With 5 features, direct visualization is impossible. But notice: Return and Beta move together, Volatility and Beta move together. PCA finds the best linear combinations that capture the most variance. The first principal component might be a "risk" factor, the second a "size" factor.

The first principal component (PC1) points in the direction of maximum variance. Projecting onto this direction loses the least information. PC2 is perpendicular to PC1 and captures the remaining variance. Together they form a new coordinate system aligned with the data's natural axes.

The scree plot shows explained variance per component. Look for the "elbow" -- where the curve flattens. Components after the elbow add little. A common rule: keep enough components to explain 90-95% of total variance. This is dimensionality reduction -- fewer features, nearly the same information.

PCA reconstruction: $\hat{X} = X_{\text{reduced}} \cdot W^T$. With fewer components, fine details are lost but major patterns are preserved. Acceptable when: the lost variance is noise, or you need speed/storage savings. This is similar to JPEG compression -- lose some detail, keep the essence.

t-SNE is a nonlinear method that preserves local neighborhoods -- similar points stay close. But unlike PCA, distances between distant clusters are NOT meaningful. The perplexity parameter controls how many neighbors each point considers: low perplexity = tight clusters, high perplexity = more global structure.

PCA: linear, fast, invertible, preserves global structure -- use for feature reduction, preprocessing, or when you need to transform new data. t-SNE: nonlinear, slow, not invertible, reveals clusters -- use for visualization only. You cannot apply a t-SNE mapping to new points without rerunning on the full dataset.