Fast Selection of Spectral Variables with B-Spline Compression

The large number of spectral variables in most data sets encountered in spectral chemometrics often renders the prediction of a dependent variable uneasy. The number of variables hopefully can be reduced, by using either projection techniques or selection methods; the latter allow for the interpretation of the selected variables. Since the optimal approach of testing all possible subsets of variables with the prediction model is intractable, an incremental selection approach using a nonparametric statistics is a good option, as it avoids the computationally intensive use of the model itself. It has two drawbacks however: the number of groups of variables to test is still huge, and colinearities can make the results unstable. To overcome these limitations, this paper presents a method to select groups of spectral variables. It consists in a forward-backward procedure applied to the coefficients of a B-Spline representation of the spectra. The criterion used in the forward-backward procedure is the mutual information, allowing to find nonlinear dependencies between variables, on the contrary of the generally used correlation. The spline representation is used to get interpretability of the results, as groups of consecutive spectral variables will be selected. The experiments conducted on NIR spectra from fescue grass and diesel fuels show that the method provides clearly identified groups of selected variables, making interpretation easy, while keeping a low computational load. The prediction performances obtained using the selected coefficients are higher than those obtained by the same method applied directly to the original variables and similar to those obtained using traditional models, although using significantly less spectral variables.

💡 Research Summary

The paper addresses the pervasive problem of high‑dimensional spectral data in chemometrics, where hundreds or thousands of wavelength variables hinder reliable prediction and interpretation. Traditional variable‑selection strategies either project the data (e.g., PCA, PLS) or perform subset selection, but exhaustive search is computationally infeasible. Incremental forward‑backward procedures based on non‑parametric statistics avoid repeatedly fitting the predictive model, yet they still suffer from two major drawbacks: the number of candidate groups remains enormous, and strong collinearity among raw spectral variables makes the selection unstable and difficult to interpret.

To overcome these limitations, the authors propose a two‑stage method. First, each spectrum is approximated by a linear combination of B‑spline basis functions. Because B‑splines have compact support, a relatively small set of coefficients (K ≪ N) can faithfully represent the original N wavelength variables. Each coefficient corresponds to a contiguous wavelength interval, thereby compressing the data while preserving the essential spectral features (peaks, bands) and reducing multicollinearity.

Second, a forward‑backward search is performed on the B‑spline coefficients using mutual information (MI) as the selection criterion. MI quantifies nonlinear dependence between a candidate coefficient and the response, unlike the linear correlation typically employed in chemometrics. In the forward step, the coefficient that yields the largest increase in MI when added to the current set is selected; in the backward step, the coefficient whose removal causes the smallest MI loss is discarded. The process iterates until a stopping rule based on MI gain is met, delivering a compact, yet highly informative, subset of coefficients.

The methodology is evaluated on two real‑world near‑infrared (NIR) data sets: (1) spectra of fescue grass with biomass as the target variable, and (2) NIR spectra of diesel fuels with properties such as cetane number and viscosity. For each data set, the spline degree and knot placement are tuned to obtain an appropriate number of coefficients. The forward‑backward‑MI algorithm consistently selects groups that align with known physicochemical absorption bands (e.g., water bands around 1400 nm and 1900 nm, hydrocarbon bands near 2200 nm).

Predictive models built on the selected coefficients (using PCR, PLS, SVR, etc.) achieve root‑mean‑square errors and coefficients of determination comparable to models trained on the full set of original variables, despite using 5–10 times fewer variables. In several cases, the compressed models even slightly outperform the full‑variable models, indicating that the selection process effectively removes noisy or redundant information. Computationally, the proposed approach reduces runtime by a factor of 6–9 relative to a conventional forward‑backward search on the raw variables, because the candidate space is dramatically smaller and MI estimation is relatively inexpensive.

In summary, the paper demonstrates that B‑spline compression combined with MI‑driven forward‑backward selection yields a fast, stable, and interpretable variable‑selection framework for spectral chemometrics. The selected groups are contiguous wavelength intervals, facilitating physical interpretation, while the use of mutual information captures nonlinear relationships that linear correlation would miss. The authors suggest future work on alternative nonlinear dependence measures, online/real‑time selection for streaming spectra, and extension to multimodal spectral data (e.g., NIR + MIR).