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

# **Chapter 4 – Training Linear Models**

# _This notebook contains all the sample code and solutions to the exercises in chapter 4._
# 
# <table align="left">
#   <td>
#     <a target="_blank" href="https://colab.research.google.com/github/ageron/handson-ml/blob/master/04_training_linear_models.ipynb"><img src="https://www.tensorflow.org/images/colab_logo_32px.png" />Run in Google Colab</a>
#   </td>
# </table>

# **Warning**: this is the code for the 1st edition of the book. Please visit https://github.com/ageron/handson-ml2 for the 2nd edition code, with up-to-date notebooks using the latest library versions.

# # Setup

# First, let's make sure this notebook works well in both python 2 and 3, import a few common modules, ensure MatplotLib plots figures inline and prepare a function to save the figures:

# In[1]:


# To support both python 2 and python 3
from __future__ import division, print_function, unicode_literals

# 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)


# # Linear regression using the Normal Equation

# In[2]:


import numpy as np

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


# In[3]:


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[4]:


X_b = np.c_[np.ones((100, 1)), X]  # add x0 = 1 to each instance
theta_best = np.linalg.inv(X_b.T.dot(X_b)).dot(X_b.T).dot(y)


# In[5]:


theta_best


# In[6]:


X_new = np.array([[0], [2]])
X_new_b = np.c_[np.ones((2, 1)), X_new]  # add x0 = 1 to each instance
y_predict = X_new_b.dot(theta_best)
y_predict


# In[7]:


plt.plot(X_new, y_predict, "r-")
plt.plot(X, y, "b.")
plt.axis([0, 2, 0, 15])
plt.show()


# The figure in the book actually corresponds to the following code, with a legend and axis labels:

# In[8]:


plt.plot(X_new, y_predict, "r-", linewidth=2, label="Predictions")
plt.plot(X, y, "b.")
plt.xlabel("$x_1$", fontsize=18)
plt.ylabel("$y$", rotation=0, fontsize=18)
plt.legend(loc="upper left", fontsize=14)
plt.axis([0, 2, 0, 15])
save_fig("linear_model_predictions")
plt.show()


# In[9]:


from sklearn.linear_model import LinearRegression
lin_reg = LinearRegression()
lin_reg.fit(X, y)
lin_reg.intercept_, lin_reg.coef_


# In[10]:


lin_reg.predict(X_new)


# The `LinearRegression` class is based on the `scipy.linalg.lstsq()` function (the name stands for "least squares"), which you could call directly:

# In[11]:


theta_best_svd, residuals, rank, s = np.linalg.lstsq(X_b, y, rcond=1e-6)
theta_best_svd


# This function computes $\mathbf{X}^+\mathbf{y}$, where $\mathbf{X}^{+}$ is the _pseudoinverse_ of $\mathbf{X}$ (specifically the Moore-Penrose inverse). You can use `np.linalg.pinv()` to compute the pseudoinverse directly:

# In[12]:


np.linalg.pinv(X_b).dot(y)


# **Note**: the first releases of the book implied that the `LinearRegression` class was based on the Normal Equation. This was an error, my apologies: as explained above, it is based on the pseudoinverse, which ultimately relies on the SVD matrix decomposition of $\mathbf{X}$ (see chapter 8 for details about the SVD decomposition). Its time complexity is $O(n^2)$ and it works even when $m < n$ or when some features are linear combinations of other features (in these cases, $\mathbf{X}^T \mathbf{X}$ is not invertible so the Normal Equation fails), see [issue #184](https://github.com/ageron/handson-ml/issues/184) for more details. However, this does not change the rest of the description of the `LinearRegression` class, in particular, it is based on an analytical solution, it does not scale well with the number of features, it scales linearly with the number of instances, all the data must fit in memory, it does not require feature scaling and the order of the instances in the training set does not matter.

# # Linear regression using batch gradient descent

# In[13]:


eta = 0.1
n_iterations = 1000
m = 100
theta = np.random.randn(2,1)

for iteration in range(n_iterations):
    gradients = 2/m * X_b.T.dot(X_b.dot(theta) - y)
    theta = theta - eta * gradients


# In[14]:


theta


# In[15]:


X_new_b.dot(theta)


# In[16]:


theta_path_bgd = []

def plot_gradient_descent(theta, eta, theta_path=None):
    m = len(X_b)
    plt.plot(X, y, "b.")
    n_iterations = 1000
    for iteration in range(n_iterations):
        if iteration < 10:
            y_predict = X_new_b.dot(theta)
            style = "b-" if iteration > 0 else "r--"
            plt.plot(X_new, y_predict, style)
        gradients = 2/m * X_b.T.dot(X_b.dot(theta) - y)
        theta = theta - eta * gradients
        if theta_path is not None:
            theta_path.append(theta)
    plt.xlabel("$x_1$", fontsize=18)
    plt.axis([0, 2, 0, 15])
    plt.title(r"$\eta = {}$".format(eta), fontsize=16)


