Identifiable Calibration with a Holdout

These examples and illustrative results are independently authored teaching materials, not measured model results.

Use case

Positioning an annotation in a document editor requires converting screen locations to document coordinates. Teaching y references are document 100→screen 250 and document 300→screen 650, with independent document 200→screen 450. Assume unrotated aligned axes, but unknown scale and origin.

Mechanism

Confirm page identity, zoom and the same anchor in both systems. For each unknown axis obtain two pairs with distinct document coordinates, then compute scale=(screen2-screen1)/(document2-document1) and origin=screen1-scale×document1. Reject degenerate, nonfinite or contract-invalid scales. Verify an additional reference within a defined tolerance, then convert targets using document=(screen-origin)/scale. Establish x separately rather than guessing from y. Invalidate after layout or zoom changes.

Bad example

Use document 100→screen 250 alone to assume scale 2.5 and place screen 500 at document 200, without checking origin or another reference.

Good example

Solve y using 100→250 and 300→650. Verify independent 200→450 within a teaching tolerance of 1 screen unit before converting target screen 550. Report parameters, residual, page and zoom identity. Do not position in two dimensions until x is established.

Why the change matters

One pair cannot identify both scale and offset. Fitting points do not independently validate the fit; another reference can expose incorrect anchors or a model unsuitable for the layout.

Observable expectation

Teaching calculation gives scale=2 and origin=50. The independent prediction is 450 with residual 0; screen 550 maps to document 250. If the independent observation is 460, residual 10 exceeds this example’s tolerance 1 and positioning stops. That tolerance is illustrative, not universal accuracy.

Limits

Rotation, shear and nonlinear transforms require another model. Identical coordinates on an axis cannot identify its scale. Share scales only with independently established uniform scaling. Unknown tolerance, changed scrolling origin or the wrong page invalidates reuse.