我想在使用PCA预处理输入数据的模型中解释回归模型权重。实际上，我有100个高度相关的输入维度，因此我知道PCA很有用。但是，为便于说明，我将使用虹膜数据集。
下面的sklearn代码说明了我的问题：
import numpy as np
import sklearn.datasets, sklearn.decomposition
from sklearn.linear_model import LinearRegression
# load data
X = sklearn.datasets.load_iris().data
w = np.array([0.3, 10, -0.1, -0.01])
Y = np.dot(X, w)
# set number of components to keep from PCA
n_components = 4
# reconstruct w
reg = LinearRegression().fit(X, Y)
w_hat = reg.coef_
print(w_hat)
# apply PCA
pca = sklearn.decomposition.PCA(n_components=n_components)
pca.fit(X)
X_trans = pca.transform(X)
# reconstruct w
reg_trans = LinearRegression().fit(X_trans, Y)
w_trans_hat = np.dot(reg_trans.coef_, pca.components_)
print(w_trans_hat)
运行此代码，您可以看到权重得到很好的再现。
但是，如果我将分量的数量设置为3（即
n_components = 3），则打印出的权重将与真实的分量大不相同。我是否误解了如何转换这些权重？还是由于PCA的信息丢失从4个变为3个？
最佳答案
我认为这很好，只是我在查看w_trans_hat而不是重构的Y：
import numpy as np
import sklearn.datasets, sklearn.decomposition
from sklearn.linear_model import LinearRegression
# load data
X = sklearn.datasets.load_iris().data
# create fake loadings
w = np.array([0.3, 10, -0.1, -0.01])
# centre X
X = np.subtract(X, np.mean(X, 0))
# calculate Y
Y = np.dot(X, w)
# set number of components to keep from PCA
n_components = 3
# reconstruct w using linear regression
reg = LinearRegression().fit(X, Y)
w_hat = reg.coef_
print(w_hat)
# apply PCA
pca = sklearn.decomposition.PCA(n_components=n_components)
pca.fit(X)
X_trans = pca.transform(X)
# regress Y on principal components
reg_trans = LinearRegression().fit(X_trans, Y)
# reconstruct Y using regressed weights and transformed X
Y_trans = np.dot(X_trans, reg_trans.coef_)
# show MSE to original Y
print(np.mean((Y - Y_trans) ** 2))
# show w implied by reduced model in original space
w_trans_hat = np.dot(reg_trans.coef_, pca.components_)
print(w_trans_hat)