Statistical PCA: coordinates, reconstruction and scaling
Three original data-based lessons derive centered covariance, scores and reconstruction, then test scaling and degenerate directions. Public worked synthetic cases and written review tasks do not establish population structure or independently assessed statistical proficiency.
Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.
Readiness and preparation
Loading saved mathematics work…
PCA evidence: reconstruct, rescale and validate
The mixed assessment has separate response history and review criteria; it does not change lesson participation records.
Lesson sequence
Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Center data and construct principal-component scores
Lesson details and objectives
Compute the actual covariance, directions and coordinates of a small dataset.
- Construct centered data and sample covariance with a declared divisor.
- Derive score variance and eigenvector selection from a quadratic form.
- Distinguish centering, loading signs and numerical certificates from statistical validation.
6 practice tasks.
Reconstruct observations and account for lost variation
Lesson details and objectives
Prove the projection loss and compare mean reconstruction with new-data performance.
- Restore the mean after low-rank projection.
- Prove the reconstruction-loss identity and optimality with explicit metric.
- Keep training explained variance separate from validation error.
6 practice tasks.
Choose scales and diagnose unidentified directions
Lesson details and objectives
Compare covariance and correlation PCA without hiding zero variance or tied eigenvalues.
- Derive standardized covariance and restore physical coordinates.
- Diagnose constant columns, repeated eigenvalues and rank-one data.
- Compare output metrics and avoid treating rounded degeneracy as exact mathematics.
6 practice tasks.