Data fitting with curve_fit not correct - python-3.x

Question

I have some experimental data which needs to be fitted so we can elucidate x value for certain y value.
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy.interpolate import interp1d
#from xlrd import open_workbook
points = np.array([(0, -0.0142294), (20, 0.0308458785714286), (50,
0.1091054), (100
,0.2379176875), (200, 0.404354166666667)])
x = points[:,0]
y = points[:,1]
def func(x, p1,p2):
return p1*(1-np.e**(-p2*x))
popt, pcov = curve_fit(func, x, y)
p1 = popt[0]
p2 = popt[1]
curvex=np.linspace(0,200,1000)
fit = func(curvex, p1, p2)
plt.plot(x, y, 'yo', label='data')
f = interp1d(fit, curvex, kind = 'nearest')
print (f(100))
plt.plot(curvex,fit,'r', linewidth=1)
plt.plot(x,y,'x',label = 'Xsaved')
plt.show()
Data is not fitted correctly. Help would be much appreciated.

Answer 1

Here is an example graphical fitter using your data and equation, with scipy's differential_evolution genetic algorithm used to supply initial parameter estimates. The scipy implementation of Differential Evolution ises the Latin Hypercube algorithm to ensure a thorough search of parameter space, and this requires bounds within which to search. In this example I have used the data maximum and minimum values as search bounds, this seems to work in this case. Note that it is much easier to find ranges within which to search than specific values.
import numpy, scipy, matplotlib
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy.optimize import differential_evolution
import warnings
points = numpy.array([(0, -0.0142294), (20, 0.0308458785714286), (50, 0.1091054), (100 ,0.2379176875), (200, 0.404354166666667)])
x = points[:,0]
y = points[:,1]
# rename to match previous example code below
xData = x
yData = y
def func(x, p1,p2):
return p1*(1-numpy.exp(-p2*x))
# function for genetic algorithm to minimize (sum of squared error)
def sumOfSquaredError(parameterTuple):
warnings.filterwarnings("ignore") # do not print warnings by genetic algorithm
val = func(xData, *parameterTuple)
return numpy.sum((yData - val) ** 2.0)
def generate_Initial_Parameters():
# min and max used for bounds
maxX = max(xData)
minX = min(xData)
maxY = max(yData)
minY = min(yData)
minAllData = min(minX, minY)
maxAllData = min(maxX, maxY)
parameterBounds = []
parameterBounds.append([minAllData, maxAllData]) # search bounds for p1
parameterBounds.append([minAllData, maxAllData]) # search bounds for p2
# "seed" the numpy random number generator for repeatable results
result = differential_evolution(sumOfSquaredError, parameterBounds, seed=3)
return result.x
# by default, differential_evolution completes by calling curve_fit() using parameter bounds
geneticParameters = generate_Initial_Parameters()
# now call curve_fit without passing bounds from the genetic algorithm,
# just in case the best fit parameters are aoutside those bounds
fittedParameters, pcov = curve_fit(func, xData, yData, geneticParameters)
print('Fitted parameters:', fittedParameters)
print()
modelPredictions = func(xData, *fittedParameters)
absError = modelPredictions - yData
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(yData))
print()
print('RMSE:', RMSE)
print('R-squared:', Rsquared)
print()
##########################################################
# graphics output section
def ModelAndScatterPlot(graphWidth, graphHeight):
f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)
axes = f.add_subplot(111)
# first the raw data as a scatter plot
axes.plot(xData, yData, 'D')
# create data for the fitted equation plot
xModel = numpy.linspace(min(xData), max(xData))
yModel = func(xModel, *fittedParameters)
# now the model as a line plot
axes.plot(xModel, yModel)
axes.set_xlabel('X Data') # X axis data label
axes.set_ylabel('Y Data') # Y axis data label
plt.show()
plt.close('all') # clean up after using pyplot
graphWidth = 800
graphHeight = 600
ModelAndScatterPlot(graphWidth, graphHeight)