GTWR介绍

本博客主要介绍GTWR模型及其扩展以及传统的GWR。

1.GTWR介绍

Huang在GWR模型的基础上提出GTWR模型，GWR模型中自变量的回归参数随空间地理位置的变化而变化，而GTWR模型中自变量的回归参数是随着时空位置的变化而变化。因此该模型相对GWR模型能较好地描述解释变量与因变量之间的时空关系，GTWR模型的基本表达式如下所示：

其中，(ui,vi)表示第i个样本点的经纬度坐标，ti表示观测时间，yi表示第i个样本点的因变量值，Xik表示第i个样本点的第k个解释变量。 为模型误差项,β0(ui,vi,ti)代表第i个样本点的回归常数，βk(ui,vi,ti)表示第i个样本点的第k个解释变量的回归系数。
估计回归系数的表达式如下所示：

上式中Wiuvt表示时空位置i的权重，GTWR模型通过构建时空权重矩阵来决定其他样本点的值对回归样本点的影响比重，因此时空权重矩阵在GTWR模型的计算过程中起到核心作用，其形式为一个对角矩阵，表示为W(ui, vi, ti) = diag (W1uvt, W2uvt, …, Wnuvt)。构建基于时空距离权重矩阵W(ui, vi, ti)的过程如下所示：首先，运用欧式距离公式 计算样本点间的空间距离，同理，运用 计算样本间的时间距离。由于时间距离和空间距离的测量单位不同，空间距离一般以米或千米为单位，时间距离通常以小时或天为单位。测量单位不同对结果造成影响，因此，时空距离应如下计算：

权重函数一般选择高斯函数或bi-square函数，代入后可转化权重函数,并计算得到权重矩阵如下所示：

其中：$\tau =\mu /\lambda$

2.代码实现

由于代码实现部分是直接从Jupyter Notebook上拷贝下来的，因此序号是重新开始的，懒得修改了，从下面开始当做一个完整的文档来看。

---

import pandas as pd
import geopandas as gpd
import matplotlib.pyplot as plt
import numpy as np
from mgtwr.gtwr import GTWR, MGTWR,GTWRResults

1.所需要的数据格式

np.random.seed(10)
u = np.array([(i-1)%12 for i in range(1,1729)]).reshape(-1,1)
v = np.array([((i-1)%144)//12 for i in range(1,1729)]).reshape(-1,1)
t = np.array([(i-1)//144 for i in range(1,1729)]).reshape(-1,1)
x1 = np.random.uniform(0,1,(1728,1))
x2 = np.random.uniform(0,1,(1728,1))
epsilon = np.random.randn(1728,1)
beta0 = 5
beta1 = 3 + (u + v + t)/6
beta2 = 3+((36-(6-u)**2)*(36-(6-v)**2)*(36-(6-t)**2))/128
y = beta0 + beta1 * x1 + beta2 * x2 + epsilon
coords = np.hstack([u,v])
X = np.hstack([x1,x2])

df = np.hstack([coords,t,X,y])
df = pd.DataFrame(df)
df.rename(columns={0:'lng',1:'lat',2:'t',3:'x1',4:'x2',5:'y'},inplace=True)
df.to_csv('Data_example.csv',encoding='gbk',index=False)

1.1 关于经纬度及距离度量的一些源码分析

# 打印出计算距离的向量
# 重新载入修改后的代码
import numpy as np
import pandas as pd
import sys
sys.path.append('./mgtwr-1.0.4/')
import new_mgtwr
from new_mgtwr import gtwr
import importlib
importlib.reload(gtwr)

np.random.seed(10)
u = np.array([(i-1)%12 for i in range(1,1729)]).reshape(-1,1)
v = np.array([((i-1)%144)//12 for i in range(1,1729)]).reshape(-1,1)
t = np.array([(i-1)//144 for i in range(1,1729)]).reshape(-1,1)
x1 = np.random.uniform(0,1,(1728,1))
x2 = np.random.uniform(0,1,(1728,1))
epsilon = np.random.randn(1728,1)
beta0 = 5
beta1 = 3 + (u + v + t)/6
beta2 = 3+((36-(6-u)**2)*(36-(6-v)**2)*(36-(6-t)**2))/128
y = beta0 + beta1 * x1 + beta2 * x2 + epsilon
coords = np.hstack([u,v])
X = np.hstack([x1,x2])
# 拟合GTWR
gtwr = gtwr.GTWR(coords=coords, t=t, y=y, X=X, bw=0.8, tau=0.7, kernel='gaussian', fixed=True,constant=True).fit()
print(gtwr.R2)

