 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iii/9/b/iii/solution

The principal component scores are the projections of centered observations onto the loading directions:

Z = X_{c} V, Z_{ij} = k = 1 \sum p (X_{ik} - \overline{X}_{k}) V_{kj} .

X

already denotes centered data, this simplifies to

Z = X V

. The principal component loadings are directions; the principal component scores are coordinates of individual observations in those directions.

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