### Planar Data Classification with One Hidden Layer

In this notebook

• I will build a simple neural network, which will have one hidden layer. You will see a big difference between this model and the one you implemented using logistic regression.

Goals of this Notebook are to:

• Implement a 2-class classification neural network with a single hidden layer
• Use units with a non-linear activation function, such as tanh
• Compute the cross entropy loss
• Implement forward and backward propagation

### 1 - Packages

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

• numpy is the fundamental package for scientific computing with Python.
• sklearn provides simple and efficient tools for data mining and data analysis.
• matplotlib is a library for plotting graphs in Python.
• testCases provides some test examples to assess the correctness of your functions
• planar_utils provide various useful functions used in this assignment
# Package imports
import numpy as np
import matplotlib.pyplot as plt
from testCases_v2 import *
import sklearn
import sklearn.datasets
import sklearn.linear_model

%matplotlib inline

np.random.seed(1) # set a seed so that the results are consistent


### 2 - Dataset

First, let’s get the dataset we will work on. The following code will load a “flower” 2-class dataset into variables X and Y.

X, Y = load_planar_dataset()


Visualise the dataset using matplotlib. The data looks like a “flower” with some red (label y=0) and some blue (y=1) points. Our goal is to build a model to fit this data.

# Visualise the data:
plt.scatter(X[0, :], X[1, :], c=Y, s=40, cmap=plt.cm.Spectral);


We have:

• a numpy-array (matrix) X that contains your features (x1, x2)
• a numpy-array (vector) Y that contains your labels (red:0, blue:1).

Lets first get a better sense of what our data is like.

Let’s explore how many training examples we have. In addition, what is the shape of the variables X and Y?

shape_X = X.shape
shape_Y = Y.shape
m = X.shape[1]

print ('The shape of X is: ' + str(shape_X))
print ('The shape of Y is: ' + str(shape_Y))
print ('I have m = %d training examples!' % (m))

The shape of X is: (2, 400)
The shape of Y is: (1, 400)
I have m = 400 training examples!


### 3 - Simple Logistic Regression

Before building a full neural network, let’s first see how logistic regression performs on this problem. We can use sklearn’s built-in functions to do that. Let’s run the code below to train a logistic regression classifier on the dataset.

# Train the logistic regression classifier
clf = sklearn.linear_model.LogisticRegressionCV();
clf.fit(X.T, Y.T);


We can now plot the decision boundary of these models. Let’s run the code below.

# Plot the decision boundary for logistic regression
plot_decision_boundary(lambda x: clf.predict(x), X, Y)
plt.title("Logistic Regression")

# Print accuracy
LR_predictions = clf.predict(X.T)
print ('Accuracy of logistic regression: %d ' % float((np.dot(Y,LR_predictions) + np.dot(1-Y,1-LR_predictions))/float(Y.size)*100) +
'% ' + "(percentage of correctly labelled datapoints)")

Accuracy of logistic regression: 47 % (percentage of correctly labelled datapoints)


Interpretation: The dataset is not linearly separable, so logistic regression doesn’t perform well. Hopefully a neural network will do better. Let’s try this now!

### 4 - Neural Network model

Logistic regression did not work well on the “flower dataset”. Therefore, we are going to train a Neural Network with a single hidden layer.

Here is our model:

Mathematically:

For one example $x^{(i)}$: $z^{[1] (i)} = W^{[1]} x^{(i)} + b^{[1]}\tag{1}$ $a^{[1] (i)} = \tanh(z^{[1] (i)})\tag{2}$ $z^{[2] (i)} = W^{[2]} a^{[1] (i)} + b^{[2]}\tag{3}$ $\hat{y}^{(i)} = a^{[2] (i)} = \sigma(z^{ [2] (i)})\tag{4}$ $% 0.5 \\ 0 & \mbox{otherwise } \end{cases}\tag{5} %]]>$

Given the predictions on all the examples, you can also compute the cost $J$ as follows: $J = - \frac{1}{m} \sum\limits_{i = 0}^{m} \large\left(\small y^{(i)}\log\left(a^{[2] (i)}\right) + (1-y^{(i)})\log\left(1- a^{[2] (i)}\right) \large \right) \small \tag{6}$

The general methodology to build a Neural Network is to:

1. Define the neural network structure ( # of input units, # of hidden units, etc).
2. Initialize the model’s parameters
3. Loop:
• Implement forward propagation
• Compute loss
• Implement backward propagation to get the gradients

We often build helper functions to compute steps 1-3 and then merge them into one function called nn_model(). Once we’ve built nn_model() and learnt the right parameters, we can make predictions on new data.

