Beyond Linearity: Identifying and Managing Nonlinear Effects in Spectroscopic Data

Abstract

Spectroscopic calibration relies on robust models linking spectral measurements to chemical concentrations. Linear methods such as PLS have been central to chemometrics, yet real-world systems often exhibit nonlinear effects due to concentration saturation, matrix interactions, scattering, and detector response deviations. This tutorial reviews nonlinear calibration approaches for spectroscopy. Beginning with the standard linear regression model in matrix form, we highlight limitations in nonlinear conditions and introduce extensions, including polynomial regression, kernel methods, Gaussian processes, and neural networks. Mathematical derivations are presented with emphasis on kernel-based reformulations that retain computational efficiency. Applications are drawn from vibrational, electronic, and atomic spectroscopies. The tutorial concludes with a discussion of interpretability, validation, and future research needs in nonlinear chemometric modeling.

1. Introduction

Multivariate calibration enables the transformation of complex spectral data into quantitative predictions of chemical composition or physical properties. Traditionally, calibration assumes a linear relationship between spectral absorbances and analyte concentrations, consistent with the Beer–Lambert law. Linear regression and PLS regression are widely applied because they balance interpretability with predictive performance (1).

However, linearity is often violated in practice. Deviations occur due to:

• Chemical effects: spectral band saturation at high concentration, molecular interactions, and hydrogen bonding.
• Physical effects: scattering and path length variations in diffuse reflectance NIR spectroscopy (2).
• Instrumental effects: wavelength misalignments, detector nonlinearity, stray light, and temperature sensitivity.

Detecting and correcting nonlinearities is essential to improving prediction accuracy, especially when models must be transferred between instruments or applied to new samples. This tutorial introduces the mathematical frameworks underpinning nonlinear calibration, focusing on approaches used in spectroscopy.

2. Theory of Linearity and Nonlinearity in Spectroscopic Calibration

2.1 The Linear Multivariate Regression Model

The baseline assumption in chemometric calibration is that analyte responses can be expressed as a linear function of spectral variables (1):

In practice, B is estimated using methods such as ordinary least squares (OLS) or PLS regression. The linear model assumes additivity and proportionality between absorbance and concentration, which breaks down when nonlinear effects dominate.

2.2 Sources of Nonlinearity

Several mechanisms produce nonlinear calibration relationships:

1. Beer–Lambert deviations
   At high analyte concentrations, absorption bands saturate, producing nonlinear absorbance–concentration curves.
2. Scattering in diffuse reflectance
   In NIR, particle size distributions cause nonlinear multiplicative effects, often requiring scatter correction (multiplicative scatter correction, standard normal variate) (2).
3. Instrumental nonlinearities
   Detector saturation, stray light, and wavelength shifts introduce nonlinear contributions unrelated to chemistry.
4. Chemical interactions
   Hydrogen bonding, pH effects, and conformational changes alter band positions and intensities in nonlinear ways.

2.3 General Nonlinear Regression Model

The nonlinear calibration model generalizes the linear form (2):

3. Methods for Modeling Nonlinearities

3.1 Polynomial Regression

The simplest extension is polynomial regression, where higher-order and interaction terms are included (2):

• Strengths: simple, interpretable, effective for mild nonlinearities.
• Limitations: exponential growth in terms of high-dimensional spectra; prone to overfitting.

3.2 Kernel Partial Least Squares (K-PLS)

Kernel methods extend linear algorithms by mapping data into a high-dimensional feature space, Φ(X), where linear relations hold. In kernel PLS, regression is performed on the kernel matrix (3):

K-PLS performs PLS regression using K rather than X.

• Strengths: captures complex nonlinearities; avoids explicit computation of Φ(X).
• Limitations: kernel selection and parameter tuning are critical.

3.3 Gaussian Process Regression (GPR)

GPR is a Bayesian nonparametric approach that models functions as distributions (4):

Where the mean prediction and variance are derived from the kernel-defined covariance matrix.

• Strengths: provides uncertainty estimates; interpretable in probabilistic terms.
• Limitations: computationally expensive for large spectral datasets.

3.4 Neural Networks

Artificial neural networks (ANNs) model nonlinear mappings through multiple layers of weighted transformations. A one-hidden-layer feedforward network is (5):

• Strengths: highly flexible; suitable for hyperspectral imaging.
• Limitations: It requires large datasets and is prone to overfitting; interpretability is limited.

4. Discussion and Future Research

The choice of a nonlinear model in spectroscopy depends on the type of nonlinearity, data size, and interpretability requirements. Polynomial regression works well for mild nonlinearities, while kernel methods provide robust modeling for complex but structured nonlinear effects. GPR is especially valuable when uncertainty quantification is needed. Neural networks excel with very large, high-dimensional datasets such as hyperspectral images.

Future research directions include:

1. Hybrid physical–statistical models: Combining radiative transfer theory with machine learning to ensure interpretability and generalization.
2. Transferable nonlinear models: Addressing calibration transfer between instruments without requiring full recalibration.
3. Explainable Artificial Intelligence (AI): Enhancing neural networks and kernel methods with interpretability tools (Shapley values, spectral contribution analysis) (5).
4. Efficient algorithms for large datasets: Scaling GPR and deep learning approaches to high-throughput spectroscopy.

References

(1) Wold, S.; Sjöström, M.; Eriksson, L. PLS-Regression: A Basic Tool of Chemometrics. Chemom. Intell. Lab. Syst. 2001, 58 (2), 109–130. DOI: 10.1016/S0169-7439(01)00155-1.

(2) Martens, H.; Næs, T. Multivariate Calibration; Wiley: Chichester, UK, 1989.

(3) Rosipal, R.; Trejo, L. Kernel Partial Least Squares Regression in Reproducing Kernel Hilbert Space. J. Mach. Learn. Res. 2001, 2, 97–123. http://jmlr.org/papers/v2/rosipal01a.html (accessed 2025-09-12).

(4) Rasmussen, C. E.; Williams, C. K. I. Gaussian Processes for Machine Learning; MIT Press: Cambridge, MA, 2006. http://www.gaussianprocess.org/gpml/ (accessed 2025-09-12).

(5) Goodfellow, I.; Bengio, Y.; Courville, A. Deep Learning; MIT Press: Cambridge, MA, 2016. https://www.deeplearningbook.org/ (accessed 2025-09-12).

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This article was partially constructed with the assistance of a generative AI model and has been carefully edited and reviewed for accuracy and clarity.