Suggest next point

A c-optimal suggestion for \(T_c\) on a synthetic order-parameter trend (generic \(T_c = 100\) K). The shaded band is the utility curve; the dashed line marks the suggested x. The result line reports the events factor and the Monte-Carlo-calibrated post-fit sigma.

Once a trend model is fitted (Parameter trending), the natural next question at the instrument is where to put the next run, and how long to count it, to constrain that model the most. The Suggest next point section of the model-fit dialog answers this from the fit you already have — its parameter values and their covariance — rather than from a rule of thumb.

How it works

Treating the fit’s parameters as approximately Gaussian around their fitted values (the Laplace approximation), a hypothetical new measurement at some \(x\) reduces the uncertainty in the model parameters by an amount that depends on how sensitive the model is to each parameter there, and how precisely the new point would be measured. Two standard design criteria turn that into a single number per candidate \(x\):

c-optimal (the default) — minimise the posterior variance of one parameter of interest, e.g. “pin down \(T_c\)”. You choose the target parameter from the model’s free parameters.
D-optimal (“all parameters”) — shrink the whole parameter covariance at once, without favouring any single one.

Both are evaluated as an expected information gain, using the fit’s covariance matrix and the model’s sensitivity to each parameter (a numerical derivative at the fitted values) — full derivations are not needed to use the feature; what matters in practice is that the result is a curve over :math:`x`, not a bare number, so you see why one region is more informative than another before committing beam time to it. The whole curve is shown, together with its maximum, so you retain judgement over the suggestion rather than being handed a single instruction.

A worked example

Fit a trend model as usual (here the OrderParameter form on a synthetic \(\nu(T)\)-like trend through a generic \(T_c = 100\) K), then open the Suggest next point section beneath the fit result:

Target. Choose the parameter you most want to pin down — Tc, in the screenshot above — or All parameters (D-optimal) to reduce the whole covariance ellipsoid instead.
Candidate range. Defaults to the measured x span; widen it if you are willing to measure outside the runs you already have (extrapolated candidates are drawn in a visually distinct style, see below).
Click Suggest. The utility curve appears as a shaded band on the preview with a dashed line at the suggested x, and the result line reads, for example:
```
Measure at x = 97 × 0 of a typical run's statistics
→ σ(Tc) ≈ 0.603 (MC-calibrated)
```
For this order-parameter trend the suggestion lands just below the fitted \(T_c\), where \(\partial y/\partial T_c\) is steepest — exactly where classical optimal-design theory places the information for a critical-point model. The “× 0” here means the precision goal below is already met at the existing statistics — see the next step.
Precision goal. With a single-parameter target, type a target sigma (e.g. 1.0 for \(\sigma(T_c) \le 1\) K) in Precision goal. The result line’s events factor is how many times a reference run’s event count (the existing runs’ own statistics) you would need at the suggested x to reach that goal, found by solving the posterior variance for \(N\) in closed form. A factor of 0 means the goal is already met without a new point at all; a factor below 1 means less counting than a typical run is enough; above 1, more.
Typical run / rate. Fill in your instrument’s typical run size (Mevents) to convert the events factor into an absolute Mevents figure, and optionally a count rate (Mevents/h) to additionally show the equivalent counting time. Both fields are display-only: they convert the recommendation into units you can act on at the instrument, but never feed back into the suggestion itself — events, not wall-clock time, are what set the statistics, and beam conditions vary too much for a time estimate to be part of the calculation.

Why the sigma is “MC-calibrated”

The rank-one update used to rank candidates is exact for a linear model, but for the curved models trending most often fits — order parameters, critical divergences — it can underestimate the realised post-fit uncertainty near the critical point, because the model’s local-linear approximation breaks down exactly where the sensitivity is largest. Rather than show that optimistic figure, the dialog runs a short Monte Carlo pass off-thread the moment you request a suggestion: it simulates adding the proposed point (drawn from the fitted model plus the empirical noise at the target statistics), refits, and repeats a few dozen times. The median realised sigma is what the result line reports, labelled (MC-calibrated); if that pass has not finished yet the line briefly reads “calibrating…” and falls back to the analytic figure (labelled (approximate)) if the calibration itself fails. The ranking of candidates — which \(x\) is better than which — is reliable either way; it is only the absolute predicted sigma and event count that the calibration corrects.

