6 b] Explain Principal Component Analysis.
Principal Component Analysis (PCA) is a dimensionality reduction technique used to simplify a dataset while retaining most of the important information. PCA transforms the data into a new coordinate system, where the greatest variance by any projection of the data lies on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on.
Here’s how PCA works:
- Feature Transformation:
PCA identifies a new set of axes (called principal components) that are linear combinations of the original features. The principal components are uncorrelated and ranked by the amount of variance in the data that they explain. - Dimensionality Reduction:
By projecting the data onto a smaller number of principal components, PCA reduces the complexity of the dataset. For example, instead of having 100 features, PCA might allow you to reduce it to 10 principal components that still capture most of the variance. - Optimization:
PCA finds these components by solving an optimization problem that minimizes the reconstruction error (similar to the idea of minimizing squared error in a recommendation system). This ensures that the first few principal components capture the largest portion of the variance. - Orthogonality:
The principal components are orthogonal to each other, ensuring that there’s no redundancy in the information carried by each component.
Advantages of PCA:
- It reduces the dimensionality of the data, making computations more efficient.
- It eliminates correlation between features, which can improve model performance.
Disadvantages:
- PCA can sometimes lose interpretability because the new components are combinations of original features, making them less meaningful in a real-world context.