Fitting Models Exercise 2

Imports



In [2]:

    
%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 [3]:

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



In [20]:

    
plt.figure(figsize=(8,6))
plt.errorbar(tdata, ydata, dy, fmt='.k', ecolor='lightgray')
plt.tick_params(axis='x', direction='out', top='off')
plt.tick_params(axis='y', direction='out', right='off')
plt.xlabel('t'), plt.ylabel('y'), plt.title('Oscillation Raw Data');



In [15]:

    
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 [16]:

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



In [17]:

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



In [18]:

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









    



A = -4.896 +/- 0.063
λ = 0.094 +/- 0.003
ω = -1.001 +/- 0.001
δ = 0.027 +/- 0.014



In [21]:

    
tfit = np.linspace(0,20)
yfit = model(tfit, theta_best[0], theta_best[1], theta_best[2], theta_best[3])
plt.figure(figsize=(8,6))
plt.plot(tfit, yfit)
plt.plot(tdata, ydata, 'k.')
plt.tick_params(axis='x', direction='out', top='off')
plt.tick_params(axis='y', direction='out', right='off')
plt.xlabel('t'), plt.ylabel('y'), plt.title('Oscillation Curve Fitting');



In [10]:

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