MCR-ALS Lite: Multivariate Curve Resolution - Alternating Least Squares

Lite implementation of MCR-ALS for resolving spectral mixtures into pure component profiles.

📖 Overview

MCR-ALS Lite is the fundamental, lightweight implementation of Multivariate Curve Resolution using Alternating Least Squares. This Lite version provides the core algorithm with essential non-negativity constraints and spectral normalization. It serves as the foundation for more advanced MCR variants that may be added to this repository in the future.

MCR-ALS is a powerful soft-modeling technique for decomposing complex multivariate data into chemically meaningful components. It resolves mixtures by iteratively estimating concentration profiles and pure spectral profiles from experimental data.

The Bilinear Model

MCR-ALS decomposes a data matrix according to the bilinear model:

D = C × S + E

where:

• D (I × J): Experimental data matrix (e.g., spectra collected over time)
• C (I × N): Concentration profiles of N components across I samples
• S (N × J): Pure spectral profiles of N components across J variables
• E (I × J): Residual matrix (experimental noise and model error)

Key Features of the Lite Version

• ✅ Flexible initialization with C_init, S_init, both, or neither
• ✅ Explicit component specification (N parameter)
• ✅ Optional non-negativity constraints on C and/or S (individually controllable)
• ✅ Spectral normalization (unit Euclidean norm) to prevent scale ambiguity
• ✅ Real-time visualization of convergence progress
• ✅ Lack of Fit (LOF) monitoring at each iteration
• ✅ Simple, lightweight implementation with minimal dependencies
• ✅ Foundation for future MCR variants in this repository

Note: This is the Lite version. Future additions to this repository may include MCR-ALS with additional constraints (unimodality, closure, selectivity), MCR-BANDS, weighted MCR-ALS, and other advanced variants.

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Applications

MCR-ALS is widely used in:

• Spectroscopy: UV-Vis, NIR, Raman, FTIR, NMR
• Chromatography: HPLC-DAD, GC-MS, LC-MS
• Process monitoring: Reaction kinetics, process analytical technology (PAT)
• Environmental analysis: Mixture quantification
• Quality control: Pharmaceutical, food, and chemical industries

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📊 Algorithm

Alternating Least Squares (ALS)

1. Initialize concentration profiles C and/or spectral profiles S (with N components specified)

2. Iterate until convergence:

   If C_init provided:

   • Step 1: Fix C, solve for S: min ||D - C×S||² (with non-negativity if nonnegS = true)
   • Step 2: Normalize each row of S to unit norm, compensate in C
   • Step 3: Fix S, solve for C: min ||D - C×S||² (with non-negativity if nonnegC = true)

   If S_init provided:

   • Step 1: Fix S, solve for C: min ||D - C×S||² (with non-negativity if nonnegC = true)

   • Step 2: Fix C, solve for S: min ||D - C×S||² (with non-negativity if nonnegS = true)

   • Step 3: Normalize each row of S to unit norm, compensate in C

   • Step 4: Calculate Lack of Fit: LOF = 100 × ||D - C×S||_F / ||D||_F

   • Step 5: Check convergence: if |LOF(i-1) - LOF(i)| < tol, stop

3. Return optimized C, S, and LOF history

Normalization Strategy

Each spectral component (row of S) is normalized by its Euclidean norm:

% Normalize spectral row i and compensate in C S(i,:) = S(i,:) / norm(S(i,:), 2); C(:,i) = C(:,i) * norm(S(i,:), 2);

This ensures:

• Unique scaling between C and S
• Spectral profiles have unit intensity
• Concentration profiles carry the scaling information

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🚀 Installation

Prerequisites

• MATLAB R2016a or later
• No additional toolboxes required (uses custom fnnls implementation)

Setup

1. Clone or download the repository
2. Add the MCR-ALS Lite folder to your MATLAB path:

addpath('path/to/Codes/MCR-ALS Lite');

3. Verify installation:

which MCR_ALS_Lite

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📝 Usage

Basic Example - Initialize with Concentration Profiles

% Load your data matrix D (samples × variables) % For example: D could be 50 spectra × 200 wavelengths

% Specify number of components and initialize concentration profiles nComponents = 3; C_init = rand(size(D,1), nComponents); % Random initialization

% Run MCR-ALS Lite (C_init provided, S_init is []) [C, S, lof] = MCR_ALS_Lite(D, nComponents, C_init, [], 100, 1e-6);

% C: Concentration profiles (samples × components) % S: Spectral profiles (components × variables) % lof: Lack of fit per iteration (%)

Basic Example - Initialize with Spectral Profiles

% Load your data matrix D (samples × variables) % For example: D could be 50 spectra × 200 wavelengths

% Specify number of components and initialize spectral profiles nComponents = 3; S_init = rand(nComponents, size(D,2)); % Random initialization

% Run MCR-ALS Lite (C_init is [], S_init provided) [C, S, lof] = MCR_ALS_Lite(D, nComponents, [], S_init, 100, 1e-6);

% C: Concentration profiles (samples × components) % S: Spectral profiles (components × variables) % lof: Lack of fit per iteration (%)

Example - Using Optional Non-Negativity Constraints

% Full non-negativity (default behavior, backward compatible) [C, S, lof] = MCR_ALS_Lite(D, 3, C_init, [], 100, 1e-6, [true true]);

% Only constrain concentrations (C) to be non-negative [C, S, lof] = MCR_ALS_Lite(D, 3, C_init, [], 100, 1e-6, [true false]);

% Only constrain spectra (S) to be non-negative [C, S, lof] = MCR_ALS_Lite(D, 3, C_init, [], 100, 1e-6, [false true]);

