投影数据并从主成分重建
您将把数据集变换到主成分空间，直观展示二维投影，并重建原始特征以量化信息损失。
投影数据并从主成分重建 是 CoddyKit 上的免费 Machine Learning Academy 课时。 这是第 2 节课，共 4 节。 你可以在下方免费阅读本课时的完整内容 — 然后在浏览器中使用内置代码编辑器和全天候 AI 导师进行实践。 这是 Machine Learning Academy 学习路径的一部分，你的进度在网页和 CoddyKit 应用中同步。 Machine Learning Academy 课程共包含 4 节课。
本课时的部分内容尚未翻译，以英文显示。
Projection: From High-D to Low-D
After PCA finds the principal components, projection transforms each data point into the new component space. The projected coordinates are called scores. If you keep only 2 components from 64 original features, each 64-dimensional point becomes a 2-dimensional score. This is achieved by multiplying the centred data matrix by the matrix of eigenvectors (the loadings matrix).
The transform Method in sklearn
In scikit-learn, pca.fit(X) learns the components and pca.transform(X) projects the data. The convenience method pca.fit_transform(X) does both in one call. The result is a matrix of shape (n_samples, n_components) — each row is a point in the reduced space.
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits
X, y = load_digits(return_X_y=True) # 1797 x 64
X_scaled = StandardScaler().fit_transform(X)
pca = PCA(n_components=10)
X_reduced = pca.fit_transform(X_scaled)
print('Original shape:', X_scaled.shape)
print('Reduced shape:', X_reduced.shape)
print('Variance retained:', pca.explained_variance_ratio_.sum().round(4))Visualising the 2D Projection
Projecting to 2 components gives a scatter plot where class separation is often visible even though labels were never used during PCA. This is an important exploratory tool: if classes are well-separated in 2D PCA space, a simple linear classifier may perform well in the full-dimensional space.
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits
X, y = load_digits(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)
pca2 = PCA(n_components=2)
X_2d = pca2.fit_transform(X_scaled)
plt.figure(figsize=(8, 6))
for digit in range(10):
mask = y == digit
plt.scatter(X_2d[mask, 0], X_2d[mask, 1], label=str(digit), s=10, alpha=0.6)
plt.legend(title='Digit', bbox_to_anchor=(1, 1))
plt.title('MNIST digits in 2D PCA space')
plt.tight_layout()
plt.show()Reconstruction: Going Back to Original Space
Reconstruction reverses the projection: multiply the reduced scores by the transpose of the loadings matrix and add back the mean. The result is an approximation of the original data in the original feature space. Perfect reconstruction is only possible if you kept all components; retaining fewer introduces reconstruction error.
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits
import numpy as np
X, _ = load_digits(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)
pca = PCA(n_components=20)
X_reduced = pca.fit_transform(X_scaled)
# Reconstruct back to 64 dimensions
X_reconstructed = pca.inverse_transform(X_reduced)
print('Reconstruction shape:', X_reconstructed.shape)
# Mean squared reconstruction error
mse = np.mean((X_scaled - X_reconstructed) ** 2)
print(f'MSE: {mse:.4f}')Visualising Reconstruction Quality
For image data, you can plot original and reconstructed images side by side. With more components retained, the reconstruction looks sharper. With very few components, digits become blurry blobs. This visual comparison is a powerful communication tool for showing stakeholders the trade-off between compression and information loss.
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits
X, _ = load_digits(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)
fig, axes = plt.subplots(3, 5, figsize=(12, 7))
component_counts = [1, 2, 5, 10, 30]
for col, nc in enumerate(component_counts):
pca = PCA(n_components=nc)
X_r = pca.inverse_transform(pca.fit_transform(X_scaled))
# Un-standardise for display (approximate)
axes[0, col].imshow(X[0].reshape(8, 8), cmap='gray')
axes[0, col].set_title(f'Original' if col == 0 else '')
axes[1, col].imshow(X_r[0].reshape(8, 8), cmap='gray')
axes[1, col].set_title(f'n={nc}')
plt.tight_layout()
plt.show()Reconstruction Error vs Number of Components
Plot reconstruction MSE against the number of components to see the information-loss curve. This is the quantitative version of the visual comparison. A sharp decrease in MSE as you add the first few components mirrors the scree plot, confirming that most information lives in a small subspace.
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits
import numpy as np
X, _ = load_digits(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)
components = [1, 2, 5, 10, 20, 30, 40, 50, 64]
mse_values = []
for nc in components:
pca = PCA(n_components=nc)
X_r = pca.inverse_transform(pca.fit_transform(X_scaled))
mse_values.append(np.mean((X_scaled - X_r) ** 2))
plt.plot(components, mse_values, marker='o')
plt.xlabel('Number of components')
plt.ylabel('Reconstruction MSE')
plt.title('Information Loss vs Compression')
plt.show()Interpretting Reconstruction Error
At zero components, reconstruction error equals the total variance of the data. At full components, error is zero. The ratio 1 - explained_variance_ratio.sum() tells you the fraction of variance discarded. For most practical ML pipelines, keeping 95–99% of variance (and discarding 1–5%) loses very little predictive signal while significantly reducing feature count and training time.
