Fitting Models Exercise 1

Imports



In [3]:

    
%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 [4]:

    
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 [15]:

    
t=np.linspace(-5,5,30)
def quadr(a,b,c):
    x=np.linspace(-5,5,30)
    
    return a*x**2+b*x+c
y=quadr(0.5,2.0,-4.0)+np.random.normal(0,2,30)
plt.scatter(t,y)
plt.tight_layout()
plt.xlabel('$x$')
plt.ylabel('$y(x)$')
plt.title('$y(x)$ vs $x$')









    Out[15]:





<matplotlib.text.Text at 0x7f087009dac8>



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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.



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# YOUR CODE HERE
raise NotImplementedError()



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