Dimensionality Reduction in Clay

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.