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Background models and estimation

A background contributes an additive baseline to a powder calculation. It is separate from excluded observations, peak-profile tails and the phase scale.

Line-segment model

For anchors \((x_k,b_k)\) and \((x_{k+1},b_{k+1})\),

\[b(x)=b_k+(b_{k+1}-b_k)\frac{x-x_k}{x_{k+1}-x_k}.\]

Outside the anchor range the endpoint intensity is retained. Anchor positions are fixed nodes; the selected intensities can be fit parameters. This is not a linearly extrapolated edge.

Polynomial model

\[b(x)=\sum_m c_m u^m,\qquad u=x/x_0-1.\]

origin is the nonzero normalisation coordinate \(x_0\) in the pattern's axis units. It is not an intensity intercept. The currently admitted polynomial term orders are 0 through 5.

Chebyshev model

\[b(x)=\sum_m c_mT_m(u),\qquad u=\frac{2x-x_{\min}-x_{\max}}{x_{\max}-x_{\min}},\]
\[T_0=1,\quad T_1=u,\quad T_{m+1}=2uT_m-T_{m-1}.\]

x_min and x_max define the normalisation domain. The admitted Chebyshev term orders are 0 through 23. These are true Chebyshev coefficients, not powers of the unnormalised angle or time coordinate. Preserve the origin/domain when moving coefficients between projects.

Automatic estimation

The estimator proposes a baseline and then editable anchors or series coefficients. The method is distinct from the final fitted background type. Supported method selectors are auto, arpls, snip and fabc; auto resolves to arpls.

arPLS

An iterated penalised smoothing solve uses

\[(W+\lambda D_2^{\mathsf T}D_2)\mathbf z=W\mathbf y,\]
\[w_i=\left[1+\exp\left(\frac{2}{s}[r_i-(2s-m)]\right)\right]^{-1},\qquad\mathbf r=\mathbf y-\mathbf z.\]

\(m,s\) describe the negative residual subset; the reweighting suppresses peak contributions. Smoothness controls the penalty rather than a crystallographic parameter.

SNIP

Peak clipping in log-log-square transformed intensity uses

\[v_i=\log[\log(\sqrt{y_i+1}+1)+1],\qquad v_i\leftarrow\min(v_i,(v_{i-w}+v_{i+w})/2).\]

The window and traversal settings determine the sequence of updates; the transform is inverted afterwards. This is an estimation algorithm, not an extra Bragg contribution.

FABC and anchor reduction

FABC classifies baseline versus peak regions using a wavelet/noise criterion, then smooths through the baseline points. Anchor reduction converts the dense baseline to a smaller line-segment model using a noise-scaled vertical tolerance. in_peaks controls whether proposed anchors are moved out of or dropped from peak windows; use_model determines whether calculated peaks are subtracted before estimation.

Inspect the estimate. A missing phase or unsuitable calculated pattern can affect a model-assisted baseline proposal. Fitting the proposed background is a separate step.

Excluded observations

An excluded powder interval removes observations from the fitting objective and agreement-factor sums. It does not add a background component or change the definition of the calculated pattern. Retain excluded ranges with the experiment and justify them physically; masking a mismatch does not improve its model. Single-crystal reflection exclusion is a separate planned capability.

References

  1. Baek, S.-J., Park, A., Ahn, Y.-J. & Choo, J. (2015). Baseline correction using asymmetrically reweighted penalized least squares smoothing. Analyst, 140, 250–257. DOI. arPLS baseline estimation.
  2. Ryan, C. G., Clayton, E., Griffin, W. L., Sie, S. H. & Cousens, D. R. (1988). SNIP, a statistics-sensitive background treatment for the quantitative analysis of PIXE spectra in geoscience applications. Nuclear Instruments and Methods in Physics Research B, 34, 396–402. DOI. SNIP peak clipping.
  3. Cobas, J. C., Bernstein, M. A., Martín-Pastor, M. & García Tahoces, P. (2006). A new general-purpose fully automatic baseline-correction procedure for 1D and 2D NMR data. Journal of Magnetic Resonance, 183, 145–151. DOI. FABC baseline classification.