Fitting a power law with known exponent and extracting the coefficient in Python

Here is a graphical fitter example using scipy's curve_fit():

import numpy, scipy, matplotlib import matplotlib.pyplot as plt from scipy.optimize import curve_fit xData = numpy.array([1.1, 2.2, 3.3, 4.4, 5.0, 6.6, 7.7]) yData = numpy.array([1.1, 20.2, 30.3, 60.4, 50.0, 60.6, 70.7]) def func(x, a): return (a * numpy.square(x)) # same as the scipy default initialParameters = numpy.array([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)