Correction Spectra MCR-ALS Extension

Overview

The Correction Spectra extension enables a specialized MCR-ALS model where fixed spectral shapes are refined by paired correction spectra. This is useful when:

• You have known/reference spectral shapes but they may not perfectly fit your data
• You want to quantify spectral deviations from these references
• You need explicit control over spectral components (fixed + corrections + free)

Model Structure

With k = 2n + a components:

Spectra Set	Count	Role	Constraints
Fixed (S_fix)	n	Known/reference shapes	Hard-constrained, normalized to 1
Corrections (S_corr)	n	Deviations from fixed	Orthogonal to S_fix, zero-mean, small
Free (S_free)	a	Additional components	Standard ALS (nonnegative, etc.)

Concentrations:

• C_fix: Standard kinetics for fixed spectra
• C_corr: Proportional to C_fix (same time profile, different amplitude)
• C_free: Independent kinetics

Mathematical Model

The decomposition is:

X ≈ [C_fix  C_corr  C_free] × [S_fix^T]
                               [S_corr^T]
                               [S_free^T]

With constraints:

• Pairwise coupling: C_corr[:, i] = α_i × C_fix[:, i]
• Spectral orthogonality: S_corr[, i] ⟂ S_fix[, i]
• Zero-mean: mean(S_corr[, i]) = 0

• Regularization: min

  S_corr

  ² (Tikhonov)

How to Use

Step 1: Load Fixed Spectra

1. Go to the ALS Constraints → Spectra panel
2. Upload reference/fixed spectral shapes using “Fix spectral shape(s)”
3. Select which fixed spectra to include in the model (checkboxes)

Step 2: Enable Correction Spectra

1. A new panel “Correction Spectra” appears after fixed spectra are loaded
2. Check “Enable correction spectra”
3. Options appear:
   • Correction penalty (λ): Controls regularization strength
     • λ = -4: weak penalty, larger corrections allowed
     • λ = -2: moderate (default)
     • λ = 0: strong penalty, very small corrections
   • Zero-mean corrections: Recommended enabled

Step 3: Run ALS

1. Set total number of dimensions in ”# Spectra” box
   • For example: 2 fixed + 2 corrections + 1 free = 5 dimensions
2. Run ALS as usual
3. Output:
   • S_1, S_2: Fixed spectra (unchanged)
   • S_3, S_4: Correction spectra (paired with S_1, S_2)
   • S_5: Free spectrum
   • Kinetics automatically structured accordingly

Interpretation

Example: 2 Fixed Spectra + Corrections

Input:

• 2 reference spectra files (species A and B)

Output dimensions (k=5):

• S_1 / C_1: Species A (pure reference kinetics)
• S_2 / C_2: Species B (pure reference kinetics)
• S_3 / C_3: Correction for A (explains deviations from pure A)
• S_4 / C_4: Correction for B (explains deviations from pure B)
• S_5 / C_5: Free component (unknown species/baseline)

Physical meaning:

• If correction S_3 is near zero → reference A is good
• If correction S_3 has structure → reference A has systematic errors (e.g., peak shift, shoulders)
• C_3 has same temporal behavior as C_1 but different amplitude

Parameters Explained

Correction Penalty (λ)

Controls how aggressively corrections are shrunk toward zero:

penalty_factor = 1 / (1 + λ × ||S_corr||₂)

• λ = 10^-4 ≈ 0.0001: Weak; corrections can be as large as you want
• λ = 10^-2 = 0.01: Moderate; corrections stay small-ish
• λ = 10^0 = 1: Strong; corrections are heavily damped
• λ = 10^-3 (default slider -3): Good starting point

Zero-Mean Corrections

If enabled:

S_corr,i ← S_corr,i - mean(S_corr,i)

Why?

• Prevents corrections from becoming a baseline shift
• Forces them to explain structured deviations (peaks, troughs)
• Recommended: keep enabled

Advanced Usage

Interpretation of Correction Magnitude

# In output spectra S_3 vs S_1 (correction vs fixed):
correction_ratio <- max(abs(S_corr)) / max(abs(S_fix))

• < 1%: Reference is nearly perfect
• 1–10%: Minor systematic deviations (acceptable)
• 10–50%: Significant deviations (model needs review)
• > 50%: Reference may be wrong; check data

Comparing Solutions

Standard ALS (k=3):

X ≈ C_1 S_1^T + C_2 S_2^T + C_3 S_3^T

Result: 3 unknown spectra

Coupled Correction ALS (k=3 with 1 fixed):

X ≈ C_fix S_fix^T + C_corr S_corr^T + C_free S_free^T

Result: 1 known, 1 correction, 1 free

Advantage: Interpretability (fixed component is constrained to known shape).

Troubleshooting

“Correction Spectra” panel doesn’t appear

• Cause: No fixed spectra loaded
• Fix: Go to “Fix spectral shape(s)” and upload a reference file

Corrections are zero or very small

• Cause: Reference spectra fit well
• Fix: This is good! It means your references are accurate.
• Or: Increase λ (move slider to -4)

Corrections are too large (> 50% of fixed)

• Cause: Reference spectra don’t match your data
• Fix:
  • Try different reference files
  • Enable “Soft constraint” for fixed spectra (smaller weight)
  • Reduce λ to allow larger corrections

Convergence is slow

• Cause: Over-regularization or conflicting constraints
• Fix: Try λ = -2 or -1; run more iterations

Output Files

When you save ALS results with corrections enabled:

• Spectra file: S_1 (fixed), S_2 (fixed), S_3 (correction A), S_4 (correction B), S_5 (free), …
• Kinetics file: C_1, C_2 (independent), C_3, C_4 (proportional to C_1, C_2), C_5 (independent), …

Note: C_3 ∝ C_1 and C_4 ∝ C_2 by design.

References

• Tauler, R. (1995). Multivariate curve resolution applied to second order data. Chemometrics Intell. Lab. Syst., 30, 133–146.
• Jaumot, J., Gargallo, R., Juan, A. D., & Tauler, R. (2005). A graphical user-friendly interface of MCR-ALS. Chemometrics Intell. Lab. Syst., 76, 101–110.

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Code Files

• server_files/ALS_CorrectionSpectra.R – Core functions (getC_Coupled, getS_Coupled, myals_Coupled, als_Coupled)
• ui_files/ALSInputConstraintsCorrectionSpectra.R – UI panel
• server_files/ALS_CorrectionSpectra_Server.R – Integration logic with main ALS