当 fit_intercept=False 时，为什么 Sklearn R 平方与 statsmodels 不同？

我正在使用 Sklearn 和 statsmodels 执行线性回归。

我知道 Sklearn 和 statsmodels 会产生相同的结果。如下所示，Sklearn 和 statsmodels 得到了相同的结果，但即使截距为零时系数相同，结果也不同fit_intercept=False在 Sklearn 中。

你能解释一下原因吗？或者我使用时给我任何方法fit_intercept=False在 Sklearn 中。

import numpy as np
import statsmodels.api as sm
from sklearn.linear_model import LinearRegression

# dummy data:
y = np.array([1,3,4,5,2,3,4])
X = np.array(range(1,8)).reshape(-1,1) # reshape to column

# intercept is not zero : the result are the same
# scikit-learn:
lr = LinearRegression()
lr.fit(X,y)
print(lr.score(X,y))
# 0.16118421052631582

# statsmodels
X_ = sm.add_constant(X)
model = sm.OLS(y,X_)
results = model.fit()
print(results.rsquared)
# 0.16118421052631582

# intercept is zero : the result are different
# scikit-learn:
lr = LinearRegression(fit_intercept=False)
lr.fit(X,y)
print(lr.score(X,y))
# -0.4309210526315792

# statsmodels
model = sm.OLS(y,X)
results = model.fit()
print(results.rsquared)
# 0.8058035714285714

不一致的原因是statsmodels根据模型是否包含截距，使用不同的公式来计算 R 平方。如果包括截距，statsmodels将残差平方和除以中心总平方和，而如果不包括截距，statsmodels将残差平方和除以非中心总平方和。这意味着statsmodels使用以下公式计算 R 平方，可在文档 https://www.statsmodels.org/dev/generated/statsmodels.regression.linear_model.RegressionResults.rsquared.html:

import numpy as np

def rsquared(y_true, y_pred, fit_intercept=True):
    '''
    Statsmodels R-squared, see https://www.statsmodels.org/dev/generated/statsmodels.regression.linear_model.RegressionResults.rsquared.html.
    '''
    if fit_intercept:
        return 1 - np.sum((y_true - y_pred) ** 2) / np.sum((y_true - np.mean(y_true)) ** 2)
    else:
        return 1 - np.sum((y_true - y_pred) ** 2) / np.sum(y_true ** 2)

另一方面，sklearn始终使用分母处的中心平方和，无论截距是否实际包含在模型中（即无论是否fit_intercept=True or fit_intercept=False）。也可以看看这个答案 https://stackoverflow.com/a/54618898/11989081.

import numpy as np
import statsmodels.api as sm
from sklearn.linear_model import LinearRegression

def rsquared(y_true, y_pred, fit_intercept=True):
    '''
    Statsmodels R-squared, see https://www.statsmodels.org/dev/generated/statsmodels.regression.linear_model.RegressionResults.rsquared.html.
    '''
    if fit_intercept:
        return 1 - np.sum((y_true - y_pred) ** 2) / np.sum((y_true - np.mean(y_true)) ** 2)
    else:
        return 1 - np.sum((y_true - y_pred) ** 2) / np.sum(y_true ** 2)

# dummy data:
y = np.array([1, 3, 4, 5, 2, 3, 4])
X = np.array(range(1, 8)).reshape(-1, 1) # reshape to column

# intercept is not zero: the result are the same
# scikit-learn:
lr = LinearRegression(fit_intercept=True)
lr.fit(X, y)
print(lr.score(X, y))
# 0.16118421052631582
print(rsquared(y, lr.predict(X), fit_intercept=True))
# 0.16118421052631582

# statsmodels
X_ = sm.add_constant(X)
model = sm.OLS(y, X_)
results = model.fit()
print(results.rsquared)
# 0.16118421052631582
print(rsquared(y, results.fittedvalues, fit_intercept=True))
# 0.16118421052631593

# intercept is zero: the result are different
# scikit-learn:
lr = LinearRegression(fit_intercept=False)
lr.fit(X, y)
print(lr.score(X, y))
# -0.4309210526315792
print(rsquared(y, lr.predict(X), fit_intercept=True))
# -0.4309210526315792
print(rsquared(y, lr.predict(X), fit_intercept=False))
# 0.8058035714285714

# statsmodels
model = sm.OLS(y, X)
results = model.fit()
print(results.rsquared)
# 0.8058035714285714
print(rsquared(y, results.fittedvalues, fit_intercept=False))
# 0.8058035714285714