We have a dataset of 50 chemical elements ($X$) found in clay samples. To find the geographic source, we project this down to 2 dimensions using Principal Component Analysis.
We seek a linear combination that maximizes variance:
$$ \mathbf{w}_{(1)} = \underset{\|\mathbf{w}\|=1}{\operatorname{arg\,max}} \, \left\{ \sum_i (t_1)_{(i)}^2 \right\} $$Where our target matrix $T$ relates to the original data $X$ and weights $W$ by:
$$ T = X W $$This reveals that the Al/Ti ratio is the strongest predictor of the kiln site.