神经网络伪代码 神经网络编程

参考书籍：《Python神经网络编程》

神经网络基本编程

采用Sigmod激活函数，建立一个包含一层隐藏层的神经网络通用代码：

import numpy
# scipy.special for the sigmoid function expit()
import scipy.special

class neuralNetwork:   
    # initialise the neural network
    def __init__(self, inputnodes, hiddennodes, outputnodes, learningrate):
        # set number of nodes in each input, hidden, output layer
        self.inodes = inputnodes
        self.hnodes = hiddennodes
        self.onodes = outputnodes        
        # link weight matrices, wih and who
        # weights inside the arrays are w_i_j, where link is from node i to node j in the next layer
        # w11 w21
        # w12 w22 etc 
        self.wih = numpy.random.normal(0.0, pow(self.inodes, -0.5), (self.hnodes, self.inodes))
        self.who = numpy.random.normal(0.0, pow(self.hnodes, -0.5), (self.onodes, self.hnodes))
        # learning rate
        self.lr = learningrate 
        # activation function is the sigmoid function
        self.activation_function = lambda x: scipy.special.expit(x)        

    
    # train the neural network
    def train(self, inputs_list, targets_list):
        # convert inputs list to 2d array
        inputs = numpy.array(inputs_list, ndmin=2).T
        targets = numpy.array(targets_list, ndmin=2).T        
        # calculate signals into hidden layer
        hidden_inputs = numpy.dot(self.wih, inputs)
        # calculate the signals emerging from hidden layer
        hidden_outputs = self.activation_function(hidden_inputs)      
        # calculate signals into final output layer
        final_inputs = numpy.dot(self.who, hidden_outputs)
        # calculate the signals emerging from final output layer
        final_outputs = self.activation_function(final_inputs)       
        # output layer error is the (target - actual)
        output_errors = targets - final_outputs
        # hidden layer error is the output_errors, split by weights, recombined at hidden nodes
        hidden_errors = numpy.dot(self.who.T, output_errors) 
        
        # update the weights for the links between the hidden and output layers
        self.who += self.lr * numpy.dot((output_errors * final_outputs * (1.0 - final_outputs)), numpy.transpose(hidden_outputs))
        
        # update the weights for the links between the input and hidden layers
        self.wih += self.lr * numpy.dot((hidden_errors * hidden_outputs * (1.0 - hidden_outputs)), numpy.transpose(inputs))

    
    # query the neural network
    def query(self, inputs_list):
        # convert inputs list to 2d array
        inputs = numpy.array(inputs_list, ndmin=2).T
        
        # calculate signals into hidden layer
        hidden_inputs = numpy.dot(self.wih, inputs)
        # calculate the signals emerging from hidden layer
        hidden_outputs = self.activation_function(hidden_inputs)
        
        # calculate signals into final output layer
        final_inputs = numpy.dot(self.who, hidden_outputs)
        # calculate the signals emerging from final output layer
        final_outputs = self.activation_function(final_inputs)
        
        return final_outputs

# number of input, hidden and output nodes
input_nodes = 3
hidden_nodes = 3
output_nodes = 3
# learning rate is 0.3
learning_rate = 0.3
# create instance of neural network
n = neuralNetwork(input_nodes,hidden_nodes,output_nodes, learning_rate)    
n.query([1.0, 0.5, -1.5])

这里神经网络的训练是基于梯度下降的算法的。通过误差的反向传递最终得到一个最优解。公式为：
$w_{jk} = w_{jk} - \alpha \frac{dE}{dw_{jk}}$

$E = \sum_n (t_n - o_n)^2$

其中α代表学习率，决定了训练的速度。后面的求导代表$w_{jk}$在误差中的梯度，通过这个式子可以使得输出的值和真实值相比误差不断减小。算得
$\frac{dE}{dw_{jk}} = - 2(t_k - o_k) * sigmoid(\sum_j w_{jk}o_j)(1-sigmoid(\sum_j w_{jk}o_j)) * o_j$
其中$t_k$为该权重对应输出节点的真实值，$o_k$为现在该输出节点预测值，$o_j$是隐藏层的输出值。可以写成：
$\frac{dE}{dw_{jk}} = - e_k * sigmoid(\sum_j w_{jk}o_j)(1-sigmoid(\sum_j w_{jk}o_j)) * o_j$

$\Delta w_{jk} = \alpha * \frac{dE}{dw_{jk}}$

识别手写数字

代码及相关资源来源：https://www.epubit.com/bookDetails?id=N34292&typeName=%E6%90%9C%E7%B4%A2

https://link.zhihu.com/?target=http%3A//www.pjreddie.com/media/files/mnist_train.csv

https://link.zhihu.com/?target=http%3A//www.pjreddie.com/media/files/mnist_test.csv

