以下代码来自Deep Learning for Computer Vision with Python第十章。
本例程需要在同一文件内新建四个文件。分别是1、perceptron.py；2、perceptron_or.py；3、perceptron_and.py；4、perceptron_xor.py。
1、perceptron.py
# import the necessary packages
import numpy as np
class Perceptron:
def __init__(self, N, alpha=0.1):
# initialize the weight matrix and store the learning rate
self.W = np.random.randn(N + 1) / np.sqrt(N)
self.alpha = alpha
def step(self, x):
# apply the step function
return 1 if x > 0 else 0
def fit(self, X, y, epochs=10):
# insert a column of 1's as the last entry in the feature
# matrix -- this little trick allows us to treat the bias
# as a trainable parameter within the weight matrix
X = np.c_[X, np.ones((X.shape[0]))]
# loop over the desired number of epochs
for epoch in np.arange(0, epochs):
# loop over each individual data point
for (x, target) in zip(X, y):
# take the dot product between the input features
# and the weight matrix, then pass this value
# through the step function to obtain the prediction
p = self.step(np.dot(x, self.W))
#print("[training] self.W={}, x={}, target={}".format(self.W, x, target))
# only perform a weight update if our prediction
# does not match the target
if p != target:
# determine the error
error = p - target
# update the weight matrix
self.W += -self.alpha * error * x
def predict(self, X, addBias=True):
# ensure our input is a matrix
X = np.atleast_2d(X)
# check to see if the bias column should be added
if addBias:
# insert a column of 1's as the last entry in the feature
# matrix (bias)
X = np.c_[X, np.ones((X.shape[0]))]
# take the dot product between the input features and the
# weight matrix, then pass the value through the step
# function
return self.step(np.dot(X, self.W))
分析：Perception类是一个（3-1）结构的神经网络，（3-1）代表有输入层有3个神经元（其中两个神经元用于处理输入参数x1和x2，另外一个神经元输入固定为1），输出层有1个神经元。示意图见下图。
神经网络权重文件除了权重（w1和w2），还加上了偏置（b）。本身输入参数只有两个，对应的权值是w1和w2，输入层神经元为3的目的是把偏置b也加入权重矩阵中。当训练权重矩阵时，偏置b也得以更新。输出参数的表达式是y=step(x1*w1+x2*w2+b)。
神经元的激活函数是Step函数。Step函数包含一个输入参数和一个输出参数。当输入小于0，则输出为0；当输入大于0，则输出1。
fit函数用于训练，使用的是随机梯度下降法。predict函数作用是测试样品，把目标样品经过本神经网络，获得预测结果。
2、perceptron_or.py
# import the necessary packages
from perceptron import Perceptron
import numpy as np
# construct the OR dataset
X = np.array([[0, 0], [0, 1], [1, 0], [1, 1]])
y = np.array([[0], [1], [1], [1]])
# define our perceptron and train it
print("[INFO] training perceptron...")
p = Perceptron(X.shape[1], alpha=0.1)
p.fit(X, y, epochs=20)
# now that our perceptron is trained we can evaluate it
print("[INFO] testing perceptron...")
# now that our network is trained, loop over the data points
for (x, target) in zip(X, y):
# make a prediction on the data point and display the result
# to our console
pred = p.predict(x)
print("[INFO] data={}, ground-truth={}, pred={}".format(
x, target[0], pred))
在或运算中，两个输入参数和一个输出参数的关系见下表。
| 输入参数1：x1 | 输入参数2：x2 | 输出参数：y |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
Perceptron函数用于新建一个2层神经网络。第一个输入参数X.shape[1]是X中每个样品的参数个数。alpha是梯度下降、更新权值的速度。越接近1，速度越快，但是越大越容易错过局部最大值。
fit函数第一个参数是样品的输入参数矩阵，第二个参数是样品的输入参数的输出矩阵（真实值），第三个参数是迭代次数。data是样品的输入参数，ground-truth表示真实值，pred是预测结果。
用python运行perceptron_or.py，可得到以下结果：
============= RESTART: E:\FENG\workspace_python\perceptron_or.py =============
[INFO] training perceptron...
[INFO] testing perceptron...
[INFO] data=[0 0], ground-truth=0, pred=0
[INFO] data=[0 1], ground-truth=1, pred=1
[INFO] data=[1 0], ground-truth=1, pred=1
[INFO] data=[1 1], ground-truth=1, pred=13、perceptron_and.py
# import the necessary packages
from perceptron import Perceptron
import numpy as np
# construct the OR dataset
X = np.array([[0, 0], [0, 1], [1, 0], [1, 1]])
y = np.array([[0], [0], [0], [1]])
# define our perceptron and train it
print("[INFO] training perceptron...")
p = Perceptron(X.shape[1], alpha=0.1)
p.fit(X, y, epochs=20)
# now that our perceptron is trained we can evaluate it
print("[INFO] testing perceptron...")
# now that our network is trained, loop over the data points
for (x, target) in zip(X, y):
# make a prediction on the data point and display the result
# to our console
pred = p.predict(x)
print("[INFO] data={}, ground-truth={}, pred={}".format(
x, target[0], pred))
这个文件和上面那份文件差别不大，因此不分析了。
4、perceptron_xor.py
# import the necessary packages
from perceptron import Perceptron
import numpy as np
# construct the OR dataset
X = np.array([[0, 0], [0, 1], [1, 0], [1, 1]])
y = np.array([[0], [1], [1], [0]])
# define our perceptron and train it
print("[INFO] training perceptron...")
p = Perceptron(X.shape[1], alpha=0.1)
p.fit(X, y, epochs=20)
# now that our perceptron is trained we can evaluate it
print("[INFO] testing perceptron...")
# now that our network is trained, loop over the data points
for (x, target) in zip(X, y):
# make a prediction on the data point and display the result
# to our console
pred = p.predict(x)
print("[INFO] data={}, ground-truth={}, pred={}".format(
x, target[0], pred))
结果如下：
============ RESTART: E:\FENG\workspace_python\perceptron_xor.py ============
[INFO] training perceptron...
[INFO] testing perceptron...
[INFO] data=[0 0], ground-truth=0, pred=1
[INFO] data=[0 1], ground-truth=1, pred=1
[INFO] data=[1 0], ground-truth=1, pred=0
[INFO] data=[1 1], ground-truth=0, pred=0可见，异或的预测结果并不准确。主要因为，只具有2层神经元、而不具备隐含层的神经网络并无法非线性的对样品分类。
上图说明的是，与和或样本的空间分布，因为可以用一条直线把输出0和输出1的样本分类，因此比较简单。经过实践发现，使用2层神经网络的分类器也能实现预期效果。但是，异或的样本逻辑比较复杂。
为了正确分类异或样本，必须改变神经网络结构，进一步增加隐含层，尝试重新训练以及测试。