深入解析神经网络从原理到实现

def sigmoid(z):
  return 1.0/(1.0+np.exp(-z))

def sigmoid_prime(z):
  return sigmoid(z)*(1-sigmoid(z))

def feedforward(self, a):
  for b, w in zip(self.biases, self.weights):
    a = sigmoid(np.dot(w, a)+b)
  return a

biases = [np.random.randn(y, 1) for y in sizes[1:]]
weights = [np.random.randn(y, x) for x, y in zip(sizes[:-1], sizes[1:])]

def backprop(self, x, y):
  """返回一个元组(nabla_b, nabla_w)代表目标函数的梯度."""
  nabla_b = [np.zeros(b.shape) for b in self.biases]
  nabla_w = [np.zeros(w.shape) for w in self.weights]
  # feedforward
  activation = x
  activations = [x] # list to store all the activations, layer by layer
  zs = [] # list to store all the z vectors, layer by layer
  for b, w in zip(self.biases, self.weights):
    z = np.dot(w, activation)+b
    zs.append(z)
    activation = sigmoid(z)
    activations.append(activation)
  # backward pass
  delta = self.cost_derivative(activations[-1], y) * \
    sigmoid_prime(zs[-1])
  nabla_b[-1] = delta
  nabla_w[-1] = np.dot(delta, activations[-2].transpose())
  """l = 1 表示最后一层神经元，l = 2 是倒数第二层神经元, 依此类推."""
  for l in xrange(2, self.num_layers):
    z = zs[-l]
    sp = sigmoid_prime(z)
    delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
    nabla_b[-l] = delta
    nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
  return (nabla_b, nabla_w)

# -*- coding: utf-8 -*-

import random
import numpy as np

class Network(object):

  def __init__(self, sizes):
  """参数sizes表示每一层神经元的个数，如[2,3,1],表示第一层有2个神经元，第二层有3个神经元，第三层有1个神经元."""
    self.num_layers = len(sizes)
    self.sizes = sizes
    self.biases = [np.random.randn(y, 1) for y in sizes[1:]]
    self.weights = [np.random.randn(y, x)
            for x, y in zip(sizes[:-1], sizes[1:])]

  def feedforward(self, a):
    """前向传播"""
    for b, w in zip(self.biases, self.weights):
      a = sigmoid(np.dot(w, a)+b)
    return a

  def SGD(self, training_data, epochs, mini_batch_size, eta,
      test_data=None):
    """随机梯度下降"""
    if test_data: 
      n_test = len(test_data)
    n = len(training_data)
    for j in xrange(epochs):
      random.shuffle(training_data)
      mini_batches = [
        training_data[k:k+mini_batch_size]
        for k in xrange(0, n, mini_batch_size)]
      for mini_batch in mini_batches:
        self.update_mini_batch(mini_batch, eta)
      if test_data:
        print "Epoch {0}: {1} / {2}".format(j, self.evaluate(test_data), n_test)
      else:
        print "Epoch {0} complete".format(j)

  def update_mini_batch(self, mini_batch, eta):
    """使用后向传播算法进行参数更新.mini_batch是一个元组(x, y)的列表、eta是学习速率"""
    nabla_b = [np.zeros(b.shape) for b in self.biases]
    nabla_w = [np.zeros(w.shape) for w in self.weights]
    for x, y in mini_batch:
      delta_nabla_b, delta_nabla_w = self.backprop(x, y)
      nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
      nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]
    self.weights = [w-(eta/len(mini_batch))*nw
            for w, nw in zip(self.weights, nabla_w)]
    self.biases = [b-(eta/len(mini_batch))*nb
            for b, nb in zip(self.biases, nabla_b)]

  def backprop(self, x, y):
    """返回一个元组(nabla_b, nabla_w)代表目标函数的梯度."""
    nabla_b = [np.zeros(b.shape) for b in self.biases]
    nabla_w = [np.zeros(w.shape) for w in self.weights]
    # 前向传播
    activation = x
    activations = [x] # list to store all the activations, layer by layer
    zs = [] # list to store all the z vectors, layer by layer
    for b, w in zip(self.biases, self.weights):
      z = np.dot(w, activation)+b
      zs.append(z)
      activation = sigmoid(z)
      activations.append(activation)
    # backward pass
    delta = self.cost_derivative(activations[-1], y) * sigmoid_prime(zs[-1])
    nabla_b[-1] = delta
    nabla_w[-1] = np.dot(delta, activations[-2].transpose())
    """l = 1 表示最后一层神经元，l = 2 是倒数第二层神经元, 依此类推."""
    for l in xrange(2, self.num_layers):
      z = zs[-l]
      sp = sigmoid_prime(z)
      delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
      nabla_b[-l] = delta
      nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
    return (nabla_b, nabla_w)

  def evaluate(self, test_data):
    """返回分类正确的个数"""
    test_results = [(np.argmax(self.feedforward(x)), y) for (x, y) in test_data]
    return sum(int(x == y) for (x, y) in test_results)

  def cost_derivative(self, output_activations, y):
    return (output_activations-y)

def sigmoid(z):
  return 1.0/(1.0+np.exp(-z))

def sigmoid_prime(z):
  """sigmoid函数的导数"""
  return sigmoid(z)*(1-sigmoid(z))

# -*- coding: utf-8 -*-

from network import *

def vectorized_result(j,nclass):
  """离散数据进行one-hot"""
  e = np.zeros((nclass, 1))
  e[j] = 1.0
  return e

def get_format_data(X,y,isTest):
  ndim = X.shape[1]
  nclass = len(np.unique(y))
  inputs = [np.reshape(x, (ndim, 1)) for x in X]
  if not isTest:
    results = [vectorized_result(y,nclass) for y in y]
  else:
    results = y
  data = zip(inputs, results)
  return data

#随机生成数据
from sklearn.datasets import *
np.random.seed(0)
X, y = make_moons(200, noise=0.20)
ndim = X.shape[1]
nclass = len(np.unique(y))

#划分训练、测试集
from sklearn.cross_validation import train_test_split
train_x,test_x,train_y,test_y = train_test_split(X,y,test_size=0.2,random_state=0)

training_data = get_format_data(train_x,train_y,False)
test_data = get_format_data(test_x,test_y,True)

net = Network(sizes=[ndim,10,nclass])
net.SGD(training_data=training_data,epochs=5,mini_batch_size=10,eta=0.1,test_data=test_data)