### Deep Neural Network for Image Classification: Application

I will use the functions I’ve implemented in the previous assignment to build a deep network, and apply it to cat vs non-cat classification. Hopefully, we will see an improvement in accuracy relative to our previous logistic regression implementation.

Goals of this Notebook is to:

• Build and apply a deep neural network to supervised learning.

Let’s get started!

### 1 - Packages

Let’s first import all the packages that I need during this assignment.

• numpy is the fundamental package for scientific computing with Python.
• matplotlib is a library to plot graphs in Python.
• h5py is a common package to interact with a dataset that is stored on an H5 file.
• PIL and scipy are used here to test your model with your own picture at the end.
• dnn_app_utils provides the functions implemented in the “Building your Deep Neural Network: Step by Step” assignment to this notebook.
• np.random.seed(1) is used to keep all the random function calls consistent.
import time
import numpy as np
import h5py
import matplotlib.pyplot as plt
import scipy
from PIL import Image
from scipy import ndimage
from dnn_app_utils_v3 import *

%matplotlib inline
plt.rcParams['figure.figsize'] = (5.0, 4.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'

np.random.seed(1)


### 2 - Dataset

I will use the same “Cat vs non-Cat” dataset as in “Logistic Regression as a Neural Network” (Assignment 2). The model I had built had 70% test accuracy on classifying cats vs non-cats images. Hopefully, my new model will perform a better!

Problem Statement: The dataset (“data.h5”) contains: - a training set of m_train images labelled as cat (1) or non-cat (0) - a test set of m_test images labelled as cat and non-cat - each image is of shape (num_px, num_px, 3) where 3 is for the 3 channels (RGB).

Let’s get more familiar with the dataset. Load the data by running the cell below.

train_x_orig, train_y, test_x_orig, test_y, classes = load_data()


The following code will show us an image in the dataset. Feel free to change the index and re-run the cell multiple times to see other images.

# Example of a picture
index = 10
plt.imshow(train_x_orig[index])
print ("y = " + str(train_y[0,index]) + ". It's a " + classes[train_y[0,index]].decode("utf-8") +  " picture.")

y = 0. It's a non-cat picture.


# Explore your dataset
m_train = train_x_orig.shape[0]
num_px = train_x_orig.shape[1]
m_test = test_x_orig.shape[0]

print ("Number of training examples: " + str(m_train))
print ("Number of testing examples: " + str(m_test))
print ("Each image is of size: (" + str(num_px) + ", " + str(num_px) + ", 3)")
print ("train_x_orig shape: " + str(train_x_orig.shape))
print ("train_y shape: " + str(train_y.shape))
print ("test_x_orig shape: " + str(test_x_orig.shape))
print ("test_y shape: " + str(test_y.shape))

Number of training examples: 209
Number of testing examples: 50
Each image is of size: (64, 64, 3)
train_x_orig shape: (209, 64, 64, 3)
train_y shape: (1, 209)
test_x_orig shape: (50, 64, 64, 3)
test_y shape: (1, 50)


As usual, I will reshape and standardise the images before feeding them to the network. The code is in the cell below.

Figure 1: Image to vector conversion.
# Reshape the training and test examples
train_x_flatten = train_x_orig.reshape(train_x_orig.shape[0], -1).T   # The "-1" makes reshape flatten the remaining dimensions
test_x_flatten = test_x_orig.reshape(test_x_orig.shape[0], -1).T

# Standardize data to have feature values between 0 and 1.
train_x = train_x_flatten/255.
test_x = test_x_flatten/255.

print ("train_x's shape: " + str(train_x.shape))
print ("test_x's shape: " + str(test_x.shape))


train_x's shape: (12288, 209)
test_x's shape: (12288, 50)


$12,288$ equals $64 \times 64 \times 3$ which is the size of one reshaped image vector.

### 3 - Architecture of our model

Now that we are familiar with the dataset, it is time to build a deep neural network to distinguish cat images from non-cat images.

I will build two different models in this notebook:

• A 2-layer neural network
• An L-layer deep neural network

I will then compare the performance of these models, and also try out different values for $L$.

Let’s look at the two architectures.

### 3.1 - 2-layer neural network

Figure 2: 2-layer neural network.
The model can be summarised as: ***INPUT -> LINEAR -> RELU -> LINEAR -> SIGMOID -> OUTPUT***.

