本文目录:
1. 感知器
2. 感知器的训练法则
3. 梯度下降和delta法则
4. python实现
1. 感知器
人工神经网络以感知器(perceptron)为基础。感知器以一个实数值向量作为输入,计算这些输入的线性组合,然后如果结果大于某个阈值,就输出1,否则输出-1(或0)。更精确地,如果输入为$x_1$到$x_n$,那么感知器计算的输出为:
其中,$w_i$是实数常量,叫做权值,用来决定输入$x_i$对感知器输出的贡献率。因为仅以一个阈值来决定输出,我们有时也把这种感知器叫做硬限幅感知器,当输出为1和-1时,也叫做sgn感知器(符号感知器)。
2. 感知器的训练法则
感知器的学习任务是决定一个权向量,它可以是感知器对于给定的训练样例输出正确的1或-1。为得到可接受的权向量,一种办法是从随机的权值开始,然后反复应用这个感知器到每一个训练样例,只要它误分类样例就修改感知器的权值。重复这个过程,直到感知器正确分类所有的训练样例。每一步根据感知器训练法则(perceptron Iraining rule) 来修改权值:${w_{i + 1}} \leftarrow {w_i} + \Delta {w_i}$,其中$\Delta {w_i} = \eta (t - o){x_i}$,$\eta$是学习速率,用来缓和或者加速每一步调整权值的程度。
3. 梯度下降和delta法则
4. python实现
训练数据:总共500个训练样本,链接https://pan.baidu.com/s/1qWugzIzdN9qZUnEw4kWcww,提取码:ncuj
损失函数:均方误差(MSE)
代码如下:
import numpy as np
import matplotlib.pyplot as plt class hardlim():
def __init__(self, path):
self.path = path def file2matrix(self, delimiter):
fp = open(self.path, 'r')
content = fp.read() # content现在是一行字符串,该字符串包含文件所有内容
fp.close()
rowlist = content.splitlines() # 按行转换为一维表
# 逐行遍历
# 结果按分隔符分割为行向量
recordlist = [list(map(float, row.split(delimiter))) for row in rowlist if row.strip()]
return np.mat(recordlist) def drawScatterbyLabel(self, dataSet):
m, n = dataSet.shape
target = np.array(dataSet[:, -1])
target = target.squeeze() # 把二维数据变为一维数据
for i in range(m):
if target[i] == 0:
plt.scatter(dataSet[i, 0], dataSet[i, 1], c='blue', marker='o')
if target[i] == 1:
plt.scatter(dataSet[i, 0], dataSet[i, 1], c='red', marker='o') def buildMat(self, dataSet):
m, n = dataSet.shape
dataMat = np.zeros((m, n))
dataMat[:, 0] = 1
dataMat[:, 1:] = dataSet[:, :-1]
return dataMat def classfier(self, x):
x[x >= 0.5] = 1
x[x < 0.5] = 0
return x if __name__ == '__main__':
hardlimit = hardlim('testSet.txt') print('1. 导入数据')
inputData = hardlimit.file2matrix('\t')
target = inputData[:, -1]
m, n = inputData.shape
print('size of input data: {} * {}'.format(m, n)) print('2. 按分类绘制散点图')
hardlimit.drawScatterbyLabel(inputData) print('3. 构建系数矩阵')
dataMat = hardlimit.buildMat(inputData) alpha = 0.1 # learning rate
steps = 600 # total iterations
weights = np.ones((n, 1)) # initialize weights
weightlist = [] print('4. 训练模型')
for k in range(steps):
output = hardlimit.classfier(dataMat * np.mat(weights))
errors = target - output
print('iteration: {} error_norm: {}'.format(k, np.linalg.norm(errors)))
weights = weights + alpha*dataMat.T*errors # 梯度下降
weightlist.append(weights) print('5. 画出训练过程')
X = np.linspace(-5, 15, 301)
weights = np.array(weights)
length = len(weightlist)
for idx in range(length):
if idx % 100 == 0:
weight = np.array(weightlist[idx])
Y = -(weight[0] + X * weight[1]) / weight[2]
plt.plot(X, Y)
plt.annotate('hplane:' + str(idx), xy=(X[0], Y[0]))
plt.show() print('6. 应用模型到测试数据中')
testdata = np.mat([-0.147324, 2.874846]) # 测试数据
m, n = testdata.shape
testmat = np.zeros((m, n+1))
testmat[:, 0] = 1
testmat[:, 1:] = testdata
result = sum(testmat * (np.mat(weights)))
if result < 0.5:
print(0)
else:
print(1)
训练结果如下:
【参考文献】
《机器学习》Mitshell,第四章
《机器学习算法原理与编程实践》郑捷,第五章5.2.2