感知机算法

第2章 感知机

二分类模型

$f(x)=sign(w\ast x+b)$

损失函数 $L(w,b)=-\Sigma y_i(w\ast x_i+b)$

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算法

随即梯度下降法 Stochastic Gradient Descent

随机抽取一个误分类点使其梯度下降。

$w=w+\eta y_i{x}_i$

$b=b+\eta y_i$

当实例点被误分类，即位于分离超平面的错误侧，则调整w, b的值，使分离超平面向该无分类点的一侧移动，直至误分类点被正确分类

最后计算求得的超平面方程是$w_1{x}_1+w_2{x}_2+b=0$，我们如果知道x1可以反解除x2，方便进行绘图

拿出iris数据集中两个分类的数据和[sepal length，sepal width]作为特征

import pandas as pd
import numpy as np
from sklearn.datasets import load_iris
import matplotlib.pyplot as plt
%matplotlib inline

# load 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']
df.label.value_counts()

2    50
1    50
0    50
Name: label, dtype: int64

plt.scatter(df[:50]['sepal length'], df[:50]['sepal width'], label='0')
plt.scatter(df[50:100]['sepal length'], df[50:100]['sepal width'], label='1')
plt.xlabel('sepal length')
plt.ylabel('sepal width')
plt.legend()

data = np.array(df.iloc[:100, [0, 1, -1]])

X, y = data[:,:-1], data[:,-1]

y = np.array([1 if i == 1 else -1 for i in y])

Perceptron

# 数据线性可分，二分类数据
# 此处为一元一次线性方程
class Model:
    def __init__(self):
        self.w = np.ones(len(data[0])-1, dtype=np.float32)
        self.b = 0
        self.l_rate = 0.1
        # self.data = data
    
    def sign(self, x, w, b):
        y = np.dot(x, w) + b
        return y
    
    # 随机梯度下降法
    def fit(self, X_train, y_train):
        is_wrong = False
        while not is_wrong:
            wrong_count = 0
            for d in range(len(X_train)):
                X = X_train[d]
                y = y_train[d]
                if y * self.sign(X, self.w, self.b) <= 0:
                    self.w = self.w + self.l_rate*np.dot(y, X)
                    self.b = self.b + self.l_rate*y
                    wrong_count += 1
            if wrong_count == 0:
                is_wrong = True
        return 'Perceptron Model!'
        
    def score(self):
        pass

perceptron = Model()
perceptron.fit(X, y)

'Perceptron Model!'

perceptron.b

-12.099999999999973

x_points = np.linspace(4, 7,10)
y_ = -(perceptron.w[0]*x_points + perceptron.b)/perceptron.w[1]
plt.plot(x_points, y_)  #这里的y_是对应的x2，而x_point对应的是x1。y_是根据x1反解出来的，方便画图

plt.plot(data[:50, 0], data[:50, 1], 'bo', color='blue', label='0')
plt.plot(data[50:100, 0], data[50:100, 1], 'bo', color='orange', label='1')
plt.xlabel('sepal length')
plt.ylabel('sepal width')
plt.legend()

scikit-learn Perceptron

from sklearn.linear_model import Perceptron

clf = Perceptron(fit_intercept=False, max_iter=1000, shuffle=False)
clf.fit(X, y)

Perceptron(alpha=0.0001, class_weight=None, eta0=1.0, fit_intercept=False,
      max_iter=1000, n_iter=None, n_jobs=1, penalty=None, random_state=0,
      shuffle=False, tol=None, verbose=0, warm_start=False)

# Weights assigned to the features.
print(clf.coef_)

[[  74.6 -127.2]]

# 截距 Constants in decision function.
print(clf.intercept_)

[0.]

x_ponits = np.arange(4, 8)
y_ = -(clf.coef_[0][0]*x_ponits + clf.intercept_)/clf.coef_[0][1]
plt.plot(x_ponits, y_) #给出对应的x和y的点，默认画出来的是直线

plt.plot(data[:50, 0], data[:50, 1], 'bo', color='blue', label='0')
plt.plot(data[50:100, 0], data[50:100, 1], 'bo', color='orange', label='1')
plt.xlabel('sepal length')
plt.ylabel('sepal width')
plt.legend()

原文代码作者：https://github.com/wzyonggege/statistical-learning-method