去除影响图中的高残差和高杠杆点？

Question

我正在使用波士顿房价数据进行一些线性回归。

看，有多个高残值点和几个高杠杆点。

如何删除高残差和高杠杆点，以便重新运行线性回归模型并重新绘制影响图和 Q-Q 图？

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输入：

m = ols('PRICE ~ CRIM + RM + PTRATIO',bos).fit()
print(m.summary())

截断输出：

                 coef    std err          t      P>|t|      [0.025      0.975]
Intercept     -3.3066      4.038     -0.819      0.413     -11.240       4.627
CRIM          -0.2021      0.032     -6.301      0.000      -0.265      -0.139
RM             7.3816      0.402     18.360      0.000       6.592       8.171
PTRATIO       -1.0742      0.133     -8.081      0.000      -1.335      -0.813

影响图：

QQ剧情：

Answer 1

对于学生化残差，有经验法则将观察值标记为可能的异常值：

Studentized Residual value for any observation > |3|

您可以使用

statsmodels

库轻松找到这些观察结果。 下面的代码可以帮助您找到任何观察的学生化残差值 > |3|

的观察

from statsmodels.regression.linear_model import OLS
from statsmodels.stats.outliers_influence import OLSInfluence as olsi
import seaborn as sb
import matplotlib.pyplot as plt
%matplotlib inline

lrmodel = OLS(y_train, x_train)
results = lrmodel.fit()

studentized_residuals = olsi(results).resid_studentized
keep_observ_at_indx = [i for i in studentized_residuals if abs(i) > 3] # applying the above mentioned thumb rule
leverage_pts = olsi(results).hat_matrix_diag        # this will give the array of leverage values
sb.residplot(x = studentized_residuals, y = leverage_pts, color = 'brown')
plt.show()

a.）现在，我们终于有了 Studentized_residuals > 3 的索引，用这些索引获取观察结果。

b.) 据我从互联网上了解到的信息，我认为库克距离将帮助我们消除高杠杆点。但我不确定多大才算“太大”！所以不能对此过多评论。以下是计算“库克距离”的方法

cook_dist = dict(olsi(result).cooks_distance[0])

# {key(index) : value(cook's distance)}

Answer 2

from statsmodels.stats.outliers_influence import OLSInfluence

model = sm.ols('y ~ x', data=df) # df is the data with columns x, y
model = model.fit()

studentized_residuals = OLSInfluence(model).summary_frame().student_resid
leverage = OLSInfluence(model).summary_frame().hat_diag

studentized_residual_threshold = 3
p = 2             # p is the number of model parameters including the intercept
n = df.shape[0]   # n is the number of observations
leverage_threshold = 3 * (p/n)
outlier_index = list(set(list(studentized_residuals[abs(studentized_residuals) > studentized_residual_threshold].index) + list(leverage[leverage > leverage_threshold].index))) 

# remove the outliers
df.drop(index=outlier_index, inplace=True)