目录
介绍：
Activation Function（激活函数）是一种非线性函数，应用在神经网络的每个节点（神经元）上，用来引入非线性变换，增加神经网络的表达能力。
在神经网络中，每个节点的输入是通过加权和计算得到的，然后通过激活函数进行非线性变换，得到输出。激活函数可以将输入的范围映射到一个固定的范围内，常用的范围是[0, 1]或[-1, 1]。激活函数的引入可以使神经网络具有更强的表达能力，能够处理更复杂的输入数据。
常见的激活函数有：
- Sigmoid函数：将输入映射到[0, 1]的范围内，具有平滑的非线性特性，但存在梯度消失的问题。
- ReLU函数：将输入小于0的部分映射为0，大于0的部分保持不变，具有较好的非线性特性，但存在神经元死亡的问题。
- Tanh函数：将输入映射到[-1, 1]的范围内，具有平滑的非线性特性，但也存在梯度消失的问题。
- Leaky ReLU函数：在ReLU函数的基础上，将输入小于0的部分乘以一个小的斜率，解决了神经元死亡的问题。
选择合适的激活函数取决于具体的任务和数据特点，不同的激活函数在不同的情况下会有不同的表现。
数据集：
# scatter plot of the circles dataset with points colored by class
from sklearn.datasets import make_circles
from numpy import where
from matplotlib import pyplot
# generate circles
X, y = make_circles(n_samples=1000, noise=0.1, random_state=1)
# select indices of points with each class label
for i in range(2):
samples_ix = where(y == i)
pyplot.scatter(X[samples_ix, 0], X[samples_ix, 1], label=str(i))
pyplot.legend()
pyplot.show()
模型一（tanh） ：
# mlp for the two circles classification problem
from sklearn.datasets import make_circles
from sklearn.preprocessing import MinMaxScaler
from keras.layers import Dense
from keras.models import Sequential
from keras.optimizers import SGD
from keras.initializers import RandomUniform
from matplotlib import pyplot
# generate 2d classification dataset
X, y = make_circles(n_samples=1000, noise=0.1, random_state=1)
# scale input data to [-1,1]
scaler = MinMaxScaler(feature_range=(-1, 1))
X = scaler.fit_transform(X)
# split into train and test
n_train = 500
trainX, testX = X[:n_train, :], X[n_train:, :]
trainy, testy = y[:n_train], y[n_train:]
# define model
model = Sequential()
init = RandomUniform(minval=0, maxval=1)
model.add(Dense(5, input_dim=2, activation='tanh', kernel_initializer=init))
model.add(Dense(1, activation='sigmoid', kernel_initializer=init))
# compile model
opt = SGD(lr=0.01, momentum=0.9)
model.compile(loss='binary_crossentropy', optimizer=opt, metrics=['accuracy'])
# fit model
history = model.fit(trainX, trainy, validation_data=(testX, testy), epochs=500, verbose=0)
# evaluate the model
_, train_acc = model.evaluate(trainX, trainy, verbose=0)
_, test_acc = model.evaluate(testX, testy, verbose=0)
print('Train: %.3f, Test: %.3f' % (train_acc, test_acc))
# plot training history
pyplot.plot(history.history['accuracy'], label='train')
pyplot.plot(history.history['val_accuracy'], label='test')
pyplot.legend()
pyplot.show()
模型二（relu）：
# define model
model = Sequential()
init = RandomUniform(minval=0, maxval=1)
model.add(Dense(5, input_dim=2, activation='relu', kernel_initializer=init))
model.add(Dense(1, activation='sigmoid', kernel_initializer=init))
# compile model
opt = SGD(lr=0.01, momentum=0.9)
model.compile(loss='binary_crossentropy', optimizer=opt, metrics=['accuracy'])
# fit model
history = model.fit(trainX, trainy, validation_data=(testX, testy), epochs=500, verbose=0)
# evaluate the model
_, train_acc = model.evaluate(trainX, trainy, verbose=0)
_, test_acc = model.evaluate(testX, testy, verbose=0)
print('Train: %.3f, Test: %.3f' % (train_acc, test_acc))
# plot training history
pyplot.plot(history.history['accuracy'], label='train')
pyplot.plot(history.history['val_accuracy'], label='test')
pyplot.legend()
pyplot.show()
模型三（sigmoid） ：
# define model
model = Sequential()
init = RandomUniform(minval=0, maxval=1)
model.add(Dense(5, input_dim=2, activation='sigmoid', kernel_initializer=init))
model.add(Dense(1, activation='sigmoid', kernel_initializer=init))
# compile model
opt = SGD(lr=0.01, momentum=0.9)
model.compile(loss='binary_crossentropy', optimizer=opt, metrics=['accuracy'])
# fit model
history = model.fit(trainX, trainy, validation_data=(testX, testy), epochs=500, verbose=0)
# evaluate the model
_, train_acc = model.evaluate(trainX, trainy, verbose=0)
_, test_acc = model.evaluate(testX, testy, verbose=0)
print('Train: %.3f, Test: %.3f' % (train_acc, test_acc))
# plot training history
pyplot.plot(history.history['accuracy'], label='train')
pyplot.plot(history.history['val_accuracy'], label='test')
pyplot.legend()
pyplot.show()
模型四（多层tanh）：
# define model
init = RandomUniform(minval=0, maxval=1)
model = Sequential()
model.add(Dense(5, input_dim=2, activation='tanh', kernel_initializer=init))
model.add(Dense(5, activation='tanh', kernel_initializer=init))
model.add(Dense(5, activation='tanh', kernel_initializer=init))
model.add(Dense(5, activation='tanh', kernel_initializer=init))
#Initializers define the way to set the initial random weights of Keras layers. The keyword arguments used for passing
#initializers to layers depends on the layer.
model.add(Dense(5, activation='tanh', kernel_initializer=init))
model.add(Dense(1, activation='sigmoid', kernel_initializer=init))
# compile model
opt = SGD(lr=0.01, momentum=0.9)
model.compile(loss='binary_crossentropy', optimizer=opt, metrics=['accuracy'])
# fit model
history = model.fit(trainX, trainy, validation_data=(testX, testy), epochs=500, verbose=0)
# evaluate the model
_, train_acc = model.evaluate(trainX, trainy, verbose=0)
_, test_acc = model.evaluate(testX, testy, verbose=0)
print('Train: %.3f, Test: %.3f' % (train_acc, test_acc))
# plot training history
pyplot.plot(history.history['accuracy'], label='train')
pyplot.plot(history.history['val_accuracy'], label='test')
pyplot.legend()
pyplot.show()
模型五（多层relu）：
# define model
model = Sequential()
model.add(Dense(5, input_dim=2, activation='relu', kernel_initializer='he_uniform'))
model.add(Dense(5, activation='relu', kernel_initializer='he_uniform'))
model.add(Dense(5, activation='relu', kernel_initializer='he_uniform'))
model.add(Dense(5, activation='relu', kernel_initializer='he_uniform'))
model.add(Dense(5, activation='relu', kernel_initializer='he_uniform'))
#he_uniform . Draws samples from a uniform distribution within [-limit, limit] , where limit = sqrt(6 / fan_in)
#( fan_in is the number of input units in the weight tensor).
model.add(Dense(1, activation='sigmoid'))
# compile model
opt = SGD(lr=0.01, momentum=0.9)
model.compile(loss='binary_crossentropy', optimizer=opt, metrics=['accuracy'])
# fit model
history = model.fit(trainX, trainy, validation_data=(testX, testy), epochs=500, verbose=0)
# evaluate the model
_, train_acc = model.evaluate(trainX, trainy, verbose=0)
_, test_acc = model.evaluate(testX, testy, verbose=0)
print('Train: %.3f, Test: %.3f' % (train_acc, test_acc))
# plot training history
pyplot.plot(history.history['accuracy'], label='train')
pyplot.plot(history.history['val_accuracy'], label='test')
pyplot.legend()
pyplot.show()
