Let's tackle the two questions one by one.

Question 1: PCA vs. MDA

Given Statements:

1. I. Projection direction may not be the best in terms of minimum squared error for MDA.
2. II. Projection direction is best in terms of minimum squared error for PCA.

Analysis:

• PCA (Principal Component Analysis) focuses on finding the direction (principal components) that maximizes the variance of the data. In other words, it seeks directions that preserve the most information about the data, minimizing the squared error between the original data and its projection onto the principal components. Therefore, Statement II is correct.

• MDA (Multiple Discriminant Analysis), on the other hand, is focused on maximizing the separability between different classes. It finds directions that maximize the distance between the means of different classes while minimizing the variance within each class. MDA does not optimize for minimum squared error in the same way PCA does, so Statement I is also correct.

Conclusion:

• Both I and II are true.

The correct answer is c) Both I and II.

Question 2: Direction of Projection in MDA

In this problem, we are dealing with a two-class problem, where:

• $\mathcal{S}_W$ is the within-class scatter matrix.
• $\mathcal{S}_B$ is the between-class scatter matrix.
• $\mu_1$ and $\mu_2$ are the means of the two classes.

Analysis:

The optimal direction for projection in MDA (also known as Fisher's linear discriminant) is given by maximizing the ratio of the between-class scatter to the within-class scatter. This can be mathematically formulated as finding the vector $w$ that maximizes the following objective:

$J(w) = \frac{w^T \mathcal{S}_B w}{w^T \mathcal{S}_W w}$

The optimal direction of projection is given by:

$w^* = \mathcal{S}_W^{-1} (\mu_1 - \mu_2)$

Thus, Option c) $\mathcal{S}_W^{-1} (\mu_1 - \mu_2)$ is correct.

Conclusion:

The correct answer is c) $\mathcal{S}_W^{-1} (\mu_1 - \mu_2)$.

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Do you want more details or have any questions?

5 Related Questions:

1. What are the key differences between PCA and MDA in terms of their objectives?
2. How does PCA handle data with multiple classes compared to MDA?
3. Can PCA be used for supervised learning tasks, or is it strictly unsupervised?
4. What role does the eigenvalue decomposition play in PCA?
5. How does the choice of projection direction affect classification accuracy in MDA?

Tip:

When deciding between PCA and MDA, remember that PCA is better for dimensionality reduction based on data variance, while MDA is more suitable for supervised tasks where class separability is critical.