多元线性回归--案例分析及python实践

article/2025/8/28 10:07:20

回归分析--多元回归

介绍一下多元回归分析中的统计量

总观测值 $\large n$
总自变量 $\large k$
自由度 $df$ ：回归自由度 $\large (k)$ ，残差自由度 $\large (n-k-1)$
SST总平方和 $\large SST=\sum \left (y_{i} -\bar{y}\right )^{2}$
SSR回归平方和 $\large SSR=\sum \left (\widehat{y}_{i} -\bar{y}\right )^{2}$
SSE残差平方和 $\large SSE=\sum \left (y_{i} -\widehat{y}_{i}\right )^{2}$
MSR均方回归 $\large MSR=\frac{SSR}{k}$
MSE均方残差 $\large MSE=\frac{SSE}{n-k-1}$
判定系数R_square $\large R^{2}=\frac{SSR}{SST}=\frac{\sum \left (\widehat{y}_{i} -\bar{y}\right )^{2}}{\sum \left (y_{i} -\bar{y}\right )^{2}} = 1-\frac{\sum \left (y_{i} -\widehat{y}_{i}\right )^{2}}{\sum \left (y_{i} -\bar{y}\right )^{2}}$
调整的判定系数Adjusted_R_square $\large R_{a}^{2} = 1-(1-R^{2})(\frac{n-1}{n-k-1})$
复相关系数Multiple_R $\large MultipleR = \sqrt{R^{2}}$
估计标准误差 $\large s_{e}=\sqrt{\frac{\sum \left (y_{i} -\widehat{y}_{i}\right )^{2}}{n-k-1}} = \sqrt{\frac{SSE}{n-k-1}}=\sqrt{MSE}$
F检验统计量 $\large F = \frac{SSR/k}{SSE/\left ( n-k-1 \right )} = \frac{MSR}{MSE} \sim F\left ( k, n-k-1 \right )$
回归系数抽样分布的标准误差 $\large s_{e}(\widehat{\beta _{k}}) = \sqrt{(s_{e})^{2}(X^{T}X)_{kk}^{-1}}$
各回归系数的t检验统计量 $\large t_{k} = \frac{\widehat{\beta }_{k}}{Se(\widehat{\beta _{k}})}$
各回归系数的置信区间 $\large \widehat{\beta }_{k} \pm t_{\alpha /2}(n-k-1) *s_{e}(\widehat{\beta _{k}})$
对数似然值（log likelihood）

$\large \sigma = \sqrt{\frac{\sum (y^{(i)} -\theta ^{T}x^{(i)})^{2}}{n}}$

$\large logLik = L(\theta ) = \prod_{i=1}^{m}\frac{1}{\sqrt{2\pi }\sigma }exp^{\left ( -\frac{(y^{(i)} -\theta ^{T}x^{(i)})^{2}}{2\sigma ^{2}} \right )}$

AIC准则 $\large AIC=2k - 2logLik$
BIC准则 $\large AIC= - 2logLik + k ln(n)$

案例分析及python实践

# 导入相关包
import pandas as pd
import numpy as np
import math
import scipy
import matplotlib.pyplot as plt
from scipy.stats import t

# 构建数据
columns = {'A':"分行编号", 'B':"不良贷款(亿元)", 'C':"贷款余额(亿元)", 'D':"累计应收贷款(亿元)", 'E':"贷款项目个数", 'F':"固定资产投资额(亿元)"}
data={"A":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],"B":[0.9,1.1,4.8,3.2,7.8,2.7,1.6,12.5,1.0,2.6,0.3,4.0,0.8,3.5,10.2,3.0,0.2,0.4,1.0,6.8,11.6,1.6,1.2,7.2,3.2],"C":[67.3,111.3,173.0,80.8,199.7,16.2,107.4,185.4,96.1,72.8,64.2,132.2,58.6,174.6,263.5,79.3,14.8,73.5,24.7,139.4,368.2,95.7,109.6,196.2,102.2],"D":[6.8,19.8,7.7,7.2,16.5,2.2,10.7,27.1,1.7,9.1,2.1,11.2,6.0,12.7,15.6,8.9,0,5.9,5.0,7.2,16.8,3.8,10.3,15.8,12.0],"E":[5,16,17,10,19,1,17,18,10,14,11,23,14,26,34,15,2,11,4,28,32,10,14,16,10],"F":[51.9,90.9,73.7,14.5,63.2,2.2,20.2,43.8,55.9,64.3,42.7,76.7,22.8,117.1,146.7,29.9,42.1,25.3,13.4,64.3,163.9,44.5,67.9,39.7,97.1]}df = pd.DataFrame(data)
X = df[["C", "D", "E", "F"]]
Y = df[["B"]]

# 构建多元线性回归模型
from sklearn.linear_model import LinearRegression
lreg = LinearRegression()
lreg.fit(X, Y)x = X
y_pred = lreg.predict(X)
y_true = np.array(Y).reshape(-1,1)coef = lreg.coef_[0]
intercept = lreg.intercept_[0]

