# Implementation of linear regression in Python

I wrote an implementation for multivariate linear regression in Python, for data in this link: http://www-bcf.usc.edu/~gareth/ISL/Advertising.csv

My main focus is to avoid loops as much as possible by using vectorization, and letting numpy's functions do all the work.

How can I improve my code for performance and enhanced vectorization?

This is my code:

import numpy as np
import matplotlib.pyplot as plt

m, n = data.shape

X = data[:, 0:n-1]
Y = data[:, n-1].reshape((m, 1))

# add unity vector column with size m to the X matrix, to account for theta_0
ones = np.ones((m, 1))
X = np.hstack((ones, X))

iterations = 500        # gradient descent iterations count
alpha = 0.01            # learning rate

theta = np.random.rand(n, 1)

def plotData():
# get rid of X_0 constants column
features = X[:, 1: n]
plt.figure(1)

plt.subplot(131)
plt.plot(features[:, 0], Y, 'bo')
plt.ylabel('Sales')

plt.subplot(132)
plt.plot(features[:, 1], Y, 'ro')
plt.ylabel('Sales')

plt.subplot(133)
plt.plot(features[:, 2], Y, 'yo')
plt.ylabel('Sales')

plt.show()

def computeCost():
hypothesis = np.dot(X, theta);
delta = np.dot((hypothesis - Y).transpose(), (hypothesis - Y))
return (1 / m) * delta

# normalizeFeatures: scale-normalize all features to speed up gradient descent convergence
def normalizeFeatures():
# 1) generate Average vector mu, contains the average of each feature in the X matrix
# 2) generate Std. deviation vector sigma, contains the std. dev. of each feature in the X matrix
# 3) subtract average value and divide by the standard deviation, for each feature column
mu = np.ones((1, n))
sigma = np.ones((1, n))

# range() starts from 1 not 0, to skip the first all-ones constants column in the features matrix
for i in range(1,n):
mu[0][i] = np.mean(X[:, i])
sigma[0][i] = np.std(X[:, i])
X[:, i] = (X[:, i] - mu[0][i]) / sigma[0][i]

return mu, sigma;

# vector to keep track of progression of cost function with each iteration
J_history = np.ones((iterations, 1))

for i in range(iterations):
delta = np.dot((np.dot(X,theta) - Y).transpose(), X).transpose()
theta -= (alpha/m) * delta
J_history[i, 0] = computeCost()

plt.plot(np.linspace(0, iterations, iterations), J_history)
plt.title('Cost function against number of iterations')
plt.xlabel('Number of iterations')
plt.ylabel('Cost function J(theta)')
plt.show()
return

# normalEquation() calculates hypothesis equation coefficients analytically
def normalEquation():
A = np.linalg.pinv(np.dot(X.transpose(), X))
B = np.dot(X.transpose(), Y)
theta = np.dot(A, B)

return theta

def predict(x_vector, mu, sigma):
# scale feature vector
for i in range(1, n):
x_vector[0][i] = (x_vector[0,i]- mu[0][i]) / sigma[0][i]

return np.dot(x_vector, theta)

if __name__ == '__main__':
plotData()
mu, sigma = normalizeFeatures()
print('Hypothesis coefficients from gradient descent:\n {}'.format(theta))
print('Hypothesis coefficients from normal equation:\n {}'.format(normalEquation()))

prediction_vector = np.array([1, 40, 40, 48]).reshape(1,4)
print('Prediction for values [1, 40, 40, 48] is {}'.format(predict(prediction_vector, mu, sigma)))


Just reviewing normalizeFeatures.

1. Instead of a comment explaining what the function does, write a docstring. (Docstrings are available from the interactive interpreter via the help function.)

2. The function operates on the global variable X. This makes the function inflexible (you can't use it for anything other than modifying the particular variable X), and hard to test. It would be better if the function received the array as an argument, so that it could be used on any array:

def normalize_features(a):
"""Given array a with shape (m, n), update it so that each column
(except column 0) has mean 0 and standard deviation 1. Return
arrays mu, sigma with shape (1, n) where mu[0,i] is the original
mean of a[:,i] and sigma[0,i] is the original standard deviation
of a[:,i].

"""

3. This complicated specification can be simplified. First, we can avoid the "(except column 0)" bit and let the caller pass X[:,1:] if they want to leave column 0 alone. Second, we can return mu and sigma with shape (n,) instead of (1, n) and let the caller reshape if necessary (which it probably isn't).

def normalize_features(a):
"""Given array a with shape (m, n), update it so that each column has
mean 0 and standard deviation 1. Return arrays mu, sigma giving
the original mean and standard deviation of each column of a.

"""

4. The loop over the columns is unnecessary because numpy.mean and numpy.std take an axis argument. So the body of the function can be:

mu = a.mean(axis=0)
sigma = a.std(axis=0)
a[...] = (a - mu) / sigma
return mu, sigma

5. It might be better not to update the array X, but to compute a new array containing the normalized data (this uses twice the memory, but that's only a concern if X is really big). If you did this, then you wouldn't need to remember the original mean and standard deviation (because you still have the original data) and so the normalization becomes a one-liner:

def normalize_features(a):
"""Return copy of a with each column normalized so that it has mean 0
and standard deviation 1.

"""
return (a - a.mean(axis=0)) / a.std(axis=0)

6. Consider using scipy.stats.zscore instead of writing your own.

• Is a[...] actual NumPy syntax or did you omit something? – mkrieger1 Aug 23 '17 at 13:36
• @mkrieger1: It is actual NumPy syntax. See the documentation. – Gareth Rees Aug 23 '17 at 14:03