# Interaction between Development Intensity: An Evaluation of Alternative Spatial Weight Matrices

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Spatial Weight Matrix

^{2}and mean absolute percentage error, based on the Twin Cities metropolitan area census block data. The land use spatial model with a correlation matrix is further applied to predict density in 2022 for the Twin Cities.

#### 2.1. Adjacency Matrix

#### 2.2. Accessibility Matrix

#### 2.3. Correlation Matrix

#### 2.4. An Illustration of Three Spatial Weight Matrices

## 3. Model Specification and Statistical Technique

#### 3.1. Model Specification

#### 3.2. Statistical Technique

## 4. Data

## 5. Results

## 6. Application

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Spatial distributions of density changes between 2022 and 2017 for the Twin Cities: (

**a**) change in job density; (

**b**) change in worker density.

Variables | $\mathbf{ln}({\mathit{D}}_{\mathit{e},\mathit{i},\mathit{t}})$ | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Model 1 | Model 2 | Model 3 | Model 4 | |||||||||

Coef. | Drisc/Kraay Std. Err. | Sig. | Coef. | Drisc/Kraay Std. Err. | Sig. | Coef. | Drisc/Kraay Std. Err. | Sig. | Coef. | Drisc/Kraay Std. Err. | Sig. | |

$\mathrm{ln}({D}_{e,i,t-5})$ | 9.12 × 10^{−01} | 2.61 × 10^{−03} | *** | 9.12 × 10^{−01} | 2.32 × 10^{−03} | *** | 9.11 × 10^{−01} | 2.81 × 10^{−03} | *** | 9.42 × 10^{−01} | 1.13 × 10^{−02} | *** |

${W}^{n}\mathrm{ln}\left(\frac{{D}_{e,i,t-5}}{{D}_{e,i,t-10}}\right)$ | 3.68 × 10^{−03} | 5.46 × 10^{−03} | ||||||||||

${W}^{n}\mathrm{ln}\left(\frac{{D}_{w,i,t-5}}{{D}_{w,i,t-10}}\right)$ | 1.32 × 10^{−03} | 1.22 × 10^{−02} | ||||||||||

${W}^{a}\mathrm{ln}\left(\frac{{D}_{e,i,t-5}}{{D}_{e,i,t-10}}\right)$ | 3.22 × 10^{−03} | 4.59 × 10^{−04} | ** | |||||||||

${W}^{a}\mathrm{ln}\left(\frac{{D}_{w,i,t-5}}{{D}_{w,i,t-10}}\right)$ | −1.00 × 10^{−03} | 8.26 × 10^{−04} | ||||||||||

${W}^{c}\mathrm{ln}\left(\frac{{D}_{e,i,t-5}}{{D}_{e,i,t-10}}\right)$ | 4.08 × 10^{−04} | 1.29 × 10^{−05} | *** | |||||||||

${W}^{c}\mathrm{ln}\left(\frac{{D}_{w,i,t-5}}{{D}_{w,i,t-10}}\right)$ | 2.98 × 10^{−05} | 2.88 × 10^{−05} | ||||||||||

Const. | 9.12 × 10^{−01} | 2.61 × 10^{−03} | *** | 9.12 × 10^{−01} | 2.32 × 10^{−03} | *** | 9.11 × 10^{−01} | 2.81 × 10^{−03} | *** | 9.42 × 10^{−01} | 1.13 × 10^{−02} | *** |

Obs. | 49,260 | 49,260 | 49,260 | 49,260 | ||||||||

Group | 8210 | 8210 | 8210 | 8210 | ||||||||

Prob > F | 0.000 | 0.000 | 0.000 | 0.000 | ||||||||

R^{2} | 0.7943 | 0.7943 | 0.7944 | 0.8969 |

Variables | $\mathbf{ln}({\mathit{D}}_{\mathit{w},\mathit{i},\mathit{t}})$ | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Model 5 | Model 6 | Model 7 | Model 8 | |||||||||

Coef. | Drisc/Kraay Std. Err. | Sig. | Coef. | Drisc/Kraay Std. Err. | Sig. | Coef. | Drisc/Kraay Std. Err. | Sig. | Coef. | Drisc/Kraay Std. Err. | Sig. | |

$\mathrm{ln}({D}_{w,i,t-5})$ | 8.94 × 10^{−01} | 1.54 × 10^{−02} | *** | 8.93 × 10^{−01} | 1.67 × 10^{−02} | *** | 8.84 × 10^{−01} | 1.33 × 10^{−02} | *** | 9.36 × 10^{−01} | 8.70 × 10^{−03} | *** |

${W}^{n}\mathrm{ln}\left(\frac{{D}_{e,i,t-5}}{{D}_{e,i,t-10}}\right)$ | 9.75 × 10^{−03} | 3.68 × 10^{−03} | * | |||||||||

${W}^{n}\mathrm{ln}\left(\frac{{D}_{w,i,t-5}}{{D}_{w,i,t-10}}\right)$ | −2.82 × 10^{−02} | 4.33 × 10^{−03} | ** | |||||||||

${W}^{a}\mathrm{ln}\left(\frac{{D}_{e,i,t-5}}{{D}_{e,i,t-10}}\right)$ | 2.18 × 10^{−03} | 3.60 × 10^{−04} | ** | |||||||||