coords_new: 
 [[ 0.          0.          0.        ]
 [ 1.          0.          0.        ]
 [ 2.          0.          0.        ]
 ...
 [ 9.         11.          9.20326029]
 [10.         11.          9.20326029]
 [11.         11.          9.20326029]]
distance: 
 [ 0.          1.          2.         ... 16.93221781 17.48427865
 18.0748444 ]
coords_new: 
 [[ 0.          0.          0.        ]
 [ 1.          0.          0.        ]
 [ 2.          0.          0.        ]
 ...
 [ 9.         11.          9.20326029]
 [10.         11.          9.20326029]
 [11.         11.          9.20326029]]
distance: 
 [ 0.          1.          2.         ... 16.93221781 17.48427865
 18.0748444 ]
---------------------------------------------------------------------------

KeyboardInterrupt                         Traceback (most recent call last)

print(np.hstack([coords,t]))

[[ 0  0  0]
 [ 1  0  0]
 [ 2  0  0]
 ...
 [ 9 11 11]
 [10 11 11]
 [11 11 11]]

通过上述分析已经证明：gtwr包输入coords和t，计算时空权重矩阵时，计算方式与师姐论文中所写是一致的，即
，是直接用欧式距离计算的，因此输入经纬度是不正确的，需要先将经纬度修改成欧式距离的点输入才可以。

2.GTWR代码实现

这部分代码实现使用的是mgtwr-1.0.4这个包

2.1随机生成一些实战数据

包括坐标、自变量、因变量、时间序列

np.random.seed(10)
u = np.array([(i-1)%12 for i in range(1,1729)]).reshape(-1,1)
v = np.array([((i-1)%144)//12 for i in range(1,1729)]).reshape(-1,1)
t = np.array([(i-1)//144 for i in range(1,1729)]).reshape(-1,1)
x1 = np.random.uniform(0,1,(1728,1))
x2 = np.random.uniform(0,1,(1728,1))
epsilon = np.random.randn(1728,1)
beta0 = 5
beta1 = 3 + (u + v + t)/6
beta2 = 3+((36-(6-u)**2)*(36-(6-v)**2)*(36-(6-t)**2))/128
y = beta0 + beta1 * x1 + beta2 * x2 + epsilon
coords = np.hstack([u,v])
X = np.hstack([x1,x2])

print("coords:\n",coords,coords.shape)
print("t:\n",t,t.shape)
print("X:\n",X ,X.shape)
print("y:\n",y,y.shape)

coords:
 [[ 0  0]
 [ 1  0]
 [ 2  0]
 ...
 [ 9 11]
 [10 11]
 [11 11]] (1728, 2)
t:
 [[ 0]
 [ 0]
 [ 0]
 ...
 [11]
 [11]
 [11]] (1728, 1)
X:
 [[0.77132064 0.89509824]
 [0.02075195 0.63372887]
 [0.63364823 0.46276849]
 ...
 [0.22233723 0.05576403]
 [0.63305281 0.82517067]
 [0.05729903 0.87299008]] (1728, 2)
y:
 [[10.65655043]
 [ 5.69275413]
 [ 7.63470055]
 ...
 [ 8.84411885]
 [27.93264986]
 [16.93148075]] (1728, 1)

2.2通过黄金分割法则去搜索最佳的bw和tau两个参数

# 得到最优的bw和tau两个参数
from mgtwr.sel_bws import Sel_bws
sel = Sel_bws(coords, t, y, X, kernel = 'gaussian', fixed = True)
bw, tau = sel.search(bw_max = 40, tau_max = 5, verbose = True)
print(bw,tau)

bw, tau, score: 15.3 4.9 18744.792990631904
bw, tau, score: 9.4 4.9 18632.45981380435
bw, tau, score: 5.8 4.9 18316.139592165968
bw, tau, score: 3.6 4.9 17661.030394977093
bw, tau, score: 2.2 4.9 16352.432741417268
bw, tau, score: 1.4 3.8 14817.811620052911
bw, tau, score: 0.8 1.0 13749.050577638549
bw, tau, score: 0.8 4.9 13749.050577638549
bw, tau, score: 0.8 0.2 13749.050577638549
bw, tau, score: 0.8 2.3 13749.050577638549
bw, tau, score: 0.8 0.7 13749.050577638549
bw, tau, score: 0.8 0.7 13749.050577638549
0.8 0.7