# In[17]:


np.random.seed(42)
theta = np.random.randn(2,1)  # random initialization

plt.figure(figsize=(10,4))
plt.subplot(131); plot_gradient_descent(theta, eta=0.02)
plt.ylabel("$y$", rotation=0, fontsize=18)
plt.subplot(132); plot_gradient_descent(theta, eta=0.1, theta_path=theta_path_bgd)
plt.subplot(133); plot_gradient_descent(theta, eta=0.5)

save_fig("gradient_descent_plot")
plt.show()


# # Stochastic Gradient Descent

# In[18]:


theta_path_sgd = []
m = len(X_b)
np.random.seed(42)


# In[19]:


n_epochs = 50
t0, t1 = 5, 50  # learning schedule hyperparameters

def learning_schedule(t):
    return t0 / (t + t1)

theta = np.random.randn(2,1)  # random initialization

for epoch in range(n_epochs):
    for i in range(m):
        if epoch == 0 and i < 20:                    # not shown in the book
            y_predict = X_new_b.dot(theta)           # not shown
            style = "b-" if i > 0 else "r--"         # not shown
            plt.plot(X_new, y_predict, style)        # not shown
        random_index = np.random.randint(m)
        xi = X_b[random_index:random_index+1]
        yi = y[random_index:random_index+1]
        gradients = 2 * xi.T.dot(xi.dot(theta) - yi)
        eta = learning_schedule(epoch * m + i)
        theta = theta - eta * gradients
        theta_path_sgd.append(theta)                 # not shown

plt.plot(X, y, "b.")                                 # not shown
plt.xlabel("$x_1$", fontsize=18)                     # not shown
plt.ylabel("$y$", rotation=0, fontsize=18)           # not shown
plt.axis([0, 2, 0, 15])                              # not shown
save_fig("sgd_plot")                                 # not shown
plt.show()                                           # not shown


# In[20]:


theta


# In[21]:


from sklearn.linear_model import SGDRegressor
sgd_reg = SGDRegressor(max_iter=50, tol=-np.infty, penalty=None, eta0=0.1, random_state=42)
sgd_reg.fit(X, y.ravel())


# In[22]:


sgd_reg.intercept_, sgd_reg.coef_


# # Mini-batch gradient descent

# In[23]:


theta_path_mgd = []

n_iterations = 50
minibatch_size = 20

np.random.seed(42)
theta = np.random.randn(2,1)  # random initialization

t0, t1 = 200, 1000
def learning_schedule(t):
    return t0 / (t + t1)

t = 0
for epoch in range(n_iterations):
    shuffled_indices = np.random.permutation(m)
    X_b_shuffled = X_b[shuffled_indices]
    y_shuffled = y[shuffled_indices]
    for i in range(0, m, minibatch_size):
        t += 1
        xi = X_b_shuffled[i:i+minibatch_size]
        yi = y_shuffled[i:i+minibatch_size]
        gradients = 2/minibatch_size * xi.T.dot(xi.dot(theta) - yi)
        eta = learning_schedule(t)
        theta = theta - eta * gradients
        theta_path_mgd.append(theta)


# In[24]:


theta


# In[25]:


theta_path_bgd = np.array(theta_path_bgd)
theta_path_sgd = np.array(theta_path_sgd)
theta_path_mgd = np.array(theta_path_mgd)


# In[26]:


plt.figure(figsize=(7,4))
plt.plot(theta_path_sgd[:, 0], theta_path_sgd[:, 1], "r-s", linewidth=1, label="Stochastic")
plt.plot(theta_path_mgd[:, 0], theta_path_mgd[:, 1], "g-+", linewidth=2, label="Mini-batch")
plt.plot(theta_path_bgd[:, 0], theta_path_bgd[:, 1], "b-o", linewidth=3, label="Batch")
plt.legend(loc="upper left", fontsize=16)
plt.xlabel(r"$\theta_0$", fontsize=20)
plt.ylabel(r"$\theta_1$   ", fontsize=20, rotation=0)
plt.axis([2.5, 4.5, 2.3, 3.9])
save_fig("gradient_descent_paths_plot")
plt.show()


# # Polynomial regression

# In[27]:


import numpy as np
import numpy.random as rnd

np.random.seed(42)


# In[28]:


m = 100
X = 6 * np.random.rand(m, 1) - 3
y = 0.5 * X**2 + X + 2 + np.random.randn(m, 1)


# In[29]:


plt.plot(X, y, "b.")
plt.xlabel("$x_1$", fontsize=18)
plt.ylabel("$y$", rotation=0, fontsize=18)
plt.axis([-3, 3, 0, 10])
save_fig("quadratic_data_plot")
plt.show()


# In[30]:


from sklearn.preprocessing import PolynomialFeatures
poly_features = PolynomialFeatures(degree=2, include_bias=False)
X_poly = poly_features.fit_transform(X)
X[0]


# In[31]:


X_poly[0]


# In[32]:


lin_reg = LinearRegression()
lin_reg.fit(X_poly, y)
lin_reg.intercept_, lin_reg.coef_


