Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 47 / 3 / c / i / Solution

Covariance-based PCA uses centered protein measurements in their original gram units. Its total variance is $218.4$. Correlation-based PCA first divides each centered food measurement by its sample standard deviation, giving nine unit-variance variables and total variance nine. Thus it is principal component analysis on a correlation matrix, not an equivalent rotation of the original unscaled measurements.

The variances of cereals and milk together account for $170.9/218.4\approx 78.3\%$ of the original total. Consequently those variables have considerable influence on covariance-based PCA. Its first component is dominated by a cereal coefficient $-0.86$ opposed to milk $0.42$, broadly a cereal-versus-milk contrast. Its second is dominated by milk $0.83$, with cereals $0.40$ and fish $-0.29$. Its third is chiefly white meat $0.80$ versus fish $-0.52$, with a smaller negative milk term. Component signs may all be reversed without changing the analysis; the important features are relative signs and magnitudes.

After standardization, eggs and starch can contribute as strongly as large-variance foods. The first correlation-based component gives positive weight to meat, eggs, milk and starch, opposed principally to cereals and pulses/nuts; eggs now has coefficient $0.43$ despite its small raw variance. The second emphasizes fish and fruit/vegetables negatively, opposed especially to starch and white meat. The third contrasts white meat and fruit/vegetables with milk, fish and red meat. Thus standardization changes both the emphasis and the directions, rather than just the displayed units of the same components.

The explained variance of a principal component must use the appropriate total:Raw and standardized percentages describe different variance objectives and must not be compared as if one method were always superior.

Standardize when relative patterns across variables, rather than their absolute gram-scale variation, should have equal marginal weight. Correlation-based PCA is also insensitive to changing a variable's measurement unit, which covariance-based PCA is not. Here every food group already has the same units, so differing units do not force standardization. If large fluctuations in actual protein amounts are scientifically important, covariance-based PCA is appropriate. If dietary composition across all food groups is the focus, correlation-based PCA can be more informative. Standardization can nevertheless amplify noise in nearly constant variables; the decision should follow the scientific purpose and measurement reliability.