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits
X, _ = load_digits(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)
for nc in [5, 10, 20, 30, 40]:
pca = PCA(n_components=nc)
pca.fit(X_scaled)
retained = pca.explained_variance_ratio_.sum()
print(f'n_components={nc:2d} retained={retained:.3f} discarded={1-retained:.3f}')Using inverse_transform in Practice
pca.inverse_transform(X_reduced) is a method on the fitted PCA object. It returns the data in the original feature space but with the information from discarded components zeroed out. This is useful for anomaly detection: reconstruct training data and flag points with high reconstruction error as outliers that the PCA model could not represent well.
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
import numpy as np
# Simulated normal vs anomalous points
X_normal = np.random.randn(100, 10)
X_anomaly = np.random.randn(5, 10) * 10 # far from origin
X_all = np.vstack([X_normal, X_anomaly])
X_scaled = StandardScaler().fit_transform(X_all)
pca = PCA(n_components=5)
X_r = pca.inverse_transform(pca.fit_transform(X_scaled))
errors = np.mean((X_scaled - X_r) ** 2, axis=1)
print('Max error index:', np.argmax(errors), '(anomalies start at index 100)')Whitening: Decorrelated Components with Unit Variance
Setting PCA(whiten=True) scales the projected scores so each component has unit variance. This removes correlations between components and can improve the performance of algorithms like SVMs or neural networks that are sensitive to feature scale. Whitening is standard preprocessing before training on PCA-reduced features.
from sklearn.decomposition import PCA
from sklearn.datasets import load_iris
from sklearn.preprocessing import StandardScaler
import numpy as np
X, _ = load_iris(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)
pca_white = PCA(n_components=3, whiten=True)
X_w = pca_white.fit_transform(X_scaled)
print('Component variances (should be 1.0):', np.var(X_w, axis=0).round(4))PCA Limitations on Non-Linear Data
PCA finds only linear projections. If data lies on a curved surface — like a Swiss roll — PCA projects onto a flat plane, destroying the manifold structure. In such cases, consider Kernel PCA with an RBF kernel or non-linear alternatives like t-SNE or UMAP for exploration. For model preprocessing, however, linear PCA is usually sufficient and much faster.
Project and Reconstruct: Complete Workflow
A clean PCA pipeline always follows the same pattern: standardise, fit PCA on training data, transform train and test separately, optionally reconstruct to inspect quality.
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.model_selection import train_test_split
from sklearn.datasets import load_digits
import numpy as np
X, y = load_digits(return_X_y=True)
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=0)
scaler = StandardScaler()
X_train_s = scaler.fit_transform(X_train)
X_test_s = scaler.transform(X_test) # use train scaler
pca = PCA(n_components=0.95)
X_train_r = pca.fit_transform(X_train_s) # fit only on train
X_test_r = pca.transform(X_test_s) # transform test
print(f'Reduced: {X_train_r.shape[1]} components from 64 features')Quick Check
Test your understanding of PCA projection and reconstruction from this lesson.
Lesson Recap
In this lesson you learned: pca.transform projects data into component space with shape (n_samples, n_components), pca.inverse_transform reconstructs data in original feature space with information from discarded components lost, and reconstruction error quantifies information loss and can flag anomalies. Next up we explore t-SNE — a non-linear technique for 2D visualisation of high-dimensional data.
常见问题解答
「投影数据并从主成分重建」课时是免费的吗？
是的 — 「投影数据并从主成分重建」的完整文本可在网页上免费阅读。要进行交互式练习（内置代码编辑器和全天候 AI 导师）并解锁 Machine Learning Academy 课程的其余内容，请升级到 CoddyKit PRO。 Machine Learning Academy 课程共包含 4 节课。
「投影数据并从主成分重建」这节课中我会学到什么？
您将把数据集变换到主成分空间，直观展示二维投影，并重建原始特征以量化信息损失。 你通过在浏览器中直接运行的动手代码来练习 Machine Learning Academy，全天候 AI 导师会在你学习这节课的过程中回答你的问题。
学习 Machine Learning Academy 需要有经验吗？
无需任何先前经验。CoddyKit 上的 Machine Learning Academy 课程适合初学者到高级学习者，你可以从这里开始或从头开始，按照自己的节奏学习。 这是第 2 节课，共 4 节。
「投影数据并从主成分重建」课时需要多长时间？
大多数 CoddyKit 课程大约需要 5–10 分钟。每节课都很精短且互动，所以你能稳步进步，并在网页和应用中从离开的地方继续。
我能在这节 Machine Learning Academy 课中编写并运行代码吗？
能。每节 Machine Learning Academy 课都包含内置代码编辑器，你可以在浏览器中直接编写并运行真实代码，并获得即时 AI 反馈 — 无需本地设置。