Drift rejection by contrast orthogonality

Drift rejection by contrast orthogonality removes specified additive time trends from a load comparison by choosing contrast weights orthogonal to those trends. In an aerodynamic interference experiment, it prevents an assumed form of balance-zero drift from being confused with nonadditivity between component configurations.

For observations at times $t_r$, writeThe columns of $B$ specify nuisance trends and $\mathbf{\gamma }$ their unknown amplitudes. The drift contribution vanishes for every amplitude exactly whenFor polynomial instrumental drift through degree $p$, the columns are $1,t,\dots ,t^p$, so the conditions are ${\sum }_r{w}_r{t}_r^j=0$ for $j=0,\dots ,p$. This orthogonality is a statement about actual observation times, not merely the number of repetitions.

An original eight-run construction uses equally spaced times $t=0,1,\dots ,7$ in a fixed time unit and the configuration orderFor the factorial contrast for aerodynamic interference, chooseEach configuration occurs twice and its total weight is the coefficient required for $\Delta =D_{11}-D_{10}-D_{01}+D_{00}$. Direct summation givesThe schedule rejects constant, linear and quadratic additive drift, but a cubic term $c{t}^3$ leaves bias $-24c$. If equal run spacing changes, the moments must be recomputed. If independent residual errors have common variance ${\sigma }^2$, the contrast variance is ${\sigma }^2{\mathbf{w}}^T\mathbf{w}=2{\sigma }^2$; correlated errors instead require the full covariance matrix.

Rejection must preserve the physical signal. If a desired signal column lies in the nuisance column space, every vector rejecting that space also rejects the signal. Orthogonal projection exposes this identifiability limit; collecting more readings in the same confounded arrangement need not fix it. Gain drift, configuration-dependent tare and abrupt offsets are also outside the stated additive polynomial model.

Kenneth E. Ward's original attached- and isolated-strut measurements motivate separating combined load from component drag.^[ref] The eight-run schedule here is a constructed design, not a reconstruction of that historical experiment.