### 4.1 - Defining the neural network structure

Define three variables:

• n_x: the size of the input layer
• n_h: the size of the hidden layer (set this to 4)
• n_y: the size of the output layer

Use shapes of X and Y to find n_x and n_y. Also, hard code the hidden layer size to be 4.

def layer_sizes(X, Y):
"""
Arguments:
X -- input dataset of shape (input size, number of examples)
Y -- labels of shape (output size, number of examples)

Returns:
n_x -- the size of the input layer
n_h -- the size of the hidden layer
n_y -- the size of the output layer
"""

n_x = X.shape[0] # size of input layer
n_h = 4
n_y = Y.shape[0] # size of output layer

return (n_x, n_h, n_y)

X_assess, Y_assess = layer_sizes_test_case()
(n_x, n_h, n_y) = layer_sizes(X_assess, Y_assess)
print("The size of the input layer is: n_x = " + str(n_x))
print("The size of the hidden layer is: n_h = " + str(n_h))
print("The size of the output layer is: n_y = " + str(n_y))

The size of the input layer is: n_x = 5
The size of the hidden layer is: n_h = 4
The size of the output layer is: n_y = 2


### 4.2 - Initialise the model’s parameters

Now I will implement the function initialise_parameters(). This function will be used to:

• Make sure our parameters’ sizes are right. Refer to the neural network figure above if needed.
• You will initialize the weights matrices with random values.
• Use: np.random.randn(a,b) * 0.01 to randomly initialize a matrix of shape (a,b).
• You will initialize the bias vectors as zeros.
• Use: np.zeros((a,b)) to initialize a matrix of shape (a,b) with zeros.
def initialise_parameters(n_x, n_h, n_y):
"""
Argument:
n_x -- size of the input layer
n_h -- size of the hidden layer
n_y -- size of the output layer

Returns:
params -- python dictionary containing your parameters:
W1 -- weight matrix of shape (n_h, n_x)
b1 -- bias vector of shape (n_h, 1)
W2 -- weight matrix of shape (n_y, n_h)
b2 -- bias vector of shape (n_y, 1)
"""

np.random.seed(2) # we set up a seed so that your output matches ours although the initialization is random.

W1 = np.random.randn(n_h, n_x) * 0.01
b1 = np.zeros((n_h, 1))
W2 = np.random.randn(n_y, n_h) * 0.01
b2 = np.zeros((n_y,1))

assert (W1.shape == (n_h, n_x))
assert (b1.shape == (n_h, 1))
assert (W2.shape == (n_y, n_h))
assert (b2.shape == (n_y, 1))

parameters = {"W1": W1,
"b1": b1,
"W2": W2,
"b2": b2}

return parameters

n_x, n_h, n_y = initialize_parameters_test_case()

parameters = initialise_parameters(n_x, n_h, n_y)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))

W1 = [[-0.00416758 -0.00056267]
[-0.02136196  0.01640271]
[-0.01793436 -0.00841747]
[ 0.00502881 -0.01245288]]
b1 = [[ 0.]
[ 0.]
[ 0.]
[ 0.]]
W2 = [[-0.01057952 -0.00909008  0.00551454  0.02292208]]
b2 = [[ 0.]]


### 4.3 - The Loop

Now let’s implement forward_propagation().

• Look above at the mathematical representation of our classifier.
• We can use the function sigmoid(). It is built-in (imported) in the notebook.
• We can use the function np.tanh(). It is part of the numpy library.
• The steps I will implement are:
1. Retrieve each parameter from the dictionary “parameters” (which is the output of initialise_parameters()) by using parameters[".."].
2. Implement Forward Propagation. Compute $Z^{[1]}, A^{[1]}, Z^{[2]}$ and $A^{[2]}$ (the vector of all your predictions on all the examples in the training set).
• Values needed in the backpropagation are stored in “cache”. The cache will be given as an input to the backpropagation function.
def forward_propagation(X, parameters):
"""
Argument:
X -- input data of size (n_x, m)
parameters -- python dictionary containing your parameters (output of initialization function)

Returns:
A2 -- The sigmoid output of the second activation
cache -- a dictionary containing "Z1", "A1", "Z2" and "A2"
"""
# Retrieve each parameter from the dictionary "parameters"
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]

