Asymmetric Least Squares (ALS)

Table of contents

  1. Introduction
  2. How to use
  3. Behavior
  4. Method
  5. References

Introduction

Asymmetric Least Squares (ALS) is a baseline removal algorithm for estimating a smooth background beneath spectral peaks. SpectraGuru uses ALS when the user wants an adjustable smoothness penalty and asymmetric peak weighting.

How to use

  1. Upload data, then open Processing Page.
  2. Enable Baseline removal.
  3. In Select baseline removal function, choose ALS.
  4. Set ALS lambda, ALS asymmetry (p), ALS difference order (d), and ALS maximum iterations.
  5. Apply the processing workflow to subtract the estimated baseline.

Behavior

ALS estimates a smooth baseline below the spectral peaks by repeatedly fitting a penalized least-squares curve and reweighting residuals. The result should reduce slow background drift while preserving sharper peaks for downstream analysis.

Method

ALS solves a weighted smoothness-penalized least-squares problem:

\[\min_z \sum_i w_i(y_i-z_i)^2 + \lambda \lVert D_d z\rVert^2\]

Weights are updated asymmetrically so points above the baseline contribute less:

\[w_i = p \text{ if } y_i > z_i,\quad w_i = 1-p \text{ otherwise}\]
Parameter Tunable or fixed Implementation
ALS lambda Tunable Smoothness penalty; default 100, UI range 1 to 1e10
ALS asymmetry (p) Tunable Weighting asymmetry; default 0.001
ALS difference order (d) Tunable Difference order 1-3; default 1
ALS maximum iterations Tunable Default 50, UI range 1-200
Linear solve Fixed Sparse solve of the weighted penalty system

References

  1. Eilers, P. H. C., & Boelens, H. F. M. (2005). Baseline correction with asymmetric least squares smoothing. https://zanran_storage.s3.amazonaws.com/www.science.uva.nl/ContentPages/443199618.pdf