Sequential Orthogonalisation of a Signal Set

When a research desk has a stack of candidate signals that are all correlated with each other — several momentum variants, say, that all partly capture the same underlying effect — combining them naively double-counts whatever they share. Sequential orthogonalisation fixes this using the same idea as Gram-Schmidt: order the signals by priority (often the strongest standalone performer first), keep the first signal as-is, then for every subsequent signal, regress it on all the signals already kept and retain only the residual — the part of the new signal that isn't explainable by anything already in the set.

Concretely, with signals $x_1, x_2, x_3$ ordered by priority: $x_1$ is kept unchanged. $x_2$ is replaced by the residual of regressing $x_2$ on $x_1$, so the resulting signal is exactly uncorrelated with $x_1$. $x_3$ is replaced by the residual of regressing $x_3$ on both $x_1$ and the orthogonalised $x_2$, making it uncorrelated with both. The result is a set of signals that are mutually orthogonal by construction, each representing genuinely incremental information over everything ranked ahead of it, which can then be combined by simple addition without worrying about double-counting shared exposure.

The catch is that the ordering matters: whichever signal goes first keeps 100% of its raw information, while a signal ranked last only survives as whatever thin residual remains after everything else has claimed the shared variance — so a signal that's genuinely valuable but highly correlated with an earlier one can end up looking almost worthless after orthogonalisation, purely because of where it sat in the queue.