What is a principal component?

Question

BlackTom AI · Accepted Answer

A principal component is a linear combination of the original features that defines a new orthogonal direction capturing maximal variance.

1. Relevant knowledge points and concepts
- Principal Component Analysis (PCA) finds orthogonal directions (principal components) in feature space that sequentially maximize data variance.
- These directions are eigenvectors of the data covariance matrix; their eigenvalues indicate variance explained.
- Principal component scores are projections of the original data onto these directions and are used as new features for dimensionality reduction.

2. Complete reasoning process step by step
- Start with data matrix X of size n × p (n samples, p features). Center each column: Xc = X − 1μ^T, where μ is the p-vector of feature means.
- Compute the sample covariance matrix: S = (1/(n−1)) Xc^T Xc. S is p × p, symmetric.
- Solve the eigenvalue problem: S v = λ v. Each eigenvector v (length p) is a principal component direction; λ is the variance along that direction.
- Order eigenvectors by descending eigenvalues: v1, v2, ... . v1 points in the direction of maximum variance, v2 in the next orthogonal direction, etc.
- Compute principal component scores (new features): z_k = Xc v_k for k = 1..p. These z_k are the transformed features (new directions/features).

3. Intermediate calculation steps (symbolic)
- Centering: Xc_ij = X_ij − μ_j.
- Covariance: S_jk = (1/(n−1)) Σ_i Xc_ij Xc_ik.
- Eigen-decomposition: find v and λ such that det(S − λI) = 0 and S v = λ v.
- Projection: z = Xc V where V = [v1 v2 ...].

4. Why the correct answer is correct
- The principal component is exactly one of these new directions (an eigenvector) formed from the original features; when you project data onto it you obtain a new feature that summarizes variance along that direction. Thus it is "a new direction or feature created from the original features."

5. Why other options are incorrect
- "The center of a K Means cluster": that is a cluster centroid, unrelated to PCA components.
- "The correct label of a data point": labels are supervised targets; principal components are unsupervised feature directions.
- "The probability of belonging to one cluster": that is a clustering membership or soft assignment, not a PCA direction.

6. Conclusion
This explanation confirms the provided answer: a principal component is "A new direction or feature created from the original features."

What is a principal component?Single choice

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