Which of the following statements about Principal Component Analysis (PCA) are correct? (Select all that apply)

Question

BlackTom AI · Accepted Answer

PCA finds orthogonal directions in the data that capture maximal variance, so principal components are uncorrelated and ordered by how much variance they explain.

Relevant concepts:
- Centering and covariance: PCA begins by centering data (subtracting the mean) and computing the covariance matrix Σ.
- Eigen-decomposition: PCA computes eigenvectors v_i and eigenvalues λ_i of Σ. Eigenvectors form an orthonormal basis; eigenvalues quantify variance explained.
- Projection and dimensionality reduction: Projecting data onto the top k eigenvectors (principal components) retains the largest portion of total variance and thus minimizes reconstruction error in the least-squares sense for linear reconstructions.
- Scaling sensitivity: The covariance matrix depends on the units/scale of variables, so variables with larger scale dominate PCA unless data are standardized.

Step-by-step reasoning:
1. Center data X so each column has mean zero.
2. Compute covariance Σ = (1/(n−1)) X^T X.
3. Solve Σ v_i = λ_i v_i. Order eigenvalues λ1 ≥ λ2 ≥ ...; eigenvectors v_i are orthonormal.
4. Project: Z = X V where V contains top k eigenvectors. The covariance of Z is V^T Σ V = diag(λ1,...,λk) — a diagonal matrix, showing the projected features (PCs) are uncorrelated and each diagonal entry equals the variance captured by that PC.
5. Choosing top k components retains sum(λ1..λk) / sum(all λ) fraction of total variance, which is the basis for PCA’s dimensionality reduction with minimal information (variance) loss.

Why the correct answers are correct:
- "PCA is a dimensionality reduction technique..." — correct because PCA selects directions maximizing variance and minimizes reconstruction error for linear projections, thereby retaining most variance.
- "Normalization is important..." — correct because Σ scales with variable units; without scaling, high-variance variables dominate component directions.

Why other options are incorrect:
- Option stating PCA projects to a space where features are correlated is wrong: principal components are orthogonal and uncorrelated.
- Option claiming PCA is supervised is wrong: PCA is unsupervised and does not use labels.

Conclusion: The correct selections are the two provided answers, which match the authoritative answer.

Which of the following statements about Principal Component Analysis (PCA) are correct? (Select all that apply)多项选择题

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