Module correction 

Per-pair multiplicative correction of an electronic-stopping model, fitted to measured points, with a bootstrap uncertainty and held-out residuals.

A Correction wraps a base ElectronicStopping model and multiplies its stopping cross section by a smooth factor g(E) for one ion–target pair. The factor is

g(E) = exp( sum_{k=0}^{d} c_k T_k(x) ),   x = (2 ln E - ln E_lo - ln E_hi) / (ln E_hi - ln E_lo)

where T_k are Chebyshev polynomials of the first kind (J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, CRC Press (2003), ch. 2) and [E_lo, E_hi] is the energy span of the training points. The exponential keeps g positive. The coefficients minimise sum_i w_i (ln(S_i / B_i) - sum_k c_k T_k(x_i))^2, where S_i is the measured stopping, B_i the base model at the same point and w_i = 1 / (sigma_i / S_i)^2 when every training point states an uncertainty sigma_i > 0, otherwise w_i = 1. The normal equations are solved by Gaussian elimination with partial pivoting (G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins (2013), sec. 3.4 and 4.2). The form of the correction (a low-order polynomial of ln E for the log of the ratio) is our own choice; no published fit is reproduced.

Where the correction applies

Only inside [E_lo, E_hi] of the training points, and only for the ion (Z and mass) and target element it was fitted to. Anywhere else Correction::stopping returns the base model unchanged, because a polynomial extrapolated beyond its data has no physical basis.

Uncertainty

The nonparametric bootstrap of B. Efron, Ann. Statist. 7, 1 (1979): the training points are resampled with replacement replicates times, the correction is refitted to each resample, and the pointwise band at energy E is the central interval of g_b(E) over the replicates (linear interpolation between order statistics). A resample whose fit is singular (for example fewer distinct energies than coefficients) is discarded and counted in CorrectionFit::failed_replicates. The band covers the scatter of the points about the fitted form; it does not include any systematic error of the measurement or any bias of the polynomial form. If the measured points carry no stated uncertainty (sigma 0, as in tables read off a smoothed curve), the band says nothing about measurement error.

Reproducibility

The held-out split uses the stream (seed, 0) of crate::rng::stream and replicate b uses (seed, b + 1). Replicates run on the current rayon pool and are collected in index order; every sum is sequential, so the result is bit-identical for a given seed whatever the thread count.

Held-out points

A deterministic, seeded subset of the interior points (never the lowest and highest energy, so they stay inside the fitted span) is left out of the fit. CorrectionFit::held_out_report gives the residuals of the base model and of the corrected model on exactly those points, side by side, whether or not the correction helps.

Structs

Correction

An ElectronicStopping model multiplied by a fitted CorrectionFit.

CorrectionFit

A fitted correction: coefficients, bootstrap replicates and the split.

FitSettings

Settings of fit_correction.

HeldOutPoint

One held-out point with the base and corrected model values.

HeldOutReport

Residuals of the base and the corrected model on the held-out points.

ResidualStats

Statistics of model - measured over a set of points.

Constants

CORRECTED_MODEL_NAME

Name returned by Correction::name.

Functions

fit_correction

Fit a Correction of base to a single-system element-target dataset.