Fitting to exponential functions using python

Question

I am trying to fit a data set on exponential funtion. To do this I have created this function to recreate exponential functions:

def exponential(x,a,b,c):
    return a*(b**x)+c

I am using the module scipy . Here the code to do the fit and print it:

def fit_exponential(x_data,y_data,file):
  params,paramscov= optimize.curve_fit(exponential, x_data, y_data,p0=[1,2,3])

  #Here we calculate the Coeficent of deternination (R²)
  #It is a statistical measure of how well the regression predictions approximate the real data points.
  residuals = y_data - exponential(x_data, *params)

  ss_res = np.sum(residuals**2)
  ss_tot = np.sum((y_data-np.mean(y_data))**2)
  r_squared = 1 - (ss_res / ss_tot)
  print('R²= ',r_squared)

  result = print_exponential(*params)
  print(result)

  #Compound the chart of data and the data with a little text of results
  plt.figure(figsize=(6, 4))
  plt.plot(x_data, exponential(x_data,*params),label='Fitted function',color='m')
  plt.scatter(x_data, y_data, label='Data',color='salmon')

  texto='R²= '+str(round(r_squared,5))+'\n'+result

  plt.text(x_data[-1]*0.55, y_data[-1]*0.15, texto,verticalalignment='center',bbox=dict(facecolor='m', alpha=0.3))

  plt.legend(loc='best')
  plt.xlabel('Size N')
  plt.ylabel('Steps')

  plt.savefig(file)

What I am obtaining is this chart:

As we see the data seems to be exponential but I cant fit a function to it. I have already see some posts, but I couldn't do it.

Answer 1

Here is a graphical Python fitter with your equation and data extracted from the scatterplot, you should re-fit using the actual data.

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

xData = numpy.array([1.408e-01, 8.169e-01, 1.915e+00, 3.183e+00, 3.972e+00, 4.986e+00, 5.972e+00, 6.986e+00, 7.972e+00, 8.873e+00, 9.915e+00, 1.087e+01, 1.192e+01, 1.299e+01, 1.386e+01, 1.496e+01, 1.594e+01, 1.792e+01, 1.682e+01, 1.890e+01, 1.992e+01]) 
yData = numpy.array([8.214e-01, 8.214e-01, 8.214e-01, 8.214e-01, 6.160e-01, 8.214e-01, 8.214e-01, 4.107e-01, 1.027e+00, 1.027e+00, 8.214e-01, 1.027e+00, 1.027e+00, 1.643e+00, 1.643e+00, 3.285e+00, 5.749e+00, 2.300e+01, 1.170e+01, 4.723e+01, 9.651e+01])


def func(x, a, b, c):
    return a*(b**x)+c


# these are the same as the scipy defaults
initialParameters = numpy.array([1.0, 1.0, 1.0])

# curve fit the test data
fittedParameters, pcov = curve_fit(func, xData, yData, initialParameters)

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('Parameters:', fittedParameters)
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)