Asymmetric Least Squares Baseline Correction

Baseline correction using asymmetric least squares, accelerated with C++.

Asymmetric least squares method is a technique that can correct the baseline while adjusting the balance between the fit of the baseline to the spectrum and the smoothness of the baseline curve.

Dependencies

• Eigen (C++ template library for linear algebra)
• pybind11 (Seamless operability between C++11 and Python)
• NumPy (Fundamental package for scientific computing with Python)

Principle

In ALS, the baseline $\mathbf{Z}=\lbrace z_1, z_2, \cdots, z_n \rbrace$ for the spectrum $\mathbf{Y}=\lbrace y_1, y_2, \cdots, y_n \rbrace$ is determined to minimize the function:

$$F(\mathbf{Z}) = \sum_iw_i(y_i-z_i)^2 + \lambda\sum_i(\Delta^2z_i)^2$$

where the weights $w_i$ are used to treat each point of the spectrum asymmetrically, with a small value $p$ (such as 0.001) given by:

$$w_i = \left\{ \begin{array}{ll} p & (y_i \geq z_i) \\\ 1-p & (y_i \lt z_i) \end{array} \right.$$

This prioritizes points where the spectrum falls below the baseline $(y_i < z_i)$ in fitting, causing the baseline to fall below the spectrum. $\Delta^2$ represents the second derivative (numerical derivative) of the spectrum and is calculated as:

$$\begin{aligned} \Delta^2z_i & = (z_i - z_{i-1}) - (z_{i-1} - z_{i-2}) \\\ & = z_i - 2z_{i-1} + z_{i-2} \end{aligned}$$

The first term of $F(\mathbf{Z})$ represents the difference between the spectrum and the baseline, while the second term serves as a penalty term representing the complexity of the baseline curve, with the constant $\lambda$ determining its scale. Setting $\lambda$ to a small value yields a complex baseline curve, while a large value yields a smooth baseline curve.

References

P.H.C. Eilers, Anal. Chem., 2004, 76, 404-411.

Algorithm

The algorithm for baseline calculation using ALS is as follows. First, initialize all the weights $\mathbf{w}=\lbrace w_1, w_2, \cdots, w_n \rbrace$ to 1. Then update the baseline $\mathbf{Z}$ and weights $\mathbf{w}$ iteratively:

$$\begin{array}{ll} \mathbf{Z}\leftarrow\mathop{\rm arg~min}\limits_{\mathbf{Z}}\bigg\lbrace \sum_iw_i(y_i-z_i)^2 + \lambda\sum_i(\Delta^2z_i)^2 \bigg\rbrace & (1) \\ \\\ w_i\leftarrow\left\{ \begin{array}{ll} p & (y_i \geq z_i) \\\ 1-p & (y_i \lt z_i) \end{array} \right. & (2) \end{array}$$

until convergence.

$F(\mathbf{Z})$ can be expressed using matrices and vectors as:

$$F(\mathbf{Z}) = (\mathbf{Y}-\mathbf{Z})^T\mathbf{W}(\mathbf{Y}-\mathbf{Z}) + \lambda\mathbf{Z}^T(\mathbf{D}^T\mathbf{D})\mathbf{Z}$$

where $\mathbf{W}$ is a diagonal matrix with the weights $\mathbf{w}$ as its diagonal elements, and $\mathbf{D}$ is an $(n-2)×n$ matrix such that $\mathbf{DZ} = \Delta^2\mathbf{Z}$, defined as:

$$\mathbf{D} = \begin{pmatrix} 1 & -2 & 1 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 \\\ 0 & 1 & -2 & 1 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 \\\ & & & & & \vdots \\\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 1 & -2 & 1 & 0 \\\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 1 & -2 & 1 \\\ \end{pmatrix}$$

For equation $(1)$, the baseline $\mathbf{Z}$ is obtained when the partial derivative of $F(\mathbf{Z})$ with respect to $\mathbf{Z}$:

$$\frac{\partial F(\mathbf{Z})}{\partial \mathbf{Z}} = -2\mathbf{W}(\mathbf{Y}-\mathbf{Z}) + 2\lambda(\mathbf{D}^T\mathbf{D})\mathbf{Z}$$

is zero which occurs when:

$$(\mathbf{W} + \lambda \mathbf{D}^T\mathbf{D})\mathbf{Z} = \mathbf{WY}$$

and thus, the baseline $\mathbf{Z}$ is given by:

$$\mathbf{Z} = (\mathbf{W} + \lambda \mathbf{D}^T\mathbf{D})^{-1}\mathbf{WY}$$

References

P.H.C. Eilers, Anal. Chem., 2003, 75, 3631-3636.