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

Diagnostic attenuation model

Beyond quadratic localization: an explicit control › Diagnostic attenuation model · lines 1478–1517

En01-B@2#eq:Y

For a setting with fixed effective separation $d_{i}$ , suppose

Y_{h, i} = - lo g V_{h, i} = β_{0, h} + β_{2, h} d_{i}^{2} + hλ t Γ_{r_{c}} (d_{i}) + r_{h, i}, V_{h, i} > 0.

The common offset and quadratic coefficient can differ between fitted hypotheses, but each is shared across the settings of its hypothesis.

The constant and quadratic forms are motivated by saturated short-wavelength and long-wavelength localization, respectively Pino2018 (Sec. II). Actual gas-scattering localization and setting-dependent preparation are not assumed to obey them exactly; their departures enter $r_{h, i}$ .

Equation (eq:Y) is a controlled diagnostic, not the complete seven-stage prediction.

Choose

(d_{1}, d_{2}, d_{3}) = R (1/4, 1/2, 3/4), w = (5, - 8, 3) .

Then

i \sum w_{i} = 0, i \sum w_{i} d_{i}^{2} = 0.

Define

C_{h} = i \sum w_{i} Y_{h, i}, B_{r_{c}} = i \sum w_{i} Γ_{r_{c}} (d_{i}) .

Displayed formulas in this passage

En01-B@2#eq:Y · line 1481
$Y_{h, i} = - lo g V_{h, i} = β_{0, h} + β_{2, h} d_{i}^{2} + hλ t Γ_{r_{c}} (d_{i}) + r_{h, i}, V_{h, i} > 0.$
unlabeled · line 1500
$(d_{1}, d_{2}, d_{3}) = R (1/4, 1/2, 3/4), w = (5, - 8, 3) .$
unlabeled · line 1506
$i \sum w_{i} = 0, i \sum w_{i} d_{i}^{2} = 0.$
unlabeled · line 1512
$C_{h} = i \sum w_{i} Y_{h, i}, B_{r_{c}} = i \sum w_{i} Γ_{r_{c}} (d_{i}) .$

Nearby in this section

theorem: Theorem: Nuisance cancellation and robust margin