python R 空间自回归模型SAR 参数估计 统计模拟 实验

一、编写一次估计函数

1. 载入numpy、固定随机种子

import numpy as np 
np.random.seed(1)

2. 编写makeY函数,生成用于模拟实验的Y

def makeY(rho, sigma2true, Ysize):
    I = np.identity(Ysize)
    W = I/rho
   
   
# 使用while语句,必须保证I - rho * W可逆
    while np.linalg.det(I - rho * W) == 0:
   
       
# 生成一列epsilon
        epsilon = np.random.normal(0, sigma2true, Ysize)

        # 生成邻接矩阵 
        A = I
        for i in range(0 , Ysize) :
           for j in range(0 , Ysize) :
               if i < j:
                   A[i, j] = np.random.binomial(1, rho)
                   A[j, i] = A[i, j]

        # 标准化带权邻接矩阵A,得到W
        W = A/sum(A)

 
   
# 求所生成网络的密度,衡量其稀疏性
    Ydensity = sum(sum(A)) / (Ysize * (Ysize - 1))
 
     
   
# 利用输入参数和随机数生成样本Y
    Y = np.matmul(np.linalg.inv(I - rho * W), epsilon)
    return [Y, W, Ydensity]

 

3. 编写主函数SAR_MLE_newton

def SAR_MLE_newton (rho, sigma2true, Ysize):
    for i in range(1,1000):
        Y = makeY(rho, sigma2true, Ysize)[0]
        W = makeY(rho, sigma2true, Ysize)[1]
        Ydensity = makeY(rho, sigma2true, Ysize)[2]
        I = np.identity(Ysize)
       
       
# 使用牛顿迭代法估计对Y建立SAR模型后的参数
        rhohatm = 0
        dist1 = 0.01 
        Wf = I - (rhohatm * W)
        np.linalg.det(Wf)
       
       
while abs(dist1)>1.0E-6:
            Wf = I - (rhohatm * W)
            rhohatm = rhohatm - dist1
           
           
if np.linalg.det(Wf) ==0:
                break
               
           
else:
                Ws = np.matmul(W, np.linalg.inv(Wf))
                sigma2 = np.matmul(np.matmul(np.matmul(Y.T, Wf.T), Wf ), Y) / Ysize
                a1rhohat = -np.trace(Ws) + np.matmul(np.matmul(np.matmul(Y.T, W.T), Wf), Y) / sigma2
                a2rhohat = -np.trace(np.matmul(Ws, Ws)) - np.matmul(np.matmul(np.matmul(Y.T, W.T), W), Y) /sigma2
                dist1 = a1rhohat / a2rhohat
               
       
if abs(dist1)<1.0E-6:
            break
       
   
# 估计SE(σ
    a2rho = np.trace(np.matmul(Ws, Ws)) + np.trace(np.matmul(Ws.T, Ws))
    a2sigma = Ysize / (2 * sigma2 * sigma2)
    a2sigrho = np.trace(Ws) / sigma2
    im = np.array([[a2sigma, a2sigrho], [a2sigrho, a2rho]]) 
    sdhatm = np.sqrt(np.linalg.inv(im)[1, 1])
    
   
# 返回ρ的真实值、ρ的估计值、σ平方的真实值、σ平方的估计值
    return [rho, rhohatm, sigma2true, sigma2, sdhatm, Ydensity]

 

二、选取不同的ρ、σ、Ysize,进行比较。

1. 生成待循环值

different_rho = np.arange(0.1, 1, 0.1)
different_sigma2 = np.arange(1, 20, 2)
different_Ysize = np.arange(10, 110, 10)

2. 初始化结果输出数组

result = np.zeros((different_rho.size, different_sigma2.size, different_sigma2.size, 6))

3. 多重循环,记录结果

for ii in range(0, different_rho.size):
    for jj in range(0, different_sigma2.size):
        for kk in range(0, different_Ysize.size):
            result[ii, jj, kk, : ] = SAR_MLE_newton(different_rho[ii], different_sigma2[jj], different_Ysize[kk])

