Fitting Models Exercise 2

Imports



In [14]:

    
%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 [15]:

    
# YOUR CODE HERE
data = np.load('decay_osc.npz')
tdata = data['tdata']
ydata = data['ydata']
dy = data['dy']
data.close









    Out[15]:





<bound method NpzFile.close of <numpy.lib.npyio.NpzFile object at 0x7f98a459b198>>



In [46]:

    
plt.figure(figsize=(7,5))
plt.errorbar(tdata, ydata, dy, fmt='og', ecolor='gray')
plt.xlabel('t')
plt.ylabel('y')
plt.grid();



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 [33]:

    
# YOUR CODE HERE
def model(t, a, lamb, omega, delta):
    y = a * np.exp(-lamb * t)*np.cos(omega*t) + delta
    return y

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



In [40]:

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









    



a = -5.285 +/- 0.520
lambda = 0.071 +/- 0.014
omega = -1.012 +/- 0.520
delta = 0.229 +/- 0.014



In [45]:

    
xfit = np.linspace(0,20)
yfit = model(xfit, theta_best[0], theta_best[1], theta_best[2], theta_best[3])

plt.figure(figsize=(7,5))
plt.plot(xfit, yfit)
plt.errorbar(tdata, ydata, dy, fmt='og', ecolor='gray')
plt.xlabel('t')
plt.ylabel('y')
plt.grid();



In [ ]:

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