人工神经网络——感知机学习算法的对偶形式

from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
from sklearn.neural_network import MLPClassifier
from sklearn.datasets import load_iris

生成线性可分数据集

def creat_data(n):
    np.random.seed(1)
    x_11=np.random.randint(0,100,(n,1))
    x_12=np.random.randint(0,100,(n,1))
    x_13=20+np.random.randint(0,10,(n,1))
    x_21=np.random.randint(0,100,(n,1))
    x_22=np.random.randint(0,100,(n,1))
    x_23=10-np.random.randint(0,10,(n,1))

    new_x_12=x_12*np.sqrt(2)/2-x_13*np.sqrt(2)/2
    new_x_13=x_12*np.sqrt(2)/2+x_13*np.sqrt(2)/2
    new_x_22=x_22*np.sqrt(2)/2-x_23*np.sqrt(2)/2
    new_x_23=x_22*np.sqrt(2)/2+x_23*np.sqrt(2)/2

    plus_samples=np.hstack([x_11,new_x_12,new_x_13,np.ones((n,1))])
    minus_samples=np.hstack([x_21,new_x_22,new_x_23,-np.ones((n,1))])
    samples=np.vstack([plus_samples,minus_samples])
    np.random.shuffle(samples)
    return samples

绘制数据集

def plot_samples(ax,samples):
    Y=samples[:,-1]
    Y=samples[:,-1]
    position_p=Y==1
    position_m=Y==-1
    ax.scatter(samples[position_p,0],samples[position_p,1],samples[position_p,2],marker='+',label='+',color='b')
    ax.scatter(samples[position_m,0],samples[position_m,1],samples[position_m,2],marker='^',label='-',color='y')

fig=plt.figure()
ax=Axes3D(fig)
data=creat_data(100)
plot_samples(ax,data)
ax.legend(loc='best')
plt.show()

感知机学习算法的原始形式算法

def perceptron(train_data,eta,w_0,b_0):
    x=train_data[:,:-1]
    y=train_data[:,-1]
    length=train_data.shape[0]
    w=w_0
    b=b_0
    step_num=0
    while True:
        i=0
        while(i1
            x_i=x[i].reshape((x.shape[1],1))
            y_i=y[i]
            if y_i*(np.dot(np.transpose(w),x_i)+b)<=0:
                w+=eta*y_i*x_i
                b+=eta*y_i
                break
            else:
                i+=1
        if(i==length):
            break
    return(w,b,step_num)

生成分离超平面

def creat_hyperplane(x,y,w,b):
    return (-w[0][0]*x-w[1][0]*y-b)/w[2][0]

根据训练数据集和 α⃗  α → 得到 w⃗  w → ，因为在计算中大量地需要 w⃗  w →

def creat_w(train_data,alpha):
    x=train_data[:,:-1]
    y=train_data[:,-1]
    N=train_data.shape[0]
    w=np.zeros((x.shape[1],1))
    for i in range(0,N):
        w+=alpha[i][0]*y[i]*(x[i].reshape(x[i].size,1))
    return w

感知机学习算法的对偶形式

def perceptron_dual(train_data,eta,alpha_0,b_0):
    x=train_data[:,:-1]
    y=train_data[:,-1]
    length=train_data.shape[0]
    alpha=alpha_0
    b=b_0
    step_num=0
    while_num=0
    while True:
        i=0
        while(i1
            x_i=x[i].reshape((x.shape[1],1))
            y_i=y[i]
            w=creat_w(train_data,alpha)
            z=y_i*(np.dot(np.transpose(w),x_i)+b)
            if z<=0:
                alpha[i][0]+=eta
                b+=eta*y_i
                break
            else:
                i+=1
        if(i==length):
            break
    return (alpha,b,step_num)

原始形式和对偶形式的运行情况

data=creat_data(100)
eta,w_0,b_0=0.1,np.ones((3,1),dtype=float),1
w_1,b_1,num_1=perceptron(data,eta,w_0,b_0)
alpha,b_2,num_2=perceptron_dual(data,eta=0.1,alpha_0=np.zeros((data.shape[0]*2,1)),b_0=0)
w_2=creat_w(data,alpha)

print('w_1,b_1',w_1,b_1)
print('w_2,b_2',w_2,b_2)

fig=plt.figure()
plt.suptitle('perceptron')
ax=Axes3D(fig)

#绘制样本点
plot_samples(ax,data)

#绘制分离超平面
x=np.linspace(-30,100,100)
y=np.linspace(-30,100,100)
x,y=np.meshgrid(x,y)
z=creat_hyperplane(x,y,w_1,b_1)
z_2=creat_hyperplane(x,y,w_2,b_2)
ax.plot_surface(x,y,z,rstride=1,cstride=1,color='g',alpha=0.2)
ax.plot_surface(x,y,z,rstride=1,cstride=1,color='c',alpha=0.2)
ax.legend(loc='best')
plt.show()