selection model is to select an appropriate SU that minimizes the influence of the MAUP on the variables and models. We set four goals as:Goal 1: The spatial autocorrelation of the variables is minimal to achieve homogeneity in the region and randomness in the interval. The degree of spatial autocorrelation was measured by Moran’s indexI, with smaller values corresponding to less spatial autocorrelation.Goal 2: The collinearity degree between variables is minimal to ensure that the variables are independent. The degree of collinearity was measured using VIF, with smaller values corresponding to less collinearity.Goal 3: The model-fitting effect is the best. The fitting effect of the GWR model was measured usingR2, with larger values considered more desirable.Goal 4: The model-fitting performance, measured using the RMSE in the case of the GWR model, is optimal. Smaller RMSEs correspond to a better fit.A multiobjective function is now established according to the above objectives.The decision variables are the SU type (z).The objective function is:minIzminVzmaxRzminRSSzThe constraints are as follows:z=100m∗100m,100m∗200m,200m∗100m,…,3000m∗1000mIz, Vz, Rz, and RSSz are the mean values of Moran’s I, VIF, R2, and RMSE, respectively, for SU type z.In this study, multiple objective functions were combined into a unified function via a weighted grouping method.(3)minFz=w1Iz+w2Vz−w3Rz+w4RSSz,where w1,w2,w3,w4 are the weights of Iz, Vz, Rz, and RSSz, respectively.


### 4.3.2. Model Solution

To determine the weights while ensuring that (i) they include both subjective and objective factors, (ii) the uncertainty of subjective factors is reduced, and (iii) any discrepancy between the objective weight and the actual degree of importance is minimized, we chose a hybrid weighting method combining the analytic hierarchy process (AHP) and the entropy weight method.The calculation process of AHP is mainly divided into three steps:(i)
The comparison matrixB=aijn×n is established, where aij is a value indicating the importance of factor i relative to factor j and n is the number of scoring objects.(ii)
The largest eigenvector of matrix U was found. The maximum eigenvalue of the comparison matrixλmax is then obtained, followed by the corresponding eigenvector w using the equation Bw=λmaxw′.(iii)
A consistency test was conducted. If the random consistency ratioCR<0.1, it can be concluded that the consistency of the comparison matrix is strong, where CR=CI/RI, with CI=/λmax−n/n−1, and RI are obtained from Table 3.The entropy weighting method determines the index weights according to their degree of difference. The main calculation process is as follows.(i)
The variables are normalized and forwarded.(ii)
The information entropy of thej-th index is calculated according to the equation as:(4)Ej=−1lnn∑i=1mxij′lnxij′,wherexij′ is the value of the j-th index of the i-th sample after normalization and forward transformation.(iii)
The difference coefficientgj of the j-th index is calculated as the following type.(5)gj=1−Ej.(iv)
The weight indexwj″ is calculated as the following type.(6)wj′=gj∑j=1ngj.Table 3
Determination of matrix RI value.
n2345678910RI00.520.91.121.241.321.411.451.49Then, the combined weight is calculated as the following type:(7)wj=0.7wj′+0.3wj′.


## 4.3.1. Model Construction

In the study of the influence of BE on TSS, it should be noted that the aggregation values of variables, such as population density, land use diversity, and intersection density, are affected by the MAUP. Simultaneously, the performance of the model is also affected by MAUP, as the parameter estimates and performance indicators of different SUs are different. Therefore, multiobjective functions with both variable -and model result-related objectives should be designed. The goal of a multiobjective selection model is to select an appropriate SU that minimizes the influence of the MAUP on the variables and models. We set four goals as:Goal 1: The spatial autocorrelation of the variables is minimal to achieve homogeneity in the region and randomness in the interval. The degree of spatial autocorrelation was measured by Moran’s indexI, with smaller values corresponding to less spatial autocorrelation.Goal 2: The collinearity degree between variables is minimal to ensure that the variables are independent. The degree of collinearity was measured using VIF, with smaller values corresponding to less collinearity.Goal 3: The model-fitting effect is the best. The fitting effect of the GWR model was measured usingR2, with larger values considered more desirable.Goal 4: The model-fitting performance, measured using the RMSE in the case of the GWR model, is optimal. Smaller RMSEs correspond to a better fit.A multiobjective function is now established according to the above objectives.The decision variables are the SU type (z).The objective function is:minIzminVzmaxRzminRSSzThe constraints are as follows:z=100m∗100m,100m∗200m,200m∗100m,…,3000m∗1000mIz, Vz, Rz, and RSSz are the mean values of Moran’s I, VIF, R2, and RMSE, respectively, for SU type z.In this study, multiple objective functions were combined into a unified function via a weighted grouping method.(3)minFz=w1Iz+w2Vz−w3Rz+w4RSSz,where w1,w2,w3,w4 are the weights of Iz, Vz, Rz, and RSSz, respectively.


