A varying coefficient geographically weighted spatial lag model for compositional data
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摘要:
针对已有模型无法刻画面类型空间依赖下成分数据的空间异质性，提出模型参数可变的成分数据空间自回归模型。通过假定空间滞后参数、成分型系数、数值型系数为位置坐标的函数，允许空间效应和变量关系在全局空间上非均匀分布。基于等距对数比(ilr)变换、工具变量法和局部线性地理加权法，对模型参数进行估计。数值模拟实验表明：所提出模型的表现优于已有的成分数据空间自回归模型，并且参数估计量是有效的。基于一组实际数据，说明所提模型的实用性。
Abstract:When it comes to area data with compositional factors, existing regression models seldom ever take spatial heterogeneity into account. To solve the problem, a compositional spatial autoregressive model with varying coefficients is proposed. By assuming that the spatial lag parameter, the compositional coefficient, and the numerical coefficient are functions of the location coordinates, the new model permits spatial effects and linear interactions between covariates and response to change in space. Based on isometric log-ratio (ILR) transformation, instrumental variables and local linear geographically weighted method, the parameters are estimated. The simulation study shows that the proposed model is superior to the existing spatial autoregressive model for compositional data, and the parameters estimation are effective. The utility of the proposed model is demonstrated by a real data set.
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表 1 空间参数和数值型系数的RMSE和SSD
Table 1. RMSE and SSD for the spatial lag parameter and numerical coefficient
情形 n RMSE SSD 本文模型 已有模型[11] 本文模型 已有模型[11] $ \rho ({u_i},{v_i}) $ $ \beta ({u_i},{v_i}) $ $ \rho ({u_i},{v_i}) $ $ \beta ({u_i},{v_i}) $ $ \rho ({u_i},{v_i}) $ $ \beta ({u_i},{v_i}) $ $ \rho ({u_i},{v_i}) $ $ \beta ({u_i},{v_i}) $ 情形1 49 0.010 0.010 −0.005 0.001 0.16 0.22 0.03 0.04 144 0.008 0.006 −0.003 0.000 0.08 0.10 0.01 0.02 441 0.009) 0.004 −0.001 0.000 0.05 0.06 0.01 0.01 情形2 49 0.083 0.008 0.312 0.134 0.26 0.27 0.20 0.23 144 0.023 0.006 0.280 0.126 0.11 0.11 0.11 0.13 441 0.014 0.003 0.266 0.121 0.07 0.06 0.06 0.07 
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表 2 成分系数各成分的RMSE和整个成分系数的SSD
Table 2. RMSE for each component and mean SSD for the entire compositional coefficient
情形 n RMSE SSD 本文模型 已有模型[11] 本文模型 已有模型[11] $ \beta _1^D({u_i},{v_i}) $ $ \beta _2^D({u_i},{v_i}) $ $ \beta _3^D({u_i},{v_i}) $ $ \beta _1^D({u_i},{v_i}) $ $ \beta _2^D({u_i},{v_i}) $ $ \beta _3^D({u_i},{v_i}) $ $ \beta _2^D({u_i},{v_i}) $ $ \beta _2^D({u_i},{v_i}) $ 情形1 49 0.003 0.003 0.001 0.001 0.000 −0.001 0.050 0.009 144 0.002 0.001 0.001 −0.001 0.000 0.000 0.024 0.005 441 0.001 0.001 0.001 0.000 0.000 0.000 0.014 0.003 情形2 49 0.007 0.012 0.018 0.092 0.148 0.191 0.056 0.057 144 0.004 0.007 0.011 0.088 0.144 0.187 0.023 0.032 441 0.002 0.004 0.007 0.086 0.142 0.184 0.012 0.018
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表 3 本文模型的回归结果
Table 3. Regression results for the proposed model
城市 $ \rho $ $ \beta _1^D $ $ \beta _2^D $ $ \beta _3^D $ $ \beta $ 兴安盟 −0.10 1.09 −0.16 −3.42 3.61 通辽 −0.31 0.85 −1.31 −4.04 2.58 锡林郭勒 0.22 1.30 1.37 4.85 0.63 赤峰 −0.09 0.66 0.85 −1.26 0.38 张家口 0.12 0.32 −0.44 0.41 0.20 包头 0.13 −0.04 2.25 3.86 −0.21 呼和浩特 0.00 0.24 1.56 2.35 0.00 乌兰察布 0.05 0.16 1.05 2.50 0.04 鄂尔多斯 −0.09 0.09 1.63 3.21 −0.10 巴彦淖尔 0.75 0.84 0.61 6.50 −1.10 乌海 0.53 −0.73 0.61 3.83 −1.13 阿拉善盟 0.24 −0.70 1.36 2.51 −1.41 承德 −0.12 0.31 0.28 −2.49 0.13 北京 −0.03 0.06 −0.76 −1.09 0.01 天津 −0.14 −0.20 −0.56 −1.69 −0.20 唐山 −0.20 −0.18 0.26 −2.38 −0.12 秦皇岛 −0.34 −0.29 1.46 −3.03 −0.02 廊坊 −0.07 −0.05 −0.75 −1.32 −0.08 保定 −0.05 −0.10 −1.38 −0.20 −0.03 沧州 −0.17 −0.36 −0.90 −1.72 −0.34 衡水 −0.12 −0.38 −1.52 −0.91 −0.27 邢台 −0.11 −0.43 −1.52 −0.35 −0.25 邯郸 −0.19 −0.56 −1.27 −0.89 −0.42 长治 −0.24 −0.58 −0.07 −0.71 −0.33 晋城 −0.40 −0.75 1.29 −0.50 −0.66 运城 −0.67 −0.77 4.20 2.09 −0.63 临汾 −0.32 −0.56 1.18 0.43 −0.12 吕梁 −0.09 −0.17 −0.29 1.86 0.22 晋中 0.01 −0.14 −1.17 1.57 0.18 太原 0.02 −0.09 −1.09 1.81 0.22 阳泉 0.01 −0.15 −1.47 1.19 0.10 忻州 0.02 0.02 −0.98 2.00 0.26 石家庄 −0.03 −0.19 −1.62 0.51 0.00 朔州 −0.04 0.19 −0.10 2.06 0.25 大同 0.05 0.23 −0.06 1.53 0.20
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