Fitting Models Exercise 1

Imports



In [2]:

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

Fitting a quadratic curve

For this problem we are going to work with the following model:

$$ y_{model}(x) = a x^2 + b x + c $$

The true values of the model parameters are as follows:



In [3]:

    
a_true = 0.5
b_true = 2.0
c_true = -4.0

First, generate a dataset using this model using these parameters and the following characteristics:

For your $x$ data use 30 uniformly spaced points between $[-5,5]$.
Add a noise term to the $y$ value at each point that is drawn from a normal distribution with zero mean and standard deviation 2.0. Make sure you add a different random number to each point (see the size argument of np.random.normal).

After you generate the data, make a plot of the raw data (use points).



In [14]:

    
x = np.linspace(-5,5,30)
n = np.random.normal(0,2.0,30)
y = a_true*x*x + b_true*x + c_true
y = y + n
plt.plot(x,y,ls='None',marker='.')









    Out[14]:





[<matplotlib.lines.Line2D at 0x7fd1fe0bae80>]



In [ ]:

    
assert True # leave this cell for grading the raw data generation and plot

Now fit the model to the dataset to recover estimates for the model's parameters:

Print out the estimates and uncertainties of each parameter.
Plot the raw data and best fit of the model.



In [15]:

    
def model(x,a,b,c):
    return a*x*x + b*x + c



In [16]:

    
best, cov = opt.curve_fit(model, x, y)
print('a = {0:.3f} +/- {1:.3f}'.format(best[0], np.sqrt(cov[0,0])))
print('b = {0:.3f} +/- {1:.3f}'.format(best[1], np.sqrt(cov[1,1])))
print('c = {0:.3f} +/- {1:.3f}'.format(best[2], np.sqrt(cov[2,2])))









    



a = 0.519 +/- 0.059
b = 2.057 +/- 0.156
c = -4.212 +/- 0.700



In [ ]:

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



In [19]:

    
x_new = np.linspace(-5,5,30)
y_new  = model(x,best[0],best[1],best[2])
plt.plot(x_new,y_new)
plt.plot(x,y,ls='None',marker='.')









    Out[19]:





[<matplotlib.lines.Line2D at 0x7fd1fe039080>]



In [ ]: