這篇文章通過對花鳶尾屬植物進行分類，來學習如何利用實際數據構建一個感知機模型，（文末附GD和SGD參數更新手推公式）。
目錄
一、數據集
Iris數據集是常用的分類實驗數據集，由Fisher, 1936收集整理。Iris也稱鳶尾花卉數據集，是一類多重變量分析的數據集。數據集包含150個數據樣本，分爲3類，每類50個數據，每個數據包含4個屬性。可通過花萼長度，花萼寬度，花瓣長度，花瓣寬度4個屬性預測鳶尾花卉屬於（Setosa，Versicolour，Virginica）三個種類中的哪一類。
iris以鳶尾花的特徵作爲數據來源，常用在分類操作中。該數據集由3種不同類型的鳶尾花的各50個樣本數據構成。其中的一個種類與另外兩個種類是線性可分離的，後兩個種類是非線性可分離的。
該數據集包含了4個屬性：
& Sepal.Length（花萼長度），單位是cm;
& Sepal.Width（花萼寬度），單位是cm;
& Petal.Length（花瓣長度），單位是cm;
& Petal.Width（花瓣寬度），單位是cm;
種類：Iris Setosa（山鳶尾）、Iris Versicolour（雜色鳶尾），以及Iris Virginica（維吉尼亞鳶尾）。
二、需要導入的庫
import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
from matplotlib.colors import ListedColormap
#from sklearn.linear_model import Perceptron
import Perceptron as perceptron_class
三、讀取數據集
df = pd.read_csv('https://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data', header=None)
df.tail()
#抽取出前100條樣本，這正好是Setosa和Versicolor對應的樣本，我們將Versicolor對應的數據作爲類別1，Setosa對應的作爲-1。
# 對於特徵，我們抽取出sepal length和petal length兩維度特徵，然後用散點圖對數據進行可視化
#We extract the first 100 class labels that correspond to 50 Iris-Setosa and 50 Iris-Versicolor flowers.
y = df.iloc[0:100, 4].values
print(y)
y = np.where(y == 'Iris-setosa', -1, 1)#滿足條件(condition)，輸出x，不滿足輸出y。
print(y)
X = df.iloc[0:100, [0, 2]].values
print(X)
print(X.shape,y.shape)#(100,2) (100,)
結果：
['Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa'
'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa'
'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa'
'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa'
'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa'
'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa'
'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa'
'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa'
'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa'
'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa' 'Iris-setosa'
'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor'
'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor'
'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor'
'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor'
'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor'
'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor'
'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor'
'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor'
'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor'
'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor'
'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor'
'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor' 'Iris-versicolor'
'Iris-versicolor' 'Iris-versicolor']
[-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
-1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1]
[[5.1 1.4]
[4.9 1.4]
[4.7 1.3]
[4.6 1.5]
[5. 1.4]
[5.4 1.7]
[4.6 1.4]
[5. 1.5]
[4.4 1.4]
[4.9 1.5]
[5.4 1.5]
[4.8 1.6]
[4.8 1.4]
[4.3 1.1]
[5.8 1.2]
[5.7 1.5]
[5.4 1.3]
[5.1 1.4]
[5.7 1.7]
[5.1 1.5]
[5.4 1.7]
[5.1 1.5]
[4.6 1. ]
[5.1 1.7]
[4.8 1.9]
[5. 1.6]
[5. 1.6]
[5.2 1.5]
[5.2 1.4]
[4.7 1.6]
[4.8 1.6]
[5.4 1.5]
[5.2 1.5]
[5.5 1.4]
[4.9 1.5]
[5. 1.2]
[5.5 1.3]
[4.9 1.5]
[4.4 1.3]
[5.1 1.5]
[5. 1.3]
[4.5 1.3]
[4.4 1.3]
[5. 1.6]
[5.1 1.9]
[4.8 1.4]
[5.1 1.6]
[4.6 1.4]
[5.3 1.5]
[5. 1.4]
[7. 4.7]
[6.4 4.5]
[6.9 4.9]
[5.5 4. ]
[6.5 4.6]
[5.7 4.5]
[6.3 4.7]
[4.9 3.3]
[6.6 4.6]
[5.2 3.9]
[5. 3.5]
[5.9 4.2]