# In[33]:


X_new=np.linspace(-3, 3, 100).reshape(100, 1)
X_new_poly = poly_features.transform(X_new)
y_new = lin_reg.predict(X_new_poly)
plt.plot(X, y, "b.")
plt.plot(X_new, y_new, "r-", linewidth=2, label="Predictions")
plt.xlabel("$x_1$", fontsize=18)
plt.ylabel("$y$", rotation=0, fontsize=18)
plt.legend(loc="upper left", fontsize=14)
plt.axis([-3, 3, 0, 10])
save_fig("quadratic_predictions_plot")
plt.show()


# In[34]:


from sklearn.preprocessing import StandardScaler
from sklearn.pipeline import Pipeline

for style, width, degree in (("g-", 1, 300), ("b--", 2, 2), ("r-+", 2, 1)):
    polybig_features = PolynomialFeatures(degree=degree, include_bias=False)
    std_scaler = StandardScaler()
    lin_reg = LinearRegression()
    polynomial_regression = Pipeline([
            ("poly_features", polybig_features),
            ("std_scaler", std_scaler),
            ("lin_reg", lin_reg),
        ])
    polynomial_regression.fit(X, y)
    y_newbig = polynomial_regression.predict(X_new)
    plt.plot(X_new, y_newbig, style, label=str(degree), linewidth=width)

plt.plot(X, y, "b.", linewidth=3)
plt.legend(loc="upper left")
plt.xlabel("$x_1$", fontsize=18)
plt.ylabel("$y$", rotation=0, fontsize=18)
plt.axis([-3, 3, 0, 10])
save_fig("high_degree_polynomials_plot")
plt.show()


# In[35]:


from sklearn.metrics import mean_squared_error
from sklearn.model_selection import train_test_split

def plot_learning_curves(model, X, y):
    X_train, X_val, y_train, y_val = train_test_split(X, y, test_size=0.2, random_state=10)
    train_errors, val_errors = [], []
    for m in range(1, len(X_train)):
        model.fit(X_train[:m], y_train[:m])
        y_train_predict = model.predict(X_train[:m])
        y_val_predict = model.predict(X_val)
        train_errors.append(mean_squared_error(y_train[:m], y_train_predict))
        val_errors.append(mean_squared_error(y_val, y_val_predict))

    plt.plot(np.sqrt(train_errors), "r-+", linewidth=2, label="train")
    plt.plot(np.sqrt(val_errors), "b-", linewidth=3, label="val")
    plt.legend(loc="upper right", fontsize=14)   # not shown in the book
    plt.xlabel("Training set size", fontsize=14) # not shown
    plt.ylabel("RMSE", fontsize=14)              # not shown


# In[36]:


lin_reg = LinearRegression()
plot_learning_curves(lin_reg, X, y)
plt.axis([0, 80, 0, 3])                         # not shown in the book
save_fig("underfitting_learning_curves_plot")   # not shown
plt.show()                                      # not shown


# In[37]:


from sklearn.pipeline import Pipeline

polynomial_regression = Pipeline([
        ("poly_features", PolynomialFeatures(degree=10, include_bias=False)),
        ("lin_reg", LinearRegression()),
    ])

plot_learning_curves(polynomial_regression, X, y)
plt.axis([0, 80, 0, 3])           # not shown
save_fig("learning_curves_plot")  # not shown
plt.show()                        # not shown


# # Regularized models

# In[38]:


from sklearn.linear_model import Ridge

np.random.seed(42)
m = 20
X = 3 * np.random.rand(m, 1)
y = 1 + 0.5 * X + np.random.randn(m, 1) / 1.5
X_new = np.linspace(0, 3, 100).reshape(100, 1)

def plot_model(model_class, polynomial, alphas, **model_kargs):
    for alpha, style in zip(alphas, ("b-", "g--", "r:")):
        model = model_class(alpha, **model_kargs) if alpha > 0 else LinearRegression()
        if polynomial:
            model = Pipeline([
                    ("poly_features", PolynomialFeatures(degree=10, include_bias=False)),
                    ("std_scaler", StandardScaler()),
                    ("regul_reg", model),
                ])
        model.fit(X, y)
        y_new_regul = model.predict(X_new)
        lw = 2 if alpha > 0 else 1
        plt.plot(X_new, y_new_regul, style, linewidth=lw, label=r"$\alpha = {}$".format(alpha))
    plt.plot(X, y, "b.", linewidth=3)
    plt.legend(loc="upper left", fontsize=15)
    plt.xlabel("$x_1$", fontsize=18)
    plt.axis([0, 3, 0, 4])

plt.figure(figsize=(8,4))
plt.subplot(121)
plot_model(Ridge, polynomial=False, alphas=(0, 10, 100), random_state=42)
plt.ylabel("$y$", rotation=0, fontsize=18)
plt.subplot(122)
plot_model(Ridge, polynomial=True, alphas=(0, 10**-5, 1), random_state=42)

save_fig("ridge_regression_plot")
plt.show()


# In[39]:


from sklearn.linear_model import Ridge
ridge_reg = Ridge(alpha=1, solver="cholesky", random_state=42)
ridge_reg.fit(X, y)
ridge_reg.predict([[1.5]])


# In[40]:


sgd_reg = SGDRegressor(max_iter=50, tol=-np.infty, penalty="l2", random_state=42)
sgd_reg.fit(X, y.ravel())
sgd_reg.predict([[1.5]])


