PCA Checkpoint

Trace covariance, power iteration, deflation, and reuse of fitted components; diagnose rank and scaling mistakes.

Get every question correct to pass this checkpoint. You can review feedback, revise your answers, and submit again.

1. Centered observations are [[-2,0],[2,0]]. What is C[0][0] under this exercise's covariance convention?

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8 4 2 0

2. A fitted PCA has components stored as k rows of width d. Which projection has shape (n,k)?

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components @ (X-mean) (X-mean) @ components X @ components.T + mean (X-mean) @ components.T

3. For C=diag(9,3), start in direction [1,1]. After one power multiplication, what is the first-to-second coordinate ratio? Enter a number.

4. After extracting v=[1,0] with eigenvalue 9 from C=diag(9,3), what should deflation leave?

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diag(0,3) diag(9,0) diag(1,3) diag(0,0)

5. Rank-one data produces two identical unit component rows. Shapes and individual lengths pass. Which check exposes the defect?

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The training mean must be zero before fitting Each component must have positive entries The dot product of the two components must be near zero The transformed data must have fewer rows

6. PCA fitted mean=[2,10] and component=[[1,0]]. What scalar coordinate does transform([[5,99]], component, mean) return? Enter a number.

7. All training rows are identical. You request two components in two dimensions. Which result obeys the exercise contract?

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Two copies of any unit vector Any two orthogonal unit vectors, with zero training coordinates Two zero vectors, with undefined coordinates An arbitrary positive explained-variance percentage

8. Feature units are meaningful and a low-variance feature separates classes. Which conclusion is justified before using PCA?

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Standardize automatically; labels can never be lost Retaining 95% variance guarantees 95% accuracy Choose scaling deliberately and validate predictive performance Fit PCA on the test rows to learn their coordinate system

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