## 4.3.2. Model Solution

To determine the weights while ensuring that (i) they include both subjective and objective factors, (ii) the uncertainty of subjective factors is reduced, and (iii) any discrepancy between the objective weight and the actual degree of importance is minimized, we chose a hybrid weighting method combining the analytic hierarchy process (AHP) and the entropy weight method.The calculation process of AHP is mainly divided into three steps:(i)
The comparison matrixB=aijn×n is established, where aij is a value indicating the importance of factor i relative to factor j and n is the number of scoring objects.(ii)
The largest eigenvector of matrix U was found. The maximum eigenvalue of the comparison matrixλmax is then obtained, followed by the corresponding eigenvector w using the equation Bw=λmaxw′.(iii)
A consistency test was conducted. If the random consistency ratioCR<0.1, it can be concluded that the consistency of the comparison matrix is strong, where CR=CI/RI, with CI=/λmax−n/n−1, and RI are obtained from Table 3.The entropy weighting method determines the index weights according to their degree of difference. The main calculation process is as follows.(i)
The variables are normalized and forwarded.(ii)
The information entropy of thej-th index is calculated according to the equation as:(4)Ej=−1lnn∑i=1mxij′lnxij′,wherexij′ is the value of the j-th index of the i-th sample after normalization and forward transformation.(iii)
The difference coefficientgj of the j-th index is calculated as the following type.(5)gj=1−Ej.(iv)
The weight indexwj″ is calculated as the following type.(6)wj′=gj∑j=1ngj.Table 3
Determination of matrix RI value.
n2345678910RI00.520.91.121.241.321.411.451.49Then, the combined weight is calculated as the following type:(7)wj=0.7wj′+0.3wj′.


## 5. Case Study

In this study, the main urban area of Jinan is taken as the research area (shown in Figure4), and the specific scope is the area enclosed by the Ji-Guang Expressway, the Jinan Ring Expressway, the South Viaduct of the Second Ring Road, and the Jinan Ring Expressway, representing an area of approximately 438 square kilometers (shown in Figure 5). BE data were obtained from the POI data provided by the AutoNavi API, and road network data were obtained using OpenStreetMap. TSS data were calculated using taxi tracks and accident data shared by the traffic management department. The basic BE and TSS data are shown in Figure 6 to Figure 10, respectively. According to the design scheme of the SU scale and type, the SU ranged from 100 to 1000 m, with the five zoning types being 1 : 1.1 : 2, 1 : 3, 2 : 1, and 3 : 1, resulting in a total of 50 SU types. The 500 m × 500 m area is shown as an example in Figure 5.Figure 4
Study area.Figure 5
Population distribution of study area.Figure 6
Intersection, bus station, and CBD of the study area.Figure 7
POI of the study area.Figure 8
POI of the study area.Figure 9
Speed distribution for the study area.Figure 10
SU types corresponding for a scale of 500 m. (a) 500 m × 500 m. (b) 500 m × 1000 m. (c) 1000 m × 500 m. (d) 500 m × 1500 m. (e) 1500 m × 500 m.
(a)(b)(c)(d)(e)


## 6. Results and Discussion

### 6.1. Effects of MAUP on Moran’s I Variables

Before analyzing the scale and type of SU effects onR2 and RMSE for the application of the GWR model, an investigation of the effect of MAUP on the variables should be performed. The distributional characteristics of the variables for the different SUs are shown in Figure 11. As evident from the figure, the spatial autocorrelation of variables fluctuates with the different SU sampling scales and zoning types and shows relatively consistent regularity as follows:(i)
Larger SU scales result in weaker data differences and stronger spatial autocorrelation, whereas smaller SU scales yield weaker data similarity and spatial autocorrelation.(ii)
The effect of the SU type on spatial autocorrelation was weaker than that of the SU scale.(iii)
The Moran’s I values of the variables were all positive. This means that the variables had a positive spatial correlation. The correlation became more significant as the spatial distribution locations were aggregated.Figure 11
Moran’s I of variables. (a) Moran’s Iof HHI. (b) Moran’s I of Intersection density. (c) Moran’s I of bus station density. (d) Moran’s I of population density. (e) Moran’s I of distance to CBD. (f) Moran’s I of traffic safe. (g) Moran’s I of traffic comfortable. (h) Moran’s I of traffic efficiency.
(a)(b)(c)(d)(e)(f)(g)(h)


