GATE Data Science and Artificial Intelligence 2024 | Sample Paper | Question: 30

Question

For perfectly spherical $2\text{D}$ data centered at the origin, which of the following the pairs of vectors are possible pairs of principal components?

1. $(1,0)$ and $(0,1)$
2. $(0,-1)$ and $(-1,0)$
3. $(1,1)$ and $(1,-1)$
4. $(-1,1)$ and $(-1,-1)$

 

• A. $\text{i}$
• B. $\text{i and iii}$
• C. $\text{i, ii, and iii}$
• D. $\text{i, ii, iii and iv}$

Answer 1

Ans: D

PCA finds directions which are mutually orthogonal. So, finding the dot product of pairs of vectors in the options, all are zero.

Reference: Georgia Tech

Comments

ans should be D , for d [-1 1] [ -1 -1]^T is [0]  for orthogonality(a^t.b=0)

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Thank you :)

Answer 2

Spherical data implies equal variance in all directions: Since the data points are distributed uniformly on a sphere, there is no preferential direction with higher variance.

Principal components capture directions of maximum variance: By definition, principal components (PCs) identify directions with the highest variance in the data.

Orthogonality ensures independence: PCs are chosen to be uncorrelated (orthogonal) to avoid redundancy in capturing variance.

Therefore, all the given pairs of orthogonal unit vectors:

(1,0) and (0,1)

(0,-1) and (-1,0)

(1,1) and (1,-1)

(-1,1) and (-1,-1)