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

# In[1]:


# Python ≥3.5 is required
import sys
assert sys.version_info >= (3, 5)

# Scikit-Learn ≥0.20 is required
import sklearn
assert sklearn.__version__ >= "0.20"

# Common imports
import numpy as np
import os

# to make this notebook's output stable across runs
np.random.seed(42)

# To plot pretty figures
get_ipython().run_line_magic('matplotlib', 'inline')
import matplotlib as mpl
import matplotlib.pyplot as plt
mpl.rc('axes', labelsize=14)
mpl.rc('xtick', labelsize=12)
mpl.rc('ytick', labelsize=12)

# Where to save the figures
PROJECT_ROOT_DIR = "."
CHAPTER_ID = "training_linear_models"
IMAGES_PATH = os.path.join(PROJECT_ROOT_DIR, "images", CHAPTER_ID)
os.makedirs(IMAGES_PATH, exist_ok=True)

def save_fig(fig_id, tight_layout=True, fig_extension="png", resolution=300):
    path = os.path.join(IMAGES_PATH, fig_id + "." + fig_extension)
    print("Saving figure", fig_id)
    if tight_layout:
        plt.tight_layout()
    plt.savefig(path, format=fig_extension, dpi=resolution)


# A linear model makes a prediction by simply computing a weighted sum of the input features, plus a constant called the _bias term_ (also called the _intercept term_)
# 
# $$ \hat{y} = h_{\boldsymbol{θ}}(\boldsymbol{x}) = \boldsymbol{θ} · \boldsymbol{x}  = θ_0 + θ_1x_1 + θ_2x_2 + \ldots + θ_nx_n$$

# In[2]:


X = 2 * np.random.rand(100, 1)
y = 4 + 3 * X + np.random.randn(100, 1)

plt.plot(X, y, "b.")
plt.xlabel("$x_1$", fontsize=18)
plt.ylabel("$y$", rotation=0, fontsize=18)
plt.axis([0, 2, 0, 15])
save_fig("generated_data_plot")
plt.show()


# In[ ]: