1、线性神经元模型
这里先说一下，一般我们学习映射问题，都是先从“线性映射”开始，然后深入到“非线性映射”，之间最主要的差别就是映射函数f(x):X---->Y,到底是线性的，还是非线性的。所以，线性神经元模型的激活函数肯定就是线性的。我们首先来看下面的图：（图片均来自网络）
图1：线性神经元模型
解释一下：首先，我们输入训练样本X和初始化权重向量W，将其进行向量的点乘，然后将点乘求和的结果作用于激活函数f(x)=x(典型的线性函数),得到实际的输出O，现在我们根据实际输出O和期望输出d之间的差距error，来调整初始化的权重向量W。如此反复，直到W调整到合适的结果为止。最后在预测阶段，我们就可以使用训练好的W，进行前向传播，即X----->Output的计算，其中的Quantizer可以理解为一个数字转换器，将实际输出压缩到-1到1之间。
那我们来看看，他与单层感知器模型的差别，比较图1,2，你会发现，他们在结构上非常相似，只是神经元的激活函数不同，由感知器的sign函数变成了purelin函数（f(x)=x）：
图2：单层感知器模型
2.线性神经元的学习规则
有了上面的模型，我们可以看出，他是通过期望输出与实际输出之间的误差进行参数W的调整的，所以介绍一下LMS学习规则如下：
从公式2.26b中可以得出我们权重W的调节程度，那么W=W+△W
# Implementing an adaptive linear neuron in Python
from IPython.display import Image
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from matplotlib.colors import ListedColormap
class AdalineGD(object):
"""ADAptive LInear NEuron classifier.
Parameters
------------
eta : float
Learning rate (between 0.0 and 1.0)
n_iter : int
Passes over the training dataset.
random_state : int
Random number generator seed for random weight
initialization.
Attributes
-----------
w_ : 1d-array
Weights after fitting.
cost_ : list
Sum-of-squares cost function value in each epoch.
"""
def __init__(self, eta=0.01, n_iter=50, random_state=1):
self.eta = eta
self.n_iter = n_iter
self.random_state = random_state
def fit(self, X, y):
""" Fit training data.
Parameters
----------
X : {array-like}, shape = [n_samples, n_features]
Training vectors, where n_samples is the number of samples and
n_features is the number of features.
y : array-like, shape = [n_samples]
Target values.
Returns
-------
self : object
"""
rgen = np.random.RandomState(self.random_state)
self.w_ = rgen.normal(loc=0.0, scale=0.01, size=1 + X.shape[1])
self.cost_ = []
for i in range(self.n_iter):
net_input = self.net_input(X)
