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?

Question

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?

BlackTom AI · Accepted Answer

The first principal component is the direction of maximal variance. Compute the data mean, covariance, and the leading eigenvector to find that direction.

1) Relevant concepts
- PCA finds orthogonal directions (principal components) that capture maximal variance.
- With centered data, the first principal component is the eigenvector of the covariance matrix with the largest eigenvalue. Scaling by a positive constant (normalization) does not change the direction.

2) Step-by-step calculations
- Data points: (0,0), (2,1), (6,3), (-2,-1), (-6,-3).
- Compute the mean: x̄ = (0+2+6-2-6)/5 = 0, ȳ = (0+1+3-1-3)/5 = 0. Data are already centered.
- Compute second-moment sums:
  Σ x^2 = 0^2 + 2^2 + 6^2 + (-2)^2 + (-6)^2 = 0+4+36+4+36 = 80
  Σ y^2 = 0+1+9+1+9 = 20
  Σ xy  = 0 + 2*1 + 6*3 + (-2)*(-1) + (-6)*(-3) = 0+2+18+2+18 = 40
- Covariance matrix (using 1/5 for population form): 
  C = (1/5) [ [80, 40], [40, 20] ] = [ [16, 8], [8, 4] ].

3) Find leading eigenvector
- Observe C is rank-1 with rows proportional; try v = [2,1]:
  C [2,1]^T = [ [16,8],[8,4] ] [2,1]^T = [40,20]^T = 20 [2,1]^T.
- So [2,1] is an eigenvector with largest eigenvalue 20. The unit version is (1/√5)(2,1), but the principal direction is along (2,1).

4) Why this is correct
- The direction [2,1] maximizes variance because it is the eigenvector associated with the largest eigenvalue of the covariance matrix; projection onto this direction yields the largest spread of the data.

5) Why other options are incorrect
- (1,2) has the wrong orientation (not aligned with covariance eigenvector).
- (-2,-1) is the negative of (2,1); while it represents the same line, the provided answer specifies (2,1). A normalized version 1/√5 (2,1) would also be correct in direction but is not the exact provided answer.

Conclusion: The first principal component (direction of maximal variance) is (2,1), which matches the provided answer.

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?单项选择题

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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?单项选择题