# Implement Forward Propagation to calculate A2 (probabilities)
Z1 = np.dot(W1, X) + b1
A1 = np.tanh(Z1)
Z2 = np.dot(W2, A1) + b2
A2 = sigmoid(Z2)

assert(A2.shape == (1, X.shape[1]))

cache = {"Z1": Z1,
"A1": A1,
"Z2": Z2,
"A2": A2}

return A2, cache

X_assess, parameters = forward_propagation_test_case()
A2, cache = forward_propagation(X_assess, parameters)

# Note: we use the mean here just to make sure that your output matches ours.
print(np.mean(cache['Z1']) ,np.mean(cache['A1']),np.mean(cache['Z2']),np.mean(cache['A2']))

0.262818640198 0.091999045227 -1.30766601287 0.212877681719


Now that we have computed $A^{[2]}$ (in the Python variable “A2”), which contains $a^{2}$ for every example, we can compute the cost function as follows:

Now let’s implement compute_cost() to compute the value of the cost $J$.

• There are many ways to implement the cross-entropy loss:

but for this notebook, it will be implemented as follow:

logprobs = np.multiply(np.log(A2),Y)
cost = - np.sum(logprobs)                # no need to use a for loop!


(we can use either np.multiply() and then np.sum() or directly np.dot()).

def compute_cost(A2, Y, parameters):
"""
Computes the cross-entropy cost given in equation (13)

Arguments:
A2 -- The sigmoid output of the second activation, of shape (1, number of examples)
Y -- "true" labels vector of shape (1, number of examples)
parameters -- python dictionary containing your parameters W1, b1, W2 and b2

Returns:
cost -- cross-entropy cost given equation (13)
"""

m = Y.shape[1] # number of example

# Compute the cross-entropy cost
logprobs = np.multiply(np.log(A2),Y) +  np.multiply(np.log(1-A2), (1-Y))
cost = -1/m*np.sum(logprobs)

cost = np.squeeze(cost)     # makes sure cost is the dimension we expect.
# E.g., turns [[17]] into 17
assert(isinstance(cost, float))

return cost

A2, Y_assess, parameters = compute_cost_test_case()

print("cost = " + str(compute_cost(A2, Y_assess, parameters)))

cost = 0.693058761039


Using the cache computed during forward propagation, we can now implement backward propagation.

Now let’s implement the function backward_propagation().

Backpropagation is usually the hardest (most mathematical) part in deep learning. Below is the representation of backpropagation implementation mathermatically and programically. For this notebook, we will particularly use the right side of the slide.

Tips:

• To compute dZ1 we’ll need to compute $g^{[1]’}(Z^{[1]})$. Since $g^{[1]}(.)$ is the tanh activation function, if $a = g^{[1]}(z)$ then $g^{[1]’}(z) = 1-a^2$. So we can compute $g^{[1]’}(Z^{[1]})$ using (1 - np.power(A1, 2)).
def backward_propagation(parameters, cache, X, Y):
"""
Implement the backward propagation using the instructions above.

Arguments:
parameters -- python dictionary containing our parameters
cache -- a dictionary containing "Z1", "A1", "Z2" and "A2".
X -- input data of shape (2, number of examples)
Y -- "true" labels vector of shape (1, number of examples)

Returns:
"""
m = X.shape[1]

# First, retrieve W1 and W2 from the dictionary "parameters".
W1 = parameters["W1"]
W2 = parameters["W2"]

# Retrieve also A1 and A2 from dictionary "cache".
A1 = cache["A1"]
A2 = cache["A2"]

# Backward propagation: calculate dW1, db1, dW2, db2.
dZ2 = A2-Y
dW2 = 1./m*np.dot(dZ2, A1.T)
db2 = 1./m*np.sum(dZ2, axis = 1, keepdims=True)
dZ1 = np.dot(W2.T, dZ2) * (1 - np.power(A1, 2))
dW1 = 1./m* np.dot(dZ1, X.T)
db1 = 1./m*np.sum(dZ1, axis = 1, keepdims=True)

"db1": db1,
"dW2": dW2,
"db2": db2}


parameters, cache, X_assess, Y_assess = backward_propagation_test_case()

grads = backward_propagation(parameters, cache, X_assess, Y_assess)

dW1 = [[ 0.00301023 -0.00747267]
[ 0.00257968 -0.00641288]
[-0.00156892  0.003893  ]
[-0.00652037  0.01618243]]
db1 = [[ 0.00176201]
[ 0.00150995]
[-0.00091736]
[-0.00381422]]
dW2 = [[ 0.00078841  0.01765429 -0.00084166 -0.01022527]]
db2 = [[-0.16655712]]


Now let’s implement the update rule by using gradient descent. I will use (dW1, db1, dW2, db2) in order to update (W1, b1, W2, b2).

