解释PCA之后的OLS重量(在Python中)

Question

我想在一个模型中解释回归模型权重，其中输入数据已通过 pca 进行了预处理。实际上，我有100秒的输入维度高度相关，因此我知道PCA很有用。但是，为了插图，我将使用 iris 数据集。

sklearn 代码以下说明了我的问题：

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，可以看到权重可以很好地重现。

但是，如果我将组件的数量设置为3（即 n_components = 3），然后加权与真实偏差大大偏离。

我是否误解了如何改变这些权重？还是由于PCA的信息损失从4个组件移动到3个组件？

Answer 1

我认为这很好，只是我正在查看w_trans_hat而不是重建的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)