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Scree Plot

Formula

\[ \text{scree plot: } k \mapsto \lambda_k \quad \text{or} \quad k \mapsto \sum_{j=1}^{k}\mathrm{EVR}_j \]

Plot

type: bars
xs: 1 | 2 | 3 | 4 | 5 | 6
ys: 0.42 | 0.24 | 0.14 | 0.09 | 0.06 | 0.05
xmin: 0.5
xmax: 6.5
ymin: 0
ymax: 0.48
height: 280
title: Example scree plot (explained variance by component)

Parameters

  • \(k\): component index
  • \(\lambda_k\): ordered eigenvalue

What it means

A scree plot visualizes component importance to help choose a truncation point in PCA/factor methods.

What it's used for

  • Selecting a practical number of principal components.
  • Explaining diminishing returns from additional components.

Key properties

  • Often look for an "elbow" where gains flatten.
  • Can plot eigenvalues or cumulative explained variance.

Common gotchas

  • Elbows can be subjective.
  • Task performance should still validate the dimensionality choice.

Example

If cumulative variance jumps to 95% by component 20 and then flattens, 20 is a reasonable candidate.

How to Compute (Pseudocode)

Input: ordered eigenvalues or explained-variance ratios
Output: scree plot data points (and optionally cumulative curve)

for k from 1 to K:
  y[k] <- eigenvalue[k] or explained_variance_ratio[k]
  cumulative[k] <- sum_{j=1..k} explained_variance_ratio[j]   # optional
plot k vs y (and/or cumulative[k])
return plot data

Complexity

  • Time: \(O(K)\) once eigenvalues/EVR values are available
  • Space: \(O(K)\) for plot vectors
  • Assumptions: Cost of PCA/eigendecomposition is excluded; this card covers postprocessing/visualization preparation

See also