Detailed Architecture of figure 2:

• The input is a (64,64,3) image which is flattened to a vector of size $(12288,1)$.
• The corresponding vector: $[x_0,x_1,…,x_{12287}]^T$ is then multiplied by the weight matrix $W^{[1]}$ of size $(n^{[1]}, 12288)$.
• I then add a bias term and take its relu to get the following vector: $[a_0^{[1]}, a_1^{[1]},…, a_{n^{[1]}-1}^{[1]}]^T$.
• I then repeat the same process.
• I multiply the resulting vector by $W^{[2]}$ and add our intercept (bias).
• Finally, I will take the sigmoid of the result. If it is greater than 0.5, I will classify it to be a cat.

### 3.2 - L-layer deep neural network

It is hard to represent an L-layer deep neural network with the above representation. However, here is a simplified network representation:

Figure 3: L-layer neural network.
The model can be summarised as: ***[LINEAR -> RELU] $\times$ (L-1) -> LINEAR -> SIGMOID***

Detailed Architecture of figure 3:

• The input is a (64,64,3) image which is flattened to a vector of size (12288,1).
• The corresponding vector: $[x_0,x_1,…,x_{12287}]^T$ is then multiplied by the weight matrix $W^{[1]}$ and then I add the intercept $b^{[1]}$. The result is called the linear unit.
• Next, I will take the relu of the linear unit. This process could be repeated several times for each $(W^{[l]}, b^{[l]})$ depending on the model architecture.
• Finally, I will take the sigmoid of the final linear unit. If it is greater than 0.5, I will classify it to be a cat.

### 3.3 - General methodology

As usual I will follow the Deep Learning methodology to build the model: 1. Initialise parameters / Define hyperparameters 2. Loop for num_iterations: a. Forward propagation b. Compute cost function c. Backward propagation d. Update parameters (using parameters, and grads from backprop) 4. Use trained parameters to predict labels

Let’s now implement those two models!

## 4 - Two-layer neural network

Let’s use the helper functions I have implemented in the previous assignment to build a 2-layer neural network with the following structure: LINEAR -> RELU -> LINEAR -> SIGMOID. Below is the functions I need and their inputs are:

def initialize_parameters(n_x, n_h, n_y):
...
return parameters
def linear_activation_forward(A_prev, W, b, activation):
...
return A, cache
def compute_cost(AL, Y):
...
return cost
def linear_activation_backward(dA, cache, activation):
...
return dA_prev, dW, db
...
return parameters

### CONSTANTS DEFINING THE MODEL ####
n_x = 12288     # num_px * num_px * 3
n_h = 7
n_y = 1
layers_dims = (n_x, n_h, n_y)

def two_layer_model(X, Y, layers_dims, learning_rate = 0.0075, num_iterations = 3000, print_cost=False):
"""
Implements a two-layer neural network: LINEAR->RELU->LINEAR->SIGMOID.

Arguments:
X -- input data, of shape (n_x, number of examples)
Y -- true "label" vector (containing 1 if cat, 0 if non-cat), of shape (1, number of examples)
layers_dims -- dimensions of the layers (n_x, n_h, n_y)
num_iterations -- number of iterations of the optimization loop
learning_rate -- learning rate of the gradient descent update rule
print_cost -- If set to True, this will print the cost every 100 iterations

Returns:
parameters -- a dictionary containing W1, W2, b1, and b2
"""

np.random.seed(1)
costs = []                              # to keep track of the cost
m = X.shape[1]                           # number of examples
(n_x, n_h, n_y) = layers_dims

# Initialize parameters dictionary, by calling one of the functions you'd previously implemented
parameters = initialize_parameters(n_x, n_h, n_y)

# Get W1, b1, W2 and b2 from the dictionary parameters.
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]

for i in range(0, num_iterations):

# Forward propagation: LINEAR -> RELU -> LINEAR -> SIGMOID. Inputs: "X, W1, b1, W2, b2". Output: "A1, cache1, A2, cache2".
A1, cache1 = linear_activation_forward(X, W1, b1, activation= "relu")
A2, cache2 = linear_activation_forward(A1, W2, b2, activation= "sigmoid")

