What is PCA (Principal Component Analysis)?

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🔑 Key Ideas of PCA

1. Variance Maximization

   • PCA finds new axes (called principal components) along which the data has the highest variance.

   • The first component captures the most variance, the second captures the next most (orthogonal to the first), and so on.

2. Linear Transformation

   • Principal components are linear combinations of the original features.

   • This transformation decorrelates the data, making features independent in the new space.

3. Dimensionality Reduction

   • Instead of keeping all original features, PCA keeps only the top k principal components that explain most of the variance.

⚙️ Steps of PCA

1. Standardize the data (mean = 0, variance = 1).

2. Compute the covariance matrix of features.

3. Find eigenvalues and eigenvectors of the covariance matrix.

4. Sort eigenvectors by eigenvalues (importance).

5. Project the data onto the top k eigenvectors → reduced dimensions.

🖼 Example

• Suppose you have a dataset of 100 features describing customer behavior.

• PCA might reveal that only 10 principal components explain 90% of the variance.

• You can then work with these 10 instead of all 100, reducing noise and complexity.

✅ Benefits of PCA

• Reduces computation time and storage.

• Removes multicollinearity among features.

• Improves model generalization.

• Useful for visualization in 2D or 3D.

👉 In short: PCA is a mathematical method that converts correlated features into a smaller set of uncorrelated variables (principal components) while keeping most of the data’s essential information.

Read more :

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Explain dimensionality reduction.

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