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三層神經網絡
- 節點數：784*100*10
- 學習率： 0.1
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預測結果得分（五次）
- ?0.9512
- ?0.9497
- ?0.9506
- ?0.9505
- ?0.9464
- 平均預測得分：0.94968
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四層神經網絡
- 節點數：784*100*100*10
- 學習率 ：0.1
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預測結果得分（五次）
- ?0.9095
- ?0.9142
- ?0.9033
- ?0.9130
- ?0.9046
- 平均預測得分：0.90892
- 結論：針對這種情況，簡單的神經網絡對MNIST數據集的分析，增加神經網絡層數未能提高學習效果。
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代碼（參考 Tariq Rashid的《Python 神經網絡編程》）
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三層神經網絡?
# python notebook for Make Your Own Neural Network # code for 3-layer neural network, and code for learning the MNIST dataset # 20190603 import numpy as np import matplotlib.pyplot as plt # scipy.special for the sigmoid function expit() import scipy.special as special # ensure the plots are inside this notebook, not an external window %matplotlib inline # neural network class definition class neuralNetwork(object): # initialise the neural network def __init__(self, inputNodes, hiddenNodes, outputNodes, learningRate=0.5): # 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 # pow(x, y), 返回x的y次方 self.wih = np.random.normal(0.0, pow(self.hNodes, -0.5), (self.hNodes, self.iNodes)) self.who = np.random.normal(0.0, pow(self.oNodes, -0.5), (self.oNodes, self.hNodes)) # learning rate self.lr = learningRate # activation function is the sigmoid function # lambda x: special.expit(x) 表示接受x,返回special.expit(x)函數 self.activation_function = lambda x: special.expit(x) pass # train the neural network def train(self, inputs_list, targets_list): # convert inputs list to 2d array inputs = np.array(inputs_list, ndmin=2).T targets = np.array(targets_list, ndmin=2).T # calculate signals into hidden layer hidden_inputs = np.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 = np.dot(self.who, hidden_outputs) # calculate signals emerging from final output layer final_outputs = self.activation_function(final_inputs) # error is the (targets - final_outputs) output_errors = (targets - final_outputs) # hidden layer error is the output_errors, split by weights, recombined at # hidden nodes hidden_errors = np.dot(self.who.T, output_errors) # update the weights for the links between the hidden and output layers # np.transpose(a) 表示矩陣a的轉秩 self.who += self.lr * np.dot(output_errors * final_outputs * (1.0 - final_outputs), np.transpose(hidden_outputs)) # update the weights for the links between the input and hidden layers self.wih += self.lr * np.dot(hidden_errors * hidden_outputs * (1.0 - hidden_outputs), np.transpose(inputs)) pass # query the neural network def query(self, inputs_list): # convert inputs list to 2d array # ndmin=2 表示指定最小維數為2 # .T 表示矩陣的轉秩 inputs = np.array(inputs_list, ndmin=2).T # calculate signals into hidden layer hidden_inputs = np.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 = np.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 = 784 hidden_nodes = 100 output_nodes = 10 # learning rate is 0.3 learning_rate = 0.1 # creat 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 # go through all recordes in the training data set final_training_data = [] for record in training_data_list: # split the record by the ',' commas all_values = record.split(',') # scale and shift the inputs inputs = (np.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 = np.zeros(output_nodes) + 0.01 # all_values[0] is the target label for this record targets[int(all_values[0])] = 0.99 # train Model final_training_data.append([inputs, targets]) pass # train Model for i in range(len(final_training_data)): n.train(final_training_data[i][0], final_training_data[i][1]) # load the minist 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 the first value correct_label = int(all_values[0]) # print("Correct label : ", correct_label) # scale and shift the inputs inputs = (np.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 = np.argmax(outputs) # print("NeuralNetwork's answer : ", label) # 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 = np.asarray(scorecard) print("Performance : ", scorecard_array.sum() / scorecard_array.size)?
