在空間分析的研究議題中，若地理資料具有空間非平穩性(spatial non-stationarity)的問題，地理加權迴歸(Geographically Weighted Regression； GWR)為探討此問題經常使用的工具之一。由於GWR易於計算且容易解釋，近年來已有不少學者拓展此模型；但大多數方法僅針對單個反應變數，鮮少考慮多個反應變數。若空間資料涉及多個高度相關的反應變數，GWR則會忽略其相關性，導致結果有所偏差。鑑於此，本研究提出了地理加權多變量迴歸模型(Geographically Weighted Multivariate Multiple Regression； GWMMR)，並討論其相關建模問題，包括多變量變異數分析、拔靴法之迴歸係數檢定與蒙地卡羅空間非平穩性檢定。本研究進一步使用紐約地區之截停盤查資料，實證所提出的GWMMR技術，並將其分析結果與傳統的非空間多變量迴歸和GWR模型進行比較。研究結果顯示， GWMMR表現最佳，並能提供更有用的空間非平穩資訊。

Geographically Weighted Regression (GWR) has been a popular tool in spatial research to address the spatial aspect of data that emphasizes on the issue of spatial non-stationarity. Due to its simple implementation and ease of interpretation, various extensions of GWR have been developed to study the effects of covariates on a single outcome variable. When the geographical data involves multivariate dependent variables collected in the same regions, such a modeling approach ignores the correlations between the outcomes and then yields biased results. Additionally, there has been little study focusing on GWR developments in similar direction. In these regards, this study proposes a geographically weighted multivariate multiple regression (GWMMR) as the
methodology contribution. The associated modeling properties, which include the tests for multivariate analysis of variance, the bootstrap tests of regression coefficients, and the Monte Carlo test in spatial non-stationarity, are also discussed. We further use the “stop-and-frisk＂data published by New York Police Department to illustrate the proposed approach. The analysis results are then compared with those from the traditional non-spatial multivariate regression and GWR models. The results show that GWMMR offer better predictions, which in turn provides more detailed information about spatial non-stationairty.

目錄
1 緒論1
1.1 研究背景與動機. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 研究目的. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 研究架構. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 文獻探討4
2.1 多變量迴歸. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 估計. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 多變量變異數分析(MultiVariate Analysis of Variation Analysis；
MANOVA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.3 特殊情況. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 空間分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 繪製地圖. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 空間相關. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.3 空間迴歸. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 地理加權迴歸(Geographically Weighted Regression) . . . . . . . . . . 13
2.3.1 核函數與最佳帶寬. . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 空間非平穩性檢定. . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.3 迴歸係數檢定. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.4 拔靴法檢定. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 小結. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 研究材料19
3.1 資料介紹. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 反應變數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 解釋變數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 研究方法22
4.1 地理加權多變量迴歸(Geographically Weighted Multivariate Multiple
Regression；GWMMR) . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 最佳帶寬選擇. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 迴歸係數檢定. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3.1 MANOVA 檢定. . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3.2 拔靴法之迴歸係數檢定. . . . . . . . . . . . . . . . . . . . . . 27
4.4 蒙地卡羅之空間非平穩性檢定. . . . . . . . . . . . . . . . . . . . . . . 28
4.5 雙變量地圖. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5 研究結果34
5.1 敘述性分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.1.1 反應變數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.1.2 解釋變數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 傳統非空間迴歸模型分析. . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2.1 變數轉換. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2.2 傳統一般迴歸分析. . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2.3 多變量迴歸分析. . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2.4 空間自相關. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3 空間模型分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3.1 地理加權迴歸分析. . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3.2 地理加權多變量迴歸分析. . . . . . . . . . . . . . . . . . . . . 50
5.3.3 拔靴法檢定分析. . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.3.4 模型比較分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6 討論與建議58
6.1 總結與討論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.2 未來研究與建議. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
圖目錄
2.1 空間迴歸類型圖. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.1 GWMMR 流程圖. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 自然對數犯罪率分佈圖. . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 兩種顏色依等級分成三個區塊. . . . . . . . . . . . . . . . . . . . . . . 31
4.4 逆時針轉90 度之顏色矩陣. . . . . . . . . . . . . . . . . . . . . . . . . 31
4.5 九種顏色結果. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.6 雙變量地圖結果. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.1 截停盤查強度分配直方圖. . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 截停盤查強度散佈圖. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3 反應解釋變數地圖分佈. . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.4 解釋變數地圖分佈(一) . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.5 解釋變數地圖分佈(二) . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.6 自然對數轉換警察截停盤查強度分佈圖. . . . . . . . . . . . . . . . . . 40
5.7 自然對數轉換警察截停盤查強度Q-Q plot . . . . . . . . . . . . . . . . 40
5.8 截停盤查強度之多變量常態圖. . . . . . . . . . . . . . . . . . . . . . . 41
5.9 GWR 截停盤查強度之非西班牙裔白人所佔比例顯著分佈圖. . . . . . 47
5.10 GWR 截停盤查強度之非西班牙裔黑人所佔比例顯著分佈圖. . . . . . 48
5.11 GWR 截停盤查強度之西班牙裔所佔比例顯著分佈圖. . . . . . . . . . 48
5.12 GWR 截停盤查強度之外國出生人口所佔比例顯著分佈圖. . . . . . . 49
5.13 GWR 截停盤查強度之自然對數犯罪率顯著分佈圖. . . . . . . . . . . 49
5.14 GWMMR 之MANOVA 檢定顯著分佈圖(一) . . . . . . . . . . . . . . 53
5.15 GWMMR 之MANOVA 檢定顯著分佈圖(二) . . . . . . . . . . . . . . 53
5.16 GWMMR 之MANOVA 檢定顯著分佈圖(一) . . . . . . . . . . . . . . 55
5.17 GWMMR 之MANOVA 檢定顯著分佈圖(二) . . . . . . . . . . . . . . 55
表目錄
2.1 文獻探討相關文獻整理. . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.1 截停盤查次數與截停盤查強度之敘述性統計量. . . . . . . . . . . . . . 36
5.2 各解釋變數之敘述性統計量. . . . . . . . . . . . . . . . . . . . . . . . 37
5.3 截停強度之變數顯著性結果. . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 盤查強度之變數顯著性結果. . . . . . . . . . . . . . . . . . . . . . . . 42
5.5 截停強度與盤查強度之MANOVA 結果. . . . . . . . . . . . . . . . . 43
5.6 全域Moran＇s Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.7 截停強度GWR 迴歸係數估計結果. . . . . . . . . . . . . . . . . . . . 45
5.8 盤查強度GWR 迴歸係數估計結果. . . . . . . . . . . . . . . . . . . . 45
5.9 GWR 蒙地卡羅檢定結果. . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.10 截停強度GWMMR 迴歸係數估計結果. . . . . . . . . . . . . . . . . . 50
5.11 盤查強度GWMMR 迴歸係數估計結果. . . . . . . . . . . . . . . . . . 50
5.12 GWMMR 蒙地卡羅檢定結果. . . . . . . . . . . . . . . . . . . . . . . 51
5.13 各迴歸模型配適程度比較結果. . . . . . . . . . . . . . . . . . . . . . . 57

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