polynomial_regression.py

# This file was generated from polynomial_regression.ipynb with nbconvert
# Source: https://github.com/groda/big_data

 #!/usr/bin/env python
# coding: utf-8

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# # Polynomial regression with numpy and matplotlib
# 
# A worked out example of polynomial regression with `numpy` using `matplotlib` for visualization.

# In[8]:


import numpy as np
import matplotlib.pyplot as plt


# Set a random seed

# In[9]:


np.random.seed(12)


# Set the number of data points `N` and the degree `D` of the polynomial that is going to fit to the data.
# 
# **Note:** for `D = 1` this is _linear regression_.

# In[10]:


N = 100 # number of points
D = 2   # degree of polynomial


# ### Generate random data
# 
# To generate some random data that is suitable for polynomial regression we're going to use the following functions:
# - [`np.random.uniform`](https://docs.scipy.org/doc/numpy-1.16.0/reference/generated/numpy.random.uniform.html) draws samples from a uniform distribution
# - [`np.random.sample`](https://docs.scipy.org/doc/numpy-1.16.0/reference/generated/numpy.random.sample.html) returns random floats in the half-open interval $[0.0, 1.0)$.
# - [`np.random.normal`](https://docs.scipy.org/doc/numpy-1.16.0/reference/generated/numpy.random.normal.html) draws random samples from a normal (Gaussian) distribution.
# - [`np.polyval`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.polyval.html)  evaluates a polynomial (defined by its coefficients) at the given points

# We are first going to generate a set of $x$ values taken at random in a given range ($[-N, N[$).
# 
# The array `c` is used to store the coefficients of a random polynomial. To generate a random polynomial of degree $D$ we're going to generate $D$ random coefficients and set the coefficient of the leading term to $1$.

# In[11]:


x = np.random.uniform(-N , N, N)    # x values are uniformly distributed between -100 and +100
c = 200*np.random.sample((D,))-100  # random polynomial coefficients drawn from the interval [-100, 100)
c = np.insert(c, 0, 1)              # first coeff. of the random poly is 1
y = np.polyval([1] + c, x)
noise = np.random.normal(-0.5*np.std(y), 0.5*np.std(y), N)  # some normally distributed noise
y += noise         # add noise to values
plt.scatter(x, y)
plt.show()


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t = np.linspace(-N, N, num=N)
p = np.poly1d(np.polyfit(x, y, D))
plt.plot(x, y, 'o', t, p(t), '-')
plt.show()


# ### How well did we approximate?
# 
# Show the coefficients of the approximating polynomial `p`.

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np.set_printoptions(formatter={'float': '{: 12.6f}'.format})
print("Coefficients of p:     {}".format(p.c))
print("Original coefficients: {}".format(c))


# Visualize the original polynomial `p_orig` and its approximation `p` obtained with polynomial regression.

# In[14]:


p_orig = np.poly1d(c)
plt.plot(x, y, 'o', t, p(t), '-', t, p_orig(t), '-')
plt.show()