データの射影と主成分からの再構成
データセットを主成分空間に変換し、2D射影を可視化するとともに、元の特徴量を再構成して情報損失を定量化します。
「データの射影と主成分からの再構成」はCoddyKit上の無料Machine Learning Academyレッスンです。 これはレッスン2/4です。 下記で完全なレッスンを無料で読むことができます。その後、ブラウザ内の組み込みコードエディタと24時間対応のAIチューターでハンズオン演習できます。 これはMachine Learning Academy学習パスの一部であり、ウェブとCoddyKitアプリ全体で進捗が同期されます。 Machine Learning Academyコースには全4レッスンが含まれています。
このレッスンの一部はまだ翻訳されておらず、英語で表示されています。
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.
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よくある質問
「データの射影と主成分からの再構成」レッスンは無料ですか？
はい。「データの射影と主成分からの再構成」の完全なテキストはこのウェブで無料で読めます。インタラクティブに演習し（組み込みコードエディタと24時間対応のAIチューター）、Machine Learning Academyコースの残りをアンロックするには、CoddyKit PROにアップグレードしてください。 Machine Learning Academyコースには全4レッスンが含まれています。
「データの射影と主成分からの再構成」で何を学びますか？
データセットを主成分空間に変換し、2D射影を可視化するとともに、元の特徴量を再構成して情報損失を定量化します。 ブラウザで直接実行するハンズオンコードでMachine Learning Academyを演習し、24時間対応のAIチューターがレッスンを進める中での質問に答えます。
Machine Learning Academyを始めるのに経験は必要ですか？
事前経験は必要ありません。CoddyKitのMachine Learning Academyは初級者から上級者向けに構成されているため、ここから始めるか最初から始めて、自分のペースで進むことができます。 これはレッスン2/4です。
「データの射影と主成分からの再構成」レッスンにはどのくらい時間がかかりますか？
ほとんどのCoddyKitレッスンは約5～10分かかります。各レッスンはコンパクトでインタラクティブなので、着実に進歩し、ウェブとアプリ全体で正確に前回の場所から再開できます。
このMachine Learning Academyレッスンでコードを書いて実行できますか？
はい。すべてのMachine Learning Academyレッスンに組み込みコードエディタが含まれているため、ブラウザでリアルコードを書いて実行し、即座のAIフィードバックを取得できます。ローカル設定は不要です。