lam effects

This example shows the influence of the lam parameter that is used when fitting any algorithm that is based on Whittaker-smoothing. Note that the exact lam values used in this example are unimportant, just the changes in their scale.

For this example, the arpls() algorithm will be used, which performs well in the presence of noise.

import matplotlib.pyplot as plt
import numpy as np

from pybaselines import Baseline
from pybaselines.utils import gaussian


x = np.linspace(0, 1000, 1000)
signal = (
    gaussian(x, 9, 100, 12)
    + gaussian(x, 6, 180, 5)
    + gaussian(x, 8, 350, 11)
    + gaussian(x, 15, 400, 18)
    + gaussian(x, 6, 550, 6)
    + gaussian(x, 13, 700, 8)
    + gaussian(x, 9, 800, 9)
    + gaussian(x, 9, 880, 7)
)
baseline = 5 + 10 * np.exp(-x / 800)
noise = np.random.default_rng(0).normal(0, 0.2, len(x))
y = signal + baseline + noise

baseline_fitter = Baseline(x_data=x)

For extremely small lam values, the algorithm's output is more comparable with smoothing than baseline correction (which would be the desired result if using pure Whittaker smoothing).

plt.plot(y, label='data')
lam = 1
plt.plot(baseline_fitter.arpls(y, lam=lam)[0], label=f'lam=10$^{np.log10(lam):.0f}$')
plt.legend()

Increasing the lam value produces results that more resemble the baseline. However, the baseline is still too flexible and cuts into the larger peaks.

plt.figure()
plt.plot(y, label='data')
lam = 1e3
plt.plot(baseline_fitter.arpls(y, lam=lam)[0], label=f'lam=10$^{np.log10(lam):.0f}$')
plt.legend()

Increasing the lam value further produces the desired result and follows the expected baseline without reducing the peak area.

plt.figure()
plt.plot(y, label='data')
lam = 1e6
plt.plot(baseline_fitter.arpls(y, lam=lam)[0], label=f'lam=10$^{np.log10(lam):.0f}$')
plt.legend()

Further increasing the lam value then makes the baseline too stiff and unable to follow the curve of the data.

plt.figure()
plt.plot(y, label='data')
lam = 1e10
plt.plot(baseline_fitter.arpls(y, lam=lam)[0], label=f'lam=10$^{{{np.log10(lam):.0f}}}$')
plt.legend()

Finally, put together all the results to show the change in the baseline as lam increases.

plt.figure()
plt.plot(y, label='data')
for lam in (1, 1e3, 1e6, 1e10):
    plt.plot(baseline_fitter.arpls(y, lam=lam)[0], label=f'lam=10$^{{{np.log10(lam):.0f}}}$')
plt.legend()

plt.show()