Compute elements of principal components analysis (PCA) (e.g., loading vectors, variance explained).

A reserving analyst hands you a development triangle with 12 age-to-age factors per accident year and asks which two synthetic factors capture the bulk of the systematic movement. PCA answers exactly that question.

Why PCA exists. When you have $p$ correlated predictors, much of their joint variation lives in a lower-dimensional subspace. PCA finds the orthogonal directions that capture variance in decreasing order, giving you uncorrelated synthetic predictors ranked by informativeness.

Let $X$ be the $n\times p$ data matrix with rows $x_i$. PCA requires column-centered data:

$${\tilde{x}}_{ij}=x_{ij}-{\bar{x}}_j$$

If columns are on different units (dollars and counts, say), also divide by the sample standard deviation. This is equivalent to running PCA on the correlation matrix rather than the covariance matrix.

DECISION: Mixed units or wildly different variances ⇒ standardize (use correlation). All variables on the same comparable scale ⇒ raw covariance is fine.

The first loading vector ${\varphi }_1=({\varphi }_{11},\dots ,{\varphi }_{p1}{)}^{\top }$ is the unit vector that maximizes the sample variance of the linear combination ${\varphi }_1^{\top }{\tilde{x}}_i$:

Common mistakes

• Skipping standardization with mixed units. Running raw-covariance PCA on dollars plus counts lets dollars dominate PC1 by accident. The correct fix is the correlation matrix; the wrong "PC1 explains 99%" answer is the giveaway.
• Adding squared loadings across components instead of within a component. Each ${\varphi }_m$ is unit length individually: ${\sum }_j{\varphi }_{jm}^2=1$. It is the rows of the loading matrix that need not sum to one. Test-takers who report $\Vert {\varphi }_1{\Vert }^2+\Vert {\varphi }_2{\Vert }^2=2$ are not doing anything wrong, just not anything useful.
• Confusing eigenvalues with PVE. Reporting ${\lambda }_1=10$ as "PC1 explains 10%" is a units error. Always divide by the trace; here that gives 62.5%, not 10%.

Bottom line

• Standardize first when predictors are on different scales; otherwise the largest-variance variable dominates the first component for purely unit-based reasons, so eigendecompose the correlation matrix.
• Loading vector ${\varphi }_m$ is a unit-length eigenvector of the sample covariance (or correlation) matrix with the $m$th largest eigenvalue. The $m$th score is $z_{im}={\varphi }_m^{\top }{x}_i$.
• Variance explained by component $m$ equals ${\lambda }_m$; proportion explained is ${\mathrm{PVE}}_m={\lambda }_m/{\sum }_j{\lambda }_j$.
• Total variance equals the trace $={\sum }_m{\lambda }_m$; for correlation-matrix PCA the trace equals $p$.

Exam shortcut

If the problem gives only eigenvalues, compute trace and divide; do not try to recover the loadings. If the problem gives a loading vector and an observation, the score is just the dot product, period. If a problem mixes units without telling you to standardize, assume correlation-matrix PCA and use trace $=p$; the alternative is almost always a trick designed to inflate PC1.

The full lesson (about 2,019 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• C5

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