General gradient descent rule: $\theta = \theta - \alpha \frac{\partial J }{ \partial \theta }$ where $\alpha$ is the learning rate and $\theta$ represents a parameter.

Illustration: The gradient descent algorithm with a good learning rate (converging) and a bad learning rate (diverging). Images courtesy of Adam Harley.

def update_parameters(parameters, grads, learning_rate = 1.2):
"""

Arguments:
parameters -- python dictionary containing your parameters

Returns:
parameters -- python dictionary containing your updated parameters
"""
# Retrieve each parameter from the dictionary "parameters"
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]

# Update rule for each parameter
W1 = W1-dW1*learning_rate
b1 = b1-db1*learning_rate
W2 = W2-dW2*learning_rate
b2 = b2-db2*learning_rate

parameters = {"W1": W1,
"b1": b1,
"W2": W2,
"b2": b2}

return parameters

parameters, grads = update_parameters_test_case()

print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))

W1 = [[-0.00643025  0.01936718]
[-0.02410458  0.03978052]
[-0.01653973 -0.02096177]
[ 0.01046864 -0.05990141]]
b1 = [[ -1.02420756e-06]
[  1.27373948e-05]
[  8.32996807e-07]
[ -3.20136836e-06]]
W2 = [[-0.01041081 -0.04463285  0.01758031  0.04747113]]
b2 = [[ 0.00010457]]


### 4.4 - Integrate parts 4.1, 4.2 and 4.3 in nn_model()

Now let’s build our neural network model in nn_model().

The neural network model below will put everything we did previously in order.

def nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):
"""
Arguments:
X -- dataset of shape (2, number of examples)
Y -- labels of shape (1, number of examples)
n_h -- size of the hidden layer
num_iterations -- Number of iterations in gradient descent loop
print_cost -- if True, print the cost every 1000 iterations

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

np.random.seed(3)
n_x = layer_sizes(X, Y)[0]
n_y = layer_sizes(X, Y)[2]

# Initialize parameters, then retrieve W1, b1, W2, b2. Inputs: "n_x, n_h, n_y". Outputs = "W1, b1, W2, b2, parameters".
parameters = initialise_parameters(n_x, n_h, n_y)
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]

for i in range(0, num_iterations):

# Forward propagation. Inputs: "X, parameters". Outputs: "A2, cache".
A2, cache = forward_propagation(X, parameters)

# Cost function. Inputs: "A2, Y, parameters". Outputs: "cost".
cost = compute_cost(A2, Y, parameters)

# Backpropagation. Inputs: "parameters, cache, X, Y". Outputs: "grads".
grads = backward_propagation(parameters, cache, X, Y)

# Print the cost every 1000 iterations
if print_cost and i % 1000 == 0:
print ("Cost after iteration %i: %f" %(i, cost))

return parameters

X_assess, Y_assess = nn_model_test_case()
parameters = nn_model(X_assess, Y_assess, 4, num_iterations=10000, print_cost=True)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))

Cost after iteration 0: 0.692739
Cost after iteration 1000: 0.000218
Cost after iteration 2000: 0.000107
Cost after iteration 3000: 0.000071
Cost after iteration 4000: 0.000053
Cost after iteration 5000: 0.000042
Cost after iteration 6000: 0.000035
Cost after iteration 7000: 0.000030
Cost after iteration 8000: 0.000026
Cost after iteration 9000: 0.000023
W1 = [[-0.65848169  1.21866811]
[-0.76204273  1.39377573]
[ 0.5792005  -1.10397703]
[ 0.76773391 -1.41477129]]
b1 = [[ 0.287592  ]
[ 0.3511264 ]
[-0.2431246 ]
[-0.35772805]]
W2 = [[-2.45566237 -3.27042274  2.00784958  3.36773273]]
b2 = [[ 0.20459656]]


### 4.5 Predictions

Now let’s use our model to predict by building predict(). We will use forward propagation to predict results.