数据用CSV文件存放，数字是28*28像素，每行第一个值是标签，后面的都是灰度值。训练集有100条，测试集10条。

样式展示：

代码：

import numpy
# scipy.special for the sigmoid function expit()
import scipy.special
# library for plotting arrays
import matplotlib.pyplot
# ensure the plots are inside this notebook, not an external window
%matplotlib inline

--snip--

# number of input, hidden and output nodes
input_nodes = 784
hidden_nodes = 200
output_nodes = 10

# learning rate
learning_rate = 0.1

# create instance of neural network
n = neuralNetwork(input_nodes,hidden_nodes,output_nodes, learning_rate)

# load the mnist training data CSV file into a list
training_data_file = open("mnist_dataset/mnist_train.csv", 'r')
training_data_list = training_data_file.readlines()
training_data_file.close()

# train the neural network

# epochs is the number of times the training data set is used for training
epochs = 10

for e in range(epochs):
    # go through all records in the training data set
    for record in training_data_list:
        # split the record by the ',' commas
        all_values = record.split(',')              
        # scale and shift the inputs
        inputs = (numpy.asfarray(all_values[1:]) / 255.0 * 0.99) + 0.01
        # create the target output values (all 0.01, except the desired label which is 0.99)
        targets = numpy.zeros(output_nodes) + 0.01
        # all_values[0] is the target label for this record
        targets[int(all_values[0])] = 0.99
        n.train(inputs, targets)

# load the mnist test data CSV file into a list
test_data_file = open("mnist_dataset/mnist_test.csv", 'r')
test_data_list = test_data_file.readlines()
test_data_file.close()
# scorecard for how well the network performs, initially empty
scorecard = []

# go through all the records in the test data set
for record in test_data_list:
    # split the record by the ',' commas
    all_values = record.split(',')
    # correct answer is first value
    correct_label = int(all_values[0])
    # scale and shift the inputs
    inputs = (numpy.asfarray(all_values[1:]) / 255.0 * 0.99) + 0.01
    # query the network
    outputs = n.query(inputs)
    # the index of the highest value corresponds to the label
    label = numpy.argmax(outputs)
    # append correct or incorrect to list
    if (label == correct_label):
        # network's answer matches correct answer, add 1 to scorecard
        scorecard.append(1)
    else:
        # network's answer doesn't match correct answer, add 0 to scorecard
        scorecard.append(0)


# calculate the performance score, the fraction of correct answers
scorecard_array = numpy.asarray(scorecard)
print ("performance = ", scorecard_array.sum() / scorecard_array.size)

要注意的点：

• 这里输入节点为28*28=784，输出节点为10个，分别代表0到9这10个数字，那个输出节点数值最大那么分类结果就是这个数字。
• 激活函数不可能产生0或1的输出。因此需要使用0.01和0.99代替0和1。因此训练目标是让除了目标输出节点为0.99，其他节点为0.01。
• 查看图片的代码为

image_array = numpy.asfarray(all_values[1:]).reshape((28,28))                  
matplotlib.pyplot.imshow(image_array, cmap='Greys', interpolation='None')

• 过高和过低的学习率都是有害的。过高会使得在梯度下降的过程中的来回震动加剧，难以找到最优点。过低会导致学习缓慢，迟迟达不到最优点。
• 多次运行会因为初始点的不同，得到不同的结果，这就是多个局部的最优点，我们从这些最优点中，挑选最好的即可。
• 过多的训练会过犹不及，出现过拟合现象，导致网络在新数据上表现不佳。
• 改变隐藏层的节点数目可以优化训练结果，过少的隐藏节点会使得网络学习能力不足，过多会使得网络难以训练。

识别自己手写的数字

我们可以尝试着自己的笔迹，或者是故意做一些扭曲和噪声的处理，来测试网络的辨别能力。当我们得到这种图片后首先需要把它们处理成28*28像素的图片，你可以使用图像编辑器做到这一点，然后要转换成灰度数字表现的形式代码如下：

import scipy.misc
img_array = scipy.misc.imread(image_file_name,flatten = True)
img_data = 255.0 - img_array.reshape(784)
img_data = (img_data / 255.0 * 0.99) + 0.01

其中imread函数从图像文件中读取数据，"flatten=True"把图像变成简单的浮点数数组。如果图像是彩色的，颜色值会自动转换为灰度值。这样就可以把这些数据运用到我们的模型中了。

旋转图像可以创建新的训练数据

收集更多的手写样本固然可以提高模型的准确率，但是工作量太大了。一个很好的做法是通过顺时针或逆时针旋转它们，可以创建新的样本，通过这些新的样本进行训练，同样可以很好地提高模型的准确率。但要注意旋转的角度不能过大，否则会降低神经网络的性能，因为这意味创建了不能代表数字的图像，建议旋转10°，这个角度比较理想。