Dimensionality Reduction

次元削減

概要

次元削減手法は、重要な情報を保持しながら特徴量の数を減らし、モデルの効率を高めるとともに、高次元データの可視化を可能にします。

使用する場面

• 特徴量が多い高次元データセット
• 複雑なデータセットを2Dまたは3Dで可視化する場合
• 計算の複雑さと学習時間を削減する場合
• 冗長な特徴量や相関の高い特徴量を取り除く場合
• 機械学習モデルの過学習を防ぐ場合
• クラスタリングや分類の前にデータを前処理する場合

手法

• PCA：主成分分析
• t-SNE：t分布型確率的近傍埋め込み
• UMAP：一様多様体近似と射影
• 特徴選択：重要な特徴量を選択する
• 特徴抽出：新しい特徴量を作成する

メリット

• 計算の複雑さを軽減する
• ノイズと冗長性を取り除く
• モデルの汎化性能を向上させる
• 可視化を可能にする
• 次元の呪いを防ぐ

Pythonでの実装

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA, TruncatedSVD, FactorAnalysis
from sklearn.manifold import TSNE, MDS
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_iris
from sklearn.ensemble import RandomForestClassifier
from sklearn.feature_selection import SelectKBest, f_classif, mutual_info_classif
import seaborn as sns

# Load data
iris = load_iris()
X = iris.data
y = iris.target
feature_names = iris.feature_names

# Standardize
scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)

# PCA
pca = PCA()
pca.fit(X_scaled)

# Explained variance
explained_variance = np.cumsum(pca.explained_variance_ratio_)
print("Explained Variance Ratio by Component:")
print(pca.explained_variance_ratio_)
print(f"Cumulative Variance (first 2): {explained_variance[1]:.4f}")

# Scree plot
fig, axes = plt.subplots(1, 2, figsize=(14, 4))

axes[0].plot(range(1, len(pca.explained_variance_ratio_) + 1),
             pca.explained_variance_ratio_, 'bo-')
axes[0].set_xlabel('Principal Component')
axes[0].set_ylabel('Explained Variance Ratio')
axes[0].set_title('Scree Plot')
axes[0].grid(True, alpha=0.3)

axes[1].plot(range(1, len(explained_variance) + 1),
             explained_variance, 'go-')
axes[1].axhline(y=0.95, color='r', linestyle='--', label='95% Variance')
axes[1].set_xlabel('Number of Components')
axes[1].set_ylabel('Cumulative Explained Variance')
axes[1].set_title('Cumulative Explained Variance')
axes[1].legend()
axes[1].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

# PCA with 2 components
pca_2d = PCA(n_components=2)
X_pca_2d = pca_2d.fit_transform(X_scaled)

# PCA with 3 components
pca_3d = PCA(n_components=3)
X_pca_3d = pca_3d.fit_transform(X_scaled)

# PCA visualization
fig = plt.figure(figsize=(14, 5))

# 2D PCA
ax1 = fig.add_subplot(131)
scatter = ax1.scatter(X_pca_2d[:, 0], X_pca_2d[:, 1], c=y, cmap='viridis', alpha=0.6)
ax1.set_xlabel(f'PC1 ({pca_2d.explained_variance_ratio_[0]:.2%})')
ax1.set_ylabel(f'PC2 ({pca_2d.explained_variance_ratio_[1]:.2%})')
ax1.set_title('PCA 2D')
plt.colorbar(scatter, ax=ax1)

# 3D PCA
ax2 = fig.add_subplot(132, projection='3d')
scatter = ax2.scatter(X_pca_3d[:, 0], X_pca_3d[:, 1], X_pca_3d[:, 2],
                      c=y, cmap='viridis', alpha=0.6)
ax2.set_xlabel(f'PC1 ({pca_3d.explained_variance_ratio_[0]:.2%})')
ax2.set_ylabel(f'PC2 ({pca_3d.explained_variance_ratio_[1]:.2%})')
ax2.set_zlabel(f'PC3 ({pca_3d.explained_variance_ratio_[2]:.2%})')
ax2.set_title('PCA 3D')

# Loading plot
ax3 = fig.add_subplot(133)
loadings = pca_2d.components_.T
for i, feature in enumerate(feature_names):
    ax3.arrow(0, 0, loadings[i, 0], loadings[i, 1],
             head_width=0.05, head_length=0.05, fc='blue', ec='blue')
    ax3.text(loadings[i, 0]*1.15, loadings[i, 1]*1.15, feature, fontsize=10)
ax3.set_xlim(-1, 1)
ax3.set_ylim(-1, 1)
ax3.set_xlabel(f'PC1 ({pca_2d.explained_variance_ratio_[0]:.2%})')
ax3.set_ylabel(f'PC2 ({pca_2d.explained_variance_ratio_[1]:.2%})')
ax3.set_title('PCA Loadings')
ax3.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

