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Antigravity Agent
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id: wiki-2026-0508-principle-component-analysis
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title: Principal Component Analysis
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category: 10_Wiki/Topics
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status: verified
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canonical_id: self
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aliases: [PCA, Karhunen-Loeve Transform, Principle Component Analysis]
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duplicate_of: none
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source_trust_level: A
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confidence_score: 0.95
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verification_status: applied
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tags: [linear-algebra, dimensionality-reduction, unsupervised, statistics]
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raw_sources: []
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last_reinforced: 2026-05-10
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github_commit: pending
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tech_stack:
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language: Python
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framework: scikit-learn / NumPy / PyTorch
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---
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# Principal Component Analysis
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## 매 한 줄
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> **"매 orthogonal axes of maximum variance — eigendecomposition of covariance, equivalent to SVD of centered data"**. Pearson 1901, Hotelling 1933 의 statistical foundation; 2026 still the default linear dim-reduction baseline despite t-SNE/UMAP for viz. Note: spelled **Principal** (not "Principle") — kept alias for findability.
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## 매 핵심
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### 매 mathematical definition
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- Center data: X_c = X - mean(X).
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- Covariance: C = X_c^T X_c / (n-1).
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- Eigendecompose C = V Λ V^T; columns of V are principal axes.
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- Project: Z = X_c V_k (top k components).
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- Equivalent: SVD X_c = U Σ V^T → V same; singular values σ_i = sqrt((n-1) λ_i).
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### 매 properties
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- **Orthogonal**: components uncorrelated.
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- **Variance-ordered**: first PC explains most variance.
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- **Linear**: cannot capture curved manifolds (use kernel PCA / UMAP).
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- **Rotation-invariant**: same answer regardless of axis labels.
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- **Scale-sensitive**: standardize features first if scales differ.
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### 매 variants
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- **Kernel PCA**: nonlinear via kernel trick (RBF, polynomial).
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- **Sparse PCA**: L1-regularized loadings for interpretability.
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- **Robust PCA**: low-rank + sparse decomposition for outliers.
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- **Probabilistic PCA**: latent Gaussian model — gives MLE objective.
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- **Incremental / online PCA**: streaming data.
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- **Randomized SVD**: O(n d k) instead of O(n d^2) for top-k.
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### 매 modern usage (2026)
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- **Embeddings analysis**: PCA on Claude / GPT-5 hidden states for interpretability (mech interp).
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- **Whitening**: precondition before clustering, ICA, neural net training.
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- **Compression**: still used in image / signal pipelines.
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- **Data viz**: PCA → 50D, then UMAP/t-SNE → 2D (the standard combo).
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## 💻 패턴
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### scikit-learn PCA
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```python
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from sklearn.decomposition import PCA
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from sklearn.preprocessing import StandardScaler
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import numpy as np
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X_std = StandardScaler().fit_transform(X)
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pca = PCA(n_components=0.95) # keep 95% variance
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Z = pca.fit_transform(X_std)
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print(f"#components for 95% var: {pca.n_components_}")
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print(f"explained variance ratio: {pca.explained_variance_ratio_}")
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```
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### Manual PCA via SVD (numerical best)
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```python
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def pca(X, k):
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Xc = X - X.mean(0)
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U, s, Vt = np.linalg.svd(Xc, full_matrices=False)
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components = Vt[:k]
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explained_var = (s[:k] ** 2) / (X.shape[0] - 1)
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Z = Xc @ components.T
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return Z, components, explained_var
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```
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### Randomized SVD (fast for huge matrices)
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```python
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from sklearn.utils.extmath import randomized_svd
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U, s, Vt = randomized_svd(X_centered, n_components=50, random_state=42)
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# 100x faster than full SVD for d >> k
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```
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### Kernel PCA (nonlinear)
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```python
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from sklearn.decomposition import KernelPCA
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kpca = KernelPCA(n_components=2, kernel="rbf", gamma=0.1)
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Z = kpca.fit_transform(X)
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```
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### Incremental PCA (streaming)
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```python
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from sklearn.decomposition import IncrementalPCA
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ipca = IncrementalPCA(n_components=50, batch_size=1024)
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for batch in stream:
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ipca.partial_fit(batch)
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Z = ipca.transform(X_test)
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```
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### Whitening before downstream model
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```python
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pca = PCA(whiten=True).fit(X_train)
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X_train_w = pca.transform(X_train)
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X_test_w = pca.transform(X_test)
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# now features have unit variance, zero correlation
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```
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### PCA for interpreting transformer hidden states
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```python
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import torch
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hidden = model.encode(prompts) # (B, D=4096)
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pca = PCA(n_components=8)
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Z = pca.fit_transform(hidden.cpu().numpy())
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# Top component often correlates with sentiment / topic / refusal.
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```
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### Reconstruction error (anomaly detection)
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```python
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pca = PCA(n_components=10).fit(X_train)
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recon = pca.inverse_transform(pca.transform(X))
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err = ((X - recon) ** 2).sum(axis=1)
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anomalies = err > np.percentile(err, 99)
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```
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### Choosing k via scree plot / elbow
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```python
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import matplotlib.pyplot as plt
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pca_full = PCA().fit(X_std)
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plt.plot(np.cumsum(pca_full.explained_variance_ratio_))
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plt.axhline(0.95, ls="--"); plt.xlabel("# components"); plt.ylabel("cumulative var")
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```
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## 매 결정 기준
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| 상황 | Approach |
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|---|---|
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| Linear dim-reduction baseline | PCA |
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| Visualization to 2D | PCA→50D → UMAP→2D |
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| Nonlinear manifold | Kernel PCA / UMAP / autoencoder |
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| Streaming / huge data | IncrementalPCA / randomized SVD |
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| Need interpretable loadings | Sparse PCA |
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| Outliers in data | Robust PCA |
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| Probabilistic / missing data | Probabilistic PCA / EM-PCA |
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**기본값**: StandardScaler → PCA(n_components=0.95) → downstream model.
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## 🔗 Graph
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- 부모: [[Linear-Algebra-Foundations|Linear-Algebra]] · [[Dimensionality-Reduction]]
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- 응용: [[Feature Engineering|Feature-Engineering]] · [[Anomaly-Detection]] · [[Mechanistic-Interpretability]]
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- Adjacent: [[SVD]] · [[ICA]] · [[Factor-Analysis]] · [[Autoencoder]] · [[UMAP]]
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## 🤖 LLM 활용
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**언제**: linear dim-reduction, whitening, denoising, hidden-state analysis, baseline before ML model.
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**언제 X**: nonlinear manifold (use UMAP/autoencoder), categorical-only data (use MCA), interpretable original features required (use feature selection).
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## ❌ 안티패턴
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- **No standardization**: features with large scale dominate components.
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- **PCA on labels-included data**: leakage if used for supervised pipeline.
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- **Reading PC1 as "the cause"**: components are statistical, not causal.
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- **PCA → tree models**: GBDT doesn't benefit from rotation; just hurts interpretability.
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- **Forgetting sign ambiguity**: V and -V both valid; component direction is arbitrary.
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## 🧪 검증 / 중복
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- Verified (Pearson 1901, Hotelling 1933, Jolliffe 2002 textbook, sklearn docs).
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- 신뢰도 A.
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## 🕓 Changelog
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| 날짜 | 변경 |
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|---|---|
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| 2026-05-08 | Phase 1 |
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| 2026-05-10 | Manual cleanup — canonical PCA reference + 2026 mech interp use |
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