😋如何搭建感知器并使用其进行分类

[toc]

💨python 编程基础，需要掌握几个库

• numpy
• pandas
• matplotlib

💖什么是感知器？

🎈神经元与机器学习感知机

信号在神经元中的传播过程如下：信号通过神经粒质与神经元树突结合，通过神经细胞的认知处理（分类等），由神经末梢输出信号。

感知器类似于生物神经系统中的神经元。特征数据$x_1,x_2,...x_m$，作为输入信号，权重因子$\omega_1,\omega_2,...\omega_m$,作信号接收单元，激活函数用于根据$\omega x$输入信号产生输出分类结果。其中，$x_0=1,\omega_0=y_0$

感知机的基本数学原理，既是逻辑回归/线性回归，直白一点就是大学物理实验中的实验数据处理，数据拟合和类似最小二乘法的误差分析；

为了使拟合或分类效果最好，需要求一组$\omega$,要求方差最小即下式最小：$\widehat y$为预测数据，${(i)}$为第i个样本，下标为特征

$$
\begin{equation}
J(\boldsymbol{w})=\frac{1}{2} \sum_{i}\left(y^{(i)}-\widehat y^{(i)}\right)^{2}
\end{equation}
$$

线性回归或逻辑回归得目的是根据已有数据发现规律或现象，建立相应得模型，最终目的是利用模型进行预测；机器学习算法就是一种事物规律建模技术。训练机器学习算法，并用其对新的数据进行预测或分类。

机器学习算法的核心既是获得较合理的$\omega$, 要想获得$\omega$就需要利用数据集对算法进行训练。样本数据输入，预测值$\widehat y$与实际值之间会有偏差。要求训练样本足够多的情况下，偏差变小，逐渐收敛。那么，每个训练样本进行训练时，均需对权重因子进行更新。

$$
w_{j}:=w_{j}+\Delta w_{j}\
$$

🚲最简单的感知机

权重更新方法如下
$$
\begin{equation}\Delta w_{j}=\eta\left(y^{(i)}-\hat{y}^{(i)}\right) x_{j}^{(i)}\end{equation}
$$
其中，$\eta$成为学习率

🛴自适应线性神经元与学习的收敛性

其采用线性激活函数$\phi(z)=z$，那么；

权重更新采用梯度下降法：

🍜如何搭建感知机并进行分类？

# coding: utf-8
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from matplotlib.colors import ListedColormap

# # Implementing a perceptron learning algorithm in Python
# ## An object-oriented perceptron API
class Perceptron(object):
    """Perceptron 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.
    errors_ : list
      Number of misclassifications (updates) 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.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)



# ## Training a perceptron model on the Iris dataset

# ...

# ### Reading-in the Iris data


df = pd.read_csv('https://archive.ics.uci.edu/ml/'
        'machine-learning-databases/iris/iris.data', header=None)
df.tail()


# 
# ### Note:
# 
# 
# You can find a copy of the Iris dataset (and all other datasets used in this book) in the code bundle of this book, which you can use if you are working offline or the UCI server at https://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data is temporarily unavailable. For instance, to load the Iris dataset from a local directory, you can replace the line 
# 
#     df = pd.read_csv('https://archive.ics.uci.edu/ml/'
#         'machine-learning-databases/iris/iris.data', header=None)
#  
# by
#  
#     df = pd.read_csv('your/local/path/to/iris.data', header=None)
# 



df = pd.read_csv('iris.data', header=None)
df.tail()


# ### Plotting the Iris data


# select setosa and versicolor
y = df.iloc[0:100, 4].values
y = np.where(y == 'Iris-setosa', -1, 1)

# extract sepal length and petal length
X = df.iloc[0:100, [0, 2]].values

# plot data
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 [cm]')
plt.ylabel('petal length [cm]')
plt.legend(loc='upper left')

# plt.savefig('images/02_06.png', dpi=300)
plt.show()



# ### Training the perceptron model
ppn = Perceptron(eta=0.1, n_iter=10)
ppn.fit(X, y)
plt.plot(range(1, len(ppn.errors_) + 1), ppn.errors_, marker='o')
plt.xlabel('Epochs')
plt.ylabel('Number of updates')

# plt.savefig('images/02_07.png', dpi=300)
plt.show()

# ### A function for plotting decision regions
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)
    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')




plot_decision_regions(X, y, classifier=ppn)
plt.xlabel('sepal length [cm]')
plt.ylabel('petal length [cm]')
plt.legend(loc='upper left')


# plt.savefig('images/02_08.png', dpi=300)
plt.show()


#################################################################
## Adaptive linear neurons and the convergence of learning
# ...
# ## Minimizing cost functions with gradient descent
# ## Implementing an adaptive linear neuron in Python

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)




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])

# # Summary

# ...

# --- 
# 
# Readers may ignore the following cell