Linear regression 1

import numpy as np

import matplotlib.pyplot as plt

# ==================== Part 1: Basic Function ====================

print('Running warmUpExercise ... ')

print('5x5 Identity Matrix: ')

def warmUpExercise():

    return np.eye(5)# YOUR CODE HERE

warmUpExercise()

 

 

output

Running warmUpExercise ... 
5x5 Identity Matrix: 

array([[1., 0., 0., 0., 0.],
       [0., 1., 0., 0., 0.],
       [0., 0., 1., 0., 0.],
       [0., 0., 0., 1., 0.],
       [0., 0., 0., 0., 1.]])

# ======================= Part 2: Plotting =======================

print('Plotting Data ...')

from google.colab import drive

drive.mount('/content/drive')

data = np.loadtxt('/content/drive/MyDrive/Colab Notebooks/ex1data1.txt', delimiter=',')

data = np.array(data)

X = data[:,0]

y = data[:,1]

m = len(y)

plt.figure()

plt.scatter(X,y)

# =================== Part 3: Cost and Gradient descent ===================

X = np.column_stack((np.ones(m), X))

theta = np.zeros(2)

Plotting Data ...
Drive already mounted at /content/drive; to attempt to forcibly remount, call drive.mount("/content/drive", force_remount=True).

# Some gradient descent settings

iterations = 1500

alpha = 0.01

print('Testing the cost function ...')

# compute and display initial cost

def computeCost(X, y, theta):

    # YOUR CODE HERE

    m = len(y)

    J = np.sum((X.dot(theta) - y) ** 2) / (2 * m)

    return J

J = computeCost(X, y, theta)

print('With theta = [0 ; 0] \nCost computed = ', J)

print('Expected cost value (approx) 32.07')

Testing the cost function ... With theta = [0 ; 0] Cost computed = 32.072733877455676 Expected cost value (approx) 32.07

# further testing of the cost function

J = computeCost(X, y, [-1, 2])

print('With theta = [-1 ; 2] \n Cost computed = ', J)

print('Expected cost value (approx) 54.24')

output

With theta = [-1 ; 2] 
 Cost computed =  54.24245508201238
Expected cost value (approx) 54.24

# run gradient descent

theta = [-50,-50]

print('Running Gradient Descent ...')

def gradientDescent(X, y, theta, alpha, iterations):

    # YOUR CODE HERE

    m = len(y)

    J_record = []

    for _ in range(iterations):

        gradJ = (X.T.dot(X.dot(theta) - y))/m

        theta = theta - alpha * gradJ

        J_record.append(computeCost(X, y, theta))

    return theta, J_record

theta, J_record = gradientDescent(X, y, theta, alpha, iterations)

plt.figure()

plt.plot(J_record)

print('Theta found by gradient descent:')

print(theta)

print('Expected theta values (approx)')

print('-3.6303,  1.1664')

J = computeCost(X, y, theta)

print('J =',J)

Running Gradient Descent ...
Theta found by gradient descent:
[-6.60389588  1.46509317]
Expected theta values (approx)
-3.6303,  1.1664
J = 5.144644149724434

# Plot the linear fit

plt.figure()

plt.scatter(X[:,1], y, label='Training data')

plt.plot(X[:,1], X.dot(theta), label = 'Linear regression')

plt.legend()

plt.show()

# Predict values for population sizes of 35,000 and 70,000

predict1 = np.array([1, 3.5]).dot(theta)

print('For population = 35,000, we predict a profit of ',predict1*10000)

predict2 = np.array([1, 7]).dot(theta)

print('For population = 70,000, we predict a profit of ',predict2*10000)

For population = 35,000, we predict a profit of 390129.08230406477 For population = 70,000, we predict a profit of 221840.19017197358

# ============= Part 4: Visualizing J(theta_0, theta_1) =============

print('Visualizing J(theta_0, theta_1) ...')

# Grid over which we will calculate J

theta0_vals = np.linspace(-10, 10, 100)

theta1_vals = np.linspace(-1, 4, 100)

# initialize J_vals to a matrix of 0's

J_vals = np.zeros((len(theta0_vals), len(theta1_vals)))

# Fill out J_vals

for i in range(len(theta0_vals)):

    for j in range(len(theta1_vals)):

      t = np.array([theta0_vals[i], theta1_vals[j]])

      J_vals[i,j] = computeCost(X, y, t)

J_vals = J_vals.T

fig = plt.figure()

ax = fig.add_subplot(111, projection='3d')

theta0_vals, theta1_vals = np.meshgrid(theta0_vals, theta1_vals)

ax.plot_surface(theta0_vals, theta1_vals, J_vals, cmap='viridis')

ax.set_xlabel('theta_0')

ax.set_ylabel('theta_1')

ax.set_zlabel('J(\theta_0, \theta_1)')

plt.show()

# Contour plot

plt.figure()

contour = plt.contour(theta0_vals, theta1_vals, J_vals, levels=np.logspace(-2, 3, 20))

plt.clabel(contour, inline=1, fontsize=10)

plt.xlabel('theta_0')

plt.ylabel('theta_1')

plt.plot(theta[0], theta[1], 'rx', markersize=10, linewidth=2)

plt.show()

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  fig.canvas.print_figure(bytes_io, **kw)