感知器是一种简单的监督式的机器学习算法，也是最早的神经网络体系结构之一。它由 Rosenblatt 在 20 世纪 50 年代末提出。感知器是一种二元的线性分类器，其使用 d- 维超平面来将一组训练样本( d- 维输入向量)映射成二进制输出值。它的原理如下：
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感知器的训练可以使用梯度下降法，训练算法有不同的步骤。首先(在步骤0中)，模型的参数将被初始化。在达到指定训练次数或参数收敛前，重复以下其他步骤。
第 0 步：用 0 (或小的随机值)来初始化权重向量和偏置值
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其中，
表示学习率。
第 4 步：更新权重向量和偏置量。
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In [1]:
import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import make_blobs
from sklearn.model_selection import train_test_split
np.random.seed(123)
% matplotlib inline
数据集
In [2]:
X, y = make_blobs(n_samples=1000, centers=2)
fig = plt.figure(figsize=(8,6))
plt.scatter(X[:,0], X[:,1], c=y)
plt.title("Dataset")
plt.xlabel("First feature")
plt.ylabel("Second feature")
plt.show()
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In [3]:
y_true = y[:, np.newaxis]
X_train, X_test, y_train, y_test = train_test_split(X, y_true)
print(f'Shape X_train: {X_train.shape}')
print(f'Shape y_train: {y_train.shape})')
print(f'Shape X_test: {X_test.shape}')
print(f'Shape y_test: {y_test.shape}')
Shape X_train: (750, 2)
Shape y_train: (750, 1))
Shape X_test: (250, 2)
Shape y_test: (250, 1)
感知器分类
In [6]:
class Perceptron():
def __init__(self):
pass
def train(self, X, y, learning_rate=0.05, n_iters=100):
n_samples, n_features = X.shape
# Step 0: Initialize the parameters
self.weights = np.zeros((n_features,1))
self.bias = 0
for i in range(n_iters):
# Step 1: Compute the activation
a = np.dot(X, self.weights) + self.bias
# Step 2: Compute the output
y_predict = self.step_function(a)
# Step 3: Compute weight updates
delta_w = learning_rate * np.dot(X.T, (y - y_predict))
delta_b = learning_rate * np.sum(y - y_predict)
# Step 4: Update the parameters
self.weights += delta_w
self.bias += delta_b
return self.weights, self.bias
def step_function(self, x):
return np.array([1 if elem >= 0 else 0 for elem in x])[:, np.newaxis]
def predict(self, X):
a = np.dot(X, self.weights) + self.bias
return self.step_function(a)
初始化并训练模型
In [7]:
p = Perceptron()
w_trained, b_trained = p.train(X_train, y_train,learning_rate=0.05, n_iters=500)
测试
In [10]:
y_p_train = p.predict(X_train)
y_p_test = p.predict(X_test)
print(f"training accuracy: {100 - np.mean(np.abs(y_p_train - y_train)) * 100}%")
print(f"test accuracy: {100 - np.mean(np.abs(y_p_test - y_test)) * 100}%")
training accuracy: 100.0%
test accuracy: 100.0%
可视化决策边界
In [13]:
def plot_hyperplane(X, y, weights, bias):
"""
Plots the dataset and the estimated decision hyperplane
"""
slope = - weights[0]/weights[1]
intercept = - bias/weights[1]
x_hyperplane = np.linspace(-10,10,10)
y_hyperplane = slope * x_hyperplane + intercept
fig = plt.figure(figsize=(8,6))
plt.scatter(X[:,0], X[:,1], c=y)
plt.plot(x_hyperplane, y_hyperplane, '-')
plt.title("Dataset and fitted decision hyperplane")
plt.xlabel("First feature")
plt.ylabel("Second feature")
plt.show()
In [14]:
plot_hyperplane(X, y, w_trained, b_trained)
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