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import numpy as np
def sigmoid(z):
"""
Compute the sigmoid of z
Arguments:
z -- A scalar or numpy array of any size.
Return:
s -- sigmoid(z)
"""
s = 1 / (1 + np.exp(-z))
return s
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_y -- the size of the output layer
"""
n_x = X.shape[0] # size of input layer
n_y = Y.shape[0] # size of output layer
return (n_x, n_y)
def initialize_parameters(n_x, n_h, n_y,verbose):
"""
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)
"""
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))
costs = []
if(verbose):
print()
print("W1.shape", W1.shape)
print("b1.shape", b1.shape)
print("W2.shape", W2.shape)
print("b2.shape", b2.shape)
print()
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
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
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((1 - Y), (np.log(1 - A2)))
cost = - np.sum(logprobs) / m
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".
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 = np.dot(dZ2, A1.T) / m
db2 = np.sum(dZ2, axis=1, keepdims=True) / m
dZ1 = np.dot(W2.T, dZ2) * (1 - A1 * A1)
dW1 = np.dot(dZ1, X.T) / m
db1 = np.sum(dZ1, axis=1, keepdims=True) / m
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"
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]
# Retrieve each gradient from the dictionary "grads"
dW1 = grads["dW1"]
db1 = grads["db1"]
dW2 = grads["dW2"]
db2 = grads["db2"]
# Update rule for each parameter
W1 = W1 - (learning_rate * dW1)
b1 = b1 - (learning_rate * db1)
W2 = W2 - (learning_rate * dW2)
b2 = b2 - (learning_rate * db2)
parameters = {"W1": W1,
"b1": b1,
"W2": W2,
"b2": b2,}
return parameters
def nn_model(X, Y, Xtest, Ytest, n_h, num_iterations=10000, learning_rate = 0.01, print_cost=False, verbose = True):
"""
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.
"""
n_x = layer_sizes(X, Y)[0]
n_y = layer_sizes(X, Y)[1]
return_costs = []
# Initialize parameters, then retrieve W1, b1, W2, b2. Inputs: "n_x, n_h,
# n_y". Outputs = "W1, b1, W2, b2, parameters".
parameters = initialize_parameters(n_x, n_h, n_y, False)
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]
# Loop (gradient descent)
for i in range(0, num_iterations +1):
# 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, learning_rate)
# Print the cost every 1000 iterations
if print_cost and i % print_cost == 0 and verbose:
print("\t%f = Cost after iteration %i" % (cost,i))
if i % (num_iterations / 10) == 0:
return_costs.append(cost)
#adding cost and printing cost for last value
# Predict test/train set examples
Y_prediction_train = predict(parameters, X)
Y_prediction_test = predict(parameters, Xtest)
# Print train/test Errors
if verbose:
print("\ttrain accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_train - Y)) * 100))
print("\ttest accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_test - Ytest)) * 100))
#adding final parameters
parameters["sample_costs"] = return_costs
parameters["train score"] = (100 - np.mean(np.abs(Y_prediction_train - Y)) * 100)
parameters["test score"] = (100 - np.mean(np.abs(Y_prediction_test - Ytest)) * 100)
return parameters
def predict(parameters, X):
m = X.shape[1]
Y_prediction = np.zeros((1, m))
A2, cache = forward_propagation(X, parameters)
for i in range(A2.shape[1]):
# Convert probabilities A[0,i] to actual predictions p[0,i]
if A2[0, i] < 0.5:
Y_prediction[0, i] = 0
else:
Y_prediction[0, i] = 1
assert(Y_prediction.shape == (1, m))
return Y_prediction