Linear Regression with multiple variables

<Linear Regression with multiple variables>

    1. Multiple features

    2. Feature Scaling

    3. Features and polynomial regression

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Multiple features

 

 

Multiple features (variables)

X₁ Size	X₂ Number of bedrooms	X₃ Number of floors	X₄ Age of home	y (target) Price
2104	5	1	45	460
1416	3	2	40	232
1534	3	2	30	315
852	2	1	36	178
...	...	...	...	...

• n = feature 개수
• X(i) = i번째 input feature의 training example
• Xj(i) = i번째 training example의 feature 값
• Hypothesis: Hθ(x) = θ0 + θ₁X₁ + θ₂X₂ + θ₃X₃ + θ₄X₄ + ...

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Feature Scaling

 

Feature Scaling

모든 feature들을 similar scale로 변환

ex) X₁ = SIZE (0 ~ 2000 feet)

     X₂ = 침실 개수 (1 ~ 5개)

→ X₁ = size(feet)  / 2000

    X₂ = 침실 개수 / 5

→ 0 ≤ X₁ ≤ 1

    0 ≤ X₂ ≤ 1

Feature Scaling

 

Nomalization

• Minimum, Maximum scaling
• 0 ≤ x ≤ 1 scaling: x = 3, -1(Minimum → 0), 7(Maximum → 1) → x´ = 1/2, 0, 1
• -1 ≤ x ≤ 1 scaling: x = 3, -1(Minimum → -1), 7(Maximum → 1) → x´ = 0, -1, 1
• Outlier 문제 발생 有

Standardization

• 평균, 분산 scaling
• 평균 = 2000, 분산 = 600^2 → 평균 = 0, 분산 = 1로 scaling
• x´ = (x - 2000) / 600^2

Standardization from:&nbsp;https://towardsdatascience.com/normalization-vs-standardization-quantitative-analysis-a91e8a79cebf

 

 

 

Gradient Descent New algorithm (n ≥ 1)

From: Andrew Ng

→ θ update

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polynomial regression

 

Polynomial regression (다항 회귀)

• 비선형 data
• Linear regression으로 표현할 수 없음

Polynomial regresion from: Adnrew Ng

Hθ(X) = θ0 + θ₁X₁ + θ₂X₂ + θ₃X₃

         = θ0 + θ₁(size) + θ₂(size)^2 + θ₃(size)^3 

• X₁ = (size)
• X₂ = (size)^2
• X₃ = (size)^3