# 自定义函数
def log_like(y_true, y_pred):"""y_true: 真实值y_pred：预测值"""sig = np.sqrt(sum((y_true - y_pred)**2)[0] / len(y_pred)) # 残差标准差δy_sig = np.exp(-(y_true - y_pred) ** 2 / (2 * sig ** 2)) / (math.sqrt(2 * math.pi) * sig)loglik = sum(np.log(y_sig))return loglikdef param_var(x):"""x:只含自变量宽表"""n = len(x)beta0 = np.ones((n,1))df_to_matrix = x.as_matrix()concat_matrix = np.hstack((beta0, df_to_matrix))         # 矩阵合并transpose_matrix = np.transpose(concat_matrix)           # 矩阵转置dot_matrix = np.dot(transpose_matrix, concat_matrix)     # (X.T X)^(-1)inv_matrix = np.linalg.inv(dot_matrix)                   # 求(X.T X)^(-1) 逆矩阵diag = np.diag(inv_matrix)                               # 获取矩阵对角线,即每个参数的方差return diagdef param_test_stat(x, Se, intercept, coef, alpha=0.05):n = len(x)k = len(x.columns)beta_array = param_var(x)beta_k = beta_array.shape[0]coef = [intercept] + list(coef)std_err = []t_Stat = []P_value = []t_intv = []coefLower = []coefupper = []for i in range(beta_k):se_belta = np.sqrt(Se**2 * beta_array[i])            # 回归系数的抽样标准误差t = coef[i] / se_belta                               # 用于检验回归系数的t统计量， 即检验统计量tp_value = scipy.stats.t.sf(np.abs(t), n-k-1)*2       # 用于检验回归系数的P值（P_value）t_score = scipy.stats.t.isf(alpha/2, df = n-k-1)     # t临界值coef_lower = coef[i] - t_score * se_belta            # 回归系数(斜率)的置信区间下限coef_upper = coef[i] + t_score * se_belta            # 回归系数(斜率)的置信区间上限std_err.append(round(se_belta, 3))t_Stat.append(round(t,3))P_value.append(round(p_value,3))t_intv.append(round(t_score,3))coefLower.append(round(coef_lower,3))coefupper.append(round(coef_upper,3))dict_ = {"coefficients":list(map(lambda x:round(x, 4), coef)), 'std_err':std_err, 't_Stat':t_Stat, 'P_value':P_value, 't临界值':t_intv, 'Lower_95%':coefLower, 'Upper_95%':coefupper}index = ["intercept"] + list(x.columns)stat = pd.DataFrame(dict_, index=index)return stat

# 自定义函数（计算输出各回归分析统计量）
def get_lr_stats(x, y_true, y_pred, coef, intercept, alpha=0.05):n   = len(x)k   = len(x.columns)ssr = sum((y_pred - np.mean(y_true))**2)[0]  # 回归平方和 SSRsse = sum((y_true - y_pred)**2)[0]           # 残差平方和 SSEsst = ssr + sse                              # 总平方和   SSTmsr = ssr / k                                # 均方回归   MSRmse = sse / (n-k-1)                          # 均方残差   MSER_square = ssr / sst                                   # 判定系数R^2Adjusted_R_square = 1-(1-R_square)*((n-1) / (n-k-1))   # 调整的判定系数Multiple_R = np.sqrt(R_square)                         # 复相关系数   Se = np.sqrt(sse/(n - k - 1))                          # 估计标准误差loglike = log_like(y_true, y_pred)[0]AIC = 2*(k+1) - 2 * loglike                  # (k+1) 代表k个回归参数或系数和1个截距参数BIC = -2*loglike + (k+1)*np.log(n)  # 线性关系的显著性检验F  = (ssr / k) / (sse / ( n - k - 1 ))            # 检验统计量F (线性关系的检验)pf = scipy.stats.f.sf(F, k, n-k-1)                # 用于检验的显著性F,即Significance FFa = scipy.stats.f.isf(alpha, dfn=k, dfd=n-k-1)   # F临界值# 回归系数的显著性检验stat = param_test_stat(x, Se, intercept, coef, alpha=alpha)# 输出各回归分析统计量print('='*80)print('df_Model:{}  df_Residuals:{}'.format(k, n-k-1), '\n')print('loglike:{}  AIC:{}  BIC:{}'.format(round(loglike,3), round(AIC,1), round(BIC,1)), '\n')print('SST:{}  SSR:{}  SSE:{}  MSR:{}  MSE:{}  Se:{}'.format(round(sst,4),round(ssr,4),round(sse,4),round(msr,4),round(mse,4),round(Se,4)), '\n')print('Multiple_R:{}  R_square:{}  Adjusted_R_square:{}'.format(round(Multiple_R,4),round(R_square,4),round(Adjusted_R_square,4)), '\n')print('F:{}  pf:{}  Fa:{}'.format(round(F,4), pf, round(Fa,4)))print('='*80)print(stat)print('='*80)return 0

输出结果如下：