使用神經網絡識別手寫數字：

import numpy

# scipy.special for the sigmoid function expit()，即S函數

import scipy.special

# library for plotting arrays

import matplotlib.pyplot

# ensure the plots are inside this notebook, not an external window

%matplotlib inline // 在notebook上繪圖，而不是獨立窗口

# neural network class definition

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

# numpy.random.normal(loc,scale,size) loc：概率分布的均值；scale：概率分布的方差；size：輸出的shape

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

# 使用lambda創建的函數是沒有名字的

self.activation_function = lambda x: scipy.special.expit(x)

pass

# 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))

pass

# 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

# 選擇784個輸入節點是28*28的結果，即組成手寫數字圖像的像素個數

input_nodes = 784

# 選擇使用100個隱藏層不是通過使用科學的方法得到的。通過選擇使用比輸入節點的數量小的值，強制網絡嘗試總結輸入的主要特點。

# 但是，如果選擇太少的隱藏層節點，會限制網絡的能力，使網絡難以找到足夠的特征或模式。

# 同時，還要考慮到輸出層節點數10。

# 這里應該強調一點。對于一個問題，應該選擇多少個隱藏層節點，并不存在一個最佳方法。同時，我們也沒有最佳方法選擇需要幾層隱藏層。

# 就目前而言，最好的辦法是進行實驗，直到找到適合你要解決的問題的一個數字。

hidden_nodes = 200

output_nodes = 10

# learning rate，需要多次嘗試，0.2是最佳值

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

# 就像調整學習率一樣，需要使用幾個不同的世代進行實驗并繪圖，以可視化這些效果。直覺告訴我們，所做的訓練越多，所得到的的性能越好。

# 但太多的訓練實際上會過猶不及，這是由于網絡過度擬合訓練數據。

# 在大約5或7個世代時，有一個甜蜜點。在此之后，性能會下降，這可能是過度擬合的效果。

# 性能在6個世代的情況下下降，這可能是運行中出了問題，導致網絡在梯度下降過程中被卡在了一個局部的最小值中。

# 事實上，由于沒有對每個數據點進行多次實驗，無法減小隨機過程的影響。

# 神經網絡的學習過程其核心是隨機過程，有時候工作得不錯，有時候很糟。

# 另一個可能的原因是，在較大數目的世代情況下，學習率可能設置過高了。在更多世代的情況下，減小學習率確實能夠得到更好的性能。

# 如果打算使用更長的時間(多個世代)探索梯度下降，那么可以采用較短的步長(學習率)，總體上可以找到更好的路徑。

# 要正確、科學地選擇這些參數，必須為每個學習率和世代組合進行多次實驗，盡量減少在梯度下降過程中隨機性的影響。

# 還可嘗試不同的隱藏層節點數量，不同的激活函數。

epochs = 5

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

# 輸入值需要避免0，輸出值需要避免1

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)

pass

pass

# 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()

# test the neural network

# 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)

pass

pass

# calculate the performance score, the fraction of correct answers

scorecard_array = numpy.asarray(scorecard)

print ("performance = ", scorecard_array.sum() / scorecard_array.size)

# performance = 0.9712

總結

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