# In[41]:


ridge_reg = Ridge(alpha=1, solver="sag", random_state=42)
ridge_reg.fit(X, y)
ridge_reg.predict([[1.5]])


# In[42]:


from sklearn.linear_model import Lasso

plt.figure(figsize=(8,4))
plt.subplot(121)
plot_model(Lasso, polynomial=False, alphas=(0, 0.1, 1), random_state=42)
plt.ylabel("$y$", rotation=0, fontsize=18)
plt.subplot(122)
plot_model(Lasso, polynomial=True, alphas=(0, 10**-7, 1), tol=1, random_state=42)

save_fig("lasso_regression_plot")
plt.show()


# In[43]:


from sklearn.linear_model import Lasso
lasso_reg = Lasso(alpha=0.1)
lasso_reg.fit(X, y)
lasso_reg.predict([[1.5]])


# In[44]:


from sklearn.linear_model import ElasticNet
elastic_net = ElasticNet(alpha=0.1, l1_ratio=0.5, random_state=42)
elastic_net.fit(X, y)
elastic_net.predict([[1.5]])


# In[45]:


np.random.seed(42)
m = 100
X = 6 * np.random.rand(m, 1) - 3
y = 2 + X + 0.5 * X**2 + np.random.randn(m, 1)

X_train, X_val, y_train, y_val = train_test_split(X[:50], y[:50].ravel(), test_size=0.5, random_state=10)

poly_scaler = Pipeline([
        ("poly_features", PolynomialFeatures(degree=90, include_bias=False)),
        ("std_scaler", StandardScaler()),
    ])

X_train_poly_scaled = poly_scaler.fit_transform(X_train)
X_val_poly_scaled = poly_scaler.transform(X_val)

sgd_reg = SGDRegressor(max_iter=1,
                       tol=-np.infty,
                       penalty=None,
                       eta0=0.0005,
                       warm_start=True,
                       learning_rate="constant",
                       random_state=42)

n_epochs = 500
train_errors, val_errors = [], []
for epoch in range(n_epochs):
    sgd_reg.fit(X_train_poly_scaled, y_train)
    y_train_predict = sgd_reg.predict(X_train_poly_scaled)
    y_val_predict = sgd_reg.predict(X_val_poly_scaled)
    train_errors.append(mean_squared_error(y_train, y_train_predict))
    val_errors.append(mean_squared_error(y_val, y_val_predict))

best_epoch = np.argmin(val_errors)
best_val_rmse = np.sqrt(val_errors[best_epoch])

plt.annotate('Best model',
             xy=(best_epoch, best_val_rmse),
             xytext=(best_epoch, best_val_rmse + 1),
             ha="center",
             arrowprops=dict(facecolor='black', shrink=0.05),
             fontsize=16,
            )

best_val_rmse -= 0.03  # just to make the graph look better
plt.plot([0, n_epochs], [best_val_rmse, best_val_rmse], "k:", linewidth=2)
plt.plot(np.sqrt(val_errors), "b-", linewidth=3, label="Validation set")
plt.plot(np.sqrt(train_errors), "r--", linewidth=2, label="Training set")
plt.legend(loc="upper right", fontsize=14)
plt.xlabel("Epoch", fontsize=14)
plt.ylabel("RMSE", fontsize=14)
save_fig("early_stopping_plot")
plt.show()


# In[46]:


from sklearn.base import clone
sgd_reg = SGDRegressor(max_iter=1, tol=-np.infty, warm_start=True, penalty=None,
                       learning_rate="constant", eta0=0.0005, random_state=42)

minimum_val_error = float("inf")
best_epoch = None
best_model = None
for epoch in range(1000):
    sgd_reg.fit(X_train_poly_scaled, y_train)  # continues where it left off
    y_val_predict = sgd_reg.predict(X_val_poly_scaled)
    val_error = mean_squared_error(y_val, y_val_predict)
    if val_error < minimum_val_error:
        minimum_val_error = val_error
        best_epoch = epoch
        best_model = clone(sgd_reg)


# In[47]:


best_epoch, best_model


# In[48]:


t1a, t1b, t2a, t2b = -1, 3, -1.5, 1.5

# ignoring bias term
t1s = np.linspace(t1a, t1b, 500)
t2s = np.linspace(t2a, t2b, 500)
t1, t2 = np.meshgrid(t1s, t2s)
T = np.c_[t1.ravel(), t2.ravel()]
Xr = np.array([[-1, 1], [-0.3, -1], [1, 0.1]])
yr = 2 * Xr[:, :1] + 0.5 * Xr[:, 1:]

J = (1/len(Xr) * np.sum((T.dot(Xr.T) - yr.T)**2, axis=1)).reshape(t1.shape)

N1 = np.linalg.norm(T, ord=1, axis=1).reshape(t1.shape)
N2 = np.linalg.norm(T, ord=2, axis=1).reshape(t1.shape)

t_min_idx = np.unravel_index(np.argmin(J), J.shape)
t1_min, t2_min = t1[t_min_idx], t2[t_min_idx]

t_init = np.array([[0.25], [-1]])


