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from IPython.display import display
from PIL import Image
path="D:\Regression\equation_lr_multi.png"
display(Image.open(path))
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#Import les libraries
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
#from sklearn.linear_model import LinearRegression
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#Import Dataset
dataset = pd.read_csv("d:\Regression\Advertising.csv")
dataset
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X = np.array(dataset[['TV','Radio','Newspaper']])
y = np.array(dataset['Sales'])
print(y.shape)
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plt.scatter(X[:,0], y)
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plt.scatter(X[:,1], y)
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plt.scatter(X[:,2], y)
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graph = plt.figure()
ax = graph.add_subplot(111, projection = '3d')
ax.scatter(dataset["TV"], dataset["Radio"], dataset["Sales"], c = 'r', marker = '^')
ax.set_xlabel('TV')
ax.set_ylabel('Radio')
ax.set_zlabel('Sales')
plt.show
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graph = plt.figure()
ax = graph.add_subplot(111, projection = '3d')
ax.scatter(dataset["TV"], dataset["Newspaper"], dataset["Sales"], c = 'g', marker = '^')
ax.set_xlabel('TV')
ax.set_ylabel('Newspaper')
ax.set_zlabel('Sales')
plt.show
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graph = plt.figure()
ax = graph.add_subplot(111, projection = '3d')
ax.scatter(dataset["Radio"], dataset["Newspaper"], dataset["Sales"], c = 'b', marker = '^')
ax.set_xlabel('Radio')
ax.set_ylabel('Newspaper')
ax.set_zlabel('Sales')
plt.show
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#We add an additional column composed by 1 on X matrix
X = np.array(dataset[['TV','Radio','Newspaper']])
X = np.c_[X, np.ones(X.shape[0])]
X
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# We are checking that the dimension of X is m x (n + 1) ; m = 200, n = 3
print(X.shape)
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path="D:\Regression\model_ftheta_2.png"
display(Image.open(path))
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# Initialisation du vecteur Theta
theta = np.random.randn(4,1)
theta
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# Dimension of theta is (n + 1) x 1; n = 3
print(theta.shape)
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def model(X, theta):
return X.dot(theta)
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# We are checking that dimension of X.theta is m x 1; m = 200
F = model(X, theta)
print(F.shape)
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path="D:\Regression\cost_function_2.png"
display(Image.open(path))
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print(y.shape) #We can notice that, we don't have a second dimension of y
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y = y.reshape(y.shape[0], 1) # We can use reshape function to add a second dimension to y vector.
#We used it to be able to calculate X.theta - y
print(y.shape)
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G = model(X, theta) - y
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print(G.shape)
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def cost_function(X, y, theta):
m = len(y)
n = 1/(2*m) * np.sum((model(X, theta) - y)**2)
return n
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n = cost_function (X, y, theta)
n
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path="D:\Regression\gradients.png"
display(Image.open(path))
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def grad(X, y, theta):
m = len(y)
return 1/m * X.T.dot(model(X, theta) - y)
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print(X.T.shape) #Dimension of X.T must be (n+ 1) X m
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z = grad(X, y, theta) # Dimension of grad(X, y, theta) must be (n +1) x 1
print(z.shape)
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path="D:\Regression\gradient_descent_2.png"
display(Image.open(path))
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def grandient_descent(X, y, theta, learning_rate, n_iterations):
cost_history = np.zeros(n_iterations)
for i in range(0, n_iterations):
theta = theta - learning_rate * grad(X, y, theta)
cost_history[i] = cost_function(X, y, theta)
return theta,cost_history
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n_it = 4000
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theta_final, cost_history = grandient_descent(X, y, theta, learning_rate = 0.00001, n_iterations = n_it)
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plt.plot(range(n_it), cost_history)
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# We can see that from 4000 iterations the model converges
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predictions = model(X, theta_final)
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my_pred = model(np.array([151.5,41.3,58.5,1]), theta_final)
my_pred
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path="D:\Regression\determination_coef.png"
display(Image.open(path))
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#This rate allow to mesure the percentage of
def determination_rate(y, pred):
u = ((y - pred)**2).sum()
v = ((y - y.mean())**2).sum()
return 1 - u/v
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determination_rate(y, predictions)
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graph = plt.figure()
ax = graph.add_subplot(111, projection = '3d')
ax.set_xlabel('TV')
ax.set_ylabel('Radio')
ax.set_zlabel('Sales')
ax.scatter(dataset["TV"], dataset["Radio"], dataset["Sales"], c = 'b', marker = '^')
ax.scatter(dataset["TV"], dataset["Radio"], predictions, c = 'r', marker = '^')
plt.show()
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graph = plt.figure()
ax = graph.add_subplot(111, projection = '3d')
ax.set_xlabel('TV')
ax.set_ylabel('Newspaper')
ax.set_zlabel('Sales')
ax.scatter(dataset["TV"], dataset["Newspaper"], dataset["Sales"], c = 'g', marker = '^')
ax.scatter(dataset["TV"], dataset["Newspaper"], predictions, c = 'r', marker = '^')
plt.show()
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graph = plt.figure()
ax = graph.add_subplot(111, projection = '3d')
ax.set_xlabel('Radio')
ax.set_ylabel('Newspaper')
ax.set_zlabel('Sales')
ax.scatter(dataset["Radio"], dataset["Newspaper"], dataset["Sales"], c = 'y', marker = '^')
ax.scatter(dataset["Radio"], dataset["Newspaper"], predictions, c = 'r', marker = '^')
plt.show()
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