Extended Multiplicative Signal Correction

Extended Multiplicative Signal Correction is a model-based spectral preprocessing framework introduced as an extension of multiplicative signal correction.

Traditional EMSC represents each measured spectrum using a reference spectrum together with terms that describe unwanted variation, commonly polynomial baseline components. The model coefficients are estimated by least squares. The modeled baseline contribution can then be removed and the spectrum scaled relative to the reference.

The same framework can be extended with additional known spectral profiles representing constituents or interferents. This page focuses on traditional single-reference EMSC with polynomial baseline terms.

MSC models an individual spectrum mainly through an additive offset and a multiplicative scaling relative to a reference. EMSC extends that idea by allowing additional known sources of variation to be included explicitly in the model.

Polynomial basis functions can represent wavelength-dependent baseline variation. Known interferent or constituent spectra can also be included when scientifically justified. This formulation allows different modeled sources of variation to be estimated separately.

The method can help separate modeled baseline and scaling effects from residual spectral structure. It does not guarantee perfect physical separation, and it should not be assumed to improve every predictive model.

For each spectrum the measured vector is fitted as a linear combination of a reference and of chosen basis vectors.

1. Select a reference spectrum.
2. Create basis vectors for the effects that should be modeled.
3. For basic EMSC, use the reference together with polynomial baseline vectors.
4. Fit the measured spectrum as a linear combination of those vectors by least squares.
5. Estimate the reference scaling coefficient.
6. Estimate the polynomial baseline coefficients.
7. Subtract the modeled baseline contribution.
8. Divide by the reference scaling coefficient.
9. Return the corrected spectrum.

If only a constant baseline term and the reference are included, the correction is MSC-like. Additional linear or quadratic terms allow more structured baseline variation to be represented.

Let $\mathbf{x}$ be one measured spectrum and let $\mathbf{r}$ be a chosen reference. Let ${\mathbf{v}}_0,\dots ,{\mathbf{v}}_d$ be polynomial baseline basis vectors, commonly constructed from a normalized coordinate $t_j\in [-1,1]$ with ${\mathbf{v}}_0=1$, ${\mathbf{v}}_1=\mathbf{t}$ and ${\mathbf{v}}_2={\mathbf{t}}^2$. Higher degrees may be included when justified. Degree 2 is a common demonstration choice, not a universally optimal setting.

The method is called extended because known spectral profiles can be added as further columns of $B$. In that case $\mathbf{x}=b\mathbf{r}+{\sum }_k{c}_k{\mathbf{v}}_k+{\sum }_q{d}_q{\mathbf{w}}_q+\mathbf{e}$, where ${\mathbf{w}}_q$ may represent a known interferent or constituent spectrum. The algorithm and code below use only the reference and polynomial baseline terms.

The same rows include multiplicative scale, offset, and a quadratic baseline. MSC removes only intercept and slope relative to the mean reference (dashed). Degree-2 EMSC also models that polynomial term.

The functions below are SPARKS implementations of Equations 1 to 5. They are not copied from the cited papers. Both operate on a matrix with rows as samples and columns as spectral variables. The first column of the basis is the reference. The remaining columns are polynomial terms in a coordinate running from $-1$ to $1$. A reference scaling coefficient near zero is rejected. If no reference is supplied, the mean spectrum of the supplied matrix is used.

import numpy as np def emsc(X, reference=None, degree=2):    X = np.asarray(X, dtype=float)     if X.ndim != 2:        raise ValueError("X must be a 2D array of samples by spectral variables.")     n_samples, n_variables = X.shape     # reference=None uses mean(X) of the supplied matrix. Pass the    # calibration mean explicitly when transforming new spectra.    if reference is None:        reference = X.mean(axis=0)     reference = np.asarray(reference, dtype=float)     if reference.shape != (n_variables,):        raise ValueError(            "reference must contain one value per spectral variable."        )     t = np.linspace(-1.0, 1.0, n_variables)    polynomial_terms = [t ** k for k in range(degree + 1)]    B = np.column_stack([reference] + polynomial_terms)     X_emsc = np.empty_like(X)     for i, spectrum in enumerate(X):        coefficients, *_ = np.linalg.lstsq(B, spectrum, rcond=None)        scale = coefficients[0]        baseline_coefficients = coefficients[1:]         if np.isclose(scale, 0.0):            raise ValueError(                "EMSC reference scaling coefficient is too close to zero."            )         baseline = sum(            c * v            for c, v in zip(baseline_coefficients, polynomial_terms)        )         X_emsc[i] = (spectrum - baseline) / scale     return X_emsc 

1. 1.

   Martens, H., & Stark, E. (1991). Extended multiplicative signal correction and spectral interference subtraction: new preprocessing methods for near infrared spectroscopy. Journal of Pharmaceutical and Biomedical Analysis, 9(8), 625-635.

   doi:10.1016/0731-7085(91)80188-F
2. 2.

   Afseth, N. K., & Kohler, A. (2012). Extended multiplicative signal correction in vibrational spectroscopy, a tutorial. Chemometrics and Intelligent Laboratory Systems, 117, 92-99.

   doi:10.1016/j.chemolab.2012.03.004
3. 3.

   Solheim, J. H., Zimmermann, B., Tafintseva, V., Dzurendová, S., Shapaval, V., & Kohler, A. (2022). The Use of Constituent Spectra and Weighting in Extended Multiplicative Signal Correction in Infrared Spectroscopy. Molecules, 27(6), 1900.

   doi:10.3390/molecules27061900