Data Space
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Principal Component Analysis
PCA finds the directions of maximum variance in data and projects it onto a lower-dimensional subspace. The principal components are orthogonal eigenvectors of the covariance matrix.
How to Use
- Click canvas to add data points
- Press Compute to find principal components
- Select 1 component to see 2D→1D projection
- Toggle projections to see projection lines
- Try different datasets to see how PC directions change
PCA Steps
- Center the data (subtract mean)
- Compute covariance matrix
C = (1/n)XTX - Find eigenvectors and eigenvalues of C
- Sort by eigenvalue (descending)
- Project data onto top k eigenvectors
Covariance Matrix
C = (1/n) Σᵢ (xᵢ − μ)(xᵢ − μ)T
Measures how features vary together.
Eigendecomposition
Cv = λv
Eigenvectors v are the principal directions, eigenvalues λ are the variances.
Explained Variance Ratio
ratio_k = λ_k / Σᵢ λᵢ
Fraction of total variance captured by component k.
Reconstruction
x̂ = W_k W_kT(x − μ) + μ
Approximate original data using k components.
PCA Metrics
| Points | 0 |
| PC1 Variance | - |
| PC2 Variance | - |
| Explained (PC1) | - |
| Explained (PC1+2) | - |
| Mean | - |
| Status | Add points to begin |