# Compute cost
cost = compute_cost(A2, Y)

# Initializing backward propagation
dA2 = - (np.divide(Y, A2) - np.divide(1 - Y, 1 - A2))

# Backward propagation. Inputs: "dA2, cache2, cache1". Outputs: "dA1, dW2, db2; also dA0 (not used), dW1, db1".
dA1, dW2, db2 = linear_activation_backward(dA2, cache2, activation = "sigmoid")
dA0, dW1, db1 = linear_activation_backward(dA1, cache1, activation = "relu")

# Update parameters.

# Retrieve W1, b1, W2, b2 from parameters
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]

# Print the cost every 100 training example
if print_cost and i % 100 == 0:
print("Cost after iteration {}: {}".format(i, np.squeeze(cost)))
if print_cost and i % 100 == 0:
costs.append(cost)

# plot the cost

plt.plot(np.squeeze(costs))
plt.ylabel('cost')
plt.xlabel('iterations (per hundreds)')
plt.title("Learning rate =" + str(learning_rate))
plt.show()

return parameters


Run the cell below to train our parameters and observe if our model runs correctly. The cost should be decreasing. It may take up to 5 minutes to run 2500 iterations. Check if the “Cost after iteration 0” matches the expected output below, if not click on the square (⬛) on the upper bar of the notebook to stop the cell and try to find your error.

parameters = two_layer_model(train_x, train_y, layers_dims = (n_x, n_h, n_y), num_iterations = 2500, print_cost=True)

Cost after iteration 0: 0.6930497356599888
Cost after iteration 100: 0.6464320953428849
Cost after iteration 200: 0.6325140647912677
Cost after iteration 300: 0.6015024920354665
Cost after iteration 400: 0.5601966311605747
Cost after iteration 500: 0.515830477276473
Cost after iteration 600: 0.4754901313943325
Cost after iteration 700: 0.4339163151225749
Cost after iteration 800: 0.4007977536203887
Cost after iteration 900: 0.3580705011323798
Cost after iteration 1000: 0.3394281538366412
Cost after iteration 1100: 0.3052753636196264
Cost after iteration 1200: 0.27491377282130164
Cost after iteration 1300: 0.24681768210614846
Cost after iteration 1400: 0.19850735037466116
Cost after iteration 1500: 0.1744831811255664
Cost after iteration 1600: 0.17080762978096148
Cost after iteration 1700: 0.11306524562164734
Cost after iteration 1800: 0.09629426845937152
Cost after iteration 1900: 0.08342617959726863
Cost after iteration 2000: 0.07439078704319081
Cost after iteration 2100: 0.0663074813226793
Cost after iteration 2200: 0.0591932950103817
Cost after iteration 2300: 0.053361403485605585
Cost after iteration 2400: 0.04855478562877016


Good thing that we built a vectorized implementation! Otherwise it might have taken 10 times longer to train this NN.

Now, we can use the trained parameters to classify images from the dataset. To see our predictions on the training and test sets, run the cell below.

predictions_train = predict(train_x, train_y, parameters)

Accuracy: 1.0

predictions_test = predict(test_x, test_y, parameters)

Accuracy: 0.72


Note: Notice that running the model on fewer iterations (say 1500) gives better accuracy on the test set. This is called “early stopping” and I will talk about it later. Early stopping is a way to prevent overfitting.

Yay! It seems that our 2-layer neural network has better performance (72%) than the logistic regression implementation (70%, assignment week 2). Let’s see if we can do even better with an $L$-layer model.

### 5 - L-layer Neural Network

Now let’s use the helper functions I have implemented previously to build an $L$-layer neural network with the following structure: [LINEAR -> RELU]$\times$(L-1) -> LINEAR -> SIGMOID. The functions I need and their inputs are:

def initialize_parameters_deep(layers_dims):
...
return parameters
def L_model_forward(X, parameters):
...
return AL, caches
def compute_cost(AL, Y):
...
return cost
def L_model_backward(AL, Y, caches):
...
...
return parameters

### CONSTANTS ###
layers_dims = [12288, 20, 7, 5, 1] #  4-layer model

def L_layer_model(X, Y, layers_dims, learning_rate = 0.0075, num_iterations = 3000, print_cost=False):#lr was 0.009
"""
Implements a L-layer neural network: [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID.