4 .绘制描述标准误分布的盒图,寻找规律

result1 = result[0,:,:,5]
result2 = result[:,0,:,5]
result3 = result[:,:,0,5]

for ii in range(1, different_rho.size):
    result1 = result1 + result[ii,:,:,5]
   
for jj in range(0, different_sigma2.size):
    result2 = result2 + result[:,jj,:,5]
   
for kk in range(0, different_Ysize.size):
    result3 = result3 + result[:,:,kk,5]
   
import matplotlib.pyplot as plt
plt.boxplot(result1
/different_rho.size)
plt.boxplot(result1.T/different_rho.size)
plt.boxplot(result2/different_sigma2.size)
plt.boxplot(result2.T/different_sigma2.size)
plt.boxplot(result3/different_Ysize.size)
plt.boxplot(result3.T/different_Ysize.size)

python R 空间自回归模型SAR 参数估计 统计模拟 实验_第1张图片 python R 空间自回归模型SAR 参数估计 统计模拟 实验_第2张图片 python R 空间自回归模型SAR 参数估计 统计模拟 实验_第3张图片 python R 空间自回归模型SAR 参数估计 统计模拟 实验_第4张图片 python R 空间自回归模型SAR 参数估计 统计模拟 实验_第5张图片 python R 空间自回归模型SAR 参数估计 统计模拟 实验_第6张图片

随着ρ的增大,预测标准误的均值会增大;

随着样本网络大小Y.size的增大,预测标准误会的均值和方差都会减少;

随着σ平方的增大,预测标准误差没有显著性差异。

 

 

R代码(R的运算慢,写完后发现跑不起,所以换成了Python

simulate_SAR_MLE_newtown <- function(rho, sigma2true, Ysize){
 
# 第一步:生成Y
  I <- diag(1, Ysize)
 
# 生成一列epsilon
  epsilon <- rnorm(Ysize, 0, sigma2true)
 
# 生成邻接矩阵 
  A <- I
  for (i in 1 : Ysize){
    for (j in 1 : Ysize){
      if (i < j){
        A[i, j] <- rbinom(1, 1, rho)
        A[j, i] <- A[i, j]
      }
    }
  }
 

# 求所生成网络的密度,衡量其稀疏性
  Ydensity <- sum(A) / (Ysize * (Ysize - 1))
 
# 标准化带权邻接矩阵A,得到W
  W <- t(t(A)/colSums(A))
 
# 利用输入参数和随机数生成样本Y
  Y <- solve(I - rho * W) %*% epsilon
 
# 使用牛顿迭代法估计对Y建立SAR模型后的参数
  rhohatm <- 0
  dist1 <- 0.01
  I <- diag(1, Ysize)  
  while(abs(dist1)>1.0E-6){
    rhohatm <- as.numeric(rhohatm - dist1)
    Wf <- I - (rhohatm * W)
    Ws <- W %*% solve(Wf)
    sigma2 <- t(Y) %*% t(Wf) %*% Wf %*% Y / Ysize
    a1rhohat <- -sum(diag(Ws)) + t(Y) %*% t(W) %*% Wf %*% Y / sigma2
    a2rhohat <- -sum(diag(Ws ^ 2)) - sigma2 ^ (-1) %*% t(Y) %*% t(W) %*% W %*%
    dist1 <- a1rhohat / a2rhohat
  }
  a2rho <-
sum(diag(Ws ^ 2)) + sum(diag(t(Ws) %*% Ws))
  a2sigma <- Ysize / (2 * sigma2 ^ 2)
  a2sigrho <- sum(diag(Ws)) / sigma2
  im <- matrix(c(a2sigma, a2sigrho, a2sigrho, a2rho), 2, 2
  sdhatm <- sqrt(solve(im)[2, 2])
  return(list(rhotrue = rho, rhohatm = rhohatm, sigma2true = sigma2true, sigma2hat = sigma2,sdhatm = sdhatm))
}

 

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