The question behind the work.
How can a small set of orthogonal directions retain the important variation in a dataset?
Dataset & environment
Academic PCA work; this site includes a separate, explicitly synthetic three-dimensional dataset.
From the idea to the method.
- Center and, when feature scales differ, standardize the data.
- Compute the covariance matrix and its eigenvectors.
- Sort components by explained variance and project the centered observations.
- Inspect information retained and how preprocessing changes the interpretation.
What the explorer computes
The synthetic observations are centered. The covariance matrix is decomposed into orthogonal eigenvectors with a Jacobi eigensolver; observations are projected onto the leading components. The variance bars are calculated from this dataset's eigenvalues.
X_centered = X - X.mean(axis=0)
covariance = X_centered.T @ X_centered / (len(X) - 1)
eigenvalues, eigenvectors = np.linalg.eigh(covariance)
order = eigenvalues.argsort()[::-1]
projection = X_centered @ eigenvectors[:, order[:2]]What this study explores.
- Academic focus: scaling, covariance, eigenvectors, and dimensionality reduction.
- Interpretation of principal components and explained variance.
How to assess it carefully.
Report the cumulative explained variance and reconstruction error. For supervised follow-up tasks, fit PCA only on the training fold.
What is documented.
Experimental artifacts and verified numerical results have not been published. The interactive visuals explain the method; they are not benchmark results.
Training curves, accuracy comparisons, and repository links will be added alongside the corresponding artifacts. No unverified benchmark numbers are shown.
A foundation for the next experiment.
- Maximum variance is not necessarily maximum class separation.
- Feature scaling affects the covariance structure and therefore the projection.
Future updates can include code, screenshots, reproducibility instructions, and verified outcomes.