工具包

sklearn包：提供简单有效的数据挖掘和数据分析。
加载工具包部分的代码：

import numpy as np
import matplotlib.pyplot as plt
from testCases import *
import sklearn
import sklearn.datasets
import sklearn.linear_model

数据集

需要使用的包下载
加载数据集和绘制数据集部分的代码：

from planar_utils import plot_decision_boundary, sigmoid, load_planar_dataset, load_extra_datasets
X, Y = load_planar_dataset()
plt.scatter(X[0, :], X[1, :], c=Y, s=40, cmap=plt.cm.Spectral)
plt.show()

shape_X = X.shape
shape_Y = Y.shape
m = shape_X[1]  # training set size

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))

数据集如下所示：

数据集

数据集其它的信息：

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

逻辑回归的表现

在建立全连接之前，我们首先来看一下逻辑回归对于该问题的表现，可以使用sklearn的内建函数来实现：

clf = sklearn.linear_model.LogisticRegressionCV()
clf.fit(X.T, Y.T.ravel())

plot_decision_boundary(lambda x: clf.predict(x), X, Y)
plt.title("Logistic Regression")

# 输出准确性
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)")
plt.show()

效果：

逻辑回归的表现

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

神经网络模型

由上面可以看出logistic模型对于解决“flower dataset”效果并不好，这里我们创建一个隐藏层的神经网络，下图是我们使用的网络模型：

神经网络模型

定义神经网络结构

定义三个变量：

• n_x：输入层单元数目
• n_h：隐藏层单元数目
• n_y：输出层单元数目

def layer_size(X,Y):
    n_x = X.shape[0]
    n_h = 4
    n_y = Y.shape[0]
    return (n_x,n_h,n_y)

X_assess,Y_assess = layer_sizes_test_case()
(n_x,n_h,n_y) = layer_size(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))

输出：

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

初始化模型参数

随机初始化权重参数，偏置参数初始化为0：

np.random.seed(1) # set a seed so that the results are consistent
def initialize_parameters(n_x,n_h,n_y):
    np.random.seed(2)

    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

前向传播

def forward_propagation(X, parameters):
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]

    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

交叉熵计算：

def compute_cost(A2, Y, parameters):
    m = Y.shape[1] # number of example   
    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)
    assert(isinstance(cost, float))
    return cost

反向传播

# GRADED FUNCTION: backward_propagation
def backward_propagation(parameters, cache, X, Y):
    m = X.shape[1]
  
    W1 = parameters["W1"]
    W2 = parameters["W2"]
 
    A1 = cache["A1"]
    A2 = cache["A2"]
    
    dZ2 = A2 - Y    # (n_y,1)
    dW2 = 1 / m * np.dot(dZ2, A1.T)   # (n_y, 1) .* (1, n_h)
    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)
    grads = {"dW1": dW1,
             "db1": db1,
             "dW2": dW2,
             "db2": db2}

    return grads

参数更新

def update_parameters(parameters, grads, learning_rate = 1.2):
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]

    dW1 = grads["dW1"]
    db1 = grads["db1"]
    dW2 = grads["dW2"]
    db2 = grads["db2"]

    W1 -= learning_rate * dW1
    b1 -= learning_rate * db1
    W2 -= learning_rate * dW2
    b2 -= learning_rate * db2

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

    return parameters

组成网络模型

def nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):
    np.random.seed(3)
    n_x = layer_sizes(X, Y)[0]
    n_y = layer_sizes(X, Y)[2]

    parameters = initialize_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):        
        A2, cache = forward_propagation(X, parameters)
        cost = compute_cost(A2, Y, parameters)   
        grads = backward_propagation(parameters, cache, X, Y)        
        parameters = update_parameters(parameters, grads, learning_rate = 1.2)       
        if print_cost and i % 1000 == 0:
            print ("Cost after iteration %i: %f" %(i, cost))

    return parameters

推测

def predict(parameters, X):
    A2, cache = forward_propagation(X, parameters)
    predictions = (A2 > 0.5)

    return predictions

执行模型

parameters = nn_model(X, Y, n_h = 4, num_iterations = 10000, print_cost=True)
plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y)
plt.title("Decision Boundary for hidden layer size " + str(4))
plt.show()

4个隐藏单元时的输出

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: 0.219507
Cost after iteration 9000: 0.218621

输出准确性：

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

输出：

Accuracy: 90%

调整隐藏层大小

plt.figure(figsize=(16, 32))
hidden_layer_sizes = [1, 2, 3, 4, 5, 10, 20]
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))
plt.show()

不同隐藏单元时的效果

输出：

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 10 hidden units: 90.25 %
Accuracy for 20 hidden units: 90.5 %