吳恩達 神經網絡與深度學習 第二課第三周課後作業

Let us build one hidden layer neutral network.

1-packages

# Package imports
import numpy as np
import matplotlib.pyplot as plt
from testCases import *
from operator import mod
import sklearn
import sklearn.datasets
import sklearn.linear_model
from planar_utils import plot_decision_boundary, sigmoid, load_planar_dataset, load_extra_datasets
%matplotlib inline
np.random.seed(1) # set a seed so that the results are consistent

2-Neutral Network Model

2.1 Defining the neutral network structure

Exercise: Define three variables:

• n_x: 輸入層的個數
• n_h: 隱層個數 (set this to 4)
• n_y: 輸出層的個數

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

2.2 Initialize the model’s parameters

Exercise: Implement the function initialize_parameters().

Instructions:
Make sure your 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.

#GRADED FUNCTION: initialize_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

2.3 實現前向傳播

Instructions:
Look above at the mathematical representation of your classifier.
輸出層使用 sigmoid().
hidden layer使用 np.tanh().
實現步驟：
首先從parameters中取出需要的參數
計算Z[1],A[1],Z[2]Z[1],A[1],Z[2] and A[2]A[2]

#GRADED FUNCTION: forward_propagation
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)#y hat
    ### END CODE HERE ### 
    assert(A2.shape == (1, X.shape[1]))
    cache = {"Z1": Z1,
             "A1": A1,
             "Z2": Z2,
             "A2": A2}  
    return A2, cache

2.4 計算cost function

J=−1m∑i=0m(y(i)log(a2)+(1−y(i))log(1−a2))
其中，我們計算−∑i=0my(i)log(a2)−∑i=0my(i)log⁡(a2):

可以用函數

logprobs = np.multiply(np.log(A2),Y)
cost = - np.sum(logprobs) # 此處使用numpy不需要loop
(也可以直接用函數np.dot()，相乘後相加).

#GRADED FUNCTION: compute_cost
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 w e expect. 
                                # E.g., turns [[17]] into 17 
    assert(isinstance(cost, float))    
    return cost

2.5 實現後向傳播

//缺公式

# GRADED FUNCTION: backward_propagation

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.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

2.5 梯度下降算法實現

θ=θ−α∂J/∂θ 其中 α 代表學習率and θ 代表參數.

# GRADED FUNCTION: update_parameters

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 -= learning_rate*dW1
    b1 -= learning_rate*db1
    W2 -= learning_rate*dW2
    b2 -= learning_rate*db2
    ### END CODE HERE ### 
    parameters = {"W1": W1,
                  "b1": b1,
                  "W2": W2,
                  "b2": b2} 
    return parameters

3-NN_model() Implement

# GRADED FUNCTION: 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)
        #print("cache:"+str(cache)) 
        # 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("grads:"+str(grads))
        # Gradient descent parameter update. Inputs: "parameters, grads". Outputs: "parameters".
        parameters = update_parameters(parameters,grads)
        #print("parameters:"+str(parameters))
        ### 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-預測

predictions = yprediction={activation > 0.5}={1，0
if activation>0.5otherwise

    A2, cache = forward_propagation(X,parameters)
    predictions = (A2>0.5)
    parameters = nn_model(X, Y, n_h = 4, num_iterations = 10000, print_cost=True)
    predictions = predict(parameters, X)