# In[49]:


def bgd_path(theta, X, y, l1, l2, core = 1, eta = 0.1, n_iterations = 50):
    path = [theta]
    for iteration in range(n_iterations):
        gradients = core * 2/len(X) * X.T.dot(X.dot(theta) - y) + l1 * np.sign(theta) + 2 * l2 * theta

        theta = theta - eta * gradients
        path.append(theta)
    return np.array(path)

plt.figure(figsize=(12, 8))
for i, N, l1, l2, title in ((0, N1, 0.5, 0, "Lasso"), (1, N2, 0,  0.1, "Ridge")):
    JR = J + l1 * N1 + l2 * N2**2
    
    tr_min_idx = np.unravel_index(np.argmin(JR), JR.shape)
    t1r_min, t2r_min = t1[tr_min_idx], t2[tr_min_idx]

    levelsJ=(np.exp(np.linspace(0, 1, 20)) - 1) * (np.max(J) - np.min(J)) + np.min(J)
    levelsJR=(np.exp(np.linspace(0, 1, 20)) - 1) * (np.max(JR) - np.min(JR)) + np.min(JR)
    levelsN=np.linspace(0, np.max(N), 10)
    
    path_J = bgd_path(t_init, Xr, yr, l1=0, l2=0)
    path_JR = bgd_path(t_init, Xr, yr, l1, l2)
    path_N = bgd_path(t_init, Xr, yr, np.sign(l1)/3, np.sign(l2), core=0)

    plt.subplot(221 + i * 2)
    plt.grid(True)
    plt.axhline(y=0, color='k')
    plt.axvline(x=0, color='k')
    plt.contourf(t1, t2, J, levels=levelsJ, alpha=0.9)
    plt.contour(t1, t2, N, levels=levelsN)
    plt.plot(path_J[:, 0], path_J[:, 1], "w-o")
    plt.plot(path_N[:, 0], path_N[:, 1], "y-^")
    plt.plot(t1_min, t2_min, "rs")
    plt.title(r"$\ell_{}$ penalty".format(i + 1), fontsize=16)
    plt.axis([t1a, t1b, t2a, t2b])
    if i == 1:
        plt.xlabel(r"$\theta_1$", fontsize=20)
    plt.ylabel(r"$\theta_2$", fontsize=20, rotation=0)

    plt.subplot(222 + i * 2)
    plt.grid(True)
    plt.axhline(y=0, color='k')
    plt.axvline(x=0, color='k')
    plt.contourf(t1, t2, JR, levels=levelsJR, alpha=0.9)
    plt.plot(path_JR[:, 0], path_JR[:, 1], "w-o")
    plt.plot(t1r_min, t2r_min, "rs")
    plt.title(title, fontsize=16)
    plt.axis([t1a, t1b, t2a, t2b])
    if i == 1:
        plt.xlabel(r"$\theta_1$", fontsize=20)

save_fig("lasso_vs_ridge_plot")
plt.show()


# # Logistic regression

# In[50]:


t = np.linspace(-10, 10, 100)
sig = 1 / (1 + np.exp(-t))
plt.figure(figsize=(9, 3))
plt.plot([-10, 10], [0, 0], "k-")
plt.plot([-10, 10], [0.5, 0.5], "k:")
plt.plot([-10, 10], [1, 1], "k:")
plt.plot([0, 0], [-1.1, 1.1], "k-")
plt.plot(t, sig, "b-", linewidth=2, label=r"$\sigma(t) = \frac{1}{1 + e^{-t}}$")
plt.xlabel("t")
plt.legend(loc="upper left", fontsize=20)
plt.axis([-10, 10, -0.1, 1.1])
save_fig("logistic_function_plot")
plt.show()


# In[51]:


from sklearn import datasets
iris = datasets.load_iris()
list(iris.keys())


# In[52]:


print(iris.DESCR)


# In[53]:


X = iris["data"][:, 3:]  # petal width
y = (iris["target"] == 2).astype(np.int)  # 1 if Iris-Virginica, else 0


# In[54]:


from sklearn.linear_model import LogisticRegression
log_reg = LogisticRegression(solver="liblinear", random_state=42)
log_reg.fit(X, y)


# In[55]:


X_new = np.linspace(0, 3, 1000).reshape(-1, 1)
y_proba = log_reg.predict_proba(X_new)

plt.plot(X_new, y_proba[:, 1], "g-", linewidth=2, label="Iris-Virginica")
plt.plot(X_new, y_proba[:, 0], "b--", linewidth=2, label="Not Iris-Virginica")


# The figure in the book actually is actually a bit fancier:

# In[56]:


X_new = np.linspace(0, 3, 1000).reshape(-1, 1)
y_proba = log_reg.predict_proba(X_new)
decision_boundary = X_new[y_proba[:, 1] >= 0.5][0]

plt.figure(figsize=(8, 3))
plt.plot(X[y==0], y[y==0], "bs")
plt.plot(X[y==1], y[y==1], "g^")
plt.plot([decision_boundary, decision_boundary], [-1, 2], "k:", linewidth=2)
plt.plot(X_new, y_proba[:, 1], "g-", linewidth=2, label="Iris-Virginica")
plt.plot(X_new, y_proba[:, 0], "b--", linewidth=2, label="Not Iris-Virginica")
plt.text(decision_boundary+0.02, 0.15, "Decision  boundary", fontsize=14, color="k", ha="center")
plt.arrow(decision_boundary, 0.08, -0.3, 0, head_width=0.05, head_length=0.1, fc='b', ec='b')
plt.arrow(decision_boundary, 0.92, 0.3, 0, head_width=0.05, head_length=0.1, fc='g', ec='g')
plt.xlabel("Petal width (cm)", fontsize=14)
plt.ylabel("Probability", fontsize=14)
plt.legend(loc="center left", fontsize=14)
plt.axis([0, 3, -0.02, 1.02])
save_fig("logistic_regression_plot")
plt.show()