Warnings you may see

The section always shows the utility curve when one can be computed, but flags the cases where you should trust it less rather than hiding it:

“Fit is barely constrained… consider a coarse scan.” Too few points for the number of free parameters means the fitted covariance itself is unreliable, and a model-driven suggestion can confidently point at the wrong place. This is the single strongest lesson from the autonomous scattering literature the method is grounded in: model-based suggestion only helps once a feature has been localised by an ordinary scan: it is not a substitute for one.
“Fit covariance is ill-conditioned (strong parameter correlations); utilities are approximate.” Two or more parameters are strongly correlated (a large condition number on the covariance matrix), which inflates the numerical sensitivity of the suggestion to the exact fitted values.
“The suggested point sits near a model domain boundary; its utility is step-sensitive and approximate.” Some models change behaviour sharply at a boundary (an order parameter vanishing above \(T_c\)); right at that edge, the numerical derivative the acquisition relies on is sensitive to the step size used to compute it, so treat the ranking there as approximate.
“The precision goal cannot be reached with a single new point…” The posterior variance of the target parameter has a floor set by the other parameters’ uncertainty — no amount of counting at one point drives it below that floor. The dialog reports the floor sigma rather than suggesting an absurd event count.
Extrapolated candidates (outside the measured x span) are drawn in a visually distinct style on the preview, so a suggestion that reaches beyond your data is obvious at a glance rather than a warning you have to read.

Compare against: which model fits better?

The Suggest next point section with a PowerLaw alternative fitted and the discrimination overlay showing the best discriminating point

The same trend with an alternative PowerLaw model fitted for comparison. The AIC evidence line reports the alternative’s weight is decisively lower than the order-parameter leader’s, and the second (tan) overlay shows where a new point would best tell the two models apart.

Pinning down one model’s parameters is a different question from asking whether that model is even the right one. The Compare against row lets you fit an alternative model over the same masked data as the active range — pick a component from the list, or Edit… to build a composite the same way you build the primary model — and click Fit & compare. Each successful alternative is added to a running list beneath the button, with:

AIC evidence. Each candidate’s Akaike weight \(w_i \propto \exp(-\mathrm{AIC}_i/2)\), and its ratio against the leading model. A ratio above 100 is called decisive — the standard Jeffreys-scale convention for “the data overwhelmingly prefer one model over the other”. In the screenshot, an alternative PowerLaw fit is decisively disfavoured against the OrderParameter leader for this trend, as expected of data actually generated from an order-parameter form.
Best discriminating point. A second overlay, drawn in a visually distinct style from the refinement band, showing where a new measurement would best separate the worst-agreeing alternative from the current leader — the point of maximum disagreement between the two curves, weighted by the expected measurement noise there. This is a genuinely different question from “where pins down \(T_c\) best”, and can suggest a different \(x\).

With two or more candidate models that already agree with each other everywhere in the candidate range, the dialog reports that no discriminating point exists rather than pointing at an arbitrary location — there is nothing in the current range that would tell the models apart.

Cost weighting

Weight by measurement cost (off by default) accounts for the fact that moving the instrument and counting both take time, and that time is not the same in every direction — cooling and warming a cryostat, or ramping a magnet up and down, rarely take equally long. Supplying a counting time per point, a move-time rate for increasing and decreasing \(x\), and the instrument’s current position re-weights the utility curve by a cost model (\(\mathrm{utility}^{0.7}/\mathrm{time}\), following the exponent used in the autonomous-scattering literature this method draws on) and moves the marker to whichever \(x\) is most informative per unit time, not per measurement.

Cost weighting changes where the marker sits, never the underlying precision figures: because those figures (the predicted sigma, the events factor) describe the unweighted best point, they are dropped from the result line whenever weighting moves the marker elsewhere, replaced by a “(cost-weighted)” note. Toggling the checkbox or editing the cost fields re-renders instantly from the last computed suggestion — it never re-runs the underlying calculation or the Monte Carlo calibration.

Limitations

The suggestion assumes the fitted model form is correct. It tells you where to measure to constrain this model best, not whether a different model would fit the data better — use Compare against for that question, and treat a persistently poor \(\chi^2_r\) as a sign that the refinement suggestion itself rests on shaky ground.
The analytic post-fit sigma is a rank-one, locally linear approximation, and is known to be optimistic near the sharp features (critical points, boundaries) that these models often have — this is exactly why the Monte Carlo calibration pass exists, and the calibrated figure is the one to trust for planning.
Errors set to None or Estimate from scatter disable the section entirely (the target selector, goal field, and Suggest button are all greyed out). Both modes carry no real per-point noise estimate to interpolate for a hypothetical new point — a unit-weight fit has no absolute noise scale, and a scatter-rescaled fit only recovers one after the fact — so there is nothing physical for the acquisition to predict a new measurement’s precision from. Switch to Column, Percent, or Absolute errors to use the feature.
The section itself is only available once a range has a successful fit with a covariance: if HESSE did not run or did not converge, there is no covariance to compute a sensitivity-weighted suggestion from.