[6. 4. ]
[6.1 4.7]
[5.6 3.6]
[6.7 4.4]
[5.6 4.5]
[5.8 4.1]
[6.2 4.5]
[5.6 3.9]
[5.9 4.8]
[6.1 4. ]
[6.3 4.9]
[6.1 4.7]
[6.4 4.3]
[6.6 4.4]
[6.8 4.8]
[6.7 5. ]
[6. 4.5]
[5.7 3.5]
[5.5 3.8]
[5.5 3.7]
[5.8 3.9]
[6. 5.1]
[5.4 4.5]
[6. 4.5]
[6.7 4.7]
[6.3 4.4]
[5.6 4.1]
[5.5 4. ]
[5.5 4.4]
[6.1 4.6]
[5.8 4. ]
[5. 3.3]
[5.6 4.2]
[5.7 4.2]
[5.7 4.2]
[6.2 4.3]
[5.1 3. ]
[5.7 4.1]]
(100, 2) (100,)
四、數據散點圖可視化
plt.scatter(X[:50, 0], X[:50, 1],color='red', marker='o', label='setosa')
plt.scatter(X[50:100, 0], X[50:100, 1],color='blue', marker='x', label='versicolor')
plt.xlabel('sepal length')
plt.ylabel('petal length')
plt.legend(loc='upper left')
plt.show()
五、利用感知機模型進行線性分類
#這裏是對於感知機模型進行訓練
ppn = perceptron_class.Perceptron(eta=0.1, n_iter=10)
ppn.fit(X, y) #畫出分界線
#To train our perceptron algorithm, plot the misclassification error
plt.plot(range(1,len(ppn.errors_) + 1), ppn.errors_, marker='o')
plt.xlabel('Epochs')
plt.ylabel('Number of misclassifications')
plt.show()
#Visualize the decision boundaries for 2D datasets
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))])
# plot the decision surface
x1_min, x1_max = X[:, 0].min() - 1, X[:, 0].max() + 1
x2_min, x2_max = X[:, 1].min() - 1, X[:, 1].max() + 1
xx1, xx2 = np.meshgrid(np.arange(x1_min, x1_max, resolution), np.arange(x2_min, x2_max, resolution))
Z = classifier.predict(np.array([xx1.ravel(), xx2.ravel()]).T)
Z = Z.reshape(xx1.shape)
print(xx1)
plt.contourf(xx1, xx2, Z, alpha=0.4, 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)):#除其中重複的元素，並按元素由大到小返回一個新的無元素重複的元組或者列表
plt.scatter(x=X[y == cl, 0], y=X[y == cl, 1],alpha=0.8, c=cmap(idx), marker=markers[idx],label=cl)
plot_decision_regions(X, y, classifier=ppn)
plt.xlabel('sepal length [cm]')
plt.ylabel('petal length [cm]')
plt.legend(loc='upper left')
plt.show()
分類誤差：
分類結果：
六、不同學習率損失可視化對比
#Adaptive linear neurons and the convergence of learning Implementing an Adaptive Linear Neuron in Python
fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(8,4))
ada1 = perceptron_class.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 = perceptron_class.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.show()
七、歸一化後分類
#standardization
X_std = np.copy(X)
X_std[:,0] = (X_std[:,0] - X_std[:,0].mean()) / X_std[:,0].std()
X_std[:,1] = (X_std[:,1] - X_std[:,1].mean()) / X_std[:,1].std()
ada = perceptron_class.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.show()
plt.plot(range(1, len(ada.cost_) + 1), ada.cost_, marker='o')
plt.xlabel('Epochs')
plt.ylabel('Sum-squared-error')
plt.show()
八、隨機梯度下降
#Large scale machine learning and stochastic gradient descent
ada = perceptron_class.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.show()
plt.plot(range(1, len(ada.cost_) + 1), ada.cost_, marker='o')
plt.xlabel('Epochs')
plt.ylabel('Average Cost')
plt.show()
九、完整的代碼
import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
from matplotlib.colors import ListedColormap
#from sklearn.linear_model import Perceptron
import Perceptron as perceptron_class
df = pd.read_csv('https://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data', header=None)
df.tail()
#抽取出前100條樣本，這正好是Setosa和Versicolor對應的樣本，我們將Versicolor對應的數據作爲類別1，Setosa對應的作爲-1。
# 對於特徵，我們抽取出sepal length和petal length兩維度特徵，然後用散點圖對數據進行可視化
#We extract the first 100 class labels that correspond to 50 Iris-Setosa and 50 Iris-Versicolor flowers.