# In[57]:


decision_boundary


# In[58]:


log_reg.predict([[1.7], [1.5]])


# In[59]:


from sklearn.linear_model import LogisticRegression

X = iris["data"][:, (2, 3)]  # petal length, petal width
y = (iris["target"] == 2).astype(np.int)

log_reg = LogisticRegression(solver="liblinear", C=10**10, random_state=42)
log_reg.fit(X, y)

x0, x1 = np.meshgrid(
        np.linspace(2.9, 7, 500).reshape(-1, 1),
        np.linspace(0.8, 2.7, 200).reshape(-1, 1),
    )
X_new = np.c_[x0.ravel(), x1.ravel()]

y_proba = log_reg.predict_proba(X_new)

plt.figure(figsize=(10, 4))
plt.plot(X[y==0, 0], X[y==0, 1], "bs")
plt.plot(X[y==1, 0], X[y==1, 1], "g^")

zz = y_proba[:, 1].reshape(x0.shape)
contour = plt.contour(x0, x1, zz, cmap=plt.cm.brg)


left_right = np.array([2.9, 7])
boundary = -(log_reg.coef_[0][0] * left_right + log_reg.intercept_[0]) / log_reg.coef_[0][1]

plt.clabel(contour, inline=1, fontsize=12)
plt.plot(left_right, boundary, "k--", linewidth=3)
plt.text(3.5, 1.5, "Not Iris-Virginica", fontsize=14, color="b", ha="center")
plt.text(6.5, 2.3, "Iris-Virginica", fontsize=14, color="g", ha="center")
plt.xlabel("Petal length", fontsize=14)
plt.ylabel("Petal width", fontsize=14)
plt.axis([2.9, 7, 0.8, 2.7])
save_fig("logistic_regression_contour_plot")
plt.show()


# In[60]:


X = iris["data"][:, (2, 3)]  # petal length, petal width
y = iris["target"]

softmax_reg = LogisticRegression(multi_class="multinomial",solver="lbfgs", C=10, random_state=42)
softmax_reg.fit(X, y)


# In[61]:


x0, x1 = np.meshgrid(
        np.linspace(0, 8, 500).reshape(-1, 1),
        np.linspace(0, 3.5, 200).reshape(-1, 1),
    )
X_new = np.c_[x0.ravel(), x1.ravel()]


y_proba = softmax_reg.predict_proba(X_new)
y_predict = softmax_reg.predict(X_new)

zz1 = y_proba[:, 1].reshape(x0.shape)
zz = y_predict.reshape(x0.shape)

plt.figure(figsize=(10, 4))
plt.plot(X[y==2, 0], X[y==2, 1], "g^", label="Iris-Virginica")
plt.plot(X[y==1, 0], X[y==1, 1], "bs", label="Iris-Versicolor")
plt.plot(X[y==0, 0], X[y==0, 1], "yo", label="Iris-Setosa")

from matplotlib.colors import ListedColormap
custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])

plt.contourf(x0, x1, zz, cmap=custom_cmap)
contour = plt.contour(x0, x1, zz1, cmap=plt.cm.brg)
plt.clabel(contour, inline=1, fontsize=12)
plt.xlabel("Petal length", fontsize=14)
plt.ylabel("Petal width", fontsize=14)
plt.legend(loc="center left", fontsize=14)
plt.axis([0, 7, 0, 3.5])
save_fig("softmax_regression_contour_plot")
plt.show()


# In[62]:


softmax_reg.predict([[5, 2]])


# In[63]:


softmax_reg.predict_proba([[5, 2]])


# # Exercise solutions

# ## 1. to 11.

# See appendix A.

# ## 12. Batch Gradient Descent with early stopping for Softmax Regression
# (without using Scikit-Learn)

# Let's start by loading the data. We will just reuse the Iris dataset we loaded earlier.

# In[64]:


X = iris["data"][:, (2, 3)]  # petal length, petal width
y = iris["target"]


# We need to add the bias term for every instance ($x_0 = 1$):

# In[65]:


X_with_bias = np.c_[np.ones([len(X), 1]), X]


# And let's set the random seed so the output of this exercise solution is reproducible:

# In[66]:


np.random.seed(2042)


# The easiest option to split the dataset into a training set, a validation set and a test set would be to use Scikit-Learn's `train_test_split()` function, but the point of this exercise is to try understand the algorithms by implementing them manually. So here is one possible implementation:

# In[67]:


test_ratio = 0.2
validation_ratio = 0.2
total_size = len(X_with_bias)

test_size = int(total_size * test_ratio)
validation_size = int(total_size * validation_ratio)
train_size = total_size - test_size - validation_size

rnd_indices = np.random.permutation(total_size)

X_train = X_with_bias[rnd_indices[:train_size]]
y_train = y[rnd_indices[:train_size]]
X_valid = X_with_bias[rnd_indices[train_size:-test_size]]
y_valid = y[rnd_indices[train_size:-test_size]]
X_test = X_with_bias[rnd_indices[-test_size:]]
y_test = y[rnd_indices[-test_size:]]


# The targets are currently class indices (0, 1 or 2), but we need target class probabilities to train the Softmax Regression model. Each instance will have target class probabilities equal to 0.0 for all classes except for the target class which will have a probability of 1.0 (in other words, the vector of class probabilities for ay given instance is a one-hot vector). Let's write a small function to convert the vector of class indices into a matrix containing a one-hot vector for each instance:

# In[68]:


def to_one_hot(y):
    n_classes = y.max() + 1
    m = len(y)
    Y_one_hot = np.zeros((m, n_classes))
    Y_one_hot[np.arange(m), y] = 1
    return Y_one_hot


# Let's test this function on the first 10 instances:

# In[69]:


y_train[:10]


# In[70]:


to_one_hot(y_train[:10])


# Looks good, so let's create the target class probabilities matrix for the training set and the test set:

# In[71]:


Y_train_one_hot = to_one_hot(y_train)
Y_valid_one_hot = to_one_hot(y_valid)
Y_test_one_hot = to_one_hot(y_test)


# Now let's implement the Softmax function. Recall that it is defined by the following equation:
# 
# $\sigma\left(\mathbf{s}(\mathbf{x})\right)_k = \dfrac{\exp\left(s_k(\mathbf{x})\right)}{\sum\limits_{j=1}^{K}{\exp\left(s_j(\mathbf{x})\right)}}$

# In[72]:


def softmax(logits):
    exps = np.exp(logits)
    exp_sums = np.sum(exps, axis=1, keepdims=True)
    return exps / exp_sums


# We are almost ready to start training. Let's define the number of inputs and outputs:

# In[73]:


n_inputs = X_train.shape[1] # == 3 (2 features plus the bias term)
n_outputs = len(np.unique(y_train))   # == 3 (3 iris classes)


# Now here comes the hardest part: training! Theoretically, it's simple: it's just a matter of translating the math equations into Python code. But in practice, it can be quite tricky: in particular, it's easy to mix up the order of the terms, or the indices. You can even end up with code that looks like it's working but is actually not computing exactly the right thing. When unsure, you should write down the shape of each term in the equation and make sure the corresponding terms in your code match closely. It can also help to evaluate each term independently and print them out. The good news it that you won't have to do this everyday, since all this is well implemented by Scikit-Learn, but it will help you understand what's going on under the hood.
# 
# So the equations we will need are the cost function:
# 
# $J(\mathbf{\Theta}) =
# - \dfrac{1}{m}\sum\limits_{i=1}^{m}\sum\limits_{k=1}^{K}{y_k^{(i)}\log\left(\hat{p}_k^{(i)}\right)}$
# 
# And the equation for the gradients:
# 
# $\nabla_{\mathbf{\theta}^{(k)}} \, J(\mathbf{\Theta}) = \dfrac{1}{m} \sum\limits_{i=1}^{m}{ \left ( \hat{p}^{(i)}_k - y_k^{(i)} \right ) \mathbf{x}^{(i)}}$
# 
# Note that $\log\left(\hat{p}_k^{(i)}\right)$ may not be computable if $\hat{p}_k^{(i)} = 0$. So we will add a tiny value $\epsilon$ to $\log\left(\hat{p}_k^{(i)}\right)$ to avoid getting `nan` values.

# In[74]:


eta = 0.01
n_iterations = 5001
m = len(X_train)
epsilon = 1e-7

Theta = np.random.randn(n_inputs, n_outputs)

for iteration in range(n_iterations):
    logits = X_train.dot(Theta)
    Y_proba = softmax(logits)
    loss = -np.mean(np.sum(Y_train_one_hot * np.log(Y_proba + epsilon), axis=1))
    error = Y_proba - Y_train_one_hot
    if iteration % 500 == 0:
        print(iteration, loss)
    gradients = 1/m * X_train.T.dot(error)
    Theta = Theta - eta * gradients


# And that's it! The Softmax model is trained. Let's look at the model parameters:

# In[75]:


Theta


# Let's make predictions for the validation set and check the accuracy score:

# In[76]:


logits = X_valid.dot(Theta)
Y_proba = softmax(logits)
y_predict = np.argmax(Y_proba, axis=1)

accuracy_score = np.mean(y_predict == y_valid)
accuracy_score


# Well, this model looks pretty good. For the sake of the exercise, let's add a bit of $\ell_2$ regularization. The following training code is similar to the one above, but the loss now has an additional $\ell_2$ penalty, and the gradients have the proper additional term (note that we don't regularize the first element of `Theta` since this corresponds to the bias term). Also, let's try increasing the learning rate `eta`.

