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 = fig.add_subplot(111) 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()

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