• Principal Component Analysis (PCA):
• Statement (I): Principal Component Analysis is a supervised learning method. $\implies$ Incorrect.
• PCA is an unsupervised learning method because it operates strictly on feature vectors without utilizing any target labels or class response variables.
• Statement (II): Principal Component Analysis is a dimension reduction method. $\implies$ Correct.
• PCA reduces the dimensionality of large multivariate datasets by finding orthogonal linear combinations (principal components) that maximize variance along successive axes.
• Conclusion: Statement (I) is incorrect but Statement (II) is correct (Option 4).

• Solving Transportation Problems to Optimality:
• Initial Basic Feasible Solution (IBFS) Methods:
• North-West Corner Cell Method: Quick allocation from top-left, ignores shipping costs.
• Vogel's Approximation Method (VAM): Heuristic penalty method providing an initial feasible solution.
• Optimality Determination Methods:
• Modified Distribution (MODI) Method (or $u-v$ method): Calculates dual variables $u_i + v_j = c_{ij}$ for basic cells and computes net evaluation $\Delta_{ij} = c_{ij} - (u_i + v_j)$ for non-basic cells to iteratively attain the minimum cost optimal solution.
• Stepping Stone Method: Evaluates closed loops to test optimality.
• Conclusion: Minimum cost solution is obtained by Modified distribution method (Option 4).

Correct option: (4) Modified distribution method

• Invariance of Regression Coefficients to Change of Origin:
• Let $Y$ and $X$ be the dependent and independent variables with regression coefficient:

$$b_{YX} = \frac{\text{Cov}(X, Y)}{\text{Var}(X)}$$

• If a constant $C$ is subtracted from both variables:

$$u = X - C, \quad v = Y - C$$

• The variances and covariance under linear translation remain identical:

$$\text{Var}(u) = \text{Var}(X - C) = \text{Var}(X)$$ $$\text{Cov}(u, v) = \text{Cov}(X - C, Y - C) = \text{Cov}(X, Y)$$

• The new regression coefficient is:

$$b_{vu} = \frac{\text{Cov}(u, v)}{\text{Var}(u)} = \frac{\text{Cov}(X, Y)}{\text{Var}(X)} = b_{YX}$$

• Conclusion: The regression coefficient Remains unchanged (Option 3).

Correct option: (3) Remains unchanged