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Calibration Pipeline: Fit, Diagnose, Defend

Build a small calibration report that explains what a linear model can identify, what it approximates and how its answers respond to uncertain data. Budget 12–18 hours. Use the offline kit; a reproducible script plus written derivations is sufficient. A general SVD implementation is outside scope.

Fixture and required outputs​

Observations are y=(1,2,2) at t=(0,1,2), with design columns one and t.

  1. Attempt an exact solve and produce an inconsistency witness.
  2. Fit c+mt by QR. Record coefficients, fitted values, residual, squared error, factor reconstruction and orthogonality. Prove projection optimality.
  3. Add feature 2t. Explain the new null direction and nonunique coefficients; show that the best fitted vector is unchanged. Use a supplied analytic argument to obtain the minimum-norm coefficients, instead of forcing the full-rank QR API.
  4. Examine A=[[1,1],[1,1.0001]] with b=(2,2) and b'=(2,2.0001). Solve exactly and report infinity-norm sensitivity. Distinguish uncertain input from rounding.
  5. For diag(4,1), use the supplied identity singular vectors to report rank-one error and pseudoinverse behavior. Repeat with diag(1,10^-8) and a declared cutoff.
  6. Compare power iteration on [[2,1],[1,2]] from (1,0) and (1,−1). Explain why a zero residual in the second run does not establish dominance.

Changed assumptions​

Change the middle observation to three and refit. Then assign it weight four, transforming both design and response. Derive the new objective and explain why its residual need not be orthogonal in the original unweighted metric. Finally add ridge penalty lambda*||x||² to the duplicate-feature design; report the penalty convention and show how the fit changes. A numerically accurate answer to an unstated new objective fails the project.

Calibration values — consult after attempting​

The original exact system is inconsistent; y=(1,−2,1) annihilates the design and gives −1 against the observations. The ordinary fit is (c,m)=(7/6,1/2), residual=(−1/6,1/3,−1/6), squared error 1/6. For columns (1,t,2t), all fits have intercept 7/6 and slope coefficients u+2v=1/2; minimum norm gives (c,u,v)=(7/6,1/10,1/5). The null direction is (0,−2,1).

With observations (1,3,2), the unweighted fit is (3/2,1/2). With weights (1,4,1), it is (2,1/2), fitted (2,5/2,3), residual (−1,1/2,−1). Weighted residual sums and weighted t-products vanish. They need not vanish without the weights. Derive these results rather than copying the values.

Acceptance and carry-forward​

Submit a model specification, derivations, script, exact certificates, numerical checks with scales, changed-assumption results and a short decision memo. All ten competency gates must meet their standard; neither a plot nor a library return value replaces a missing proof.

Record first attempts, assistance, reviewer disagreement and one failure that changed your approach. Carry these artifacts into the semester evidence packet. State which results depend on exact data, tolerances or an intentional model change.