# Please note that the "activation" method has no effect
# in the code since it is simply an identity function. We
# could write `output = self.net_input(X)` directly instead.
# The purpose of the activation is more conceptual, i.e.,
# in the case of logistic regression (as we will see later),
# we could change it to
# a sigmoid function to implement a logistic regression classifier.
output = self.activation(net_input)
errors = (y - output)
self.w_[1:] += self.eta * X.T.dot(errors)
self.w_[0] += self.eta * errors.sum()
cost = (errors**2).sum() / 2.0
self.cost_.append(cost)
return self
def net_input(self, X):
"""Calculate net input"""
return np.dot(X, self.w_[1:]) + self.w_[0]
def activation(self, X):
"""Compute linear activation"""
return X
def predict(self, X):
"""Return class label after unit step"""
return np.where(self.activation(self.net_input(X)) >= 0.0, 1, -1)
def plot_decision_regions(X, y, classifier, resolution=0.02):
# setup marker generator and color map
markers = ('s', 'x', 'o', '^', 'v')
colors = ('red', 'blue', 'lightgreen', 'gray', 'cyan')
cmap = ListedColormap(colors[:len(np.unique(y))]) #根据分类结果数确定颜色数量
# plot the decision surface：找出所有数据的最大最小边界值
x1_min, x1_max = X[:, 0].min() - 1, X[:, 0].max() + 1
x2_min, x2_max = X[:, 1].min() - 1, X[:, 1].max() + 1
# 将整个范围区域的面划分为网格，每个网格代表一个点
xx1, xx2 = np.meshgrid(np.arange(x1_min, x1_max, resolution),
np.arange(x2_min, x2_max, resolution))
# 判断所有点的预测结果，并根据结果画颜色，这样就把分界线找出了
# .T表示矩阵转置
Z = classifier.predict(np.array([xx1.ravel(), xx2.ravel()]).T)
Z = Z.reshape(xx1.shape)
plt.contourf(xx1, xx2, Z, alpha=0.3, cmap=cmap)
plt.xlim(xx1.min(), xx1.max())
plt.ylim(xx2.min(), xx2.max())
# plot class samples
for idx, cl in enumerate(np.unique(y)):
plt.scatter(x=X[y == cl, 0],
y=X[y == cl, 1],
alpha=0.8,
c=colors[idx],
marker=markers[idx],
label=cl,
edgecolor='black')
df = pd.read_csv('/Users/wenhanmac/Documents/Aartificial Intelligence/yunbaidu/机器学习/iris.data', header=None)
df.tail()
# select setosa and versicolor
y = df.iloc[0:100, 4].values #df.iloc[0:100, 4] 从dataframe中选择0-100行的第4列，就是花的类型值
y = np.where(y == 'Iris-setosa', -1, 1)
# extract sepal length and petal length
X = df.iloc[0:100, [0, 2]].values #第0列和第2列对应花的两个特征属性
fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(10, 4))
ada1 = AdalineGD(n_iter=10, eta=0.01).fit(X, y)
ax[0].plot(range(1, len(ada1.cost_) + 1), np.log10(ada1.cost_), marker='o')
ax[0].set_xlabel('Epochs')
ax[0].set_ylabel('log(Sum-squared-error)')
ax[0].set_title('Adaline - Learning rate 0.01')
ada2 = AdalineGD(n_iter=10, eta=0.0001).fit(X, y)
ax[1].plot(range(1, len(ada2.cost_) + 1), ada2.cost_, marker='o')
ax[1].set_xlabel('Epochs')
ax[1].set_ylabel('Sum-squared-error')
ax[1].set_title('Adaline - Learning rate 0.0001')
# plt.savefig('images/02_11.png', dpi=300)
plt.show()
# Improving gradient descent through feature scaling
# standardize features
X_std = np.copy(X)
X_std[:, 0] = (X[:, 0] - X[:, 0].mean()) / X[:, 0].std()
X_std[:, 1] = (X[:, 1] - X[:, 1].mean()) / X[:, 1].std()
ada = AdalineGD(n_iter=15, eta=0.01)
ada.fit(X_std, y)
plot_decision_regions(X_std, y, classifier=ada)
plt.title('Adaline - Gradient Descent')
plt.xlabel('sepal length [standardized]')
plt.ylabel('petal length [standardized]')
plt.legend(loc='upper left')
plt.tight_layout()
# plt.savefig('images/02_14_1.png', dpi=300)
plt.show()
plt.plot(range(1, len(ada.cost_) + 1), ada.cost_, marker='o')
plt.xlabel('Epochs')
plt.ylabel('Sum-squared-error')
plt.tight_layout()
# plt.savefig('images/02_14_2.png', dpi=300)
plt.show()
# ## Large scale machine learning and stochastic gradient descent
class AdalineSGD(object):
"""ADAptive LInear NEuron classifier.
Parameters
------------
eta : float
Learning rate (between 0.0 and 1.0)
n_iter : int
Passes over the training dataset.
shuffle : bool (default: True)
Shuffles training data every epoch if True to prevent cycles.
random_state : int
Random number generator seed for random weight
initialization.
Attributes
-----------
w_ : 1d-array
Weights after fitting.
cost_ : list
Sum-of-squares cost function value averaged over all
training samples in each epoch.
