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Polynomial Baselines

Introduction

In 2D, a polynomial can be expressed as

p(x, z) = \sum\limits_{i = 0}^{d_r} \sum\limits_{j = 0}^{d_c} {\beta_{i, j} x^i z^j}

where \beta is the matrix of coefficients for the polynomial and d_r and d_c are the polynomial degrees for the rows (x) and columns (z), respectively.

For regular polynomial fitting, the polynomial coefficients that best fit data are gotten from minimizing the least-squares:

\sum\limits_{i}^M \sum\limits_{j}^N w_{ij}^2 (y_{ij} - p(x_i, z_j))^2

where y_{ij}, x_i, and z_j are the measured data, p(x_i, z_j) is the polynomial estimate at x_i, and z_j and w_{ij} is the weighting.

However, since only the baseline of the data is desired, the least-squares approach must be modified. For polynomial-based algorithms, this is done by 1) only fitting the data in regions where there is only baseline, 2) modifying the y-values being fit each iteration, or 3) penalyzing outliers.

Note

For two dimensional data, polynomial algorithms take a single poly_order parameter that can either be a single number, in which case both the rows and columns will use the same polynomial degree, ie. d_r = d_c, or a sequence of two numbers (d_r, d_c) to use different polynomials along the rows and columns. Further, max_cross can be set to limit the polynomial coefficients for the cross terms.

Algorithms

poly (Regular Polynomial)

:meth:`~.Baseline2D.poly`: :ref:`explanation for the algorithm <algorithms/polynomial:poly (Regular Polynomial)>`. No plot will be shown since it is just a simple least-squares polynomial fitting.

modpoly (Modified Polynomial)

:meth:`~.Baseline2D.modpoly`: :ref:`explanation for the algorithm <algorithms/polynomial:modpoly (Modified Polynomial)>`.

.. plot::
   :align: center
   :context: reset

    import numpy as np
    import matplotlib.pyplot as plt
    from pybaselines.utils import gaussian2d
    from pybaselines import Baseline2D


    def create_data():
        x = np.linspace(-20, 20, 80)
        z = np.linspace(-20, 20, 80)
        X, Z = np.meshgrid(x, z, indexing='ij')
        signal = (
            gaussian2d(X, Z, 12, -9, -9)
            + gaussian2d(X, Z, 11, 3, 3)
            + gaussian2d(X, Z, 13, 11, 11)
            + gaussian2d(X, Z, 8, 5, -11, 1.5, 1)
            + gaussian2d(X, Z, 16, -8, 8)
        )
        baseline = 0.1 + 0.08 * X - 0.05 * Z + 0.005 * (Z + 20)**2
        noise = np.random.default_rng(0).normal(scale=0.1, size=signal.shape)
        y = signal + baseline + noise

        return x, z, y, baseline


    def create_plots(y, fit_baseline):
        X, Z = np.meshgrid(
            np.arange(y.shape[0]), np.arange(y.shape[1]), indexing='ij'
        )

        # 4 total plots: 2 countours and 2 projections
        row_names = ('Raw Data', 'Baseline Corrected')
        for i, dataset in enumerate((y, y - fit_baseline)):
            fig = plt.figure(layout='constrained', figsize=plt.figaspect(0.5))
            fig.suptitle(row_names[i])
            ax = fig.add_subplot(1, 2, 2)
            ax.contourf(X, Z, dataset, cmap='coolwarm')
            ax.set_xticks([])
            ax.set_yticks([])
            ax_2 = fig.add_subplot(1, 2, 1, projection='3d')
            ax_2.plot_surface(X, Z, dataset, cmap='coolwarm')
            ax_2.set_xticks([])
            ax_2.set_yticks([])
            ax_2.set_zticks([])

    x, z, y, real_baseline = create_data()
    baseline_fitter = Baseline2D(x, z, check_finite=False)

    baseline, params = baseline_fitter.modpoly(y, poly_order=(1, 2), max_cross=0)
    create_plots(y, baseline)

imodpoly (Improved Modified Polynomial)

:meth:`~.Baseline2D.imodpoly`: :ref:`explanation for the algorithm <algorithms/polynomial:imodpoly (Improved Modified Polynomial)>`.

.. plot::
   :align: center
   :context: close-figs

    # to see contents of create_data function, look at the top-most algorithm's code
    baseline, params = baseline_fitter.imodpoly(y, poly_order=(1, 2), max_cross=0)
    create_plots(y, baseline)

penalized_poly (Penalized Polynomial)

:meth:`~.Baseline2D.penalized_poly`: :ref:`explanation for the algorithm <algorithms/polynomial:penalized_poly (Penalized Polynomial)>`.

.. plot::
   :align: center
   :context: close-figs

    # to see contents of create_data function, look at the top-most algorithm's code
    baseline, params = baseline_fitter.penalized_poly(y, poly_order=(1, 2), max_cross=0)
    create_plots(y, baseline)

quant_reg (Quantile Regression)

:meth:`~.Baseline2D.quant_reg`: :ref:`explanation for the algorithm <algorithms/polynomial:quant_reg (Quantile Regression)>`.

.. plot::
   :align: center
   :context: close-figs

    # to see contents of create_data function, look at the top-most algorithm's code
    baseline, params = baseline_fitter.quant_reg(
        y, poly_order=(1, 2), max_cross=0, quantile=0.3
    )
    create_plots(y, baseline)