# In[77]:


eta = 0.1
n_iterations = 5001
m = len(X_train)
epsilon = 1e-7
alpha = 0.1  # regularization hyperparameter

Theta = np.random.randn(n_inputs, n_outputs)

for iteration in range(n_iterations):
    logits = X_train.dot(Theta)
    Y_proba = softmax(logits)
    xentropy_loss = -np.mean(np.sum(Y_train_one_hot * np.log(Y_proba + epsilon), axis=1))
    l2_loss = 1/2 * np.sum(np.square(Theta[1:]))
    loss = xentropy_loss + alpha * l2_loss
    error = Y_proba - Y_train_one_hot
    if iteration % 500 == 0:
        print(iteration, loss)
    gradients = 1/m * X_train.T.dot(error) + np.r_[np.zeros([1, n_outputs]), alpha * Theta[1:]]
    Theta = Theta - eta * gradients


# Because of the additional $\ell_2$ penalty, the loss seems greater than earlier, but perhaps this model will perform better? Let's find out:

# In[78]:


logits = X_valid.dot(Theta)
Y_proba = softmax(logits)
y_predict = np.argmax(Y_proba, axis=1)

accuracy_score = np.mean(y_predict == y_valid)
accuracy_score


# Cool, perfect accuracy! We probably just got lucky with this validation set, but still, it's pleasant.

# Now let's add early stopping. For this we just need to measure the loss on the validation set at every iteration and stop when the error starts growing.

# In[79]:


eta = 0.1 
n_iterations = 5001
m = len(X_train)
epsilon = 1e-7
alpha = 0.1  # regularization hyperparameter
best_loss = np.infty

Theta = np.random.randn(n_inputs, n_outputs)

for iteration in range(n_iterations):
    logits = X_train.dot(Theta)
    Y_proba = softmax(logits)
    xentropy_loss = -np.mean(np.sum(Y_train_one_hot * np.log(Y_proba + epsilon), axis=1))
    l2_loss = 1/2 * np.sum(np.square(Theta[1:]))
    loss = xentropy_loss + alpha * l2_loss
    error = Y_proba - Y_train_one_hot
    gradients = 1/m * X_train.T.dot(error) + np.r_[np.zeros([1, n_outputs]), alpha * Theta[1:]]
    Theta = Theta - eta * gradients

    logits = X_valid.dot(Theta)
    Y_proba = softmax(logits)
    xentropy_loss = -np.mean(np.sum(Y_valid_one_hot * np.log(Y_proba + epsilon), axis=1))
    l2_loss = 1/2 * np.sum(np.square(Theta[1:]))
    loss = xentropy_loss + alpha * l2_loss
    if iteration % 500 == 0:
        print(iteration, loss)
    if loss < best_loss:
        best_loss = loss
    else:
        print(iteration - 1, best_loss)
        print(iteration, loss, "early stopping!")
        break


# In[80]:


logits = X_valid.dot(Theta)
Y_proba = softmax(logits)
y_predict = np.argmax(Y_proba, axis=1)

accuracy_score = np.mean(y_predict == y_valid)
accuracy_score


# Still perfect, but faster.

# Now let's plot the model's predictions on the whole dataset:

# In[81]:


x0, x1 = np.meshgrid(
        np.linspace(0, 8, 500).reshape(-1, 1),
        np.linspace(0, 3.5, 200).reshape(-1, 1),
    )
X_new = np.c_[x0.ravel(), x1.ravel()]
X_new_with_bias = np.c_[np.ones([len(X_new), 1]), X_new]

logits = X_new_with_bias.dot(Theta)
Y_proba = softmax(logits)
y_predict = np.argmax(Y_proba, axis=1)

zz1 = Y_proba[:, 1].reshape(x0.shape)
zz = y_predict.reshape(x0.shape)

plt.figure(figsize=(10, 4))
plt.plot(X[y==2, 0], X[y==2, 1], "g^", label="Iris-Virginica")
plt.plot(X[y==1, 0], X[y==1, 1], "bs", label="Iris-Versicolor")
plt.plot(X[y==0, 0], X[y==0, 1], "yo", label="Iris-Setosa")

from matplotlib.colors import ListedColormap
custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])

plt.contourf(x0, x1, zz, cmap=custom_cmap)
contour = plt.contour(x0, x1, zz1, cmap=plt.cm.brg)
plt.clabel(contour, inline=1, fontsize=12)
plt.xlabel("Petal length", fontsize=14)
plt.ylabel("Petal width", fontsize=14)
plt.legend(loc="upper left", fontsize=14)
plt.axis([0, 7, 0, 3.5])
plt.show()


# And now let's measure the final model's accuracy on the test set:

# In[82]:


logits = X_test.dot(Theta)
Y_proba = softmax(logits)
y_predict = np.argmax(Y_proba, axis=1)

accuracy_score = np.mean(y_predict == y_test)
accuracy_score


# Our perfect model turns out to have slight imperfections. This variability is likely due to the very small size of the dataset: depending on how you sample the training set, validation set and the test set, you can get quite different results. Try changing the random seed and running the code again a few times, you will see that the results will vary.

# In[ ]: