吴恩达神经网络与深度学习——浅层神经网络习题3
浅层神经网络
实现具有单个隐藏层的两分类神经网络
使用具有非线性激活，如tanh
计算交叉熵损失
实现向前和向后传播
包
numpy
matplotlib
sklearn
# Package imports
import numpy as np
import matplotlib.pyplot as plt
from testCases import *
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
数据集
#planar_utils.py
import matplotlib.pyplot as plt
import numpy as np
import sklearn
import sklearn.datasets
import sklearn.linear_model
def plot_decision_boundary(model, X, y):
# Set min and max values and give it some padding
x_min, x_max = X[0, :].min() - 1, X[0, :].max() + 1
y_min, y_max = X[1, :].min() - 1, X[1, :].max() + 1
h = 0.01
# Generate a grid of points with distance h between them
xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
# np.meshgrid函数将参数1当做第1个结果的每一行, 并且一共有参数2的长度个行。
#第2个结果的每一列为参数2的内容, 并且重复参数1的长度个列。
# Predict the function value for the whole grid
Z = model(np.c_[xx.ravel(), yy.ravel()])
#np.ravel将多维数组降为一维,返回的是视图，修改时会影响原始矩阵
#np.r_是按列连接两个矩阵，就是把两矩阵上下相加，要求列数相等，类似于pandas中的concat()
#np.c_是按行连接两个矩阵，就是把两矩阵左右相加，要求行数相等，类似于pandas中的merge()。
Z = Z.reshape(xx.shape)
# Plot the contour and training examples
plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
plt.ylabel('x2')
plt.xlabel('x1')
plt.scatter(X[0, :], X[1, :], c=y.ravel(), cmap=plt.cm.Spectral)
def sigmoid(x):
s = 1/(1+np.exp(-x))
return s
def load_planar_dataset():
np.random.seed(1)
m = 400 # number of examples
N = int(m/2) # number of points per class
D = 2 # dimensionality
X = np.zeros((m,D)) # data matrix where each row is a single example
Y = np.zeros((m,1), dtype='uint8') # labels vector (0 for red, 1 for blue)
a = 4 # maximum ray of the flower
for j in range(2):
ix = range(N*j,N*(j+1))
t = np.linspace(j*3.12,(j+1)*3.12,N) + np.random.randn(N)*0.2 # theta
r = a*np.sin(4*t) + np.random.randn(N)*0.2 # radius
X[ix] = np.c_[r*np.sin(t), r*np.cos(t)]
Y[ix] = j
X = X.T
Y = Y.T
return X, Y
def load_extra_datasets():
N = 200
noisy_circles = sklearn.datasets.make_circles(n_samples=N, factor=.5, noise=.3)
## datasets.make_circles()专门用来生成圆圈形状的二维样本.
#factor表示里圈和外圈的距离之比.每圈共有n_samples/2个点
# 里圈代表一个类，外圈也代表一个类.noise表示有0.3的点是异常点
noisy_moons = sklearn.datasets.make_moons(n_samples=N, noise=.2)
#生成半环形图
blobs = sklearn.datasets.make_blobs(n_samples=N, random_state=5, n_features=2, centers=6)
#make_blobs会根据用户指定的特征数量、中心点数量、范围等来生成几类数据，这些数据可用于测试聚类算法的效果。
#n_samples是待生成的样本的总数。
#n_features是每个样本的特征数。
#centers表示类别数。
#cluster_std表示每个类别的方差，例如我们希望生成2类数据，其中一类比另一类具有更大的方差，可以将cluster_std设置为[1.0,3.0]。
gaussian_quantiles = sklearn.datasets.make_gaussian_quantiles(mean=None, cov=0.5, n_samples=N, n_features=2, n_classes=2, shuffle=True, random_state=None)
#利用高斯分位点区分不同数据
no_structure = np.random.rand(N, 2), np.random.rand(N, 2)
return noisy_circles, noisy_moons, blobs, gaussian_quantiles, no_structure
import numpy as np
def layer_sizes_test_case():
np.random.seed(1)
X_assess = np.random.randn(5, 3)
Y_assess = np.random.randn(2, 3)
return X_assess, Y_assess
def initialize_parameters_test_case():
n_x, n_h, n_y = 2, 4, 1
return n_x, n_h, n_y
def forward_propagation_test_case():
np.random.seed(1)
X_assess = np.random.randn(2, 3)
parameters = {'W1': np.array([[-0.00416758, -0.00056267],
[-0.02136196, 0.01640271],
[-0.01793436, -0.00841747],
[ 0.00502881, -0.01245288]]),
'W2': np.array([[-0.01057952, -0.00909008, 0.00551454, 0.02292208]]),
