Fitting Models Exercise 1

Imports



In [1]:

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

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 [62]:

    
a_true = 0.5
b_true = 2.0
c_true = -4.0
dy = 2.0
x = np.linspace(-5,5,30)

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 [63]:

    
ydata = a_true*x**2 + b_true*x + c_true



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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:



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In [17]:

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



In [64]:

    
plt.plot(x, ydata, 'k.')
plt.xlabel('x')
plt.ylabel('y')
plt.xlim(-5,5)
;









    Out[64]:





''



In [65]:

    
plt.errorbar(x, ydata, dy,
             fmt='.k', ecolor='lightgray')









    Out[65]:





<Container object of 3 artists>



In [66]:

    
def exp_model(x, A, B, C):
    return A*np.exp(x*B) + C
yfit = exp_model(x, a_true, b_true, c_true)



In [73]:

    
plt.plot(x, yfit)
plt.plot(x, ydata, 'k.')
plt.xlabel('x')
plt.ylabel('y')
plt.ylim(-20,100);



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