Title Page
Contents
1. Introduction 13
2. Spatial Regression Models 21
2.1. Simultaneous Autoregressive Model 21
2.2. Conditional Autoregressive Model 23
3. Inference 26
3.1. Classical Inference 26
3.2. Novel Interval Inference 30
3.3. Extending to Inference for Functions of Model Parameters 35
4. Prediction 37
5. Application 39
5.1. Simulation Study 39
5.2. Real Data Analysis 61
5.2.1. COVID-19 in Seoul 61
5.2.2. House Price in Seoul 73
6. Conclusions 83
References 86
요약 91
Table 1. CPs(ALs) of 95% CIs for σ² in the SAR and CAR models with p＝1 for a 5 × 5 regular lattice. 42
Table 2. CPs(ALs) of 95% CIs for β in the SAR and CAR models with p＝1 for a 5 × 5 regular lattice. 42
Table 3. CPs(ALs) of 95% CIs for σ² in the SAR and CAR models with p＝3 for a 5 × 5 regular lattice. 44
Table 4. CPs(ALs) of 95% CIs for β in the SAR and CAR models with p＝3 for a 5 × 5 regular lattice. 44
Table 5. CPs(ALs) of 95% CIs for σ² in the SAR and CAR models with p＝1 for a 10 × 10 regular lattice. 46
Table 6. CPs(ALs) of 95% CIs for β in the SAR and CAR models with p＝1 for a 10 × 10 regular lattice. 46
Table 7. CPs(ALs) of 95% CIs for σ² in the SAR and CAR models with p＝3 for a10 × 10 regular lattice. 48
Table 8. CPs(ALs) of 95% CIs for β in the SAR and CAR models with p＝3 for a 10 × 10 regular lattice. 48
Table 9. MSEs(biases) of estimators for σ² and ρ in the SAR and CAR models with p＝1 for a 5 × 5 regular lattice. 50
Table 10. MSEs(biases) of estimators for β in the SAR and CAR models with p＝1 for a 5 × 5 regular lattice. 50
Table 11. MSEs(biases) of estimators for σ² and ρ in the SAR and CAR models with p＝3 for a 5 × 5 regular lattice. 52
Table 12. MSEs(biases) of estimators for β in the SAR and CAR models with p＝3 for a 5 × 5 regular lattice. 52
Table 13. MSEs(biases) of estimators for σ² and ρ in the SAR and CAR models with p＝1 for a 10 × 10 regular lattice. 54
Table 14. MSEs(biases) of estimators for β in the SAR and CAR models with p＝1 for a 10 × 10 regular lattice. 54
Table 15. MSEs(biases) of estimators for σ² and ρ in the SAR and CAR models with p＝3 for a 10 × 10 regular lattice. 56
Table 16. MSEs(biases) of estimators for β in the SAR and CAR models with p＝3 for a 10 × 10 regular lattice. 56
Table 17. Descriptions and summary statistics for the observations of the covariates in the COVID-19 dataset. 63
Table 18. Estimates for ρ in the COVID-19 dataset. 64
Table 19. Estimates and 95% CIs for σ² in the COVID-19 dataset. 64
Table 20. Estimates and 95% CIs for β in the COVID-19 dataset. 64
Table 21. Descriptions and summary statistics for the observations of the covariates in the house price dataset. 75
Table 22. Estimates for ρ in the house price dataset. 75
Table 23. Estimates and 95% CIs for σ² in the house price dataset. 75
Table 24. Estimates and 95% CIs for β in the house price dataset. 75
Figure 1. Approximate population across 925 cities of the United Kingdom in January 2006. 15
Figure 2. Maximum speeds at the major road types (motorway, trunk, primary, secondary, tertiary) in Essex, United Kingdom. 16
Figure 3. Number of the SIDS in North Carolina state during 1974. 17
Figure 4. (a) CPs and (b) ALs of 95% CIs for σ² and β in the SAR and CAR models with p＝1 for a 5 × 5 regular lattice. 43
Figure 5. (a) CPs and (b) ALs of 95% CIs for σ² and β in the SAR and CAR models with p＝3 for a 5 × 5 regular lattice 45
Figure 6. (a) CPs and (b) ALs of 95% CIs for σ² and β in the SAR and CAR models with p＝1 for a 10 × 10 regular lattice. 47
Figure 7. (a) CPs and (b) ALs of 95% CIs for σ² and β in the SAR and CAR models with p＝3 for a 10 × 10 regular lattice. 49
Figure 8. (a) MSEs and (b) biases of estimators in the SAR and CAR models with p＝1 for a 5 × 5 regular lattice. 51
Figure 9. (a) MSEs and (b) biases of estimators in the SAR and CAR models with p＝3 for a 5 × 5 regular lattice. 53
Figure 10. (a) MSEs and (b) biases of estimators in the SAR and CAR models with p＝1 for a 10 × 10 regular lattice. 55
Figure 11. (a) MSEs and (b) biases of estimators in the SAR and CAR models with p＝3 for a 10 × 10 regular lattice. 57
Figure 12. Boxplots illustrating CPs of the 95% intervals for Y* across a 5×5 regular lattice for (a) p＝1 and (b) p＝3. 59
Figure 13. Boxplots illustrating CPs of the 95% intervals for Y* across a 10×10 regular lattice for (a) p＝1 and (b) p＝3. 60
Figure 14. Map for district-level confirmed rates of COVID-19 in Seoul between 02 April 2022 and 08 April 2022. 62
Figure 15. District-level boxplots for the replicated data of the confirmed rates of COVID-19. 66
Figure 16. (a) Map and (b) resulting boxplots of unobserved districts for r＝2 in the COVID-19 dataset. 68
Figure 17. (a) Map and (b) resulting boxplots of unobserved districts for r＝3 in the COVID-19 dataset. 69
Figure 18. (a) Map and (b) resulting boxplots of unobserved districts for r＝4 in the COVID-19 dataset. 70
Figure 19. (a) Map and (b) resulting boxplots of unobserved districts for r＝5 in the COVID-19 dataset. 71
Figure 20. (a) Map and (b) resulting boxplots of unobserved districts for r＝6 in the COVID-19 dataset. 72
Figure 21. Map for district-level average transaction price per square meter for apartments in Seoul for July 2020. 74
Figure 22. District-level boxplots for the replicated data of the average transaction price per square meter for apartments. 77
Figure 23. (a) Map and (b) resulting boxplots of unobserved districts for r＝2 in the house price dataset. 78
Figure 24. (a) Map and (b) resulting boxplots of unobserved districts for r＝3 in the house price dataset. 79
Figure 25. (a) Map and (b) resulting boxplots of unobserved districts for r＝4 in the house price dataset. 80
Figure 26. (a) Map and (b) resulting boxplots of unobserved districts for r＝5 in the house price dataset. 81
Figure 27. (a) Map and (b) resulting boxplots of unobserved districts for r＝6 in the house price dataset. 82