我们对高维空间的给定数据集进行主成分分析 (PCA)。以下哪项是正确的？

（选择所有适用的选项。）

We perform Principal Component Analysis (PCA) on a given data set from a high-dimensional space.

Which of the following is true?

(Select all that apply.)

Question

我们对 高维空间的给定数据集进行主成分分析 (PCA)。以下哪项是正确的？

（选择所有适用的选项。）

We perform Principal Component Analysis (PCA) on a given data set from a high-dimensional space.

Which of the following is true?

(Select all that apply.)

BlackTom AI · Accepted Answer

PCA diagonalizes the data covariance so that variance along principal directions equals the eigenvalues of the covariance matrix, ordered from largest to smallest.

1) Relevant knowledge points and concepts
- Compute covariance matrix Σ of centered data. PCA finds eigenvectors v_i and eigenvalues λ_i satisfying Σ v_i = λ_i v_i.
- Eigenvalues λ_i quantify the variance captured by each principal component.
- For a real symmetric matrix (covariance matrix), eigenvectors corresponding to distinct eigenvalues are orthogonal.

2) Step-by-step reasoning
- Perform eigen-decomposition: Σ = V Λ V^T, where Λ = diag(λ1, λ2, ..., λd) with λ1 ≥ λ2 ≥ ... ≥ λd ≥ 0.
- The fraction of total variance explained by component i is λi / (Σ_j λj).
- Because λ1 ≥ λ2 ≥ ..., the first few components (those with largest λ) will account for larger shares of total variance than later components.

3) Intermediate calculation example
- If eigenvalues are λ1 = 60, λ2 = 20, λ3 = 10, rest sum = 10, total = 100. Then the first component explains 60/100 = 60% variance; first two explain 80%, etc. In an extreme case λ1 = 95, others sum 5, then the first component alone explains 95%.

4) Why the correct answers are correct
- “The first few principal components are more likely to capture more of the original variance...” follows directly from ordering eigenvalues: top components have largest λ and therefore larger explained-variance ratios.
- “The first principal component is orthogonal to all other principal components.” Eigenvectors of the real symmetric covariance matrix form an orthonormal set, so v1 is orthogonal to v2, v3, ..., vd.

5) Why the other options are incorrect
- “If we keep fewer than 5 components, it is not possible to capture >95% variance” is false: whether <5 components can capture >95% depends on the data (e.g., a dominant first eigenvalue can capture >95%).
- “We can only keep 1 principal component in a classification task” is false: PCA is an unsupervised dimensionality-reduction step and you may retain any number of components suitable for the classification task.

6) Conclusion
The correct selections are the first two statements. This matches the provided answer.

Similar Questions

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Question at position 17 Which image is post PCA transformation? (A) (B) image.png BA

假设我们有下列 2-d空间数据点 (0,0)、(2,1)、(6,3)、(-2,-1)、(-6,-3)。第一个主成分是什么？ Suppose we have the following data points in 2-d space (0,0)、(2,1)、(6,3)、(-2,-1)、(-6,-3)。 What is the first principal component?

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

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