import math
import numpy as np
Python Basics with Numpy (optional assignment)
Welcome to your first assignment. This exercise gives you a brief introduction to Python. Even if you’ve used Python before, this will help familiarize you with functions we’ll need.
Instructions:
1、You will be using Python 3.
2、Avoid using for-loops and while-loops, unless you are explicitly told to do so.
3、Do not modify the (# GRADED FUNCTION [function name]) comment in some cells. Your work would not be graded if you change this. Each cell containing that comment should only contain one function.
4、After coding your function, run the cell right below it to check if your result is correct.
After this assignment you will:
1、Be able to use iPython Notebooks
2、Be able to use numpy functions and numpy matrix/vector operations
3、Understand the concept of “broadcasting”
4、Be able to vectorize code
1. Output the value of test
START CODE HERE ### (≈ 1 line of code)
test = “Hello World”
END CODE HERE
print (“test: ” + test)
2. Output the value of a single sigmod
def basic_sigmoid(x):
#Compute sigmoid of x.
#Arguments:
#x -- A scalar
#Return:
#s -- sigmoid(x)
### START CODE HERE ### (≈ 1 line of code)
s = 1/(1 + np.math.exp(-x))
### END CODE HERE ###
return s
print(“basic_sigmoid(3):”,basic_sigmoid(3))
basic_sigmoid(3): 0.9525741268224334
Actually, we rarely use the “math” library in deep learning because the inputs of the functions are real numbers.
In deep learning we mostly use matrices and vectors. This is why numpy is more useful.
3. Output the value of the 3-element array sigmod
x = [1, 2, 3] #Vector cannot participate in operation
print(“basic_sigmoid(x):”,basic_sigmoid(x)) # you will see this give an error when you run it, because x is a vector.
4. Output the value of exp
example of np.exp
x = np.array([1, 2, 3])
print(“np.exp(x):”,np.exp(x)) # result is (exp(1), exp(2), exp(3))
np.exp(x):[ 2.71828183 7.3890561 20.08553692]
5. The result of adding the output vectors
example of vector operation
x = np.array([1, 2, 3])
print (“x + 3:”,x + 3)
x + 3: [4 5 6]
6、 Implement the sigmoid function using numpy
def sigmoid(x):
“””
Compute the sigmoid of x
Arguments:
x -- A scalar or numpy array of any size
Return:
s -- sigmoid(x)
"""
### START CODE HERE ### (≈ 1 line of code)
s = 1 / (1 + np.exp(-x))
### END CODE HERE ###
return s
x = np.array([1, 2, 3])
print(“sigmoid(x):”,sigmoid(x))
sigmoid(x): [ 0.73105858 0.88079708 0.95257413]
7、Sigmoid gradient
Implement the function sigmoid_grad() to compute the gradient
of the sigmoid function with respect to its input x. The formula is:
sigmoid_derivative (x) = σ ′ (x) = σ (x) (1 − σ (x))
def sigmoid_derivative(x):
“””
Compute the gradient (also called the slope or derivative) of the sigmoid function with respect to its input x.
You can store the output of the sigmoid function into variables and then use it to calculate the gradient.
Arguments:
x -- A scalar or numpy array
Return:
ds -- Your computed gradient.
"""
### START CODE HERE ### (≈ 2 lines of code)
s = sigmoid(x)
ds = s * (1 - s)
### END CODE HERE ###
return ds
x = np.array([1, 2, 5])
print (“sigmoid_derivative(x) = ” + str(sigmoid_derivative(x)))
sigmoid_derivative(x) = [ 0.19661193 0.10499359 0.00664806]
8、Reshaping arrays
Exercise: Implement image2vector() that takes an input of shape (length, height, 3) and returns a vector of shape (length*height*3, 1). For example, if you would like to reshape an array v of shape (a, b, c) into a vector of shape (a*b,c) you would do:
v = v.reshape((v.shape[0]*v.shape[1], v.shape[2])) # v.shape[0] = a ; v.shape[1] = b ; v.shape[2] = c
def image2vector(image):
“””
Argument:
image – a numpy array of shape (length, height, depth)
Returns:
v -- a vector of shape (length*height*depth, 1)
"""
### START CODE HERE ### (≈ 1 line of code)
v = image.reshape(image.shape[0] * image.shape[1] * image.shape[2], 1)
### END CODE HERE ###
return v
This is a 3 by 3 by 2 array, typically images will be (num_px_x, num_px_y,3)
where 3 represents the RGB values
image = np.array([[[ 0.67826139, 0.29380381],
[ 0.90714982, 0.52835647],
[ 0.4215251 , 0.45017551]],
[[ 0.92814219, 0.96677647],
[ 0.85304703, 0.52351845],
[ 0.19981397, 0.27417313]],
[[ 0.60659855, 0.00533165],
[ 0.10820313, 0.49978937],
[ 0.34144279, 0.94630077]]])
print (“image2vector(image) = ” + str(image2vector(image)))
image2vector(image) = [[ 0.67826139]
#[ 0.29380381]
#….]
9、Normalizing rows
def normalizeRows(x):
“””
Implement a function that normalizes each row of the matrix x (to have unit length).
Argument:
x -- A numpy matrix of shape (n, m)
Returns:
x -- The normalized (by row) numpy matrix. You are allowed to modify x.
"""
### START CODE HERE ### (≈ 2 lines of code)
# Compute x_norm as the norm 2 of x. Use np.linalg.norm(..., ord = 2, axis = ..., keepdims = True)
x_norm = np.linalg.norm(x, axis = 1, keepdims = True)
# Divide x by its norm.
x = x / x_norm
### END CODE HERE ###
return x
x = np.array([
[0, 3, 4],
[1, 6, 4]])
print(“normalizeRows(x) = ” + str(normalizeRows(x)))
normalizeRows(x) = [[ 0. 0.6 0.8 ]
#[ 0.13736056 0.82416338 0.54944226]]
10、Broadcasting and the softmax function
A very important concept to understand in numpy is “broadcasting”. It is very useful for
performing mathematical operations between arrays of different shapes. For the full details
on broadcasting, you can read the official [broadcasting documentation]
def softmax(x):
“”“Calculates the softmax for each row of the input x.
Your code should work for a row vector and also for matrices of shape (n, m).
Argument:
x -- A numpy matrix of shape (n,m)
Returns:
s -- A numpy matrix equal to the softmax of x, of shape (n,m)
"""
# Apply exp() element-wise to x. Use np.exp(...).
x_exp = np.exp(x)
# Create a vector x_sum that sums each row of x_exp. Use np.sum(..., axis = 1, keepdims = True).
x_sum = np.sum(x_exp, axis = 1, keepdims = True)
# Compute softmax(x) by dividing x_exp by x_sum. It should automatically use numpy broadcasting.
s = x_exp / x_sum
return s
x = np.array([
[9, 2, 5, 0, 0],
[7, 5, 0, 0 ,0]])
print(“softmax(x) = ” + str(softmax(x)))