I am using principal component analysis for dimensionality reduction on a data set with many correlated variables, and then using the resulting principal component scores to perform LDA to discriminate samples into different groups.
We collect one scale from hundreds of fish, and for each scale, distance and count measurements are collected on the scale rings (much like tree rings). That is, for each scale belonging to an individual fish, we record several different variables. This gives us information on fish growth, age, and how much time they spend in freshwater vs saltwater environments, for example. Some of these measurements may be correlated because they overlap with one another (see image below- a distance of line a overlaps with the distance of line b). Measurements have been done this way for many years to facilitate biological interpretation. The scale variables can tell roughly which geographical region a fish comes from.
PCA is done on these measured scale variables to reduce the dimensionality and co-linearity between variables. Next, LDA is performed on new scale samples. The LDA discriminate functions are derived from PCA training data component scores. Discriminate functions are applied to transformed test (new) data (i.e. transformed using the rotation matrix generated from PCA training data) to get Bayesian posterior probabilities for each scale sample belonging to a certain geographical region.
I am wondering whether more information can be gained if we used derived variables in the PCA, compared to the actual measured variables. What I mean is, instead of using the distance of line a and distance of line b in the PCA, what if we divided the line segments such that no line segments overlapped and included 3 different non-overlapping distances in the PCA instead? If the overlapping line segment between line a and b was indeed distinct depending on geographical region, would we be able to increase our ability to distinguish different scales among geological regions by separating it into its own variable?
If no additional information, i.e., the variance that distinguishes scales by region, can be gained by using derived estimates from measured variables, why?
