https://file.aiquickdraw.com/mathbot/file/1725462507693xw37v5yc.jpg

Let's analyze each question from your assignment:

### Question 1:
**Choose the correct option regarding PCA:**

I. Direction of projection in PCA is given by the eigenvector of the covariance matrix corresponding to the largest eigenvalues.

II. PCA is also sometimes called KL transformation.

- **Option a)** Only I
- **Option b)** Only II
- **Option c)** Neither I nor II
- **Option d)** Both I and II

**Explanation:**
- **Statement I** is correct. In PCA (Principal Component Analysis), the direction of projection is determined by the eigenvectors of the covariance matrix, with the eigenvector corresponding to the largest eigenvalue being the direction of the greatest variance in the data.
- **Statement II** is also correct. PCA is sometimes referred to as the Karhunen-Loève (KL) transformation, particularly in the context of signal processing.

**Correct Answer:** **d) Both I and II**

---

### Question 2:
**Dimensionality reduction aims at:**

I. Map feature vectors to a higher-dimensional space.

II. Ensure that features in the new space are orthogonal or uncorrelated.

- **Option a)** Only I
- **Option b)** Only II
- **Option c)** Neither I nor II
- **Option d)** Both I and II

**Explanation:**
- **Statement I** is incorrect. The goal of dimensionality reduction is to reduce the number of dimensions, i.e., mapping feature vectors to a lower-dimensional space, not a higher-dimensional one.
- **Statement II** is correct. Dimensionality reduction techniques, like PCA, aim to ensure that the features in the new space are uncorrelated (orthogonal).

**Correct Answer:** **b) Only II**

---

### Question 3:
**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.**

(Unfortunately, the graph is not fully visible in the image provided, so I cannot analyze this question accurately.)

**Tip:** In general, the direction of projection in PCA is along the direction of maximum variance, which corresponds to the principal component.

---

Do you want further details on any of these questions? Here are some related questions that can deepen your understanding:

1. How does PCA handle dimensionality reduction while retaining maximum variance in the data?
2. What are the practical applications of the Karhunen-Loève (KL) transformation?
3. Can you explain the significance of orthogonality in the context of PCA?
4. What are the limitations of PCA when dealing with non-linear data structures?
5. How does PCA differ from Linear Discriminant Analysis (LDA)?

**Tip:** Always ensure you understand the fundamental principles behind

Choose the correct option regarding PCA and dimensionality reduction.

Learn how Principal Component Analysis (PCA) identifies the direction of maximum variance using eigenvectors and eigenvalues, and how dimensionality reduction ensures orthogonality in the transformed space. This content is suitable for students at the undergraduate and graduate levels.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation