Machine_Learning（线性回归习题：python版）

我就是我 2023-03-03 08:49 257阅读 0赞

## 第六章习题 ##

1.  ①创建一个ex1.txt文件，内容为eye(5)  
    ②文件重命名为ex1.m，在octave中将当前路径改为文件ex1.m的路径下  
    ③octave中输入run(‘ex1.m’)

2.**单变量线性回归**  
假设函数：hθ(x) = θ0 + θ1x1+…+θnxn  
损失函数：J(θ) = 1/2m∑(1:m)( hθ(x) - y )^2 \#m为样本数量  
梯度下降函数：θ = θ - (X.T@(X@θ - y))\*α/m

①读取文件

import numpy as np
    import pandas as pd
    import matplotlib.pyplot as plt
    data = pd.read_csv('E:\ML_exercise\ex1\ex1data1.txt',names=['population','profit'])#读取文件，添加列名
    data.head()#查看前5行
    data.tail()#查看后5行
    data.describe()
    data.info()
    
    #画散点图
    data.plot.scatter('population','profit',label='population')#绘制散点图
    plt.show()

②构造数据集

data.insert(0,'ones',1)#插入第一列为1
    data.head()
    X = data.iloc[:,0:-1]#切片
    X.head()
    y = data.iloc[:,-1]
    y.head()
    X = X.values#转换为矩阵
    X.shape#查看X的维度
    y = y.values
    y.shape
    y = y.reshape(97,1)
    y.shape

③构造损失函数

def costFunction(X,y,theta):#定义损失函数
        inner = np.power(X @ theta - y,2)#矩阵相乘用@
        return np.sum(inner)/(2*len(X))
        
    theta = np.zeros((2,1))#定义theta为一个2*1的零矩阵
    theta.shape
    cost_init = costFunction(X,y,theta)
    print(cost_init)#打印初始损失函数

④构造梯度下降函数

def gradientDescent(X,y,theta,alpha,iters):#定义梯度下降函数
        costs = []
        for i in range(iters):
            theta = theta - (X.T @ (X @ theta - y))*alpha/len(X)
            cost = costFunction(X,y,theta)#计算损失函数
            costs.append(cost)
            
            if i%100 == 0:
                print(cost)
        return theta,costs
    
    #初始化参数
    alpha = 0.02
    iters = 2000
    theta,costs = gradientDescent(X,y,theta,alpha,iters)

⑤初始化梯度下降函数

fig,ax = plt.subplots()#图形初始化
    ax.plot(np.arange(iters),costs)#曲线绘制cost
    ax.set(xlabel='iters',
          ylabel='cost',
          title='cost vs iters')
    plt.show()

⑥画回归线和预测图

x = np.linspace(y.min(),y.max())
    y_ = theta[0,0] + theta[1,0] * x
    
    fig,ax = plt.subplots()
    ax.scatter(X[:,1],y,label='training data')
    ax.plot(x,y_,'r',label='predict')
    ax.legend()
    ax.set(xlabel='population',
          ylabel='profit')
    plt.show()

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0xJdmUxQW5EMUxlQVJO_size_16_color_FFFFFF_t_70]

3.**多变量线性回归**  
①读取文件

import numpy as np
    import pandas as pd
    import matplotlib.pyplot as plt
    data = pd.read_csv('E:\ML_exercise\ex1\ex1data2.txt',names=['size','bedrooms','price'])
    data.head()

②数据预处理-特征归一化  
公式：Z = （xi - μ）/ δ

def normalize_feature(data):#特征归一化函数
        return (data-data.mean())/data.std()#mean()为均值函数，std()为标准差公式
    
    data = normalize_feature(data)
    data.head()
    
    #画散点图
    data.plot.scatter('size','price',label='size')
    plt.show()
    data.plot.scatter('bedrooms','price',label='bedrooms')
    plt.show()

③添加全为1的列在第一列

data.insert(0,'ones',1)
    data.head()

④构造数据集

X = data.iloc[:,0:-1]
    X.head()
    y = data.iloc[:,-1]
    y.head()

⑤将dataframe转成数组

X = X.values
    X.shape
    y = y.values
    y.shape
    y = y.reshape(47,1)
    y.shape

⑥构造损失函数

def cost_Function(X,y,theta):
        inner = np.power((X @ theta - y),2)
        return np.sum(inner)/(2 * len(X))
    
    #初始化θ
    theta = np.zeros((3,1))
    cost_init = cost_Function(X,y,theta)
    print(cost_init)

⑦构造梯度下降函数

def gradientDescent(X,y,theta,alpha,iters,isprint=False):
        costs = []
        
        for i in range(iters):
            theta = theta - ((X.T @ (X @ theta - y))*alpha)/len(X)
            cost = cost_Function(X,y,theta)
            costs.append(cost)
            
            if i%100 == 0:
                if isprint:
                    print(cost)
        return theta,costs

⑧看不同alpha的效果

#初始化α和迭代次数
    candidate_alpha = [0.0003,0.003,0.03,0.0001,0.001,0.01]
    iters = 2000

⑨初始化图像

fig,ax=plt.subplots()
    
    for alpha in candidate_alpha:
        _,costs = gradientDescent(X,y,theta,alpha,iters)
        ax.plot(np.arange(iters),costs,label=alpha)
        ax.legend()
    
    ax.set(xlabel='iters',
          ylabel='cost',
          title='cost vs iters')
    plt.show()

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0xJdmUxQW5EMUxlQVJO_size_16_color_FFFFFF_t_70 1]  
3.**正规方程**  
①读取文件

import numpy as np
    import pandas as pd
    import matplotlib.pyplot as plt
    data = pd.read_csv('E:\ML_exercise\ex1\ex1data2.txt',names=['size','bedrooms','price'])
    data.head()

②添加全为1的列在第一列

data.insert(0,'ones',1)
    data.head()

③构造数据集

X = data.iloc[:,0:-1]
    X.head()
    y = data.iloc[:,-1]
    y.head()

④将dataframe转成数组

X = X.values
    X.shape
    y = y.values
    y.shape
    y = y.reshape(47,1)
    y.shape

⑤构造正规方程

def normalEquation(X,y):
        theta = np.linalg.inv(X.T @ X)@ X.T @ y
        return theta

⑥打印θ

theta = normalEquation(X,y)
    print(theta)

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L0xJdmUxQW5EMUxlQVJO_size_16_color_FFFFFF_t_70 2]

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