### 6.2. Effects of MAUP on VIF of Variables

The computation of the VIF led to the conclusion that multicollinearity issues were not present, as the VIF factors were all below 10 (shown in Figure12). However, the VIF values of the independent variables fluctuated as the SU scale and type changed. In particular, the VIF values of 3 : 1 and 1 : 3 subdivisions were generally higher. The VIF value of intersection density was higher than 5 at the SU scale of 1000 m and the 1 : 3 subdivision, showing relative multicollinearity. Therefore, to reduce the multicollinearity of the variables, large SU types and scales should be avoided.Figure 12
VIF of variables. (a) VIF of Population density. (b) VIF of HHI. (c) VIF of Intersection density. (d) VIF of Distance to CBD. (e) VIF of bus station density.
(a)(b)(c)(d)(e)


### 6.3. Effects of MAUP onR2 and RMSE

The distributional characteristics of theR2 and RMSE for the GWR model are shown in Figures 13, and 14, respectively. The following characteristics can be observed in the figure.(i)
Under a unified SU scale, radial and zonal zoning had little influence on the correlation coefficient. For example, theR2 values of the 1 : 3 and 3 : 1 SU types were relatively similar, as were those of the 1 : 2 and 2 : 1 types.(ii)
Larger SU scales tend to result in fluctuations of the correlation coefficients.(iii)
Larger SU scales caused a decrease in the RMSE of GWR-TE and an increase in the RMSE of GWR-TS and GWR-TC.Figure 13
RMSE of GWR model. (a) RMSE of GWR-TE. (b) RMSE of GWR-TS. (c) RMSE of GWR-TC.
(a)(b)(c)Figure 14
R2 of GWR model. (a) R2 of GWR-TE. (b) R2 of GWR-TS. (c) R2 of GWR-TC.
(a)(b)(c)These characteristics indicate that when the SU scale and type change, the model results and goodness of fit undergo complex changes.


### 6.4. Model Solution

Nine transportation engineering experts were invited to fill in a judgment matrix questionnaire on the importance of the indicators. The weight index was calculated according to each specific judgment matrix, with that obtained from expert A (Table4) considered as an example to illustrate the calculation process.Table 4
Judgment matrix filled in by expert A.
B1B2B3B4B110.670.850.75B21.510.570.5B31.171.7510.88B41.3321.141Note. B1 represents Iz, which is the Moran’s I of SU z; B2 represents Vz, which is the VIF of SU z; B3 represents Rz, which is the R2 of the GWR model of SU z; B4 represents RMSEz, which is the RMSE of SU z.The index calculation results were as follows:λmax=4.2, CI=0.068, RI=0.075<0.1. Based on these values, the judgment matrix exhibited satisfactory consistency.Similarly, the judgment matrices of the remaining eight experts are calculated to obtain the corresponding index weight values. Finally, the weight values of all the expert scores were averaged and normalized, and the final weights of the four indices were calculated as follows:(8)W¯1=0.21W¯2=0.17W¯3=0.27W¯4=0.35.The index weights calculated using the entropy weight method were as follows:(9)w1″=0.19w2″=0.62w3″=0.13w4″=0.06.The weight values of each index were calculated, and the results were as follows:(10)w1=0.21w2=0.3w3=0.23w4=0.26.The minimum value of the objective function is obtained forz = 900 m × 900 m.


## 6.1. Effects of MAUP on Moran’s I Variables

Before analyzing the scale and type of SU effects onR2 and RMSE for the application of the GWR model, an investigation of the effect of MAUP on the variables should be performed. The distributional characteristics of the variables for the different SUs are shown in Figure 11. As evident from the figure, the spatial autocorrelation of variables fluctuates with the different SU sampling scales and zoning types and shows relatively consistent regularity as follows:(i)
Larger SU scales result in weaker data differences and stronger spatial autocorrelation, whereas smaller SU scales yield weaker data similarity and spatial autocorrelation.(ii)
The effect of the SU type on spatial autocorrelation was weaker than that of the SU scale.(iii)
The Moran’s I values of the variables were all positive. This means that the variables had a positive spatial correlation. The correlation became more significant as the spatial distribution locations were aggregated.Figure 11
Moran’s I of variables. (a) Moran’s Iof HHI. (b) Moran’s I of Intersection density. (c) Moran’s I of bus station density. (d) Moran’s I of population density. (e) Moran’s I of distance to CBD. (f) Moran’s I of traffic safe. (g) Moran’s I of traffic comfortable. (h) Moran’s I of traffic efficiency.
(a)(b)(c)(d)(e)(f)(g)(h)