'b1': np.array([[ 0.],
[ 0.],
[ 0.],
[ 0.]]),
'b2': np.array([[ 0.]])}
return X_assess, parameters
def compute_cost_test_case():
np.random.seed(1)
Y_assess = np.random.randn(1, 3)
parameters = {'W1': np.array([[-0.00416758, -0.00056267],
[-0.02136196, 0.01640271],
[-0.01793436, -0.00841747],
[ 0.00502881, -0.01245288]]),
'W2': np.array([[-0.01057952, -0.00909008, 0.00551454, 0.02292208]]),
'b1': np.array([[ 0.],
[ 0.],
[ 0.],
[ 0.]]),
'b2': np.array([[ 0.]])}
a2 = (np.array([[ 0.5002307 , 0.49985831, 0.50023963]]))
return a2, Y_assess, parameters
def backward_propagation_test_case():
np.random.seed(1)
X_assess = np.random.randn(2, 3)
Y_assess = np.random.randn(1, 3)
parameters = {'W1': np.array([[-0.00416758, -0.00056267],
[-0.02136196, 0.01640271],
[-0.01793436, -0.00841747],
[ 0.00502881, -0.01245288]]),
'W2': np.array([[-0.01057952, -0.00909008, 0.00551454, 0.02292208]]),
'b1': np.array([[ 0.],
[ 0.],
[ 0.],
[ 0.]]),
'b2': np.array([[ 0.]])}
cache = {'A1': np.array([[-0.00616578, 0.0020626 , 0.00349619],
[-0.05225116, 0.02725659, -0.02646251],
[-0.02009721, 0.0036869 , 0.02883756],
[ 0.02152675, -0.01385234, 0.02599885]]),
'A2': np.array([[ 0.5002307 , 0.49985831, 0.50023963]]),
'Z1': np.array([[-0.00616586, 0.0020626 , 0.0034962 ],
[-0.05229879, 0.02726335, -0.02646869],
[-0.02009991, 0.00368692, 0.02884556],
[ 0.02153007, -0.01385322, 0.02600471]]),
'Z2': np.array([[ 0.00092281, -0.00056678, 0.00095853]])}
return parameters, cache, X_assess, Y_assess
def update_parameters_test_case():
parameters = {'W1': np.array([[-0.00615039, 0.0169021 ],
[-0.02311792, 0.03137121],
[-0.0169217 , -0.01752545],
[ 0.00935436, -0.05018221]]),
'W2': np.array([[-0.0104319 , -0.04019007, 0.01607211, 0.04440255]]),
'b1': np.array([[ -8.97523455e-07],
[ 8.15562092e-06],
[ 6.04810633e-07],
[ -2.54560700e-06]]),
'b2': np.array([[ 9.14954378e-05]])}
grads = {'dW1': np.array([[ 0.00023322, -0.00205423],
[ 0.00082222, -0.00700776],
[-0.00031831, 0.0028636 ],
[-0.00092857, 0.00809933]]),
'dW2': np.array([[ -1.75740039e-05, 3.70231337e-03, -1.25683095e-03,
-2.55715317e-03]]),
'db1': np.array([[ 1.05570087e-07],
[ -3.81814487e-06],
[ -1.90155145e-07],
[ 5.46467802e-07]]),
'db2': np.array([[ -1.08923140e-05]])}
return parameters, grads
def nn_model_test_case():
np.random.seed(1)
X_assess = np.random.randn(2, 3)
Y_assess = np.random.randn(1, 3)
return X_assess, Y_assess
def predict_test_case():
np.random.seed(1)
X_assess = np.random.randn(2, 3)
parameters = {'W1': np.array([[-0.00615039, 0.0169021 ],
[-0.02311792, 0.03137121],
[-0.0169217 , -0.01752545],
[ 0.00935436, -0.05018221]]),
'W2': np.array([[-0.0104319 , -0.04019007, 0.01607211, 0.04440255]]),
'b1': np.array([[ -8.97523455e-07],
[ 8.15562092e-06],
[ 6.04810633e-07],
[ -2.54560700e-06]]),
'b2': np.array([[ 9.14954378e-05]])}
return parameters, X_assess
X和Y的尺寸
X, Y = load_planar_dataset()
### START CODE HERE ### (≈ 3 lines of code)
shape_X = X.shape
shape_Y = Y.shape
m = Y.size # 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))
可视化
使用matplotlib可视化数据集。数据看起来像一朵“花”，有一些红色(标记为y = 0 )和一些蓝色( y = 1 )点。你的目标是建立一个模型来拟合这些数据。
plt.scatter(X[0, :], X[1, :], c=Y[0, :], s=40, cmap=plt.cm.Spectral);
逻辑回归
# Train the logistic regression classifier
clf = sklearn.linear_model.LogisticRegressionCV(cv=5);