# t-SNE visualization
tsne = TSNE(n_components=2, random_state=42, perplexity=30)
X_tsne = tsne.fit_transform(X_scaled)

plt.figure(figsize=(8, 6))
scatter = plt.scatter(X_tsne[:, 0], X_tsne[:, 1], c=y, cmap='viridis', alpha=0.6)
plt.xlabel('t-SNE Dimension 1')
plt.ylabel('t-SNE Dimension 2')
plt.title('t-SNE Visualization')
plt.colorbar(scatter, label='Class')
plt.show()

# MDS visualization
mds = MDS(n_components=2, random_state=42)
X_mds = mds.fit_transform(X_scaled)

plt.figure(figsize=(8, 6))
scatter = plt.scatter(X_mds[:, 0], X_mds[:, 1], c=y, cmap='viridis', alpha=0.6)
plt.xlabel('MDS Dimension 1')
plt.ylabel('MDS Dimension 2')
plt.title('MDS Visualization')
plt.colorbar(scatter, label='Class')
plt.show()

# Feature Selection - SelectKBest
selector = SelectKBest(score_func=f_classif, k=2)
X_selected = selector.fit_transform(X, y)
selected_features = np.array(feature_names)[selector.get_support()]
scores = selector.scores_

feature_scores = pd.DataFrame({
    'Feature': feature_names,
    'Score': scores
}).sort_values('Score', ascending=False)

print("\nFeature Selection (F-test):")
print(feature_scores)

plt.figure(figsize=(10, 5))
plt.barh(feature_scores['Feature'], feature_scores['Score'])
plt.xlabel('F-test Score')
plt.title('Feature Importance (SelectKBest)')
plt.tight_layout()
plt.show()

# Mutual Information
selector_mi = SelectKBest(score_func=mutual_info_classif, k=2)
X_selected_mi = selector_mi.fit_transform(X, y)
scores_mi = selector_mi.scores_

feature_scores_mi = pd.DataFrame({
    'Feature': feature_names,
    'Score': scores_mi
}).sort_values('Score', ascending=False)

print("\nFeature Selection (Mutual Information):")
print(feature_scores_mi)

# Tree-based feature importance
rf = RandomForestClassifier(n_estimators=100, random_state=42)
rf.fit(X, y)
importances = rf.feature_importances_

feature_importance = pd.DataFrame({
    'Feature': feature_names,
    'Importance': importances
}).sort_values('Importance', ascending=False)

print("\nFeature Importance (Random Forest):")
print(feature_importance)

plt.figure(figsize=(10, 5))
plt.barh(feature_importance['Feature'], feature_importance['Importance'])
plt.xlabel('Importance')
plt.title('Feature Importance (Random Forest)')
plt.tight_layout()
plt.show()

# Factor Analysis
fa = FactorAnalysis(n_components=2, random_state=42)
X_fa = fa.fit_transform(X_scaled)

plt.figure(figsize=(8, 6))
scatter = plt.scatter(X_fa[:, 0], X_fa[:, 1], c=y, cmap='viridis', alpha=0.6)
plt.xlabel('Factor 1')
plt.ylabel('Factor 2')
plt.title('Factor Analysis')
plt.colorbar(scatter, label='Class')
plt.show()

# Model performance comparison
from sklearn.model_selection import cross_val_score
from sklearn.linear_model import LogisticRegression

models = {
    'Original Features': X_scaled,
    'PCA (2)': X_pca_2d,
    'PCA (3)': X_pca_3d,
    't-SNE': X_tsne,
    'Selected (2 best)': X_selected,
}

scores = {}
for name, X_reduced in models.items():
    clf = LogisticRegression(max_iter=200)
    cv_scores = cross_val_score(clf, X_reduced, y, cv=5, scoring='accuracy')
    scores[name] = {
        'Mean Accuracy': cv_scores.mean(),
        'Std Dev': cv_scores.std(),
        'Features': X_reduced.shape[1],
    }

scores_df = pd.DataFrame(scores).T
print("\nModel Performance with Different Dimensionality:")
print(scores_df)

アルゴリズムの比較

• PCA：線形、高速、解釈しやすい
• t-SNE：非線形、可視化に適している、計算コストが高い
• UMAP：非線形、局所構造と大域構造を保持する
• 特徴選択：解釈可能性を維持する
• 因子分析：統計的なアプローチ

成分数の選び方

• 説明分散：分散の95%を保持する
• エルボー法：スクリープロットで「肘」を探す
• 交差検証：後続のタスクに合わせて最適化する

成果物

• スクリープロットと累積分散
• 2D／3Dの可視化
• PCAの負荷量の解釈
• 特徴量重要度のランキング
• モデル性能の比較
• 成分の解釈