Dimensionality Reduction with Principal Component Analysis (PCA)

Principal Component Analysis (PCA) is a powerful technique for dimensionality reduction, data compression, and feature extraction. It has wide applications in various fields, particularly in data preprocessing and exploratory data analysis.

1. Objective:

• PCA is a dimensionality reduction technique used to transform high-dimensional data into a lower-dimensional representation while retaining as much of the original variance as possible.

2. Basic Idea:

• PCA identifies the directions (principal components) in which the data varies the most.
• The first principal component explains the most variance, the second explains the second most, and so on.

3. Mathematical Approach:

• PCA involves computing the eigenvectors and eigenvalues of the covariance matrix of the data.
• Eigenvectors represent the directions of maximum variance, and eigenvalues indicate the amount of variance in those directions.

4. Steps in PCA:

• Standardize the data to have zero mean and unit variance.
• Compute the covariance matrix of the standardized data.