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四層神經網絡
# python notebook for Make Your Own Neural Network # code for 4-layer neural network, and code for learning the MNIST dataset # 20190604 import numpy as np import matplotlib.pyplot as plt # scipy.special for the sigmoid function expit() import scipy.special as special # ensure the plots are inside this notebook, not an external window %matplotlib inline # neural network class definition class neuralNetwork(object): # initialise the neural network def __init__(self, inputNodes, hiddenNodes, hiddenNodes_2, outputNodes, learningRate=0.5): # set number of nodes in each input, hidden, output layer self.iNodes = inputNodes self.hNodes = hiddenNodes self.hNodes_2 = hiddenNodes_2 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 # pow(x, y), 返回x的y次方 self.wih = np.random.normal(0.0, pow(self.hNodes, -0.5), (self.hNodes, self.iNodes)) self.whh = np.random.normal(0.0, pow(int((self.hNodes + self.hNodes_2) / 2) , -0.5), (self.hNodes_2, self.hNodes)) self.who = np.random.normal(0.0, pow(self.oNodes, -0.5), (self.oNodes, self.hNodes_2)) # learning rate self.lr = learningRate # activation function is the sigmoid function # lambda x: special.expit(x) 表示接受x,返回special.expit(x)函數 self.activation_function = lambda x: special.expit(x) pass # train the neural network def train(self, inputs_list, targets_list): # convert inputs list to 2d array inputs = np.array(inputs_list, ndmin=2).T targets = np.array(targets_list, ndmin=2).T # calculate signals into hidden layer hidden_inputs = np.dot(self.wih, inputs) # calculate the signals emerging from hidden layer hidden_outputs = self.activation_function(hidden_inputs) # calculate signals into hidden_2 layer hidden_2_inputs = np.dot(self.whh, hidden_outputs) # calculate the signals emerging from hedden_2 layer hidden_2_outputs = self.activation_function(hidden_2_inputs) # calculate signals into final output layer final_inputs = np.dot(self.who, hidden_2_outputs) # calculate signals emerging from final output layer final_outputs = self.activation_function(final_inputs) # error is the (targets - final_outputs) output_errors = (targets - final_outputs) # hidden layer error is the output_errors, split by weights, recombined at # hidden nodes hidden_2_errors = np.dot(self.who.T, output_errors) # hidden_2 layer error is hidden_errors = np.dot(self.whh.T, hidden_2_errors) # update the weights for the links between the hidden and output layers # np.transpose(a) 表示矩陣a的轉秩 self.who += self.lr * np.dot(output_errors * final_outputs * (1.0 - final_outputs), np.transpose(hidden_2_outputs)) # update the weights for the links between the hideen and hidden_2 layers self.whh += self.lr * np.dot(hidden_2_errors * hidden_2_outputs * (1.0 - hidden_2_outputs), np.transpose(hidden_outputs)) # update the weights for the links between the input and hidden layers self.wih += self.lr * np.dot(hidden_errors * hidden_outputs * (1.0 - hidden_outputs), np.transpose(inputs)) pass # query the neural network def query(self, inputs_list): # convert inputs list to 2d array # ndmin=2 表示指定最小維數為2 # .T 表示矩陣的轉秩 inputs = np.array(inputs_list, ndmin=2).T # calculate signals into hidden layer hidden_inputs = np.dot(self.wih, inputs) # calculate the signals emerging from hidden layer hidden_outputs = self.activation_function(hidden_inputs) # calculate signals into hidden_2 layer hidden_2_inputs = np.dot(self.whh, hidden_outputs) # calculate the signals emerging from hidden_2 layer hidden_2_outputs = self.activation_function(hidden_2_inputs) # calculate signals into final output layer final_inputs = np.dot(self.who, hidden_2_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 = 784 hidden_nodes = 100 hidden_2_nodes = 100 output_nodes = 10 # learning rate is 0.3 learning_rate = 0.1 # creat instance of neural network n = neuralNetwork(input_nodes, hidden_nodes, hidden_2_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 # go through all recordes in the training data set final_training_data = [] for record in training_data_list: # split the record by the ',' commas all_values = record.split(',') # scale and shift the inputs inputs = (np.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 = np.zeros(output_nodes) + 0.01 # all_values[0] is the target label for this record targets[int(all_values[0])] = 0.99 # train Model final_training_data.append([inputs, targets]) pass # 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 # go through all recordes in the training data set final_training_data = [] for record in training_data_list: # split the record by the ',' commas all_values = record.split(',') # scale and shift the inputs inputs = (np.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 = np.zeros(output_nodes) + 0.01 # all_values[0] is the target label for this record targets[int(all_values[0])] = 0.99 # train Model final_training_data.append([inputs, targets]) pass # train Model for i in range(len(final_training_data)): n.train(final_training_data[i][0], final_training_data[i][1]) # load the minist 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 the first value correct_label = int(all_values[0]) # print("Correct label : ", correct_label) # scale and shift the inputs inputs = (np.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 = np.argmax(outputs) # print("NeuralNetwork's answer : ", label) # 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 = np.asarray(scorecard) print("Performance : ", scorecard_array.sum() / scorecard_array.size)?
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三層神經網絡?
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