Math Problem Statement

Consider the following vectors (1, 2), (2, 3), (3, 3), (4, 5), (5, 6), (6, 5). What will be the normalized (i.e., unit length) principal component using PCA (choose the option closest to your answer after rounding to the third decimal digit)?

Solution

To calculate the normalized principal component using Principal Component Analysis (PCA) for the given set of vectors, we will follow these steps:

Step 1: Arrange the Data in Matrix Form

The vectors given are:

1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 3 \end{pmatrix}, \begin{pmatrix} 3 \\ 3 \end{pmatrix}, \begin{pmatrix} 4 \\ 5 \end{pmatrix}, \begin{pmatrix} 5 \\ 6 \end{pmatrix}, \begin{pmatrix} 6 \\ 5 \end{pmatrix}$$ These vectors can be arranged as a matrix $$A$$, where each vector represents a row in a matrix: $$A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \\ 3 & 3 \\ 4 & 5 \\ 5 & 6 \\ 6 & 5 \end{pmatrix}$$ ### Step 2: Center the Data (Mean Subtraction) We need to subtract the mean of each column from the respective entries to center the data around the origin. ### Step 3: Compute the Covariance Matrix Next, calculate the covariance matrix for the centered data. ### Step 4: Perform Eigenvalue Decomposition Using the covariance matrix, find the eigenvectors and eigenvalues. The principal component will be the eigenvector corresponding to the largest eigenvalue. ### Step 5: Normalize the Principal Component The principal component needs to be normalized (converted into a unit vector). Let me calculate this for you.The normalized (unit length) principal component, after applying PCA to the given vectors and rounding to three decimal places, is approximately: $$\begin{pmatrix} 0.777 \\ 0.630 \end{pmatrix}$$ Would you like further clarification on any step? Here are some related questions to explore: 1. What are the steps involved in PCA and why is each step important? 2. How does the covariance matrix help in determining the principal components? 3. What is the significance of eigenvalues and eigenvectors in PCA? 4. How do we interpret the normalized principal component? 5. What happens if we use a different method for normalization? **Tip:** In PCA, always ensure the data is centered by subtracting the mean, as this directly affects the computation of the covariance matrix.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Principal Component Analysis (PCA)
Eigenvalues and Eigenvectors
Normalization

Formulas

Covariance matrix calculation
Eigenvalue and eigenvector computation
Normalization of a vector

Theorems

Principal Component Analysis (PCA)

Suitable Grade Level

College level

Related Recommendation

Calculate Normalized Principal Component Using PCA

Step-by-Step PCA for Dimensionality Reduction to k=2 Using Given Data Points

PCA: Normalized Principal Component and Projection for Binomial Vectors

Compute Eigenvectors for Matrix A = [9, -2, -6; -8, 3, 6; 10, -2, -6]

Understanding PCA Projections: Direction of Maximum Variance in Ellipse Data