How Many Principal Components?

Question

You have run PCA on a panel of standardised asset returns and obtained the eigenvalues of the covariance matrix. The mechanics are done; the judgement call remains.

How do you decide how many principal components to keep? Explain the explained-variance ratio, the cumulative-variance threshold, the scree-plot / elbow method, the Kaiser rule, and cross-validation via reconstruction error. Work through a small numeric example, and discuss the trading-specific caveats.

Accepted Answer

PCA produces a set of orthogonal directions (the eigenvectors of the covariance matrix) ranked by how much variance each captures (the eigenvalues $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_p$). The derivation of those eigenvectors is assumed here; the practical question is how many of them, $k$, to retain. There is no single correct answer — it is a bias-variance and interpretability tradeoff — so good practice is to triangulate several criteria.

1. Explained-variance ratioBecause the total variance equals the sum of the eigenvalues, the fraction of variance explained by component $j$ is \begin{equation} r_j = \frac{\lambda_j}{\sum_{i=1}^{p} \lambda_i} \end{equation} This is the single most informative number per component: it converts raw eigenvalues into interpretable percentages.

2. Cumulative-variance thresholdOrder the ratios and keep the smallest $k$ such that the cumulative explained variance crosses a target, commonly 90% or 95%: \begin{equation} \sum_{j=1}^{k} r_j \ge 0.95 \end{equation} Simple and the most widely used rule. The threshold is a judgement call: a lower target (e.g. 80%) for aggressive compression, a higher one (99%) when you cannot afford to lose information.

3. Scree plot / elbow methodPlot $\lambda_j$ (or $r_j$) against the component index $j$. Typically the curve drops steeply for the first few components and then flattens into a shallow tail of "noise" components. The elbow — where the steep part bends into the flat part — is taken as $k$. It is visual and somewhat subjective, but excellent for spotting how many components genuinely stand out from the noise floor.

4. Kaiser ruleWhen PCA is run on the correlation matrix of standardised data, the average eigenvalue is exactly 1 (they sum to $p$ and there are $p$ of them). The Kaiser rule keeps every component with $\lambda_j > 1$, i.e. every component that explains more variance than a single original standardised variable. It is a quick heuristic but tends to over-retain on large $p$ and shouldn't be used alone.

5. Cross-validation via reconstruction errorThe most principled approach. For each candidate $k$, project the data onto the top $k$ components and reconstruct it, then measure the reconstruction error \begin{equation} \frac{1}{n}\sum_{i=1}^{n} \lVert x_i - \hat{x}_i^{(k)} \rVert^2 \end{equation} on held-out data (or held-out matrix entries). On the training set this error falls monotonically with $k$, but on held-out data it eventually stops improving once added components are fitting noise. Choose the $k$ at that plateau. If PCA is a preprocessing step for a downstream predictor, even better: treat $k$ as a hyperparameter and pick the $k$ that maximises the downstream validation score.

Worked exampleSuppose six standardised return series give eigenvalues \begin{equation} \lambda = (3.2,\; 1.5,\; 0.7,\; 0.3,\; 0.2,\; 0.1) \end{equation} summing to $6.0$ (as they must for six standardised variables). The ratios and cumulative totals:

Component$\lambda_j$$r_j$Cumulative
  
    13.253.3%53.3%
    21.525.0%78.3%
    30.711.7%90.0%
    40.35.0%95.0%
    50.23.3%98.3%
    60.11.7%100%

Reading off the criteria: the 90% threshold keeps 3 components, the 95% threshold keeps 4. The Kaiser rule ($\lambda > 1$) keeps 2. The scree plot shows a clear drop from 3.2 to 1.5 to 0.7, then a flat tail from 0.3 onward — the elbow sits at 2 or 3. So the criteria roughly agree on 2 to 4 components; a reasonable choice here is 3 (90% variance, just past the elbow), with cross-validated reconstruction error used to break the tie if this feeds a predictor.

Trading caveats

Standardise firstPCA on a covariance matrix is scale-dependent. If you run it on raw returns, a single high-volatility asset can dominate the first component for no economic reason. Standardise returns (or run PCA on the correlation matrix) so every asset contributes comparably.
 	The top PC is usually "the market."For equity-return panels the first component is almost always a roughly equal-weighted long direction — the systematic market factor — typically explaining a large share of variance. Subsequent components often correspond to sector or style factors. This is useful for interpretation but means PC1 carries little idiosyncratic alpha.
 	Component instability over timeEigenvectors estimated on noisy, non-stationary returns drift and can swap order or flip sign between windows. A component that looks stable in-sample may not persist. Re-estimate on rolling windows and check that the chosen $k$ and the loadings are stable before trusting them — otherwise you are reducing dimension onto directions that will not exist next month.
 	Variance is not signalHigh-variance directions are not necessarily the predictive ones. The component that explains the most variance may be pure market beta you want to hedge out, while a low-variance component carries the alpha. Validate against the downstream objective rather than assuming more variance means more usefulness.

Title	Category	Subcategory	Difficulty	Status
Principal Component Analysis	Machine Learning	Dimensionality Reduction	Hard	Unlock
t-SNE and UMAP: Nonlinear Embeddings	Machine Learning	Dimensionality Reduction	Medium	Unlock

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How Many Principal Components?

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