Applied Spatial Statistics（九）GWR 示例
这是一个基本的示例笔记本，演示了如何使用开源“mgwr”包在 Python 中校准 GWR（Fotheringham 等人，2002）模型。mgwr 包由 Oshan 等人（2019 年）开发，托管在 https://github.com/pysal/mgwr。
Installation
pip install mgwr
我们将使用佐治亚州的一个示例数据集，以下是数据中的一些变量。
TotPop90 1990 年县的人口
PctRural 县内定义为农村的人口百分比
PctBach 县内拥有学士学位的人口百分比
PctEld 县内 65 岁或以上的人口百分比
PctFB 县内出生在美国以外的人口百分比
PctPov 县内生活在贫困线以下的人口百分比
PctBlack 县内黑人人口百分比
我们将使用几个社会人口统计变量对县级教育程度进行建模。
模型为：
PctBach = b0 + b1 * PctFB + b2 * PctBlack + b3 * PctRural
Load packages
import numpy as np
import matplotlib.pyplot as plt
import geopandas as gpd
import pandas as pd
import libpysal as ps
from libpysal.weights import Queen
from esda.moran import Moran#MGWR functions
from mgwr.gwr import GWR
from mgwr.sel_bw import Sel_BW
从 pysal 加载 Georgia 数据集
georgia_shp = gpd.read_file("https://raw.github.com/Ziqi-Li/gis5122/master/data/georgia.geojson")
佐治亚州地图和每个县的质心。
fig, ax = plt.subplots(figsize=(5,5))
georgia_shp.boundary.plot(ax=ax,aspect=1)georgia_shp.centroid.plot(ax=ax, c='black',markersize = 10,aspect=1)
/var/folders/mp/9px298sd6vs8xccb_3sql0dr0000gp/T/ipykernel_55115/3938808638.py:4: UserWarning: Geometry is in a geographic CRS. Results from 'centroid' are likely incorrect. Use 'GeoSeries.to_crs()' to re-project geometries to a projected CRS before this operation.georgia_shp.centroid.plot(ax=ax, c='black',markersize = 10,aspect=1)<Axes: >
我们可以映射因变量“PctBach”。
georgia_shp.plot(column="PctBach",legend=True,aspect=1)
plt.title("PctBach")
Text(0.5, 1.0, 'PctBach')
模型规范：
PctBach = b0 + b1 * PctFB + b2 * PctBlack + b3 * PctRural
准备要输入到 mgwr 的 Georgia 数据集
y = georgia_shp['PctBach'].valuesX = georgia_shp[['PctFB', 'PctBlack', 'PctRural']].valuesu = georgia_shp['X']
v = georgia_shp['Y']coords = list(zip(u,v))
标准化变量（可选）
- 将数据标准化，使平均值为 0，方差为 1。
- 参数估计的幅度是可比的。系数越大 -> 关系越强。
#Standardize variables
X = (X - X.mean(axis=0)) / X.std(axis=0)
y = (y - y.mean(axis=0)) / y.std(axis=0)y = y.reshape(-1,1)
两步拟合 GWR 模型
- 选择最佳带宽
- 使用最佳带宽拟合 GWR 模型
默认内核是自适应（最近邻居数）双平方。
gwr_selector = Sel_BW(coords, y, X)gwr_bw = gwr_selector.search(verbose=True)print("Selected optimal bandwidth is:", gwr_bw)
Bandwidth: 90.0 , score: 301.83
Bandwidth: 117.0 , score: 299.05
Bandwidth: 133.0 , score: 300.58
Bandwidth: 106.0 , score: 299.75
Bandwidth: 123.0 , score: 299.85
Bandwidth: 112.0 , score: 299.29
Bandwidth: 119.0 , score: 299.43
Bandwidth: 115.0 , score: 299.08
Selected optimal bandwidth is: 117.0
使用最佳 bw 拟合模型
gwr_results = GWR(coords, y, X, bw=gwr_bw).fit()
GWR 输出摘要
gwr_results.summary()
===========================================================================
Model type Gaussian
Number of observations: 159
Number of covariates: 4Global Regression Results
---------------------------------------------------------------------------
Residual sum of squares: 71.793
Log-likelihood: -162.399
AIC: 332.798
AICc: 335.191
BIC: -713.887
R2: 0.548
Adj. R2: 0.540Variable Est. SE t(Est/SE) p-value
------------------------------- ---------- ---------- ---------- ----------
X0 -0.000 0.054 -0.000 1.000
X1 0.458 0.066 6.988 0.000
X2 -0.084 0.055 -1.525 0.127
X3 -0.374 0.065 -5.734 0.000Geographically Weighted Regression (GWR) Results
---------------------------------------------------------------------------
Spatial kernel: Adaptive bisquare
Bandwidth used: 117.000Diagnostic information
---------------------------------------------------------------------------
Residual sum of squares: 51.186
Effective number of parameters (trace(S)): 11.805
Degree of freedom (n - trace(S)): 147.195
Sigma estimate: 0.590
Log-likelihood: -135.503
AIC: 296.616
AICc: 299.051
BIC: 335.913
R2: 0.678
Adjusted R2: 0.652
Adj. alpha (95%): 0.017
