https://file.aiquickdraw.com/mathbot/file/1725463518929uupxwe6c.jpg

The image contains two questions related to Principal Component Analysis (PCA) and the directions of projections. Let's analyze each of them in detail.

### Question 1:
The question asks which of the following could be the direction of projection using PCA if the data points lie on the locus of the curve shown in Figure 1. The curve in Figure 1 is an ellipse with its major axis aligned horizontally.

**Options:**
a) X-axis  
b) Y-axis  
c) Direction making an angle of 45° with the x-axis in the anticlockwise direction.  
d) Direction making an angle of 135° with the x-axis in the anticlockwise direction.

**Analysis:**
In PCA, the principal component (PC1) corresponds to the direction of maximum variance in the data. For the ellipse in Figure 1, the major axis (which is horizontal) has the maximum variance.

- The correct answer is **a) X-axis**.

### Question 2:
The question asks which of the following could be the direction of projection using PCA if the data points lie on the locus of the curve shown in Figure 2. The curve in Figure 2 is an ellipse rotated with its major axis aligned in a diagonal direction.

**Options:**
a) X-axis  
b) Y-axis  
c) Direction making an angle of 45° with the x-axis in the anticlockwise direction.  
d) Direction making an angle of 135° with the x-axis in the anticlockwise direction.

**Analysis:**
In Figure 2, the ellipse is rotated such that its major axis is aligned at approximately a 45° angle with the x-axis. Thus, the direction of maximum variance will be along this diagonal.

- The correct answer is **c) Direction making an angle of 45° with the x-axis in the anticlockwise direction**.

Would you like further details or have any additional questions?

### Five Relative Questions:
1. What is the role of the covariance matrix in PCA?
2. How do eigenvectors relate to the principal components in PCA?
3. How would the PCA change if the data were uniformly distributed in all directions?
4. What are some limitations of using PCA for dimensionality reduction?
5. How can PCA be applied to data with more than two dimensions?

### Tip:
When interpreting the results of PCA, always examine the explained variance ratio to understand how much of the total variance is captured by each principal component.

Which of the following could be the direction of projection using PCA if the data points lie on the locus of the curve shown in Figure 1? Assume X-axis is along the horizontal direction.

Options:
a) X-axis
b) Y-axis
c) Direction making an angle of 45° with the x-axis in anticlockwise direction.
d) Direction making an angle of 135° with the x-axis in anticlockwise direction.

And

Which of the following could be the direction of projection using PCA if the data points lie on the locus of the curve shown in Figure 2? Assume X-axis is along the horizontal direction.

Options:
a) X-axis
b) Y-axis
c) Direction making an angle of 45° with the x-axis in anticlockwise direction.
d) Direction making an angle of 135° with the x-axis in anticlockwise direction.

Explore how PCA determines projection directions based on data locus using elliptical shapes. Understand the role of variance and eigenvectors in PCA to identify correct directions for projection in different scenarios. Suitable for undergraduate students.

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