2.3拟合GTWR模型

gtwr = GTWR(coords=coords, t=t, y=y, X=X, bw=0.8, tau=0.7, kernel='gaussian', fixed=True,constant=True).fit()
print(gtwr.R2)
print("coords:\n",gtwr.coords,gtwr.coords.shape)
print("t:\n",gtwr.t,gtwr.t.shape)
print("X:\n",gtwr.X ,gtwr.X.shape)
print("y:\n",gtwr.y,gtwr.y.shape)
print("betas:\n",gtwr.betas , gtwr.betas.shape)
print("tvalues:\n",gtwr.betas , gtwr.tvalues.shape)

0.9827505984138646
coords:
 [[ 0  0]
 [ 1  0]
 [ 2  0]
 ...
 [ 9 11]
 [10 11]
 [11 11]] (1728, 2)
t:
 [[ 0]
 [ 0]
 [ 0]
 ...
 [11]
 [11]
 [11]] (1728, 1)
X:
 [[1.         0.77132064 0.89509824]
 [1.         0.02075195 0.63372887]
 [1.         0.63364823 0.46276849]
 ...
 [1.         0.22233723 0.05576403]
 [1.         0.63305281 0.82517067]
 [1.         0.05729903 0.87299008]] (1728, 3)
y:
 [[10.65655043]
 [ 5.69275413]
 [ 7.63470055]
 ...
 [ 8.84411885]
 [27.93264986]
 [16.93148075]] (1728, 1)
betas:
 [[ 4.51176497  4.85871553  3.74277717]
 [ 1.69616618  4.60126386 10.29018614]
 [-1.61745711  5.81373377 20.04087848]
 ...
 [ 8.61145761 -9.29987265 55.80714273]
 [ 8.70878609  8.12470695 26.84180011]
 [12.3166378  12.69657576 11.61013208]] (1728, 3)
tvalues:
 [[ 4.51176497  4.85871553  3.74277717]
 [ 1.69616618  4.60126386 10.29018614]
 [-1.61745711  5.81373377 20.04087848]
 ...
 [ 8.61145761 -9.29987265 55.80714273]
 [ 8.70878609  8.12470695 26.84180011]
 [12.3166378  12.69657576 11.61013208]] (1728, 3)

# 这个cell中的代码没有什么帮助
result = GTWRResults(gtwr.coords, gtwr.t, gtwr.y, gtwr.X, gtwr.bw, gtwr.tau, gtwr.kernel, gtwr.fixed, gtwr.constant,
                 gtwr.influ, gtwr.resid, gtwr.predy, gtwr.betas, gtwr.CCT, gtwr.tr_STS)
result

3.GWR代码实现

这部分代码实现使用的是gtwr-1.0.1这个包

3.1随机生成一些实战数据

包括坐标、自变量、因变量

from gtwr.model import GWR
np.random.seed(1)
u = np.array([(i-1)%12 for i in range(1,1729)]).reshape(-1,1)
v = np.array([((i-1)%144)//12 for i in range(1,1729)]).reshape(-1,1)
t = np.array([(i-1)//144 for i in range(1,1729)]).reshape(-1,1)
x1 = np.random.uniform(0,1,(1728,1))
x2 = np.random.uniform(0,1,(1728,1))
epsilon = np.random.randn(1728,1)
beta0 = 5
beta1 = 3 + (u + v + t)/6
beta2 = 3+((36-(6-u)**2)*(36-(6-v)**2)*(36-(6-t)**2))/128
y = beta0 + beta1 * x1 + beta2 * x2 + epsilon
coords = np.hstack([u,v])
X = np.hstack([x1,x2])

print("coords:\n",coords,coords.shape)
print("X:\n",X ,X.shape)
print("y:\n",y,y.shape)

coords:
 [[ 0  0]
 [ 1  0]
 [ 2  0]
 ...
 [ 9 11]
 [10 11]
 [11 11]] (1728, 2)
X:
 [[4.17022005e-01 8.03208194e-01]
 [7.20324493e-01 2.53933856e-01]
 [1.14374817e-04 2.90806155e-01]
 ...
 [2.14464871e-01 6.35079933e-01]
 [6.19213298e-01 6.80880527e-01]
 [1.77972335e-01 8.45248647e-01]] (1728, 2)
y:
 [[ 9.2266638 ]
 [ 7.35466982]
 [ 6.34814023]
 ...
 [24.62344484]
 [26.05928471]
 [19.80122011]] (1728, 1)

