Manuscript En01-B@2 · En01-B-v2.tex · section · f226d1eb

Moving trajectories and additional nuisance templates

Preparation and detector conditions for the control › Moving trajectories and additional nuisance templates · lines 2041–2070

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Suppose a validated branch description gives

Y_{h, i} = β_{0, h} + β_{2, h} F_{i} + hλ B_{i} + r_{h, i},

where

F_{i} = \int d_{i} (u)^{2} d u, B_{i} = \int Γ_{r_{c}} (d_{i} (u)) d u .

Choose weights with

w^{⊤} 1 = 0, w^{⊤} F = 0.

Then Theorem thm:contrast applies with $λ t B_{r_{c}}$ replaced by $λ w^{⊤} B$ .

More generally, for a matrix $N$ of declared nuisance templates, a nuisance-cancelling linear contrast exists precisely when

B \in / range N .

One choice is $w = (I - P_{N}) B$ , giving $w^{⊤} B = ∥ (I - P_{N}) B ∥^{2} > 0$ . This is linear algebra on a validated attenuation model, not proof that a branch-trajectory model describes the entire interferometer.

Displayed formulas in this passage

unlabeled · line 2044
$Y_{h, i} = β_{0, h} + β_{2, h} F_{i} + hλ B_{i} + r_{h, i},$
unlabeled · line 2049
$F_{i} = \int d_{i} (u)^{2} d u, B_{i} = \int Γ_{r_{c}} (d_{i} (u)) d u .$
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$w^{⊤} 1 = 0, w^{⊤} F = 0.$
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$B \in / range N .$