OALS: Recovery of Saturated Spectral Peaks (MATLAB)

Recovers saturated spectral signals in MATLAB by treating saturated values as missing (NaNs) and reconstructing them with Orthogonalized Alternating Least Squares (O-ALS).

Reference:
Gómez-Sánchez, Adrián, et al. Solving the missing value problem in PCA by Orthogonalized-Alternating Least Squares (O-ALS). Chemometrics and Intelligent Laboratory Systems (2024): 105153.

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Overview

The OALS function applies Orthogonalized Alternating Least Squares (O-ALS) to estimate a bilinear model of spectral data:

X ≈ T · P

where:

• T = scores (sample contributions)
• P = loadings (spectral profiles)

In the case of detector saturation:

• Saturated channels are replaced with NaN
• OALS iteratively estimates T and P under orthogonality constraints
• Full spectra are reconstructed as X̂ = T·P, providing estimates for the saturated regions

This recovery is possible because spectra lie in a low-dimensional subspace: even if some values are missing (saturated), the incomplete point still belongs to that subspace, and missing values can be inferred from it.

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Inputs

• D: Data matrix [samples × variables] with NaNs at saturated entries
• iter: Maximum number of O-ALS iterations
• nPC: Number of principal components (rank of the bilinear model)
• P (optional): Initial guess for loadings [nPC × variables]

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Outputs

• Dr: Reconstructed data matrix (saturated values recovered)
• T: Scores matrix [samples × nPC]
• P: Loadings matrix [nPC × variables]
• r2: Explained variance per iteration
• lofc: Lack of fit per iteration

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Usage Example

% Example: Recover saturated peaks in simulated spectra

% Step 1: Replace saturated values by NaN
D_withNaNs = D_saturated;
D_withNaNs(D_saturated >= ADC_max) = NaN;

% Step 2: Apply OALS with 3 components and 200 iterations
[Dr, T, P, r2, lofc] = OALS(D_withNaNs, 200, 3);

% Step 3: Substitute only missing entries in the saturated data
D_recovered = D_saturated;
missingMask = isnan(D_withNaNs);
D_recovered(missingMask) = Dr(missingMask);

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Installation / Setup

Prerequisites

• MATLAB R2016a or later

Setup

1. Save OALS.m and dependencies (ScoresLS.m, LoadingsLS.m, lofNaN.m) in a folder
2. Add path and verify:

addpath('path/to/your/functions');

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Practical Considerations

• Bilinear model and multiple samples: OALS requires several spectra; single-spectrum recovery is ill-posed.
• Coverage of saturated regions: a variable must be observed unsaturated in at least one sample; if all spectra saturate, it cannot be recovered.
• Rank selection (nPC): must be chosen or estimated beforehand; too small or too large leads to bias.
• Selective peaks: for a dominant, localized peak, restrict to that region and use nPC=1 for stability.
• Initialization and local minima: OALS is nonconvex; run multiple initializations and compare final LOF.

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License

Released under the MIT License

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Authors

• Adrián Gómez-Sánchez
• Date of Creation: September 27, 2025
• Reviewed by: Lovelace’s Square

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Changelog

• v1.0 (2025-09-27): Creation

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Keywords

• O-ALS
• PCA with missing data
• Saturation recovery
• Chemometrics
• Low-rank reconstruction