2nd/10_Wiki/Topics/AI_and_ML/Linear-Regression-Mastery.md

---
id: wiki-2026-0508-linear-regression-mastery
title: Linear Regression Mastery
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Linear Regression, OLS, Ordinary Least Squares, Ridge, Lasso]
duplicate_of: none
source_trust_level: A
confidence_score: 0.9
verification_status: applied
tags: [machine-learning, regression, statistics, sklearn, ols, ridge, lasso]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: scikit-learn/statsmodels
---

# Linear Regression Mastery

## 매 한 줄
> **"매 모든 ML의 출발점 — y = Xβ + ε"**. Linear regression은 feature와 target의 선형 관계를 OLS로 추정하는 모델. 단순함 덕분에 해석성·속도·baseline으로 강하며, regularization (Ridge/Lasso/Elastic Net)으로 high-dim에서도 살아남는다. 2026 시대에도 production tabular ML의 절반은 여전히 linear.

## 매 핵심

### 매 OLS 수식
- 모델: $y = X\beta + \varepsilon$.
- 목적: $\min_\beta \|y - X\beta\|^2$.
- 닫힌해: $\hat\beta = (X^TX)^{-1}X^Ty$ (X full-rank일 때).
- 기하적: y를 column space of X에 projection.

### 매 가정 (LINE)
- **L**inearity: y와 X의 관계가 선형.
- **I**ndependence: 잔차 i.i.d.
- **N**ormality: 잔차 ~ N(0, σ²) (소표본일 때 inference에 필요).
- **E**qual variance (homoscedasticity): 잔차 분산 일정.
- **추가**: No multicollinearity (X feature 간 상관 낮음).

### 매 Regularized 변종
- **Ridge (L2)**: $\min \|y-X\beta\|^2 + \lambda\|\beta\|_2^2$ — 모든 계수 작게.
- **Lasso (L1)**: $\min \|y-X\beta\|^2 + \lambda\|\beta\|_1$ — sparsity (feature selection).
- **Elastic Net**: L1 + L2 — 상관된 feature 그룹 처리.

### 매 진단
- R² / Adjusted R²: 설명력.
- RMSE / MAE: 예측 오차.
- VIF > 10: multicollinearity 의심.
- Residual plot: 패턴 있으면 비선형.
- QQ plot: normality 체크.
- Cook's distance: 영향력 큰 outlier.

### 매 응용
1. Tabular baseline (어떤 ML이든 첫 모델).
2. Feature 영향 해석 (coefficient).
3. Time-series trend.
4. A/B test effect size.
5. Causal inference (DiD, IV)의 backbone.

## 💻 패턴

### sklearn — 기본 OLS
```python
from sklearn.linear_model import LinearRegression
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error, r2_score
import numpy as np

X_tr, X_te, y_tr, y_te = train_test_split(X, y, test_size=0.2, random_state=42)
model = LinearRegression().fit(X_tr, y_tr)
pred = model.predict(X_te)
print("R²:", r2_score(y_te, pred), "RMSE:", np.sqrt(mean_squared_error(y_te, pred)))
print("coef:", dict(zip(feature_names, model.coef_)))
```

### Ridge / Lasso / ElasticNet — CV로 alpha 선택
```python
from sklearn.linear_model import RidgeCV, LassoCV, ElasticNetCV
from sklearn.preprocessing import StandardScaler
from sklearn.pipeline import Pipeline

ridge = Pipeline([("sc", StandardScaler()),
                  ("m", RidgeCV(alphas=np.logspace(-3, 3, 50), cv=5))]).fit(X_tr, y_tr)
lasso = Pipeline([("sc", StandardScaler()),
                  ("m", LassoCV(cv=5, max_iter=10000))]).fit(X_tr, y_tr)
en    = Pipeline([("sc", StandardScaler()),
                  ("m", ElasticNetCV(l1_ratio=[.1,.5,.7,.9,.95,1], cv=5))]).fit(X_tr, y_tr)
print("ridge alpha:", ridge.named_steps["m"].alpha_)
print("lasso non-zero:", (lasso.named_steps["m"].coef_ != 0).sum())
```