"""
def __init__(self, eta=0.01, n_iter=10, shuffle=True, random_state=None):
self.eta = eta
self.n_iter = n_iter
self.w_initialized = False
self.shuffle = shuffle
self.random_state = random_state
def fit(self, X, y):
""" Fit training data.
Parameters
----------
X : {array-like}, shape = [n_samples, n_features]
Training vectors, where n_samples is the number of samples and
n_features is the number of features.
y : array-like, shape = [n_samples]
Target values.
Returns
-------
self : object
"""
self._initialize_weights(X.shape[1])
self.cost_ = []
for i in range(self.n_iter):
if self.shuffle:
X, y = self._shuffle(X, y)
cost = []
for xi, target in zip(X, y):
cost.append(self._update_weights(xi, target))
avg_cost = sum(cost) / len(y)
self.cost_.append(avg_cost)
return self
def partial_fit(self, X, y):
"""Fit training data without reinitializing the weights"""
if not self.w_initialized:
self._initialize_weights(X.shape[1])
if y.ravel().shape[0] > 1:
for xi, target in zip(X, y):
self._update_weights(xi, target)
else:
self._update_weights(X, y)
return self
def _shuffle(self, X, y):
"""Shuffle training data"""
r = self.rgen.permutation(len(y))
return X[r], y[r]
def _initialize_weights(self, m):
"""Initialize weights to small random numbers"""
self.rgen = np.random.RandomState(self.random_state)
self.w_ = self.rgen.normal(loc=0.0, scale=0.01, size=1 + m)
self.w_initialized = True
def _update_weights(self, xi, target):
"""Apply Adaline learning rule to update the weights"""
output = self.activation(self.net_input(xi))
error = (target - output)
self.w_[1:] += self.eta * xi.dot(error)
self.w_[0] += self.eta * error
cost = 0.5 * error**2
return cost
def net_input(self, X):
"""Calculate net input"""
return np.dot(X, self.w_[1:]) + self.w_[0]
def activation(self, X):
"""Compute linear activation"""
return X
def predict(self, X):
"""Return class label after unit step"""
return np.where(self.activation(self.net_input(X)) >= 0.0, 1, -1)
ada = AdalineSGD(n_iter=15, eta=0.01, random_state=1)
ada.fit(X_std, y)
plot_decision_regions(X_std, y, classifier=ada)
plt.title('Adaline - Stochastic Gradient Descent')
plt.xlabel('sepal length [standardized]')
plt.ylabel('petal length [standardized]')
plt.legend(loc='upper left')
plt.tight_layout()
# plt.savefig('images/02_15_1.png', dpi=300)
plt.show()
plt.plot(range(1, len(ada.cost_) + 1), ada.cost_, marker='o')
