原理参考：https://zhuanlan.zhihu.com/p/77750026
SMO算法python代码
公式参考统计学习方法第7章
import numpy as np
import pandas as pd
from sklearn.datasets import load_iris
from sklearn.model_selection import train_test_split
import math
def create_data():
iris = load_iris()
df=pd.DataFrame(iris.data,columns=iris.feature_names)
df['label']=iris.target
df.columns=['sepal length', 'sepal width', 'petal length', 'petal width', 'label']
data=np.array(df.iloc[:100,[0,1,-1]])
for i in range(len(data)):
if data[i,-1]==0:
data[i,-1]=-1
return data[:,:2], data[:,-1]
X,y=create_data()
X_train,X_test,y_train,y_test=train_test_split(X,y,test_size=0.2)
# SMO算法
class SVM:
# 定义最大迭代次数，核函数
def __init__(self, max_iter, kernel='linear'):
self.max_iter = max_iter
self._kernel = kernel
# m样本量，n维度，X样本， Y样本类别，b,alpha拉格朗日乘子,E,C
def init_args(self, features, labels):
self.m, self.n = features.shape
self.X = features
self.Y = labels
self.b = 0.0
self.alpha = np.ones(self.m)
# Ei是g(x)预测值-实际值,保存至列表
self.E = [self._E(i) for i in range(self.m)]
# 惩罚参数
self.C=1.0
# 核函数
def kernel(self,x1,x2):
if self._kernel=='linear': #线性分类器 k(x,y)=x*y
return sum([x1[k]*x2[k] for k in range(self.n)])
elif self._kernel=='poly':
return (sum([x1[k]*x2[k] for k in range(self.n)])+1)**2 #d阶多项式分类器 k(x,y)={(x*y)+1}d
return 0
def _KKT(self, i): #p147 7.111~7.113
y_g = self._g(i)*self.Y[i]
if self.alpha[i]==0:
return y_g >=1
elif 0<self.alpha[i]<self.C:
return y_g ==1
else:
return y_g<=1
# g(x)预测值，输入（X[i]）
def _g(self,i):
r = self.b
for j in range(self.m):
r += self.alpha[j]*self.Y[j]*self.kernel(self.X[i], self.X[j]) # p145 公式7.105 7.117
return r
# E（x）为g(x)对输入x的预测值和实际值y的差
def _E(self, i):
return self._g(i)-self.Y[i]
def _init_alpha(self):
#外层循环首先遍历所有满足0<a<C的样本点，检验是否满足KKT P147
index_list=[i for i in range(self.m) if 0<self.alpha[i]<1]
# 否则遍历整个训练集
non_satisfy_list = [i for i in range(self.m) if i not in index_list]
index_list.extend(non_satisfy_list)
for i in index_list:
if self._KKT(i):
continue
E1=self.E[i]
# 如果E1是+，选择最小的；如果E1是负的，选择最大的
if E1 >= 0:
j = min(range(self.m), key=lambda x: self.E[x])
else:
j = max(range(self.m), key=lambda x: self.E[x])
return i, j
def _compare(self,_alpha, L, H): #7.108
if _alpha > H:
return H
elif _alpha<L:
return L
else:
return _alpha
def fit(self, features, labels):
self.init_args(features, labels)
for t in range(self.max_iter): # 迭代
# train 变量的选择 i1 i2
i1,i2 = self._init_alpha()
# 边界 p144~145
if self.Y[i1]==self.Y[i2]:
L = max(0, self.alpha[i1] + self.alpha[i2] - self.C)
H = min(self.C, self.alpha[i1] + self.alpha[i2])
else:
L = max(0, self.alpha[i2] - self.alpha[i1])
H = min(self.C, self.C + self.alpha[i2] - self.alpha[i1])
E1=self.E[i1]
E2=self.E[i2]
# eta = k11+k22-2k12 公式7.107
eta=self.kernel(self.X[i1],self.X[i1])+self.kernel(self.X[i2],self.X[i2])-2*self.kernel(self.X[i1],self.X[i2])
if eta<=0:
continue
alpha2_new_unc=self.alpha[i2]+self.Y[i2]*(E1-E2)/eta #7.106
alpha2_new = self._compare(alpha2_new_unc, L, H) #7.108
alpha1_new = self.alpha[i1] + self.Y[i1] * self.Y[i2] * (self.alpha[i2] - alpha2_new) #7.109
# 7.115
b1_new = -E1 - self.Y[i1] * self.kernel(self.X[i1], self.X[i1]) * (
alpha1_new - self.alpha[i1]) - self.Y[i2] * self.kernel(
self.X[i2],self.X[i1]) * (alpha2_new - self.alpha[i2]) + self.b
# 7.116
b2_new = -E2 - self.Y[i1] * self.kernel(self.X[i1], self.X[i2]) * (
alpha1_new - self.alpha[i1]) - self.Y[i2] * self.kernel(
self.X[i2],self.X[i2]) * (alpha2_new - self.alpha[i2]) + self.b
# p148
if 0 < alpha1_new < self.C:
b_new = b1_new
elif 0 < alpha2_new < self.C:
b_new = b2_new
else:
# 选择中点
b_new = (b1_new + b2_new) / 2
# 更新参数
self.alpha[i1] = alpha1_new
self.alpha[i2] = alpha2_new
self.b = b_new
self.E[i1] = self._E(i1)
self.E[i2] = self._E(i2)
return 'train done!'
def predict(self, data):
# g(xi)
r = self.b
for i in range(self.m):
r += self.alpha[i] * self.Y[i] * self.kernel(data, self.X[i])
return 1 if r > 0 else -1
def score(self, X_test, y_test):
right_count = 0
for i in range(len(X_test)):
result = self.predict(X_test[i])
if result == y_test[i]:
right_count += 1
return right_count / len(X_test)
svm = SVM(max_iter=200,kernel='poly')
svm.fit(X_train, y_train)
svm.score(X_test, y_test)
直接调用sklearn函数
from sklearn.svm import SVC
clf = SVC()
clf.fit(X_train, y_train)
clf.score(X_test, y_test)
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