### statsmodels — 통계적 추론 (p-value, CI)
```python
import statsmodels.api as sm
X_const = sm.add_constant(X_tr)
ols = sm.OLS(y_tr, X_const).fit()
print(ols.summary())   # coef, std-err, t, p, [95% CI]
print("Cond no:", ols.condition_number)  # >30 multicollinearity 의심
```

### 진단 — VIF + residual plot
```python
from statsmodels.stats.outliers_influence import variance_inflation_factor
import matplotlib.pyplot as plt

vif = [variance_inflation_factor(X_tr.values, i) for i in range(X_tr.shape[1])]
print(dict(zip(X_tr.columns, vif)))   # >10이면 제거 또는 PCA

resid = y_tr - model.predict(X_tr)
plt.scatter(model.predict(X_tr), resid, alpha=.4); plt.axhline(0)
plt.xlabel("fitted"); plt.ylabel("residual"); plt.show()
```

### Polynomial features (비선형 처리)
```python
from sklearn.preprocessing import PolynomialFeatures
poly = PolynomialFeatures(degree=2, interaction_only=False, include_bias=False)
Xp_tr = poly.fit_transform(X_tr)
Pipeline([("sc", StandardScaler()),
          ("m", RidgeCV())]).fit(Xp_tr, y_tr)   # 항상 Ridge — 차원 폭증
```

### Bayesian linear regression — PyMC
```python
import pymc as pm
with pm.Model() as m:
    β = pm.Normal("β", 0, 10, shape=X_tr.shape[1])
    α = pm.Normal("α", 0, 10)
    σ = pm.HalfNormal("σ", 5)
    y_obs = pm.Normal("y", α + X_tr.values @ β, σ, observed=y_tr)
    trace = pm.sample(1000, tune=1000, chains=4)
pm.summary(trace, hdi_prob=0.95)
```

### From scratch — gradient descent
```python
import numpy as np
def fit_gd(X, y, lr=1e-2, epochs=2000, l2=0.0):
    n, d = X.shape
    X_ = np.c_[np.ones(n), X]
    w = np.zeros(d + 1)
    for _ in range(epochs):
        grad = -2/n * X_.T @ (y - X_ @ w) + 2*l2 * np.r_[0, w[1:]]
        w -= lr * grad
    return w[0], w[1:]
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| Baseline 빠르게 | OLS |
| Multicollinearity / p>>n | Ridge |
| Feature selection 원함 | Lasso |
| 상관된 feature 그룹 | Elastic Net |
| 비선형 의심 | Polynomial + Ridge or move to tree |
| 통계적 추론 (p-value) | statsmodels |

**기본값**: StandardScaler + RidgeCV — 안전, 해석 가능, 빠름.

## 🔗 Graph
- 부모: [[Regression]]
- 변형: [[Ridge-Regression]], [[Elastic-Net]], [[Logistic-Regression-Foundations]]
- 응용: [[Time-Series-Analysis|Time-Series-Forecasting]], [[Causal-Inference]]
- Adjacent: [[L1-and-L2-Regularization]], [[Feature Engineering|Feature-Engineering]]

## 🤖 LLM 활용
**언제**: feature engineering ideation, residual plot 해석, statsmodels output 설명.
**언제 X**: 데이터 자체의 outlier 판단 — 도메인 지식 필요.

## ❌ 안티패턴
- **Scaling 안 함**: Ridge/Lasso는 scale에 민감.
- **VIF 무시**: coefficient 부호 뒤집힘.
- **R² 만 보고 판단**: 과적합 못 잡음 — adjusted R² 또는 CV 사용.
- **잔차 plot 안 봄**: 비선형성 놓침.
- **소표본에 polynomial deg 5+**: 폭주, overfit.

## 🧪 검증 / 중복
- Verified (ESL Hastie/Tibshirani, sklearn 1.5+, statsmodels 0.14).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — Ridge/Lasso/EN, 진단, Bayesian 추가 |