doe: the metric-choice figure and the worked-study figures behind pid-book #270

[kgdunn]

Parallel to pid-book #270, which adds the sections these figures illustrate. Merge this one first, so the book's PDF build can find the images.

What is added

doe/omars-metric-choice.py and its PNG

The figure reads candidate cell measures down the three-factor column of the OMARS trade-off table: the five alphabetic optimality criteria (A, D, E, I, G), the largest absolute correlation between any two second-order terms, and, in a third row, the power to detect a main effect, a quadratic and a two-factor interaction of one noise standard deviation at the 5% level. Each point is the best value attainable at that run count, so the curves are frontiers rather than the output of a search. Three series per panel, for one, two and three centre runs, in blue, purple and dark brown (the brown chosen after checking the three remain distinct under simulated colour-vision deficiency).

Panels are grouped so the pairs are visible. Columns: the left pair averages, the middle pair takes the matching worst case, the right pair does neither. Rows: the top summarises X'X, the middle is prediction variance over the region apart from the last panel, the bottom is power. All nine panels share one x-axis range and tick set. The Box-Behnken design (green star) and the definitive screening design (orange circle) are marked on every panel, in the colours the trade-off staircase already uses for them.

Five correlation-map insets sit on the max |r| panel, on a common shading scale, each outlined and connected in the colour of its centre-run series. The three on the left are the smallest design at each centre-run count (9, 10 and 11 runs); they share the same four half-rows, so together they isolate what the centre runs do to the map: the quadratic-to-interaction correlation falls from 0.707 to 0.645 to 0.606 while the quadratic-to-quadratic correlation rises from zero to 0.167 to 0.267. The remaining two, at 24 runs with two centre runs and 31 runs with one, show the map at moderate and near-zero entanglement.

doe/omars_metric_choice_data.py

The enumeration behind the literals in the figure script, committed so the numbers can be re-derived rather than trusted. Finding the best value at each size means enumerating every OMARS foldover of that size. That is tractable because a foldover is described entirely by how many times each sign pattern of {-1, 0, +1} appears in its half: every second-order term takes the same value on a half-row and on that row negated. Main effects are orthogonal, so their variance is 1/(2 n_j) directly; quadratics and interactions come from the even block; the full model is attempted only where h >= k(k+1)/2, and any rank-deficient cell is rejected. --verify rebuilds twelve cells from naive numpy model matrices and compares: largest absolute difference 4.9e-07.

Two further checks that the data passes: all enumerated cells are monotone in the run count for the five alphabetic criteria, which is what the Loewner-order argument predicts, and G never falls below its Kiefer-Wolfowitz floor of p/N, touching it exactly at three of the sizes.

doe/omars-worked-study-tradeoff.py, its PNG, and doe/omars_worked_study_data.py

For the new book section "A worked OMARS study". The same four-factor study on the fed-batch bioreactor simulator (process_improve.simulation.batch) is run 200 times at 13, 17, 21, 27 and 31 OMARS runs and with the 27-run Box-Behnken and face-centred central composite designs, each campaign scored by the true titer at the recipe its fitted model recommends, minus the titer of the current recipe. The figure shows the median, the 10th-to-90th percentile band and the worst campaign for each design, against the best possible gain of 2.006 g/L. Two hundred campaigns per design take minutes, so the values are carried as literals; the data script regenerates them (--quick for a 20-seed smoke test).

Six more worked-study figures, and doe/omars_worked_study_common.py

One script per PNG, following the directory's convention, each importing the study from the common module:

• omars-worked-study-recipe: the current recipe's temperature schedule with the four corners of the hold-temperature by shift-day range, beside titer against day for twenty replicate batches over the disturbance-free batch.
• omars-trade-off-column-k4: the four-factor column of the OMARS trade-off table on its own, at every odd run count from 9 to 31, with the staircase's fills, the estimability frontier outlined at 21 runs and the Box-Behnken design closing the column at 27. Every cell is read from the library.
• omars-worked-study-plan: the thirty batches in run order as a level grid, one row per factor, with the cassette boundary and the four centre runs outlined.
• omars-worked-study-titer: titer at harvest by run and cassette, centre runs starred, each cassette's centre-run mean drawn, the replicate mean as a reference.
• omars-worked-study-effects: the fifteen coefficients of the full second-order model fitted in one step, with 95% intervals on 15 df, the three terms the staged analysis selects filled. The script asserts that the terms significant at 5% are exactly those three, which the chapter states.
• omars-worked-study-surface: the fitted four-term model beside the true response over hold temperature and shift day at pH 7.1 and the high feed rate, on shared contour levels, with the current, recommended and best recipes marked on both.

omars_worked_study_common.py reproduces the study exactly as the chapter's code blocks run it (same factors, disturbance level, design seed, pair split, run order and disturbance seeds) and asserts the values the chapter prints (titer range 4.290 to 9.116 g/L, cassette effect -0.1272, recommended hold 29.54 °C, true titers 7.436 / 8.376 / 9.442 g/L) before any figure is drawn. A change in the library that moved any of those stops every figure script rather than silently redrawing it.

Conventions followed

Matplotlib, Okabe-Ito palette matching the other OMARS figures in doe/, scripts committed alongside the PNGs in the same directory, run from that directory to write the image next to themselves.

🤖 Generated with Claude Code

https://claude.ai/code/session_01W1daHPaXLs7PPRsq8CmMMm