The questions in the image refer to Principal Component Analysis (PCA) and involve identifying the direction of projections based on ellipses representing data distributions in two-dimensional space.

Question Analysis:

1. Figure 1 Analysis:

   • The ellipse is elongated along the $x_1$-axis. This suggests that the direction of maximum variance (first principal component) is along the $x_1$-axis.

   Answer:

   • Option a) $x_1$-axis is correct.

2. Figure 2 Analysis:

   • The ellipse is elongated at an angle between the $x_1$ and $x_2$ axes, indicating that the direction of maximum variance (first principal component) is along a diagonal direction.

   Answer:

   • Option c) Direction making an angle of 45° with $x_1$-axis in anticlockwise direction is correct.

Would you like a more detailed explanation or have any other questions?

Relative Questions:

1. What is Principal Component Analysis (PCA) and how is it used in data analysis?
2. How does the orientation of an ellipse indicate the direction of maximum variance in PCA?
3. What are the implications of projecting data onto the first principal component?
4. How does PCA reduce the dimensionality of data?
5. What is the significance of the eigenvectors in determining the directions of the principal components?

Tip:

In PCA, the first principal component represents the direction of maximum variance, and each subsequent component represents the next highest variance orthogonal to the previous ones.