Fitting Models Exercise 2

Imports



In [1]:

    
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import scipy.optimize as opt

Fitting a decaying oscillation

For this problem you are given a raw dataset in the file decay_osc.npz. This file contains three arrays:

tdata: an array of time values
ydata: an array of y values
dy: the absolute uncertainties (standard deviations) in y

Your job is to fit the following model to this data:

$$ y(t) = A e^{-\lambda t} \cos{\omega t + \delta} $$

First, import the data using NumPy and make an appropriately styled error bar plot of the raw data.



In [12]:

    
decay = np.load('decay_osc.npz')
tdata = decay['tdata']
ydata = decay['ydata']
dy = decay['dy']



In [14]:

    
plt.errorbar(tdata, ydata, dy,
             fmt='.k', ecolor='lightgray')
plt.xlabel('t')
plt.ylabel('y');



In [ ]:

    
assert True # leave this to grade the data import and raw data plot

Now, using curve_fit to fit this model and determine the estimates and uncertainties for the parameters:

Print the parameters estimates and uncertainties.
Plot the raw and best fit model.
You will likely have to pass an initial guess to curve_fit to get a good fit.
Treat the uncertainties in $y$ as absolute errors by passing absolute_sigma=True.



In [17]:

    
def model(xdata, A, lam, omega, delta):
    return A*np.exp(-lam*tdata)*np.cos(omega*tdata)+delta

theta_best, theta_cov = opt.curve_fit(model, tdata, ydata, sigma=dy)

print('A = {0:.3f} +/- {1:.3f}'.format(theta_best[0], np.sqrt(theta_cov[0,0])))
print('lam = {0:.3f} +/- {1:.3f}'.format(theta_best[1], np.sqrt(theta_cov[1,1])))
print('omega = {0:.3f} +/- {1:.3f}'.format(theta_best[2], np.sqrt(theta_cov[2,2])))
print('delta = {0:.3f} +/- {1:.3f}'.format(theta_best[3], np.sqrt(theta_cov[3,3])))









    



A = -4.896 +/- 0.061
lam = 0.094 +/- 0.003
omega = -1.001 +/- 0.001
delta = 0.027 +/- 0.014



In [ ]:

    
assert True # leave this cell for grading the fit; should include a plot and printout of the parameters+errors