Adj. critical t value (95%): 2.414Summary Statistics For GWR Parameter Estimates
---------------------------------------------------------------------------
Variable Mean STD Min Median Max
-------------------- ---------- ---------- ---------- ---------- ----------
X0 -0.004 0.180 -0.296 0.111 0.208
X1 0.477 0.234 0.123 0.556 0.741
X2 -0.043 0.083 -0.170 -0.053 0.100
X3 -0.328 0.060 -0.464 -0.308 -0.241
===========================================================================
局部估计值可以从“gwr_results.params”中获得，它返回一个 n x p 数组，其中 p 是模型中的预测变量的数量（包括截距）。
from matplotlib import colorsax = georgia_shp.plot(column=gwr_results.params[:,3],figsize=(10,5),legend=True, linewidth=0.0,aspect=1)plt.title("Coefficients of Pct PctRural",fontsize=12)
Text(0.5, 1.0, 'Coefficients of Pct PctRural')
编写一些绘图代码，将参数估计表面全部可视化。我们需要将 GWR 结果与县 GeoDataFrame 连接起来。
from mpl_toolkits.axes_grid1 import make_axes_locatable
from mgwr.utils import shift_colormap,truncate_colormap
from matplotlib import cm,colorsdef param_plots(result, gdf, names=[], filter_t=False):#Size of the plot. Here we have a 2 by 2 layout.fig, axs = plt.subplots(2, 2, figsize=(10, 10))axs = axs.ravel()k = gwr_results.k#The max and min values of the color bar.vmin = -0.8vmax = 0.8cmap = cm.get_cmap("coolwarm")norm = colors.BoundaryNorm(np.arange(-0.8,0.9,0.1),ncolors=256)for j in range(k):pd.concat([gdf,pd.DataFrame(np.hstack([result.params,result.bse]))],axis=1).plot(ax=axs[j],column=j,vmin=vmin,vmax=vmax,cmap="bwr",norm=norm,linewidth=0.1,edgecolor='white',aspect=1)axs[j].set_title("Parameter estimates of \n" + names[j],fontsize=10)if filter_t:rslt_filtered_t = result.filter_tvals()if (rslt_filtered_t[:,j] == 0).any():gdf[rslt_filtered_t[:,j] == 0].plot(color='grey', ax=axs[j],linewidth=0.1,edgecolor='white',aspect=1)#plt.axis('off')fig = axs[j].get_figure()cax = fig.add_axes([0.99, 0.2, 0.025, 0.6])sm = plt.cm.ScalarMappable(cmap=cmap,norm=norm)# fake up the array of the scalar mappable. Urgh...sm._A = []fig.colorbar(sm, cax=cax)
以下是从 GWR 获得的参数估计图。每个图代表每个预测因子与 PctBach 之间的空间关系。
- 正（负）关系以红色（蓝色）显示。
- 较强（较弱）的关系颜色较深（较浅）。
param_plots(gwr_results, georgia_shp, names=['intercept','PctFB', 'PctBlack', 'PctRural'])
/var/folders/mp/9px298sd6vs8xccb_3sql0dr0000gp/T/ipykernel_55115/1271372694.py:17: MatplotlibDeprecationWarning: The get_cmap function was deprecated in Matplotlib 3.7 and will be removed two minor releases later. Use ``matplotlib.colormaps[name]`` or ``matplotlib.colormaps.get_cmap(obj)`` instead.cmap = cm.get_cmap("coolwarm")
现在让我们检查一下 GWR 模型的残差。
#Here we use the Queen contiguity
w = Queen.from_dataframe(georgia_shp)#row standardization
w.transform = 'R'
/var/folders/mp/9px298sd6vs8xccb_3sql0dr0000gp/T/ipykernel_55115/876426650.py:2: FutureWarning: `use_index` defaults to False but will default to True in future. Set True/False directly to control this behavior and silence this warningw = Queen.from_dataframe(georgia_shp)
georgia_moran = Moran(gwr_results.resid_response.reshape(-1), w, permutations = 199) #199 permutations
georgia_moran.I
0.040101374209262214
我们经常看到 GWR 模型的残差具有随机模式，这表明空间异质性和自相关性都被考虑在内。局部截距在解释空间自相关性方面发挥着作用，其作用类似于空间误差模型中的自回归误差分量。
GWR 推断
局部系数显著性
以下是**显著性（p<0.05）**参数估计值的图。显著性检验已调整以解决多重检验问题。
不显著的参数用灰色遮盖。例如，您可以发现，尽管我们在回归系数中看到了这种模式，但 PctBlack 和 PctBach 之间的关联实际上并不显著。
param_plots(gwr_results, georgia_shp,names=['intercept','PctFB', 'PctBlack', 'PctRural'],filter_t=True)
/var/folders/mp/9px298sd6vs8xccb_3sql0dr0000gp/T/ipykernel_55115/1271372694.py:17: MatplotlibDeprecationWarning: The get_cmap function was deprecated in Matplotlib 3.7 and will be removed two minor releases later. Use ``matplotlib.colormaps[name]`` or ``matplotlib.colormaps.get_cmap(obj)`` instead.cmap = cm.get_cmap("coolwarm")
