1.sigmoid函数

        公式： 

         自变量取值为任意实数，值域[0,1]

        解释：将任意的输入映射到了[0,1]区间 ，我们在线性回归中可以得到一个预测值，再将该值映射到Sigmoid 函数中这样就完成了由值到概率的转换，也就是分类任务

        预测函数：

         其中，

         分类任务：

         整合：

        解释：对于二分类任务（0，1），整合后y取0只保留 ，y取1只保留

         似然函数：

         对数似然：

         此时应用梯度上升求最大值，引入

         转换为梯度下降任务

        求导过程：

参数更新： 

多分类的softmax:  

 2.代码实现

       

import numpy as np
from utils.features import prepare_for_training
from scipy.optimize import minimize
from utils.hypothesis import sigmoid

class LogisticRegression:
    def __init__(self, data, labels, polynomial_degree=0, sinusoid_degree=0, normalize_data=False):

        (data_processed, features_mean,
         features_deviation) = prepare_for_training(data, polynomial_degree, sinusoid_degree,
                                                    normalize_data)
        self.data = data_processed
        self.labels = labels
        self.unique_labels = np.unique(labels)
        self.features_mean = features_mean
        self.features_deviation = features_deviation
        self.polynomial_degree = polynomial_degree
        self.sinusoid_degree = sinusoid_degree
        self.normalize_data = normalize_data
        # 数据预处理
        num_unique_labels = len(np.unique(labels))
        num_features = self.data.shape[1]
        self.theta = np.zeros((num_unique_labels, num_features))

    def train(self, n_iterations=500):
        cost_histories = []
        num_features = self.data.shape[1]
        for label_index, unique_label in enumerate(self.unique_labels):
            current_initial_theta = np.copy(self.theta[label_index].reshape(num_features, 1))
            current_lables = (self.labels == unique_label).astype(float)
            (theta, cost_history) = LogisticRegression.gradient_descent(self.data, current_initial_theta,
                                                                          current_lables, n_iterations)
            self.theta[label_index]=theta.T
            cost_histories.append(cost_history)
        return self.theta,cost_histories

    @staticmethod
    def gradient_descent(data, current_initial_theta, current_lables, n_iterations):
        cost_history = []
        num_fratures = data.shape[1]
        result = minimize(
            # 优化的目标
            lambda x: LogisticRegression.cost_function(data, current_lables, x.reshape(num_fratures,1)),
            # 初始化的权重参数
            current_initial_theta,
            # 选择优化策略
            method='CG',
            # 梯度下降迭代计算公式
            jac=lambda x: LogisticRegression.gradient_step(data, current_lables, x.reshape(num_fratures,1)),
            callback=lambda x: cost_history.append(
                LogisticRegression.cost_function(data, current_lables, x.reshape(num_fratures,1))),
            options={
                "maxiter": n_iterations
            }
        )

        if not result.success:
            raise ArithmeticError('Can not minimize cost function' + result.message)
        theta = result.x.reshape(num_fratures,1)
        return theta,cost_history

    @staticmethod
    def cost_function(data, label, theta):
        num_examples = data.shape[0]
        prediction = LogisticRegression.hypothesis(data, theta)
        y_true_cost = np.dot(label[label == 1].T, np.log(prediction[label == 1]))
        y_false_cost = np.dot(1 - label[label == 0].T, np.log(1 - prediction[label == 0]))
        cost = (-1 / num_examples) * (y_false_cost + y_true_cost)
        return cost

    @staticmethod
    def hypothesis(data, theta):
        predictions = sigmoid(np.dot(data, theta))
        return predictions

    @staticmethod
    def gradient_step(data, label, theta):
        num_examples = data.shape[0]
        prediction = LogisticRegression.hypothesis(data, theta)
        label_diff = prediction - label
        gradients = (1 / num_examples) * np.dot(data.T, label_diff)
        return gradients.T.flatten()

    def predict(self,data):
        num_examples = data.shape[0]
        data_processed = prepare_for_training(data, self.polynomial_degree, self.sinusoid_degree,
                                              self.normalize_data)[0]
        prediction = LogisticRegression.hypothesis(data_processed, self.theta.T)
        arg = np.argmax(prediction,axis=1)
        class_prediction = np.empty(arg.shape,dtype=object)
        for index,unique_label in enumerate(self.unique_labels):
            class_prediction[arg == index] = unique_label
        return class_prediction.reshape((num_examples,1))

3.测试代码 

1.数据展示

        如图所示，我们使用了一个环形的数据分布进行测试，要想进行比较准确的分类，决策边界需要一个圆形

 2.训练数据

        我们使用类中的train函数进行训练，代码及损失变化情况如下：

 

 

训练结果分类精度达到88.9831%，结果相当不错！！！ 

 让我们绘制一下决策边界把！