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I have a function that includes x and y as independent variables and I want to fit the parameters to the data and function and plot a surface figure. I saw that if the variables have two different dimensions, I can use np.meshgrid(x,y), but then how do I find the parameters a,b,c? My code looks like this:

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

x = np.array([1,0.5,0.33,0.25,0.2])
y = np.array([1e-9,1e-8,1e-7,1e-6,1e-5,1e-4,1e-3,1e-2,1e-1,1e0,1e1,1e2,1e3,1e4,1e5])

def func(x,y,a,b,c):
 return (1-(a/(a+y)^b))*(1-np.exp(-c*x))

x,y = np.meshgrid(x,y)

Can I still use curve_fit for this type of function? If so, how can I use it to find a,b,c and also plot the 3d function?

Here is an example with 3D scatterplot, 3D surface plot, and a contour plot.

import numpy, scipy, scipy.optimize
import matplotlib
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm # to colormap 3D surfaces from blue to red
import matplotlib.pyplot as plt

graphWidth = 800 # units are pixels
graphHeight = 600 # units are pixels

# 3D contour plot lines
numberOfContourLines = 16


def SurfacePlot(func, data, fittedParameters):
 f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)

 matplotlib.pyplot.grid(True)
 axes = Axes3D(f)

 x_data = data[0]
 y_data = data[1]
 z_data = data[2]

 xModel = numpy.linspace(min(x_data), max(x_data), 20)
 yModel = numpy.linspace(min(y_data), max(y_data), 20)
 X, Y = numpy.meshgrid(xModel, yModel)

 Z = func(numpy.array([X, Y]), *fittedParameters)

 axes.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap=cm.coolwarm, linewidth=1, antialiased=True)

 axes.scatter(x_data, y_data, z_data) # show data along with plotted surface

 axes.set_title('Surface Plot (click-drag with mouse)') # add a title for surface plot
 axes.set_xlabel('X Data') # X axis data label
 axes.set_ylabel('Y Data') # Y axis data label
 axes.set_zlabel('Z Data') # Z axis data label

 plt.show()
 plt.close('all') # clean up after using pyplot or else thaere can be memory and process problems


def ContourPlot(func, data, fittedParameters):
 f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)
 axes = f.add_subplot(111)

 x_data = data[0]
 y_data = data[1]
 z_data = data[2]

 xModel = numpy.linspace(min(x_data), max(x_data), 20)
 yModel = numpy.linspace(min(y_data), max(y_data), 20)
 X, Y = numpy.meshgrid(xModel, yModel)

 Z = func(numpy.array([X, Y]), *fittedParameters)

 axes.plot(x_data, y_data, 'o')

 axes.set_title('Contour Plot') # add a title for contour plot
 axes.set_xlabel('X Data') # X axis data label
 axes.set_ylabel('Y Data') # Y axis data label

 CS = matplotlib.pyplot.contour(X, Y, Z, numberOfContourLines, colors='k')
 matplotlib.pyplot.clabel(CS, inline=1, fontsize=10) # labels for contours

 plt.show()
 plt.close('all') # clean up after using pyplot or else thaere can be memory and process problems


def ScatterPlot(data):
 f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)

 matplotlib.pyplot.grid(True)
 axes = Axes3D(f)
 x_data = data[0]
 y_data = data[1]
 z_data = data[2]

 axes.scatter(x_data, y_data, z_data)

 axes.set_title('Scatter Plot (click-drag with mouse)')
 axes.set_xlabel('X Data')
 axes.set_ylabel('Y Data')
 axes.set_zlabel('Z Data')

 plt.show()
 plt.close('all') # clean up after using pyplot or else thaere can be memory and process problems


def func(data, a, b, c):
 x = data[0]
 y = data[1]
 return a + (x**b) * (y**c)


if __name__ == "__main__":
 xData = numpy.array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0])
 yData = numpy.array([11.0, 12.1, 13.0, 14.1, 15.0, 16.1, 17.0, 18.1, 90.0])
 zData = numpy.array([1.1, 2.2, 3.3, 4.4, 5.5, 6.6, 7.7, 8.0, 9.9])

 data = [xData, yData, zData]

 initialParameters = [1.0, 1.0, 1.0] # these are the same as scipy default values in this example

 # here a non-linear surface fit is made with scipy's curve_fit()
 fittedParameters, pcov = scipy.optimize.curve_fit(func, [xData, yData], zData, p0 = initialParameters)

 ScatterPlot(data)
 SurfacePlot(func, data, fittedParameters)
 ContourPlot(func, data, fittedParameters)

 print('fitted prameters', fittedParameters)

 modelPredictions = func(data, *fittedParameters) 

 absError = modelPredictions - zData

 SE = numpy.square(absError) # squared errors
 MSE = numpy.mean(SE) # mean squared errors
 RMSE = numpy.sqrt(MSE) # Root Mean Squared Error, RMSE
 Rsquared = 1.0 - (numpy.var(absError) / numpy.var(zData))
 print('RMSE:', RMSE)
 print('R-squared:', Rsquared)