Non-linear least squares fitting of a two-dimensional data

The scipy.optimize.curve_fit routine can be used to fit two-dimensional data, but the fitted data (the ydata argument) must be repacked as a one-dimensional array first. The independent variable (the xdata argument) must then be an array of shape (2,M) where M is the total number of data points.

For a two-dimensional array of data, Z, calculated on a mesh grid (X, Y), this can be achieved efficiently using the ravel method:

xdata = np.vstack((X.ravel(), Y.ravel()))
ydata = Z.ravel()


The following code demonstrates this approach for some synthetic data set created as a sum of four Gaussian functions with some noise added:

The result can be visualized in 3D with the residuals plotted on a plane under the fitted data:

or in 2D with the fitted data contours superimposed on the noisy data:

import numpy as np
from scipy.optimize import curve_fit
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

# The two-dimensional domain of the fit.
xmin, xmax, nx = -5, 4, 75
ymin, ymax, ny = -3, 7, 150
x, y = np.linspace(xmin, xmax, nx), np.linspace(ymin, ymax, ny)
X, Y = np.meshgrid(x, y)

# Our function to fit is going to be a sum of two-dimensional Gaussians
def gaussian(x, y, x0, y0, xalpha, yalpha, A):
return A * np.exp( -((x-x0)/xalpha)**2 -((y-y0)/yalpha)**2)

# A list of the Gaussian parameters: x0, y0, xalpha, yalpha, A
gprms = [(0, 2, 2.5, 5.4, 1.5),
(-1, 4, 6, 2.5, 1.8),
(-3, -0.5, 1, 2, 4),
(3, 0.5, 2, 1, 5)
]

# Standard deviation of normally-distributed noise to add in generating
# our test function to fit.
noise_sigma = 0.1

# The function to be fit is Z.
Z = np.zeros(X.shape)
for p in gprms:
Z += gaussian(X, Y, *p)
Z += noise_sigma * np.random.randn(*Z.shape)

# Plot the 3D figure of the fitted function and the residuals.
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot_surface(X, Y, Z, cmap='plasma')
ax.set_zlim(0,np.max(Z)+2)
plt.show()

# This is the callable that is passed to curve_fit. M is a (2,N) array
# where N is the total number of data points in Z, which will be ravelled
# to one dimension.
def _gaussian(M, *args):
x, y = M
arr = np.zeros(x.shape)
for i in range(len(args)//5):
arr += gaussian(x, y, *args[i*5:i*5+5])
return arr

# Initial guesses to the fit parameters.
guess_prms = [(0, 0, 1, 1, 2),
(-1.5, 5, 5, 1, 3),
(-4, -1, 1.5, 1.5, 6),
(4, 1, 1.5, 1.5, 6.5)
]
# Flatten the initial guess parameter list.
p0 = [p for prms in guess_prms for p in prms]

# We need to ravel the meshgrids of X, Y points to a pair of 1-D arrays.
xdata = np.vstack((X.ravel(), Y.ravel()))
# Do the fit, using our custom _gaussian function which understands our
# flattened (ravelled) ordering of the data points.
popt, pcov = curve_fit(_gaussian, xdata, Z.ravel(), p0)
fit = np.zeros(Z.shape)
for i in range(len(popt)//5):
fit += gaussian(X, Y, *popt[i*5:i*5+5])
print('Fitted parameters:')
print(popt)

rms = np.sqrt(np.mean((Z - fit)**2))
print('RMS residual =', rms)

# Plot the 3D figure of the fitted function and the residuals.
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot_surface(X, Y, fit, cmap='plasma')
cset = ax.contourf(X, Y, Z-fit, zdir='z', offset=-4, cmap='plasma')
ax.set_zlim(-4,np.max(fit))
plt.show()

# Plot the test data as a 2D image and the fit as overlaid contours.
fig = plt.figure()
ax.imshow(Z, origin='bottom', cmap='plasma',
extent=(x.min(), x.max(), y.min(), y.max()))
ax.contour(X, Y, fit, colors='w')
plt.show()

Current rating: 4.9

Dominik Stańczak 2 months, 3 weeks ago

That's a pretty darn clever solution! I'm absolutely stealing this code. :)
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.29352545738220215, "perplexity": 6862.279965641667}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232254889.43/warc/CC-MAIN-20190519141556-20190519163556-00145.warc.gz"}