2 - Dataset
### START CODE HERE ### (≈ 3 lines of code) shape_X = X.shape shape_Y = Y.shape m = X.shape[1] # training set size ### END CODE HERE ###
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))
4 - Neural Network model
4.1-Defining the neural network structure 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 """ ### START CODE HERE ### (≈ 3 lines of code) n_x = X.shape[0] # size of input layer n_h = 4 n_y = Y.shape[0] # size of output layer ### END CODE HERE ### return (n_x, n_h, n_y)
4.2 - Initialize the model's parameters def initialize_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.
### START CODE HERE ### (≈ 4 lines of code) 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)) ### END CODE HERE ###
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
4.3 - The Loop 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" ### START CODE HERE ### (≈ 4 lines of code) W1 = parameters['W1'] b1 = parameters['b1'] W2 = parameters['W2'] b2 = parameters['b2'] ### END CODE HERE ###
# Implement Forward Propagation to calculate A2 (probabilities) ### START CODE HERE ### (≈ 4 lines of code) Z1 = np.dot(W1, X) + b1 A1 = np.tanh(Z1) Z2 = np.dot(W2, A1) + b2 A2 = sigmoid(Z2) ### END CODE HERE ###
assert(A2.shape == (1, X.shape[1]))
cache = {"Z1": Z1, "A1": A1, "Z2": Z2, "A2": A2}
return A2, cache
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 ### START CODE HERE ### (≈ 2 lines of code) logprobs = np.multiply(np.log(A2),Y) + np.multiply(np.log(1 - A2),1 - Y) cost = - np.sum(logprobs) / m ### END CODE HERE ###
cost = np.squeeze(cost) # makes sure cost is the dimension we expect. # E.g., turns [[17]] into 17 assert(isinstance(cost, float))
return cost
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: grads -- python dictionary containing your gradients with respect to different parameters """ m = X.shape[1]
# First, retrieve W1 and W2 from the dictionary "parameters". ### START CODE HERE ### (≈ 2 lines of code) W1 = parameters['W1'] W2 = parameters['W2'] ### END CODE HERE ###
# Retrieve also A1 and A2 from dictionary "cache". ### START CODE HERE ### (≈ 2 lines of code) A1 = cache['A1'] A2 = cache['A2'] ### END CODE HERE ###
# Backward propagation: calculate dW1, db1, dW2, db2. ### START CODE HERE ### (≈ 6 lines of code, corresponding to 6 equations on slide above) dZ2 = A2 - Y dW2 = np.dot(dZ2, A1.T) / m db2 = np.sum(dZ2, axis = 1, keepdims = True) / m dZ1 = np.multiply(np.dot(W2.T, dZ2) , (1 - np.power(A1, 2))) dW1 = np.dot(dZ1, X.T) / m db1 = np.sum(dZ1, axis = 1, keepdims = True) / m ### END CODE HERE ###
grads = {"dW1": dW1, "db1": db1, "dW2": dW2, "db2": db2}
return grads
def update_parameters(parameters, grads, learning_rate = 1.2): """ Updates parameters using the gradient descent update rule given above
Arguments: parameters -- python dictionary containing your parameters grads -- python dictionary containing your gradients
Returns: parameters -- python dictionary containing your updated parameters """ # Retrieve each parameter from the dictionary "parameters" ### START CODE HERE ### (≈ 4 lines of code) W1 = parameters['W1'] b1 = parameters['b1'] W2 = parameters['W2'] b2 = parameters['b2'] ### END CODE HERE ###
# Retrieve each gradient from the dictionary "grads" ### START CODE HERE ### (≈ 4 lines of code) dW1 = grads['dW1'] db1 = grads['db1'] dW2 = grads['dW2'] db2 = grads['db2'] ## END CODE HERE ###
# Update rule for each parameter ### START CODE HERE ### (≈ 4 lines of code) W1 = W1 - learning_rate * dW1 b1 = b1 - learning_rate * db1 W2 = W2 - learning_rate * dW2 b2 = b2 - learning_rate * db2 ### END CODE HERE ###
parameters = {"W1": W1, "b1": b1, "W2": W2, "b2": b2}
return parameters
4.4 - Integrate parts 4.1, 4.2 and 4.3 in nn_model() 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". ### START CODE HERE ### (≈ 5 lines of code) parameters = initialize_parameters(n_x, n_h, n_y) W1 = parameters['W1'] b1 = parameters['b1'] W2 = parameters['W2'] b2 = parameters['b2']
### END CODE HERE ###
# Loop (gradient descent)
for i in range(0, num_iterations):
### START CODE HERE ### (≈ 4 lines of code) # 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)
# Gradient descent parameter update. Inputs: "parameters, grads". Outputs: "parameters". parameters = update_parameters(parameters, grads)
### END CODE HERE ###
# Print the cost every 1000 iterations if print_cost and i % 1000 == 0: print ("Cost after iteration %i: %f" %(i, cost))
return parameters
4.5 Predictions 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. ### START CODE HERE ### (≈ 2 lines of code) A2, cache = forward_propagation(X, parameters) predictions = np.around(A2) ### END CODE HERE ###
return predictions
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