How do you determine the appropriate number of components in PCA?

Instruction: Describe the methods you would use to decide how many principal components to retain in a PCA analysis.

Context: This question evaluates the candidate's ability to implement dimensionality reduction techniques effectively, ensuring the retention of significant data variance.

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Official answer

Choose the number of principal components for the goal: visualization, compression or downstream prediction. Decide whether centering and scaling are appropriate before interpreting variance, since large-unit features can dominate covariance-based PCA.

Inspect the scree plot and cumulative explained variance. For proportions 0.60, 0.25, 0.10 and 0.05, two components retain 85% and three retain 95%. Neither threshold is universally correct, and high variance does not necessarily mean useful predictive signal.

The eigenvalue-greater-than-one heuristic applies to the correlation-matrix setting with standardized features; it is not an invariant rule for arbitrary raw units. Parallel analysis or held-out reconstruction can add evidence for some purposes.

For predictive use, select component count by valid downstream validation, fitting scaling and PCA inside each training fold. Retain a final test set and compare against an appropriate non-PCA baseline. Report the tradeoff in prediction, reconstruction and interpretability rather than treating an elbow as a guaranteed optimum.

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