## 6.2. Effects of MAUP on VIF of Variables

The computation of the VIF led to the conclusion that multicollinearity issues were not present, as the VIF factors were all below 10 (shown in Figure12). However, the VIF values of the independent variables fluctuated as the SU scale and type changed. In particular, the VIF values of 3 : 1 and 1 : 3 subdivisions were generally higher. The VIF value of intersection density was higher than 5 at the SU scale of 1000 m and the 1 : 3 subdivision, showing relative multicollinearity. Therefore, to reduce the multicollinearity of the variables, large SU types and scales should be avoided.Figure 12
VIF of variables. (a) VIF of Population density. (b) VIF of HHI. (c) VIF of Intersection density. (d) VIF of Distance to CBD. (e) VIF of bus station density.
(a)(b)(c)(d)(e)


## 6.3. Effects of MAUP onR2 and RMSE

The distributional characteristics of theR2 and RMSE for the GWR model are shown in Figures 13, and 14, respectively. The following characteristics can be observed in the figure.(i)
Under a unified SU scale, radial and zonal zoning had little influence on the correlation coefficient. For example, theR2 values of the 1 : 3 and 3 : 1 SU types were relatively similar, as were those of the 1 : 2 and 2 : 1 types.(ii)
Larger SU scales tend to result in fluctuations of the correlation coefficients.(iii)
Larger SU scales caused a decrease in the RMSE of GWR-TE and an increase in the RMSE of GWR-TS and GWR-TC.Figure 13
RMSE of GWR model. (a) RMSE of GWR-TE. (b) RMSE of GWR-TS. (c) RMSE of GWR-TC.
(a)(b)(c)Figure 14
R2 of GWR model. (a) R2 of GWR-TE. (b) R2 of GWR-TS. (c) R2 of GWR-TC.
(a)(b)(c)These characteristics indicate that when the SU scale and type change, the model results and goodness of fit undergo complex changes.


## 6.4. Model Solution

Nine transportation engineering experts were invited to fill in a judgment matrix questionnaire on the importance of the indicators. The weight index was calculated according to each specific judgment matrix, with that obtained from expert A (Table4) considered as an example to illustrate the calculation process.Table 4
Judgment matrix filled in by expert A.
B1B2B3B4B110.670.850.75B21.510.570.5B31.171.7510.88B41.3321.141Note. B1 represents Iz, which is the Moran’s I of SU z; B2 represents Vz, which is the VIF of SU z; B3 represents Rz, which is the R2 of the GWR model of SU z; B4 represents RMSEz, which is the RMSE of SU z.The index calculation results were as follows:λmax=4.2, CI=0.068, RI=0.075<0.1. Based on these values, the judgment matrix exhibited satisfactory consistency.Similarly, the judgment matrices of the remaining eight experts are calculated to obtain the corresponding index weight values. Finally, the weight values of all the expert scores were averaged and normalized, and the final weights of the four indices were calculated as follows:(8)W¯1=0.21W¯2=0.17W¯3=0.27W¯4=0.35.The index weights calculated using the entropy weight method were as follows:(9)w1″=0.19w2″=0.62w3″=0.13w4″=0.06.The weight values of each index were calculated, and the results were as follows:(10)w1=0.21w2=0.3w3=0.23w4=0.26.The minimum value of the objective function is obtained forz = 900 m × 900 m.


## 7. Conclusion

To the best of our knowledge, this study is the first attempt to simultaneously investigate the effect of SU scale and type on the MAUP, affecting the relationship between BE and TSS. Another capital objective was to assess how changes in SU affect the variables and model parameter data. The consequences of MAUP cannot be ignored, as the extent to which MAUP affects variables and model results is obvious, but it is difficult to explain why SU changes lead to inconsistent changes in model parameters. However, several rules can still be extracted. Longitude and latitude zoning (such as 1 : 3 or 3 : 1) has little influence on data aggregation, while SU scale variation has a great influence on both variables and model parameters. Larger SU scales resulted in weaker data differences and stronger spatial autocorrelation. These findings suggest that a change in the SU scale should prompt additional checks on the randomness of the spatial data distribution and collinearity. Comparative analysis showed that a 900m × 900 m square grid was the optimal size for investigating the relationship between BE and TSS. However, this result is only the optimal solution for the available data, and may not be valid when the data are updated. Thus, the effect of the MAUP requires further investigation under various conditions.Moreover, this study has some limitations. First, the SU scale and types were mainly based on square grids, whereas in the field of transportation, SUs generally include additional types, such as TAZ and Tyson polygons. In the future, the detection scope of SU types will be expanded to study the MAUP more comprehensively. Second, in this study, only the MAUP effect on the GWR model was investigated, but the applicability of the method to different models, such as multiple linear regression or conditional autoregressive models, for determining the effects of MAUP was not considered.



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*Source: 8288248-2022-09-14.xml*