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Principal Component Analysis (PCA) Explained
Learn centering, scaling, covariance, component scores and loadings, explained variance, component selection, and the limits of PCA in finance.
Prepared by Mark · Primary sources below
Direct answer
Principal component analysis (PCA) transforms correlated variables into a smaller set of orthogonal axes that preserve as much sample variance as possible. The first component captures the greatest variance, and each later component captures the greatest remaining variance while staying orthogonal to earlier axes. PCA can summarize risk and create predictors, but it does not automatically discover economic factors. Results depend on units, sample dates, covariance estimation, and component selection, so scores, loadings, explained variance, and out-of-sample stability belong in the same analysis.
Put the data on a deliberate baseline
PCA is applied to centered data after subtracting each variable's sample mean. Otherwise, average levels can dominate what should be a direction of variation.
Choose meaningful transformations before the calculation. A common trend in nonstationary price levels can masquerade as a useful shared risk signal.
Missing values, outliers, frequency, and the sample window all alter covariance. Data design before PCA matters more than the name assigned afterward.
Covariance and correlation answer different questions
Covariance-matrix PCA preserves original units, so a variable with greater volatility can exert more influence on the leading directions.
Correlation-matrix PCA first scales every variable to unit variance. It limits unit effects but may discard economically meaningful differences in magnitude.
The objective determines the choice. Similar-unit yield changes and a mixed collection of indicators do not necessarily call for the same matrix.
Components maximize variance in sequence
The first principal component is the unit direction whose projected observations have the greatest possible sample variance.
Every later component maximizes remaining projected variance subject to being orthogonal to all components selected before it.
Orthogonality makes component scores uncorrelated in the fitted sample. It does not prove that underlying economic shocks are independent.
Loadings and scores do different jobs
Loadings are the weights combining original variables into a component. They show which variables point with or against a particular axis.
Scores are the coordinates obtained by projecting each observation onto that axis. Loadings define a direction; scores move through time along it.
An eigenvector can be multiplied by minus one without changing its axis. A sign flip alone is not evidence that an economic relation reversed.
Explained variance is an incomplete objective
A component's eigenvalue equals its sample variance. Dividing it by the sum of all eigenvalues produces its explained-variance ratio.
A scree plot and cumulative explained variance are starting points for choosing dimension. A fixed threshold such as 80% is not universally optimal.
A low-variance direction can still contain predictive information or a binding hedge relation. Variance explained does not equal return predictability.
Select dimension outside the fitted sample
Too few components omit structure, while too many preserve sample noise alongside persistent common movement.
Compare reconstruction error, task-specific cross-validation, eigenvalue gaps, and bootstrap stability according to the intended use.
Re-estimate loadings and explained variance through rolling samples. Using full-sample axes in a historical forecast introduces future information.
Finance uses PCA to summarize common movement
Yield-curve changes may compress into level, slope, and curvature-like axes, while equity returns may reveal broad and group-specific directions.
For portfolios, the leading axes show whether risk is concentrated despite a large nominal number of holdings.
Calling a statistical axis “growth” or “liquidity” requires external variables, economic reasoning, and validation in another sample.
Report choices that make the result reproducible
Disclose variables, transformations, currencies, units, centering, scaling, missing-value treatment, outliers, and the estimation window.
State the matrix, eigenvalues, explained variance, loadings, scoring rule, and dimension criterion so another analyst can rebuild the result.
Show out-of-sample performance and rolling stability. PCA is an estimated coordinate system for chosen data, not a fixed map of reality.
Common questions
Are PCA and factor analysis the same?
No. PCA re-expresses total variance, while a common-factor model separates common covariance from variable-specific variance.
Should every variable always be standardized?
No. Standardization controls unit differences but also removes potentially meaningful differences in volatility magnitude.
Does a loading sign flip mean the structure changed?
Not necessarily. An eigenvector and its negative define the same axis, so signs must be aligned before comparing windows.
Is a high-explained-variance component a good trading signal?
No. Explained variance measures contemporaneous movement; predictive power requires a separate out-of-sample test.
Sources and further reading
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