3.2通过黄金分割法则去搜索最佳的bw参数

from gtwr.sel import search_GWRparameter
sel = search_GWRparameter(coords, y, X, kernel = 'gaussian', fixed = True)
bw = sel.search(bw_max = 40, verbose = True)
print(bw)

bw: 15.0 , score: 18694.91
bw: 10.0 , score: 18677.33
bw: 6.0 , score: 18605.29
bw: 4.0 , score: 18419.99
bw: 2.0 , score: 17795.93
bw: 2.0 , score: 17795.93
2.0

3.3拟合GWR模型

gwr = GWR(coords, y, X, 2, kernel='gaussian', fixed=True).fit()
print(gtwr.R2)

0.9827505984138646

4.mGTWR实现

这部分代码实现使用的是mgtwr-1.0.4这个包，代码思路类似于GTWR，具体不再展开了。

from mgtwr.sel_bws import Sel_bws
from mgtwr.gtwr import MGTWR
np.random.seed(10)
u = np.array([(i-1)%12 for i in range(1,1729)]).reshape(-1,1)
v = np.array([((i-1)%144)//12 for i in range(1,1729)]).reshape(-1,1)
t = np.array([(i-1)//144 for i in range(1,1729)]).reshape(-1,1)
x1 = np.random.uniform(0,1,(1728,1))
x2 = np.random.uniform(0,1,(1728,1))
epsilon = np.random.randn(1728,1)
beta0 = 5
beta1 = 3 + (u + v + t)/6
beta2 = 3+((36-(6-u)**2)*(36-(6-v)**2)*(36-(6-t)**2))/128
y = beta0 + beta1 * x1 + beta2 * x2 + epsilon
coords = np.hstack([u,v])
X = np.hstack([x1,x2])
selmulti = Sel_bws(coords, t, y, X, kernel = 'gaussian', fixed = True, multi = True)
selmulti.search(multi_bw_min=[0.1],verbose=True,tol_multi=1.0e-4)

Current iteration: 1 ,SOC: 0.0040964
Bandwidths: 0.7, 0.9, 0.5
taus: 0.3,0.1,0.1
Current iteration: 2 ,SOC: 0.0026656
Bandwidths: 0.9, 0.9, 0.5
taus: 2.8,0.1,0.1
Current iteration: 3 ,SOC: 0.0012941
Bandwidths: 0.9, 1.1, 0.5
taus: 3.0,0.1,0.1
Current iteration: 4 ,SOC: 0.0007042
Bandwidths: 0.9, 1.3, 0.5
taus: 3.0,0.0,0.1
Current iteration: 5 ,SOC: 0.0004
Bandwidths: 0.9, 1.7, 0.5
taus: 2.9,0.0,0.1
Current iteration: 6 ,SOC: 0.0002376
Bandwidths: 0.9, 2.3, 0.5
taus: 2.9,0.0,0.1
Current iteration: 7 ,SOC: 0.000144
Bandwidths: 0.9, 2.7, 0.5
taus: 2.9,0.0,0.1
Current iteration: 8 ,SOC: 9.56e-05
Bandwidths: 0.9, 3.3, 0.5
taus: 2.9,0.0,0.1





(array([0.9, 3.3, 0.5]), array([2.9, 0. , 0.1]))

mgtwr = MGTWR(coords, t, y, X, selmulti, kernel = 'gaussian', fixed = True).fit()
print(mgtwr.R2)

0.9811738693278879

mgtwr.betas

array([[ -3.87546159,   5.72441295,  13.15964215],
       [ -4.62776201,   5.84767207,  15.60722008],
       [ -6.12592788,   5.97354322,  24.98229769],
       ...,
       [-12.18185424,   6.30802687,  69.64285734],
       [-10.802428  ,   6.28437869,  48.0797626 ],
       [-13.53814487,   6.22765896,  37.8009708 ]])

3.一些说明

代码实现一节的“1.1 关于经纬度及距离度量的一些源码分析”部分是对gtwr包的源码做了修改，修改如下：修改的是源码包中的kernels这个py文件，增加了下面箭头所指的两行代码。

关于mGWR可以点击看这个链接.

参考文献

[1]周禹佳. 小汽车出行需求特征的时空影响因素建模与分析[D].浙江大学,2020.