Fitting Models Exercise 1

Imports



In [1]:

    
%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 [2]:

    
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 [8]:

    
X=np.linspace(-5,5,30)
Y=a_true*X**2+b_true*X+c_true+np.random.normal(0,2.0,size=30)
f=plt.figure(figsize=(15,10))
plt.plot(X,Y, "b*");



In [9]:

    
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 [14]:

    
opt.curve_fit?



In [31]:

    
best_guess=[0,0,0]
popt, popc=opt.curve_fit(lambda x,a,b,c:a*x**2+b*x+c,X,Y, best_guess);
print(list(zip(popt,np.diag(popc))))
f=plt.figure(figsize=(15,10))
plt.plot(X,Y,"b*");
plt.plot(X,popt[0]*X**2+popt[1]*X+popt[2],"r-");









    



[(0.48041732006096483, 0.0030596876506324383), (2.225889214458169, 0.021731907331339922), (-3.4788885195016861, 0.43638508963643552)]



In [ ]:

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