plt.xlabel('Epochs')
plt.ylabel('Average Cost')
plt.tight_layout()
# plt.savefig('images/02_15_2.png', dpi=300)
plt.show()
ada.partial_fit(X_std[0, :], y[0])
data里的数据
5.1,3.5,1.4,0.2,Iris-setosa
4.9,3.0,1.4,0.2,Iris-setosa
4.7,3.2,1.3,0.2,Iris-setosa
4.6,3.1,1.5,0.2,Iris-setosa
5.0,3.6,1.4,0.2,Iris-setosa
5.4,3.9,1.7,0.4,Iris-setosa
4.6,3.4,1.4,0.3,Iris-setosa
5.0,3.4,1.5,0.2,Iris-setosa
4.4,2.9,1.4,0.2,Iris-setosa
4.9,3.1,1.5,0.1,Iris-setosa
5.4,3.7,1.5,0.2,Iris-setosa
4.8,3.4,1.6,0.2,Iris-setosa
4.8,3.0,1.4,0.1,Iris-setosa
4.3,3.0,1.1,0.1,Iris-setosa
5.8,4.0,1.2,0.2,Iris-setosa
5.7,4.4,1.5,0.4,Iris-setosa
5.4,3.9,1.3,0.4,Iris-setosa
5.1,3.5,1.4,0.3,Iris-setosa
5.7,3.8,1.7,0.3,Iris-setosa
5.1,3.8,1.5,0.3,Iris-setosa
5.4,3.4,1.7,0.2,Iris-setosa
5.1,3.7,1.5,0.4,Iris-setosa
4.6,3.6,1.0,0.2,Iris-setosa
5.1,3.3,1.7,0.5,Iris-setosa
4.8,3.4,1.9,0.2,Iris-setosa
5.0,3.0,1.6,0.2,Iris-setosa
5.0,3.4,1.6,0.4,Iris-setosa
5.2,3.5,1.5,0.2,Iris-setosa
5.2,3.4,1.4,0.2,Iris-setosa
4.7,3.2,1.6,0.2,Iris-setosa
4.8,3.1,1.6,0.2,Iris-setosa
5.4,3.4,1.5,0.4,Iris-setosa
5.2,4.1,1.5,0.1,Iris-setosa
5.5,4.2,1.4,0.2,Iris-setosa
4.9,3.1,1.5,0.1,Iris-setosa
5.0,3.2,1.2,0.2,Iris-setosa
5.5,3.5,1.3,0.2,Iris-setosa
4.9,3.1,1.5,0.1,Iris-setosa
4.4,3.0,1.3,0.2,Iris-setosa
5.1,3.4,1.5,0.2,Iris-setosa
5.0,3.5,1.3,0.3,Iris-setosa
4.5,2.3,1.3,0.3,Iris-setosa
4.4,3.2,1.3,0.2,Iris-setosa
5.0,3.5,1.6,0.6,Iris-setosa
5.1,3.8,1.9,0.4,Iris-setosa
4.8,3.0,1.4,0.3,Iris-setosa
5.1,3.8,1.6,0.2,Iris-setosa
4.6,3.2,1.4,0.2,Iris-setosa
5.3,3.7,1.5,0.2,Iris-setosa
5.0,3.3,1.4,0.2,Iris-setosa
7.0,3.2,4.7,1.4,Iris-versicolor
6.4,3.2,4.5,1.5,Iris-versicolor
6.9,3.1,4.9,1.5,Iris-versicolor
5.5,2.3,4.0,1.3,Iris-versicolor
6.5,2.8,4.6,1.5,Iris-versicolor
5.7,2.8,4.5,1.3,Iris-versicolor
6.3,3.3,4.7,1.6,Iris-versicolor
4.9,2.4,3.3,1.0,Iris-versicolor
6.6,2.9,4.6,1.3,Iris-versicolor
5.2,2.7,3.9,1.4,Iris-versicolor
5.0,2.0,3.5,1.0,Iris-versicolor
5.9,3.0,4.2,1.5,Iris-versicolor
6.0,2.2,4.0,1.0,Iris-versicolor
6.1,2.9,4.7,1.4,Iris-versicolor
5.6,2.9,3.6,1.3,Iris-versicolor
6.7,3.1,4.4,1.4,Iris-versicolor
5.6,3.0,4.5,1.5,Iris-versicolor
5.8,2.7,4.1,1.0,Iris-versicolor
6.2,2.2,4.5,1.5,Iris-versicolor
5.6,2.5,3.9,1.1,Iris-versicolor
5.9,3.2,4.8,1.8,Iris-versicolor
6.1,2.8,4.0,1.3,Iris-versicolor
6.3,2.5,4.9,1.5,Iris-versicolor
6.1,2.8,4.7,1.2,Iris-versicolor
6.4,2.9,4.3,1.3,Iris-versicolor