Arguments:
X -- data, numpy array of shape (number of examples, num_px * num_px * 3)
Y -- true "label" vector (containing 0 if cat, 1 if non-cat), of shape (1, number of examples)
layers_dims -- list containing the input size and each layer size, of length (number of layers + 1).
learning_rate -- learning rate of the gradient descent update rule
num_iterations -- number of iterations of the optimization loop
print_cost -- if True, it prints the cost every 100 steps

Returns:
parameters -- parameters learnt by the model. They can then be used to predict.
"""

np.random.seed(1)
costs = []                         # keep track of cost

# Parameters initialisation. (≈ 1 line of code)
parameters = initialize_parameters_deep(layers_dims)

for i in range(0, num_iterations):

# Forward propagation: [LINEAR -> RELU]*(L-1) -> LINEAR -> SIGMOID.
AL, caches = L_model_forward(X, parameters)

# Compute cost.
cost = compute_cost(AL, Y)

# Backward propagation.

# Update parameters.

# Print the cost every 100 training example
if print_cost and i % 100 == 0:
print ("Cost after iteration %i: %f" %(i, cost))
if print_cost and i % 100 == 0:
costs.append(cost)

# plot the cost
plt.plot(np.squeeze(costs))
plt.ylabel('cost')
plt.xlabel('iterations (per hundreds)')
plt.title("Learning rate =" + str(learning_rate))
plt.show()

return parameters


We will now train the model as a 4-layer neural network.

Run the cell below to train our model. The cost should decrease on every iteration. It may take up to 5 minutes to run 2500 iterations. Check if the “Cost after iteration 0” matches the expected output below, if not click on the square (⬛) on the upper bar of the notebook to stop the cell and try to find our error.

parameters = L_layer_model(train_x, train_y, layers_dims, num_iterations = 2500, print_cost = True)

Cost after iteration 0: 0.771749
Cost after iteration 100: 0.672053
Cost after iteration 200: 0.648263
Cost after iteration 300: 0.611507
Cost after iteration 400: 0.567047
Cost after iteration 500: 0.540138
Cost after iteration 600: 0.527930
Cost after iteration 700: 0.465477
Cost after iteration 800: 0.369126
Cost after iteration 900: 0.391747
Cost after iteration 1000: 0.315187
Cost after iteration 1100: 0.272700
Cost after iteration 1200: 0.237419
Cost after iteration 1300: 0.199601
Cost after iteration 1400: 0.189263
Cost after iteration 1500: 0.161189
Cost after iteration 1600: 0.148214
Cost after iteration 1700: 0.137775
Cost after iteration 1800: 0.129740
Cost after iteration 1900: 0.121225
Cost after iteration 2000: 0.113821
Cost after iteration 2100: 0.107839
Cost after iteration 2200: 0.102855
Cost after iteration 2300: 0.100897
Cost after iteration 2400: 0.092878


pred_train = predict(train_x, train_y, parameters)

Accuracy: 0.985645933014

 **Train Accuracy** 0.985646
pred_test = predict(test_x, test_y, parameters)

Accuracy: 0.8


Yay! It seems that our 4-layer neural network has better performance (80%) than your 2-layer neural network (72%) on the same test set.

This is good performance for this task. Nice job!

Though in the next few notebook on “Improving deep neural networks” I will show how to obtain even higher accuracy by systematically searching for better hyperparameters (learning_rate, layers_dims, num_iterations, and others you’ll also learn in the next course).

### 6) Results Analysis

First, let’s take a look at some images the L-layer model labeled incorrectly. This will show a few mislabeled images.

print_mislabeled_images(classes, test_x, test_y, pred_test)


A few types of images the model tends to do poorly on include:

• Cat body in an unusual position
• Cat appears against a background of a similar color
• Unusual cat color and species
• Camera Angle
• Brightness of the picture
• Scale variation (cat is very large or small in image)

### 7) Test with new image

We can use our own image and see the output of our model. To do that:

1. Add your image to this Jupyter Notebook’s directory, in the “images” folder
2. Change your image’s name in the following code
3. Run the code and check if the algorithm is right (1 = cat, 0 = non-cat)!
my_image = "my_image.jpg" # change this to the name of your image file
my_label_y = [1] # the true class of your image (1 -> cat, 0 -> non-cat)

fname = "images/" + my_image