% Fully unconstrained (both C and S can be negative) [C, S, lof] = MCR_ALS_Lite(D, 3, C_init, [], 100, 1e-6, [false false]);

% Scalar input (broadcast to both modes) [C, S, lof] = MCR_ALS_Lite(D, 3, C_init, [], 100, 1e-6, true); % same as [true true]

Advanced Initialization

For better results, use informed initialization. These methods can provide either concentration (C) or spectral (S) profiles:

Method	Description	Output	When to Use
PUREST	PURity-based Evolving Self-modeling Technique	S profiles	Pure variable detection
SIMPLISMA/ESI	SIMPLe-to-use Interactive Self-modeling / Essential Spectra Identification	S profiles	Simplex geometry for pure components
Pure windows	Known regions where components are isolated	C or S	Domain knowledge of mixture
Random	Random positive values	C or S	Last resort (quick initialization)

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Running the Test Script

% Run the provided test on synthetic data test_MCR_ALS_Lite

This will:

• Generate synthetic spectral data with 3 known components
• Apply MCR-ALS Lite to recover concentration and spectral profiles
• Display convergence plots in real-time
• Compare recovered vs. true profiles

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📋 Function Reference

MCR_ALS_Lite

[C, S, lof] = MCR_ALS_Lite(D, N, C_init, S_init, maxIter, tol, nonnegMode)

Inputs:

• D — Data matrix (I × J)
• N — Number of components (positive integer)
• C_init — Initial concentration profiles (I × N) OR [] for random
• S_init — Initial spectral profiles (N × J) OR [] for random
• maxIter — Maximum iterations
• tol — (Optional) Convergence tolerance for LOF change (default: 1e-6)
• nonnegMode — (Optional) 1×2 logical vector [nonnegC, nonnegS]:
  • nonnegC = true → C constrained non-negative
  • nonnegS = true → S constrained non-negative
  • If scalar given, it is broadcast to both modes
  • Default: [true true] (fully non-negative, original behavior)

Note: Provide C_init, S_init, both, or neither (all random if both []). Provided matrices must have N components.

Outputs:

• C — Final concentration profiles (I × N)
• S — Final spectral profiles (N × J), each row normalized to unit norm
• lof — Lack of fit per iteration (%)

Dependencies:

• fnnls.m — Fast Non-Negative Least Squares solver (included)

fnnls

X = fnnls(A, B, tol, maxIter)

Fast non-negative least squares solver for multiple right-hand sides.
Solves min_X ||A*X - B||_F^2 subject to X ≥ 0, one column of B at a time.

Inputs:

• A — Design matrix (n × p)
• B — Right-hand sides (n × q)
• tol — Stationarity/zero tolerance (default: 1e-12 * ||A||_F)
• maxIter — Max active-set expansions per RHS (default: 5*p)

Outputs:

• X — Solution matrix (p × q) with non-negative entries

Notes:

• Based on the Lawson–Hanson active-set NNLS, with Bro–De Jong acceleration (reuse A'*A and A'*B across RHS).
• Falls back to pseudoinverse when a passive subset is ill-conditioned.

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🔬 References

1. Tauler, R. (1995) — Multivariate curve resolution applied to second order data. Chemometrics and Intelligent Laboratory Systems, 30(1), 133–146.
   DOI: https://doi.org/10.1016/0169-7439(95)00047-X

2. de Juan, A., & Tauler, R. (2021) — Multivariate Curve Resolution: 50 years addressing the mixture analysis problem – A review. Analytica Chimica Acta, 1145, 59–78.
   DOI: https://doi.org/10.1016/j.aca.2020.10.051

3. Jaumot, J., de Juan, A., & Tauler, R. (2015) — MCR-ALS GUI 2.0: New features and applications. Chemometrics and Intelligent Laboratory Systems, 140, 1–12.
   DOI: https://doi.org/10.1016/j.chemolab.2014.10.003

4. Lawton, W. H., & Sylvestre, E. A. (1971) — Self modeling curve resolution. Technometrics, 13(3), 617–633.
   DOI: https://doi.org/10.1080/00401706.1971.10488823

5. Bro, R., & De Jong, S. (1997) — A fast non-negativity-constrained least squares algorithm. Journal of Chemometrics, 11(5), 393–401.
   DOI: https://doi.org/10.1002/%28SICI%291099-128X%28199709/10%2911%3A5%3C393%3A%3AAID-CEM483%3E3.0.CO%3B2-L

6. Lawson, C. L., & Hanson, R. J. (1974) — Solving Least Squares Problems. Prentice–Hall. (SIAM Classics reprint)
   DOI: https://doi.org/10.1137/1.9781611971217

Additional Reading

• Tauler, R., Smilde, A., & Kowalski, B. (1995) — Selectivity, local rank, three-way data analysis and ambiguity in multivariate curve resolution. Journal of Chemometrics, 9(1), 31–58.
  DOI: https://doi.org/10.1002/cem.1180090105

• Golshan, A., Abdollahi, H., & Maeder, M. (2011) — Resolution of rotational ambiguity for three-component systems. Analytical Chemistry, 83(3), 836–841.
  DOI: https://doi.org/10.1021/ac102429q

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📄 License

Released under the MIT License.

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👤 Authors

• Adrián Gómez-Sánchez
• Date: October 30, 2025
• Reviewed by: Lovelace's Square

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📧 Contributing

Contributions are welcome! Please:

1. Fork the repository
2. Create a feature branch
3. Submit a pull request

For issues or questions, please open an issue in this repository.

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🏷️ Keywords

Multivariate Curve Resolution • MCR-ALS • Alternating Least Squares • Spectral unmixing • Chemometrics • Non-negative least squares • Bilinear decomposition • Mixture analysis • MATLAB