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linear regression
from numpy import loadtxt, zeros, ones, array, linspace, logspace
from pylab import scatter, show, title, xlabel, ylabel, plot, contour
#Evaluate the linear regression
def compute_cost(X, y, theta):
Comput cost for linear regression
#Number of training samples
m = y.size
predictions =
sqErrors = (predictions - y) ** 2
J = (1.0 / (2 * m)) * sqErrors.sum()
return J
def gradient_descent(X, y, theta, alpha, num_iters):
Performs gradient descent to learn theta
by taking num_items gradient steps with learning
rate alpha
m = y.size
J_history = zeros(shape=(num_iters, 1))
for i in range(num_iters):
predictions =
errors_x1 = (predictions - y) * X[:, 0]
errors_x2 = (predictions - y) * X[:, 1]
theta[0][0] = theta[0][0] - alpha * (1.0 / m) * errors_x1.sum()
theta[1][0] = theta[1][0] - alpha * (1.0 / m) * errors_x2.sum()
J_history[i, 0] = compute_cost(X, y, theta)
return theta, J_history
#Load the dataset
data = loadtxt('ex1data1.txt', delimiter=',')
#Plot the data
scatter(data[:, 0], data[:, 1], marker='o', c='b')
title('Profits distribution')
xlabel('Population of City in 10,000s')
ylabel('Profit in $10,000s')
X = data[:, 0]
y = data[:, 1]
#number of training samples
m = y.size
#Add a column of ones to X (interception data)
it = ones(shape=(m, 2))
it[:, 1] = X
#Initialize theta parameters
theta = zeros(shape=(2, 1))
#Some gradient descent settings
iterations = 1500
alpha = 0.01
#compute and display initial cost
print compute_cost(it, y, theta)
theta, J_history = gradient_descent(it, y, theta, alpha, iterations)
print theta
#Predict values for population sizes of 35,000 and 70,000
predict1 = array([1, 3.5]).dot(theta).flatten()
print 'For population = 35,000, we predict a profit of %f' % (predict1 * 10000)
predict2 = array([1, 7.0]).dot(theta).flatten()
print 'For population = 70,000, we predict a profit of %f' % (predict2 * 10000)
#Plot the results
result =
plot(data[:, 0], result)
#Grid over which we will calculate J
theta0_vals = linspace(-10, 10, 100)
theta1_vals = linspace(-1, 4, 100)
#initialize J_vals to a matrix of 0's
J_vals = zeros(shape=(theta0_vals.size, theta1_vals.size))
#Fill out J_vals
for t1, element in enumerate(theta0_vals):
for t2, element2 in enumerate(theta1_vals):
thetaT = zeros(shape=(2, 1))
thetaT[0][0] = element
thetaT[1][0] = element2
J_vals[t1, t2] = compute_cost(it, y, thetaT)
#Contour plot
J_vals = J_vals.T
#Plot J_vals as 15 contours spaced logarithmically between 0.01 and 100
contour(theta0_vals, theta1_vals, J_vals, logspace(-2, 3, 20))
scatter(theta[0][0], theta[1][0])

What is the use of defining J_history ?

linbug commented Jul 6, 2015

J_history is an array that allows you to remember the values of the cost function for every update. Then you can use this to plot how the cost function is changing as you update the theta parameters (if gradient descent is working properly, the cost function should be decreasing towards a minimum)

Dont get "errors_x1 = (predictions - y) * X[:, 0]" at Line 35. Why *X[:,0] is needed.?

Referring to this doc: and assuming you calculating theta0 on line 35.

Thank you

Sorry i am very noob to this. Got it !!. Its 1 anyway.

jon80 commented Dec 27, 2015

I ran the code on, and, the following error is being displayed:
Traceback (most recent call last):
File "python", line 75
print compute_cost(it, y, theta)
SyntaxError: invalid syntax

jon80 are you using Python 3.x? In these versions of Python, you need to use parentheses in print function, or you get a syntax error:


getting runtime warning in sqErrors, while running this.

NatigAli commented Apr 2, 2017

I am confused about the 15th line of your code: sqErrors = (predictions - y) ** 2.
predictions has 1xm shape and since y is defined as data[:,1] its shape is mx1.
As a result (predictions - y) returns 3x3 matrix.

So shouldn't it be as sqErrors = (predictions - y.T) ** 2? Then you get 1xm array.

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