Fitting data with SciPy

Here’s a common thing scientists need to do, and it’s easy to accomplish in python. Suppose that you have a data set consisting of temperature vs time data for the cooling of a cup of coffee. We’ll start by importing the needed libraries and defining a fitting function:

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

def fitFunc(t, a, b, c):
    return a*np.exp(-b*t) + c

Now we create some fake data as numpy arrays and add some noise; the fake data will be called noisy:

t = np.linspace(0,4,50)
temp = fitFunc(t, 2.5, 1.3, 0.5)
noisy = temp + 0.25*np.random.normal(size=len(temp))

The scipy.optimize module contains a least squares curve fit routine that requires as input a user-defined fitting function (in our case fitFunc ), ┬áthe x-axis data (in our case, t) and the y-axis data (in our case, noisy). The curve_fit routine returns an array of fit parameters, and a matrix of covariance data (the square root of the diagonal values are the 1-sigma uncertainties on the fit parameters—provided you have a reasonable fit in the first place.):

fitParams, fitCovariances = curve_fit(fitFunc, t, noisy)
print fitParams
print fitCovariance

output:

[ 2.42573207  1.38417604  0.57396876]
[[ 0.01738482  0.00815523 -0.00077816]
 [ 0.00815523  0.02509109  0.00606179]
 [-0.00077816  0.00606179  0.00292134]]

Now we plot the data points with error bars, plot the best fit curve, and label the axes:

plt.ylabel('Temperature (C)', fontsize = 16)
plt.xlabel('time (s)', fontsize = 16)
plt.xlim(0,4.1)
# plot the data as red circles with vertical errorbars
plt.errorbar(t, noisy, fmt = 'ro', yerr = 0.2)
# now plot the best fit curve and also +- 1 sigma curves
# (the square root of the diagonal covariance matrix  
# element is the uncertianty on the fit parameter.)
sigma = [fitCovariance[0,0], \
         fitCovariance[1,1], \
         fitCovariance[2,2] \
         ]
plt.plot(t, fitFunc(t, fitParams[0], fitParams[1], fitParams[2]),\
         t, fitFunc(t, fitParams[0] + sigma[0], fitParams[1] - sigma[1], fitParams[2] + sigma[2]),\
         t, fitFunc(t, fitParams[0] - sigma[0], fitParams[1] + sigma[1], fitParams[2] - sigma[2])\  
         )
# save plot to a file
savefig('dataFitted.png', bbox_inches=0)

The output file looks like this:

dataFitted

You can see a the curve fitting routine as a python script, and you can see an ipython notebook here.

 

 

 

 

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