Explained variance and visualization

The explained variance ratio of component $j$ is ${\lambda }_j/{\sum }_i{\lambda }_i$. The cumulative sum over the first $k$ components says how much of the total variance they keep.

Choosing $k$:

• Keep enough components for, say, 90–95% cumulative variance.
• Look for an elbow in the scree plot (eigenvalue vs. component number).
• Choose $k$ by cross-validating the downstream model.
• For visualisation, simply use $k=2$ or $3$.

Plotting the first two components often reveals clusters, outliers and trends; colouring the points by label shows whether classes separate.