predictions = $y_{prediction} = \mathbb 1 \text = \begin{cases} 1 & \text{if}\ activation > 0.5 0 & \text{otherwise} \end{cases}$

def predict(parameters, X):
"""
Using the learned parameters, predicts a class for each example in X

Arguments:
parameters -- python dictionary containing your parameters
X -- input data of size (n_x, m)

Returns
predictions -- vector of predictions of our model (red: 0 / blue: 1)
"""

# Computes probabilities using forward propagation, and classifies to 0/1 using 0.5 as the threshold.
A2, cache = forward_propagation(X, parameters)
predictions = A2 > 0.5

return predictions

parameters, X_assess = predict_test_case()

predictions = predict(parameters, X_assess)
print("predictions mean = " + str(np.mean(predictions)))

predictions mean = 0.666666666667


It is time to run the model and see how it performs on a planar dataset. Let’s run the following code to test our model with a single hidden layer of $n_h$ hidden units.

# Build a model with a n_h-dimensional hidden layer
parameters = nn_model(X, Y, n_h = 4, num_iterations = 10000, print_cost=True)

# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y)
plt.title("Decision Boundary for hidden layer size " + str(4))

Cost after iteration 0: 0.693048
Cost after iteration 1000: 0.288083
Cost after iteration 2000: 0.254385
Cost after iteration 3000: 0.233864
Cost after iteration 4000: 0.226792
Cost after iteration 5000: 0.222644
Cost after iteration 6000: 0.219731
Cost after iteration 7000: 0.217504
Cost after iteration 8000: 0.219471
Cost after iteration 9000: 0.218612

<matplotlib.text.Text at 0x7f613643c630>


# Print accuracy
predictions = predict(parameters, X)
print ('Accuracy: %d' % float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100) + '%')

Accuracy: 90%


Accuracy is really high compared to Logistic Regression. The model has learnt the leaf patterns of the flower! Neural networks are able to learn even highly non-linear decision boundaries, unlike logistic regression.

Now, let’s try out several hidden layer sizes.

### 4.6 - Tuning hidden layer size

Now let’s run the following code. It will take 1-2 minutes. The behaviors of the model will be different depending on various hidden layer sizes.

plt.figure(figsize=(16, 32))
hidden_layer_sizes = [1, 2, 3, 4, 5, 20, 50]
for i, n_h in enumerate(hidden_layer_sizes):
plt.subplot(5, 2, i+1)
plt.title('Hidden Layer of size %d' % n_h)
parameters = nn_model(X, Y, n_h, num_iterations = 5000)
plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y)
predictions = predict(parameters, X)
accuracy = float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100)
print ("Accuracy for {} hidden units: {} %".format(n_h, accuracy))

Accuracy for 1 hidden units: 67.5 %
Accuracy for 2 hidden units: 67.25 %
Accuracy for 3 hidden units: 90.75 %
Accuracy for 4 hidden units: 90.5 %
Accuracy for 5 hidden units: 91.25 %
Accuracy for 20 hidden units: 90.0 %
Accuracy for 50 hidden units: 90.25 %


Interpretation:

• The larger models (with more hidden units) are able to fit the training set better, until eventually the largest models overfit the data.
• The best hidden layer size seems to be around n_h = 5. Indeed, a value around here seems to fits the data well without also incurring noticeable overfitting.
• Later I will also talk about regularization, which let us use very large models (such as n_h = 50) without much overfitting.

What we have done so far:

• Build a complete neural network with a hidden layer
• Make a good use of a non-linear unit
• Implemented forward propagation and backpropagation, and trained a neural network
• See the impact of varying the hidden layer size, including overfitting.

### 5) Performance on other datasets

Now let’s rerun the whole notebook (minus the dataset part) for each of the following datasets.

# Datasets
noisy_circles, noisy_moons, blobs, gaussian_quantiles, no_structure = load_extra_datasets()

datasets = {"noisy_circles": noisy_circles,
"noisy_moons": noisy_moons,
"blobs": blobs,
"gaussian_quantiles": gaussian_quantiles}

dataset = "noisy_moons"

X, Y = datasets[dataset]
X, Y = X.T, Y.reshape(1, Y.shape[0])

# make blobs binary
if dataset == "blobs":
Y = Y%2

# Visualize the data
plt.scatter(X[0, :], X[1, :], c=Y, s=40, cmap=plt.cm.Spectral);


Reference:

last edited: 31/05/19

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