#P08 · Probability, Statistics & AI · Lv.3View in 3D Mind-Map

Pearson Correlation & Linear Regression (OLS)

Francis Galton, Karl Pearson & Carl Friedrich Gauss · 1895 CE

r_{XY} = \frac{\operatorname{Cov}(X,Y)}{\sigma_X \sigma_Y}, \quad \hat{\beta}_1 = \frac{\operatorname{Cov}(X,Y)}{\operatorname{Var}(X)} = r_{XY}\frac{\sigma_Y}{\sigma_X}

\hat{y} = \hat{\beta}_1 x + \hat{\beta}_0, \quad \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}

Quantifies linear association between two variables via Pearson correlation r ∈ [-1,1] and fits the optimal predictive line by minimizing squared residuals.

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Covariance Cov(X,Y)

Std Dev σ_X

Std Dev σ_Y

r = \frac{18}{5\times 4.5} = 0.800, \quad \hat{\beta}_1 = 0.720

피어슨 상관계수 r

0.8000

결정계수 R² (설명력)

64.0%

회귀 기울기 β̂₁

0.7200

Prerequisite Path (20 STEP)

Full Path Analyzer

Click steps in order to follow the 3D neon path from basics to target.

STEP 1#G03Pi, Radian Measure & Circular Sector

l = r\theta

STEP 2#G01Pythagorean Theorem

a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}

STEP 3#P01Permutations & Combinations

{}_nP_r = \frac{n!}{(n-r)!}

STEP 4#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 5#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 6#L01Vector Operations, L1/L2 Norms & Normalization

\|\mathbf{v}\|_2 = \sqrt{\sum_{i=1}^{n} v_i^2}

STEP 7#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 8#A06Logarithm Laws & Change of Base

\log_a(xy) = \log_a x + \log_a y

STEP 9#G04Unit Circle Trigonometry & Identities

\cos^2\theta + \sin^2\theta = 1

STEP 10#G022D/3D Euclidean Distance Formula

d(P

STEP 11#C01Limits, Continuity & Euler's Number e

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828

STEP 12#G05Law of Sines & Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos C

STEP 13#P02Conditional Probability & Chain Rule

P(A \mid B) = \frac{P(A \cap B)}{P(B)}

STEP 14#A08Binomial Theorem & Pascal's Triangle

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

STEP 15#C02Derivative Definition & Instantaneous Rate

f'(x) = \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

STEP 16#L02Dot Product & Cosine Similarity

\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta

STEP 17#P04Expected Value & Variance

E[X] = \sum_i x_i p_i

STEP 18#C03Product Rule & Chain Rule (Backpropagation Core)

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

STEP 19#C08Partial Derivatives & Gradient Vector (∇f)

\nabla f = \left(\frac{\partial f}{\partial x_1}

STEP 20#P08Pearson Correlation & Linear Regression (OLS)

r_{XY} = \frac{\operatorname{Cov}(X

Geometric & Intuitive Insight

중심화한 데이터 벡터의 내적(코사인 유사도)이 곧 피어슨 상관계수 r이며, 잔차 제곱합을 미분해 0으로 두면 회귀 기울기가 유도됩니다.

Step-by-Step Derivation

Step 1.손실함수 L(β₀, β₁) = ∑(y_i - (β₁x_i + β₀))²을 β₀, β₁에 대해 각각 편미분하여 0으로 둡니다.
Step 2.∂L/∂β₀ = 0에서 회귀직선이 반드시 무게중심 (x̄, ȳ)를 지남을 얻습니다: β₀ = ȳ - β₁x̄.
Step 3.∂L/∂β₁ = 0을 정리하면 β₁ = Cov(X,Y) / Var(X) = r (σ_Y / σ_X)가 도출됩니다.

AI & Coding Real-World Use

Foundation of supervised machine learning linear predictors, quant finance portfolio Beta (β), and feature correlation analysis.

#Linear Regression#Supervised ML#Quant Finance Beta#Feature Engineering

const slope = covXY / varX; const intercept = meanY - slope * meanX; const r = covXY / Math.sqrt(varX * varY);

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