${W}^{a}\mathrm{ln}\left(\frac{{D}_{w,i,t-5}}{{D}_{w,i,t-10}}\right)$ | −3.86 × 10^{−03} | 1.64 × 10^{−04} | *** | |||||||||

${W}^{c}\mathrm{ln}\left(\frac{{D}_{e,i,t-5}}{{D}_{e,i,t-10}}\right)$ | 2.80 × 10^{−04} | 1.97 × 10^{−05} | *** | |||||||||

${W}^{c}\mathrm{ln}\left(\frac{{D}_{w,i,t-5}}{{D}_{w,i,t-10}}\right)$ | 1.26 × 10^{−05} | 3.86 × 10^{−05} | ||||||||||

Const. | −8.36 × 10^{−01} | 1.52 × 10^{−01} | ** | −8.53 × 10^{−01} | 1.59 × 10^{−01} | ** | −9.46 × 10^{−01} | 1.26 × 10^{−01} | ** | −4.45 × 10^{−01} | 8.08 × 10^{−02} | ** |

Obs. | 49,260 | 202,230 | 202,230 | 202,230 | ||||||||

Group | 8210 | 33,705 | 33,705 | 33,705 | ||||||||

Prob > F | 0.000 | 0.000 | 0.000 | 0.000 | ||||||||

R^{2} | 0.8035 | 0.8045 | 0.8058 | 0.9027 |

Variables | ${D}_{e,i,t}$ | |||||

25 Percentile | 50 Percentile | 75 Percentile | ||||

−9.730 | −7.764 | −6.453 | ||||

Value | Elasticity | Value | Elasticity | Value | Elasticity | |

${W}^{a}\mathrm{ln}\left(\frac{{D}_{e,i,t-5}}{{D}_{e,i,t-10}}\right)$ | 2.631 | 0.003 | −0.087 | 2.796 | 0.003 | −0.116 |

${W}^{c}\mathrm{ln}\left(\frac{{D}_{e,i,t-5}}{{D}_{e,i,t-10}}\right)$ | −1274.180 | 0.000 | 5.344 | −1752.130 | 0.000 | 9.210 |

Variables | ${D}_{w,i,t}$ | |||||

25 Percentile | 50 Percentile | 75 Percentile | ||||

−8.113 | −7.437 | −6.974 | ||||

Value | Elasticity | Value | Elasticity | Value | Elasticity | |

${W}^{n}\mathrm{ln}\left(\frac{{D}_{e,i,t-5}}{{D}_{e,i,t-10}}\right)$ | 0.061 | 0.010 | −0.007 | −0.182 | 0.010 | 0.024 |

${W}^{n}\mathrm{ln}\left(\frac{{D}_{w,i,t-5}}{{D}_{w,i,t-10}}\right)$ | −0.132 | −0.028 | −0.046 | −0.389 | −0.028 | −0.147 |

${W}^{a}\mathrm{ln}\left(\frac{{D}_{e,i,t-5}}{{D}_{e,i,t-10}}\right)$ | −0.058 | 0.002 | 0.002 | 3.309 | 0.002 | −0.097 |

${W}^{a}\mathrm{ln}\left(\frac{{D}_{w,i,t-5}}{{D}_{w,i,t-10}}\right)$ | −1.091 | −0.004 | −0.052 | −0.055 | −0.004 | −0.003 |

${W}^{c}\mathrm{ln}\left(\frac{{D}_{e,i,t-5}}{{D}_{e,i,t-10}}\right)$ | −826.985 | 0.000 | 2.856 | −5.490 | 0.000 | 0.021 |

Spatial Matrices | Job Density | Worker Density | ||||
---|---|---|---|---|---|---|

MAPE | Variance | Friedman Test | MAPE | Variance | Friedman Test | |

Adjacency | 8.10 | 190.48 | 0.000 | 4.27 | 15.22 | 0.000 |

Accessibility | 8.09 | 190.82 | 4.16 | 15.00 | ||

Correlation | 6.46 | 127.79 | 4.06 | 12.19 |

Change (%) | −100–−50 | −50–0 | 0–50 | 50–100 | |
---|---|---|---|---|---|

Density | |||||

Job | 0.000 | 0.563 | 0.422 | 0.015 | |

Worker | 0.000 | 0.312 | 0.671 | 0.017 |

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**MDPI and ACS Style**

Li, M.; Cui, M.; Levinson, D.
Interaction between Development Intensity: An Evaluation of Alternative Spatial Weight Matrices. *Urban Sci.* **2023**, *7*, 22.
https://doi.org/10.3390/urbansci7010022

**AMA Style**

Li M, Cui M, Levinson D.
Interaction between Development Intensity: An Evaluation of Alternative Spatial Weight Matrices. *Urban Science*. 2023; 7(1):22.
https://doi.org/10.3390/urbansci7010022

**Chicago/Turabian Style**

Li, Manman, Mengying Cui, and David Levinson.
2023. "Interaction between Development Intensity: An Evaluation of Alternative Spatial Weight Matrices" *Urban Science* 7, no. 1: 22.
https://doi.org/10.3390/urbansci7010022