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。對於一個問題，應該選擇多少個隱藏層節點，並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並繪圖，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，這是由於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，由於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