PCA Visualization

Principal Component Analysis (PCA) reduces the complexity of many indicators into a simpler 2D visualization while preserving as much information as possible.

Instead of viewing states across 20+ individual indicators, PCA creates new composite dimensions (principal components) that capture the main patterns of variation.

PCA finds new axes (principal components) that best explain the variation in the data:

1. Center the data: Subtract the mean from each indicator
2. Find PC1: The direction of maximum variance in the data
3. Find PC2: The direction of maximum remaining variance, perpendicular to PC1
4. Continue: Each subsequent PC captures remaining variance, perpendicular to all previous

Each point on the scatter plot represents a state, positioned by its PC1 and PC2 scores:

• X-axis (PC1): Position along the first principal component
• Y-axis (PC2): Position along the second principal component
• Close states: Similar overall civic engagement profiles
• Distant states: Different profiles along these dimensions

Loadings show how much each original indicator contributes to each principal component:

• Large positive loading: Indicator strongly contributes to high scores on this PC
• Large negative loading: Indicator contributes to low scores (moves in opposite direction)
• Near-zero loading: Indicator doesn't contribute much to this PC

Example: If PC1 has high positive loadings for voting, volunteering, and group membership, then PC1 might represent "general civic engagement" — states on the right are high on all three.

Explained variance tells you how much information each component captures:

• PC1: Always captures the most variance (the dominant pattern)
• PC2: Captures the next most (the second pattern)
• Cumulative: PC1 + PC2 together show how much of the total picture you're seeing

• Overall similarity: Which states have similar civic engagement profiles
• Main dimensions: The key patterns that differentiate states
• Outliers: States with unusual combinations of indicators
• Indicator groupings: Which indicators tend to vary together (via loadings)

• Component meaning: PCs are mathematical constructs — naming them (e.g., "civic engagement") is interpretation
• Causation: Nearby states aren't similar because of any particular reason
• Raw values: Position shows relative standing, not actual participation rates
• Full picture: 2D shows PC1 + PC2 only; other dimensions may reveal different patterns
• Policy implications: States at similar positions may have arrived there through very different paths

• Naming components: Resist the urge to over-interpret PC labels. Look at what actually loads highly before naming.
• Low explained variance: If PC1 + PC2 explain only 40% of variance, the remaining 60% tells a different story
• Rotation sensitivity: The exact position of points depends on which indicators are included
• Scale dependence: We use z-scores so all indicators contribute equally; different scaling choices would change results

Best practice: Use PCA as an exploratory tool to generate hypotheses, then verify patterns using the original indicators in the Matrix or Correlation views.

Our PCA implementation:

• Uses z-score standardized data (mean=0, std=1 for each indicator)
• Computes covariance matrix from centered data
• Extracts principal components via power iteration (100 iterations)
• Deflates covariance matrix after each component
• Extracts up to 4 principal components