---
id: wiki-2026-0508-principle-component-analysis
title: Principal Component Analysis
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [PCA, Karhunen-Loeve Transform, Principle Component Analysis]
duplicate_of: none
source_trust_level: A
confidence_score: 0.95
verification_status: applied
tags: [linear-algebra, dimensionality-reduction, unsupervised, statistics]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: Python
  framework: scikit-learn / NumPy / PyTorch
---

# Principal Component Analysis

## 매 한 줄
> **"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.

## 매 핵심

### 매 mathematical definition
- Center data: X_c = X - mean(X).
- Covariance: C = X_c^T X_c / (n-1).
- Eigendecompose C = V Λ V^T; columns of V are principal axes.
- Project: Z = X_c V_k (top k components).
- Equivalent: SVD X_c = U Σ V^T → V same; singular values σ_i = sqrt((n-1) λ_i).

### 매 properties
- **Orthogonal**: components uncorrelated.
- **Variance-ordered**: first PC explains most variance.
- **Linear**: cannot capture curved manifolds (use kernel PCA / UMAP).
- **Rotation-invariant**: same answer regardless of axis labels.
- **Scale-sensitive**: standardize features first if scales differ.

### 매 variants
- **Kernel PCA**: nonlinear via kernel trick (RBF, polynomial).
- **Sparse PCA**: L1-regularized loadings for interpretability.
- **Robust PCA**: low-rank + sparse decomposition for outliers.
- **Probabilistic PCA**: latent Gaussian model — gives MLE objective.
- **Incremental / online PCA**: streaming data.
- **Randomized SVD**: O(n d k) instead of O(n d^2) for top-k.

### 매 modern usage (2026)
- **Embeddings analysis**: PCA on Claude / GPT-5 hidden states for interpretability (mech interp).
- **Whitening**: precondition before clustering, ICA, neural net training.
- **Compression**: still used in image / signal pipelines.
- **Data viz**: PCA → 50D, then UMAP/t-SNE → 2D (the standard combo).

## 💻 패턴

### scikit-learn PCA
```python
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
import numpy as np

X_std = StandardScaler().fit_transform(X)
pca = PCA(n_components=0.95)  # keep 95% variance
Z = pca.fit_transform(X_std)
print(f"#components for 95% var: {pca.n_components_}")
print(f"explained variance ratio: {pca.explained_variance_ratio_}")
```

### Manual PCA via SVD (numerical best)
```python
def pca(X, k):
    Xc = X - X.mean(0)
    U, s, Vt = np.linalg.svd(Xc, full_matrices=False)
    components = Vt[:k]
    explained_var = (s[:k] ** 2) / (X.shape[0] - 1)
    Z = Xc @ components.T
    return Z, components, explained_var
```

### Randomized SVD (fast for huge matrices)
```python
from sklearn.utils.extmath import randomized_svd
U, s, Vt = randomized_svd(X_centered, n_components=50, random_state=42)
# 100x faster than full SVD for d >> k
```

### Kernel PCA (nonlinear)
```python
from sklearn.decomposition import KernelPCA
kpca = KernelPCA(n_components=2, kernel="rbf", gamma=0.1)
Z = kpca.fit_transform(X)
```

### Incremental PCA (streaming)
```python
from sklearn.decomposition import IncrementalPCA
ipca = IncrementalPCA(n_components=50, batch_size=1024)
for batch in stream:
    ipca.partial_fit(batch)
Z = ipca.transform(X_test)
```

### Whitening before downstream model
```python
pca = PCA(whiten=True).fit(X_train)
X_train_w = pca.transform(X_train)
X_test_w  = pca.transform(X_test)
# now features have unit variance, zero correlation
```

### PCA for interpreting transformer hidden states
```python
import torch
hidden = model.encode(prompts)  # (B, D=4096)
pca = PCA(n_components=8)
Z = pca.fit_transform(hidden.cpu().numpy())
# Top component often correlates with sentiment / topic / refusal.
```

### Reconstruction error (anomaly detection)
```python
pca = PCA(n_components=10).fit(X_train)
recon = pca.inverse_transform(pca.transform(X))
err = ((X - recon) ** 2).sum(axis=1)
anomalies = err > np.percentile(err, 99)
```

### Choosing k via scree plot / elbow
```python
import matplotlib.pyplot as plt
pca_full = PCA().fit(X_std)
plt.plot(np.cumsum(pca_full.explained_variance_ratio_))
plt.axhline(0.95, ls="--"); plt.xlabel("# components"); plt.ylabel("cumulative var")
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| Linear dim-reduction baseline | PCA |
| Visualization to 2D | PCA→50D → UMAP→2D |
| Nonlinear manifold | Kernel PCA / UMAP / autoencoder |
| Streaming / huge data | IncrementalPCA / randomized SVD |
| Need interpretable loadings | Sparse PCA |
| Outliers in data | Robust PCA |
| Probabilistic / missing data | Probabilistic PCA / EM-PCA |

**기본값**: StandardScaler → PCA(n_components=0.95) → downstream model.

## 🔗 Graph
- 부모: [[Linear-Algebra-Foundations|Linear-Algebra]] · [[Dimensionality-Reduction]]
- 응용: [[Feature Engineering|Feature-Engineering]] · [[Anomaly-Detection]] · [[Mechanistic-Interpretability]]
- Adjacent: [[SVD]] · [[ICA]] · [[Factor-Analysis]] · [[Autoencoder]] · [[UMAP]]

## 🤖 LLM 활용
**언제**: linear dim-reduction, whitening, denoising, hidden-state analysis, baseline before ML model.
**언제 X**: nonlinear manifold (use UMAP/autoencoder), categorical-only data (use MCA), interpretable original features required (use feature selection).

## ❌ 안티패턴
- **No standardization**: features with large scale dominate components.
- **PCA on labels-included data**: leakage if used for supervised pipeline.
- **Reading PC1 as "the cause"**: components are statistical, not causal.
- **PCA → tree models**: GBDT doesn't benefit from rotation; just hurts interpretability.
- **Forgetting sign ambiguity**: V and -V both valid; component direction is arbitrary.

## 🧪 검증 / 중복
- Verified (Pearson 1901, Hotelling 1933, Jolliffe 2002 textbook, sklearn docs).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — canonical PCA reference + 2026 mech interp use |
