ANN神經網絡——實現異或XOR (Python實現)

一、Introduction

Perceptron can represent AND,OR,NOT

用初中的線性規划問題理解

異或的里程碑意義

想學的通透，先學歷史！

據說在人工神經網絡（artificial neural network, ANN）發展初期，由於無法實現對多層神經網絡（包括異或邏輯）的訓練而造成了一場ANN危機，到最后BP算法的出現，才讓訓練帶有隱藏層的多層神經網絡成為可能。因此異或的實現在ANN的發展史是也是具有里程碑意義的。異或之所以重要，是因為它相對於其他邏輯關系，例如與（AND）, 或（OR）等，異或是線性不可分的。如下圖：

要解決非線性可分問題，需考慮使用多層功能神經元. 例如下圖中這個
簡單的兩層感知機就能解決異或問題。在圖中，輸出層與輸入層之間的一
層神經元，被稱為隱含層(hidden layer) ，隱含層和輸出層神經元都是擁
有激活函數的功能神經元.

能解決異或問題的兩層感知機

參考周志華老師西瓜書

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二、Python 代碼實現

異或肯定是不能通過一條直線區分的，因此單層網絡無法實現異或，但兩層（包含一個隱藏層）就可以了。

在實際應用中，異或門（Exclusive-OR gate, XOR gate）是數字邏輯中實現邏輯異或的邏輯門，這一函數能實現模為2的加法。因此，異或門可以實現計算機中的二進制加法。

可以有多種方法實現Xor功能，本代碼采用的算法圖示如下

將上圖轉化為神經網絡層形式便於理解：

# ----------
#
# In this exercise, you will create a network of perceptrons that can represent
# the XOR function, using a network structure like those shown in the previous
# quizzes.
#
# You will need to do two things:
# First, create a network of perceptrons with the correct weights
# Second, define a procedure EvalNetwork() which takes in a list of inputs and
# outputs the value of this network.
#
# ----------

import numpy as np

class Perceptron:
    """
    This class models an artificial neuron with step activation function.
    """

    def __init__(self, weights = np.array([1]), threshold = 0):
        """
        Initialize weights and threshold based on input arguments. Note that no
        type-checking is being performed here for simplicity.
        """
        self.weights = weights
        self.threshold = threshold


    def activate(self, values):
        """
        Takes in @param values, a list of numbers equal to length of weights.
        @return the output of a threshold perceptron with given inputs based on
        perceptron weights and threshold.
        """
               
        # First calculate the strength with which the perceptron fires
        strength = np.dot(values,self.weights)
        
        # Then return 0 or 1 depending on strength compared to threshold  
        return int(strength >= self.threshold)#this row changed by myself
        

            
# Part 1: Set up the perceptron network
Network = [
    
    # input layer, declare input layer perceptrons here
    [ Perceptron([1,0],1),Perceptron([1,1],2),Perceptron([0,1],1) ], \
    # output node, declare output layer perceptron here
    [ Perceptron([1, -2, 1],   1) ]
 
]
# Part 2: Define a procedure to compute the output of the network, given inputs
def EvalNetwork(inputValues, Network):
    """
    Takes in @param inputValues, a list of input values, and @param Network
    that specifies a perceptron network. @return the output of the Network for
    the given set of inputs.
    """
    
    # MY MAIN CODE HERE

    # Be sure your output value is a single number
   
    #Method1 :
    return Network[1][0].activate([p.activate(inputValues) for p in Network[0]])
    # p is an instance of Perceptron.
    # inner brackets -->input layer
    # Network[1][0] -->Perceptron([1, -2, 1],   1)  -- Only one element
    
    #Method2 :
    # OutputValue = inputValues
    # for layer in Network:
    #     OutputValue = map(lambda p:p.activate(OutputValue), layer)
    # return OutputValue 
    ## but warning:this method return a list ,not a single number
    ## to review Python Grammar?

def test():
    """
    A few tests to make sure that the perceptron class performs as expected.
    """
    print "0 XOR 0 = 0?:", EvalNetwork(np.array([0,0]), Network)
    print "0 XOR 1 = 1?:", EvalNetwork(np.array([0,1]), Network)
    print "1 XOR 0 = 1?:", EvalNetwork(np.array([1,0]), Network)
    print "1 XOR 1 = 0?:", EvalNetwork(np.array([1,1]), Network)

if __name__ == "__main__":
    test()

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OUTPUT:

Running test()...
0 XOR 0 = 0?: 0
0 XOR 1 = 1?: 1
1 XOR 0 = 1?: 1
1 XOR 1 = 0?: 0
All done!

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