Planar data classification with a hidden layer
数据集
1.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))
# Predict the function value for the whole grid
Z = model(np.c_[xx.ravel(), yy.ravel()])
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, cmap=plt.cm.Spectral)
def sigmoid(x):
"""
Compute the sigmoid of x
Arguments:
x -- A scalar or numpy array of any size.
Return:
s -- 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)
noisy_moons = sklearn.datasets.make_moons(n_samples=N, noise=.2)
blobs = sklearn.datasets.make_blobs(n_samples=N, random_state=5, n_features=2, centers=6)
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)
no_structure = np.random.rand(N, 2), np.random.rand(N, 2)
return noisy_circles, noisy_moons, blobs, gaussian_quantiles, no_structure
2.testCase_v2.py
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)
b1 = np.random.randn(4, 1)
b2 = np.array([[-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': b1,
'b2': b2}
return X_assess, parameters
def compute_cost_test_case():
np.random.seed(1)
Y_assess = (np.random.randn(1, 3) > 0)
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) > 0)
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) > 0)
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
代码
import numpy as np
import matplotlib.pyplot as plt
from testCases_v2 import *
import sklearn
import sklearn.datasets
import sklearn.linear_model
from planar_utils import plot_decision_boundary, sigmoid, load_planar_dataset, load_extra_datasets
#这里要设置随机种子，保证随机生成的数据与参考答案一致
np.random.seed(1) # set a seed so that the results are consistent
#返回n[0],n[1],n[2]
def layer_sizes(X, Y):
n_x = X.shape[0] # size of input layer
n_h = 4
n_y = Y.shape[0] # size of output layer
return (n_x, n_h, n_y)
#随机化初始序列
def initialize_parameters(n_x, n_h, n_y):
#这里要设置随机种子，保证随机生成的数据与参考答案一致
np.random.seed(2)
#乘以0.01是为了防止初始参数过大，使得梯度下降得速度变慢
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
#正向传播
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) # 二元分类输出层一般使用sigmoid函数
assert (A2.shape == (1, X.shape[1]))
cache = {
"Z1": Z1,
"A1": A1,
"Z2": Z2,
"A2": A2}
return A2, cache
#计算正向传播的损失值
def compute_cost(A2, Y, parameters):
m = Y.shape[1] # number of example
logprobs = np.multiply(Y, np.log(A2)) + np.multiply(1 - Y, np.log(1 - A2))
cost = (-1 / m) * np.sum(logprobs)
cost = float(np.squeeze(cost))
assert (isinstance(cost, float))
return cost
#反向传播
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
dW2 = (1 / m) * np.dot(dZ2, A1.T)
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
#根据梯度下降来更新参数
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 = 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
#单隐藏层的神经网络的整合模块
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)
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
#预测函数，其实就是A2向量
def predict(parameters, X):
A2, cache = forward_propagation(X, parameters)
predictions = np.round(A2) # 四舍五入
return predictions
#观察单隐藏层的单元个数与识别准确率的关系
if __name__ =='__main__':
X, Y = load_planar_dataset()
plt.figure(figsize=(16, 32))
hidden_layer_sizes = [1, 2, 3, 4, 5, 20, 50] # 隐藏层单元个数
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))
4.分析
这是样本点的分布情况
以下为单隐藏层神经网络对样本点的分类情况
图1~7分别对应隐藏层单元数1, 2, 3, 4, 5, 20, 50
可以发现隐藏层单元数不是越多越好，过多的话可能会引起过拟合
所谓过拟合，是指模型在训练集中拟合的非常好，但是损失了通用性，此类模型在测试集中表现得往往不是最好