How can we rotate a feature space without throwing away its main variation?
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How can we rotate a feature space without throwing away its main variation?
PCA finds orthogonal directions that capture as much variance as possible, then represents observations using coordinates along those directions.
Building interactive view…
Understand
Build the mental model
PCA finds orthogonal directions that capture as much variance as possible, then represents observations using coordinates along those directions. Scale features when units differ meaningfully. PCA maximises variance, not task relevance or interpretability.
Click a stage to inspect what happens, what changes, and what should be checked before moving on.
Stage 1
Centre / scale
Trace the operation itself. State what information it reads, what rule it applies, and which intermediate state or rows/columns change as a consequence. Technical context for Principal Component Analysis: Eigenvectors of the covariance matrix (or SVD of centred data) define principal directions. Components are ordered by explained variance.
Practitioner checkpoint: Scale features when units differ meaningfully. PCA maximises variance, not task relevance or interpretability.
What happens if…?
Break the assumption deliberately
Rescale one feature and watch the first principal direction rotate toward it.
Move the control and explain what you expect before reading the visual.
Technical lens
Formalise what the visual is doing
Eigenvectors of the covariance matrix (or SVD of centred data) define principal directions. Components are ordered by explained variance.
Technical questionUse a tiny case to make the mechanism observable. Eigenvectors of the covariance matrix (or SVD of centred data) define principal directions. Components are ordered by explained variance. Verify one intermediate quantity, state change or mapping independently; then predict the consequence of this change: Rescale one feature and watch the first principal direction rotate toward it.
Practitioner lens
Use it responsibly
Scale features when units differ meaningfully. PCA maximises variance, not task relevance or interpretability.
Transfer testTransfer this idea to a new example and justify each decision using this practitioner rule: Scale features when units differ meaningfully. PCA maximises variance, not task relevance or interpretability. Then explain what should change if you deliberately test: Rescale one feature and watch the first principal direction rotate toward it.
Worked exploration
Use the visual as an experiment, not decoration
Standardise two strongly correlated features, centre them and draw the direction of greatest variance. Project each point onto that first component and compare the one-dimensional representation with the original cloud.
Technical lens
Eigenvectors of the covariance matrix (or SVD of centred data) define principal directions. Components are ordered by explained variance.
Practitioner check
Scale features when units differ meaningfully. PCA maximises variance, not task relevance or interpretability.
Prediction before interaction
Rescale one feature and watch the first principal direction rotate toward it.
Exploration walkthrough
Turn the interaction into an evidence trail
Standardise two strongly correlated features, centre them and draw the direction of greatest variance. Project each point onto that first component and compare the one-dimensional representation with the original cloud. Before moving the control, state your prediction. After the visual changes, name the specific state, statistic, boundary or mapping that changed and explain why that change is consistent—or inconsistent—with your prediction.
Record one observable quantity before the interaction and the same quantity afterwards.
Change one factor at a time so the causal effect of the control is inspectable.
Use an edge or failure case to discover where the concept stops behaving as the simple story suggests.
Static orientation diagram for Principal Component Analysis; use the interactive visual above to test how the relationships change.
Reference depth
Open the complete material
The flagship experience is the map. These pages contain the roads.