6.6,3.0,4.4,1.4,Iris-versicolor
6.8,2.8,4.8,1.4,Iris-versicolor
6.7,3.0,5.0,1.7,Iris-versicolor
6.0,2.9,4.5,1.5,Iris-versicolor
5.7,2.6,3.5,1.0,Iris-versicolor
5.5,2.4,3.8,1.1,Iris-versicolor
5.5,2.4,3.7,1.0,Iris-versicolor
5.8,2.7,3.9,1.2,Iris-versicolor
6.0,2.7,5.1,1.6,Iris-versicolor
5.4,3.0,4.5,1.5,Iris-versicolor
6.0,3.4,4.5,1.6,Iris-versicolor
6.7,3.1,4.7,1.5,Iris-versicolor
6.3,2.3,4.4,1.3,Iris-versicolor
5.6,3.0,4.1,1.3,Iris-versicolor
5.5,2.5,4.0,1.3,Iris-versicolor
5.5,2.6,4.4,1.2,Iris-versicolor
6.1,3.0,4.6,1.4,Iris-versicolor
5.8,2.6,4.0,1.2,Iris-versicolor
5.0,2.3,3.3,1.0,Iris-versicolor
5.6,2.7,4.2,1.3,Iris-versicolor
5.7,3.0,4.2,1.2,Iris-versicolor
5.7,2.9,4.2,1.3,Iris-versicolor
6.2,2.9,4.3,1.3,Iris-versicolor
5.1,2.5,3.0,1.1,Iris-versicolor
5.7,2.8,4.1,1.3,Iris-versicolor
6.3,3.3,6.0,2.5,Iris-virginica
5.8,2.7,5.1,1.9,Iris-virginica
7.1,3.0,5.9,2.1,Iris-virginica
6.3,2.9,5.6,1.8,Iris-virginica
6.5,3.0,5.8,2.2,Iris-virginica
7.6,3.0,6.6,2.1,Iris-virginica
4.9,2.5,4.5,1.7,Iris-virginica
7.3,2.9,6.3,1.8,Iris-virginica
6.7,2.5,5.8,1.8,Iris-virginica
7.2,3.6,6.1,2.5,Iris-virginica
6.5,3.2,5.1,2.0,Iris-virginica
6.4,2.7,5.3,1.9,Iris-virginica
6.8,3.0,5.5,2.1,Iris-virginica
5.7,2.5,5.0,2.0,Iris-virginica
5.8,2.8,5.1,2.4,Iris-virginica
6.4,3.2,5.3,2.3,Iris-virginica
6.5,3.0,5.5,1.8,Iris-virginica
7.7,3.8,6.7,2.2,Iris-virginica
7.7,2.6,6.9,2.3,Iris-virginica
6.0,2.2,5.0,1.5,Iris-virginica
6.9,3.2,5.7,2.3,Iris-virginica
5.6,2.8,4.9,2.0,Iris-virginica
7.7,2.8,6.7,2.0,Iris-virginica
6.3,2.7,4.9,1.8,Iris-virginica
6.7,3.3,5.7,2.1,Iris-virginica
7.2,3.2,6.0,1.8,Iris-virginica
6.2,2.8,4.8,1.8,Iris-virginica
6.1,3.0,4.9,1.8,Iris-virginica
6.4,2.8,5.6,2.1,Iris-virginica
7.2,3.0,5.8,1.6,Iris-virginica
7.4,2.8,6.1,1.9,Iris-virginica
7.9,3.8,6.4,2.0,Iris-virginica
6.4,2.8,5.6,2.2,Iris-virginica
6.3,2.8,5.1,1.5,Iris-virginica
6.1,2.6,5.6,1.4,Iris-virginica
7.7,3.0,6.1,2.3,Iris-virginica
6.3,3.4,5.6,2.4,Iris-virginica
6.4,3.1,5.5,1.8,Iris-virginica
6.0,3.0,4.8,1.8,Iris-virginica
6.9,3.1,5.4,2.1,Iris-virginica
6.7,3.1,5.6,2.4,Iris-virginica
6.9,3.1,5.1,2.3,Iris-virginica
5.8,2.7,5.1,1.9,Iris-virginica
6.8,3.2,5.9,2.3,Iris-virginica
6.7,3.3,5.7,2.5,Iris-virginica
6.7,3.0,5.2,2.3,Iris-virginica
6.3,2.5,5.0,1.9,Iris-virginica
6.5,3.0,5.2,2.0,Iris-virginica
6.2,3.4,5.4,2.3,Iris-virginica
5.9,3.0,5.1,1.8,Iris-virginica