clf.fit(X.T, Y.T.ravel());
# 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)")
神经网络
定义神经网络结构
# GRADED FUNCTION: layer_sizes
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)
初始化参数
# 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
n_x, n_h, n_y = initialize_parameters_test_case()
parameters = initialize_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"]))
前向传播
# 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"]
#print(W1.shape, b1.shape, W2.shape, b2.shape)
### 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
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']))
代价函数
# 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
# Retrieve W1 and W2 from parameters
### START CODE HERE ### (≈ 2 lines of code)
W1 = parameters["W1"]
W2 = parameters["W2"]
### END CODE HERE ###
# Compute the cross-entropy cost
### START CODE HERE ### (≈ 2 lines of code)
logprobs = np.multiply(np.log(A2), Y) + np.multiply((1-Y), np.log(1-A2))
### END CODE HERE ###
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)))
反向传播
# 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 = (1/m) * np.dot(dZ2, A1.T)
db2 = (1/m) * np.sum(dZ2, axis=1, keepdims=True)
dZ1 = np.multiply(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)
### END CODE HERE ###
grads = {"dW1": dW1,
"db1": db1,
"dW2": dW2,
"db2": db2}
return grads
parameters, cache, X_assess, Y_assess = backward_propagation_test_case()
grads = backward_propagation(parameters, cache, X_assess, Y_assess)
print ("dW1 = "+ str(grads["dW1"]))
print ("db1 = "+ str(grads["db1"]))
print ("dW2 = "+ str(grads["dW2"]))
print ("db2 = "+ str(grads["db2"]))
梯度下降
# 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 = 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
parameters, grads = update_parameters_test_case()
parameters = update_parameters(parameters, grads)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))
model
# 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)
# 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
X_assess, Y_assess = nn_model_test_case()
parameters = nn_model(X_assess, Y_assess, 4, num_iterations=10000, print_cost=False)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))
预测
# GRADED FUNCTION: predict
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 = (A2 > 0.5) # Vectorized
### END CODE HERE ###
return predictions
parameters, X_assess = predict_test_case()
predictions = predict(parameters, X_assess)
print("predictions mean = " + str(np.mean(predictions)))
测试
# 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))
# 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) + '%')
调整隐藏层大小
# This may take about 2 minutes to run
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))
较大的模型(具有更多隐藏单元)能够更好地适应训练集，直到过拟合。
最佳隐藏层尺寸似乎在n _ h = 5左右。事实上，这里的一个值似乎很适合数据，而不会引起明显的过度拟合。
其他数据集的性能
# 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}
### START CODE HERE ### (choose your dataset)
dataset = "noisy_moons"
### END CODE HERE ###
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.ravel(), s=40, cmap=plt.cm.Spectral);
np.meshgrid()
np,c_(),np.r_()
np.ravel()和np.flatten()
plt.scatter()
range
klearn的make_circles和make_moons生成数据
py.sklearn.datasets.make_blobs
make_gaussian_quantiles
make_gaussian_quantiles
plt.cm.Spectral
isinstance() 函数