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

Question

I have a data set that I know fits to a curve of the form:

y = a x²

and I want to extract the value of a.

What's the best way to go about this in Python (with scipy etc.) ?

Possible duplicate of [python numpy/scipy curve fitting](https://stackoverflow.com/questions/19165259/python-numpy-scipy-curve-fitting) — meowgoesthedog, Mar 01 '19 at 13:48
There are tons of posts on SO that show how to do this. Just google for `python curve_fit` or `python lmfit` and you will find plenty of examples. If you then run into issues with the implementation, post your code and data and describe the actual problem you face. — Cleb, Mar 01 '19 at 15:58

score 0 · Accepted Answer · answered Mar 01 '19 at 16:05

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)

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

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