y = df.iloc[0:100, 4].values
print(y)
y = np.where(y == 'Iris-setosa', -1, 1)#滿足條件(condition)，輸出x，不滿足輸出y。
print(y)
X = df.iloc[0:100, [0, 2]].values
print(X)
print(X.shape,y.shape)#(100,2) (100,)
plt.scatter(X[:50, 0], X[:50, 1],color='red', marker='o', label='setosa')
plt.scatter(X[50:100, 0], X[50:100, 1],color='blue', marker='x', label='versicolor')
plt.xlabel('sepal length')
plt.ylabel('petal length')
plt.legend(loc='upper left')
plt.show()
#這裏是對於感知機模型進行訓練
ppn = perceptron_class.Perceptron(eta=0.1, n_iter=10)
ppn.fit(X, y) #畫出分界線
#To train our perceptron algorithm, plot the misclassification error
plt.plot(range(1,len(ppn.errors_) + 1), ppn.errors_, marker='o')
plt.xlabel('Epochs')
plt.ylabel('Number of misclassifications')
plt.show()
#Visualize the decision boundaries for 2D datasets
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))])
# plot the decision surface
x1_min, x1_max = X[:, 0].min() - 1, X[:, 0].max() + 1
x2_min, x2_max = X[:, 1].min() - 1, X[:, 1].max() + 1
xx1, xx2 = np.meshgrid(np.arange(x1_min, x1_max, resolution), np.arange(x2_min, x2_max, resolution))
Z = classifier.predict(np.array([xx1.ravel(), xx2.ravel()]).T)
Z = Z.reshape(xx1.shape)
print(xx1)
plt.contourf(xx1, xx2, Z, alpha=0.4, 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)):#除其中重複的元素，並按元素由大到小返回一個新的無元素重複的元組或者列表
plt.scatter(x=X[y == cl, 0], y=X[y == cl, 1],alpha=0.8, c=cmap(idx), marker=markers[idx],label=cl)
plot_decision_regions(X, y, classifier=ppn)
plt.xlabel('sepal length [cm]')
plt.ylabel('petal length [cm]')
plt.legend(loc='upper left')
plt.show()
#Adaptive linear neurons and the convergence of learning Implementing an Adaptive Linear Neuron in Python
fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(8,4))
ada1 = perceptron_class.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 = perceptron_class.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.show()
#standardization
X_std = np.copy(X)
X_std[:,0] = (X_std[:,0] - X_std[:,0].mean()) / X_std[:,0].std()
X_std[:,1] = (X_std[:,1] - X_std[:,1].mean()) / X_std[:,1].std()
ada = perceptron_class.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.show()
plt.plot(range(1, len(ada.cost_) + 1), ada.cost_, marker='o')
plt.xlabel('Epochs')
plt.ylabel('Sum-squared-error')
plt.show()
#Large scale machine learning and stochastic gradient descent
ada = perceptron_class.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.show()
plt.plot(range(1, len(ada.cost_) + 1), ada.cost_, marker='o')
plt.xlabel('Epochs')
plt.ylabel('Average Cost')
plt.show()
import numpy as np
from numpy.random import seed
class Perceptron(object):
"""Perceptron classifier.
Parameters
------------
eta : float
Learning rate (between 0.0 and 1.0)
n_iter : int
Passes over the training dataset.
Attributes
-----------
w_ : 1d-array
Weights after fitting.
errors_ : list
Number of misclassifications (updates) in each epoch.
"""
def __init__(self, eta=0.01, n_iter=10):
self.eta = eta
self.n_iter = n_iter
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.w_ = np.zeros(1 + X.shape[1])
self.errors_ = []
for _ in range(self.n_iter):
errors = 0
for xi, target in zip(X, y):
update = self.eta*(target - self.predict(xi))
self.w_[1:] += update*xi
self.w_[0] += update
errors += int(update != 0.0)
self.errors_.append(errors)
return self
def net_input(self, X):
"""Calculate net input"""
return np.dot(X, self.w_[1:]) + self.w_[0]
def predict(self, X):
"""Return class label after unit step"""
return np.where(self.net_input(X) >= 0.0, 1, -1)
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.
Attributes
-------------
w_ : 1d-array
Weights after fitting.
errors_ : list
Number of misclassifications in every epoch.
"""
def __init__(self, eta=0.01, n_iter=50):
self.eta = eta
self.n_iter = n_iter
def fit(self, X, y):
""" Fit training data.
Parameters
------------
X : {array-like5}, 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.w_ = np.zeros(1 + X.shape[1])
self.cost_ = []
for i in range(self.n_iter):
output = self.net_input(X)
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 self.net_input(X)
def predict(self, X):
"""Return class label after unit step"""
return np.where(self.activation(X) >= 0.0, 1, -1)
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.
Attributes
------------
w_ : 1d-array
Weights after fitting.
cost_ : list
Number of misclassifications in every epoch.
shuffle : bool (default: True)
Shuffles training data every epoch
if True to prevent cycles.
random_state : int (default: None)
Set random state for shuffling
and initializing the weights.
"""
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
if random_state:
seed(random_state)
def fit(self, X, y):
"""Fit training data.
Parameters
------------
X : {array-like}, shape = [n_samples, n_features]
Training vector, where n_samples
is the number of samples and
n_features is the number of features.
y: arrary-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 = np.random.permutation(len(y))
return X[r], y[r]
def _initialize_weights(self, m):
"""Initialize weighs to zeros"""
self.w_ = np.zeros(1+m)
self.w_initialized = True
def _update_weights(self, xi, target):
"""Apply Adaline learning rule to update the weights"""
output = 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 self.net_input(X)
def predict(self, X):
"""Return class label after unit step"""
return np.where(self.activation(X) >= 0.0, 1, -1)
十、公式推導：
