Linear Model

机器学习

Linear Model_数据

\(x\)为学习时间，\(y\)为学习该时间能够在考试中取得的分数

在这里来为这些数据寻求一个最好的模型

线性回归

Linear Model：\(\hat{y}=x*w\)

训练损失 (误差)

MSE（Mean Squared Mean）均方误差：

\[loss=(\hat{y}-y)^2=(x*w-y)^2 \]

\(x\ (Hours)\) \(y \ (Points)\) \(y\)_\(predict\ (w=3)\) \(Loss\ (w=3)\)
1 2 3 1
2 4 6 4
3 6 9 9
\(mean=14/3\)
\(x\ (Hours)\) \(y \ (Points)\) \(y\)_\(predict\ (w=3)\) \(Loss\ (w=3)\)
1 2 4 4
2 4 8 16
3 6 12 36
\(mean=56/3\)

\[loss=(\hat{y}-y)^2=(x*w-y)^2\to cost=\frac1N\sum^N_{n=1}(\hat{y_n}-y_n)^2 \]

\(x\ (Hours)\) \(Loss\ (w=0)\) \(Loss\ (w=1)\) \(Loss\ (w=2)\) \(Loss\ (w=3)\) \(Loss\ (w=4)\)
1 4 1 0 1 4
2 16 4 0 4 16
3 36 9 0 9 36
\(MSE\) 18.7 4.7 0 4.7 18.7 
import numpy as np
import matplotlib.pyplot as plt

x_data = [1.0, 2.0, 3.0]
y_data = [2.0, 4.0, 6.0]


def forward(x):
    return x*w  # 定义线性模型：y=x*w


def loss(x, y):
    y_pred = forward(x)
    return (y_pred-y)**2  # 定义损失函数(y_pred-y)^2


w_list = []
mse_list = []
for w in np.arange(0.0, 4.1, 0.1): # w从0.0取到4.1间隔0.1
    print('w=', w)
    l_sum = 0 # 重置总损失为0
    for x_val, y_val in zip(x_data, y_data): # 成对提取x,y
        y_pred_val = forward(x_val) # 获得y_pred
        loss_val = loss(x_val, y_val) # 计算损失函数Loss
        l_sum += loss_val # 将损失添加到总损失中
        print('\t', x_val, y_val, y_pred_val, loss_val)
    print('MSE',l_sum/3) # 计算均方误差
    w_list.append(w) # 将w添加到列表中
    mse_list.append(l_sum/3) # 将均方误差添加到列表中

w= 1.9000000000000001
	 1.0 2.0 1.9000000000000001 0.009999999999999974
	 2.0 4.0 3.8000000000000003 0.0399999999999999
	 3.0 6.0 5.7 0.0899999999999999
MSE 0.046666666666666586
w= 2.0
	 1.0 2.0 2.0 0.0
	 2.0 4.0 4.0 0.0
	 3.0 6.0 6.0 0.0
MSE 0.0
w= 2.1
	 1.0 2.0 2.1 0.010000000000000018
	 2.0 4.0 4.2 0.04000000000000007
	 3.0 6.0 6.300000000000001 0.09000000000000043
MSE 0.046666666666666835
w= 2.2
	 1.0 2.0 2.2 0.04000000000000007
	 2.0 4.0 4.4 0.16000000000000028
	 3.0 6.0 6.6000000000000005 0.36000000000000065
MSE 0.18666666666666698
w= 2.3000000000000003
	 1.0 2.0 2.3000000000000003 0.09000000000000016
	 2.0 4.0 4.6000000000000005 0.36000000000000065
	 3.0 6.0 6.9 0.8100000000000006
MSE 0.42000000000000054
w= 2.4000000000000004
	 1.0 2.0 2.4000000000000004 0.16000000000000028
	 2.0 4.0 4.800000000000001 0.6400000000000011
	 3.0 6.0 7.200000000000001 1.4400000000000026
MSE 0.7466666666666679
w= 2.5
	 1.0 2.0 2.5 0.25
	 2.0 4.0 5.0 1.0
	 3.0 6.0 7.5 2.25
MSE 1.1666666666666667
w= 2.6
	 1.0 2.0 2.6 0.3600000000000001
	 2.0 4.0 5.2 1.4400000000000004
	 3.0 6.0 7.800000000000001 3.2400000000000024
MSE 1.6800000000000008
w= 2.7
	 1.0 2.0 2.7 0.49000000000000027
	 2.0 4.0 5.4 1.960000000000001
	 3.0 6.0 8.100000000000001 4.410000000000006
MSE 2.2866666666666693
w= 2.8000000000000003
	 1.0 2.0 2.8000000000000003 0.6400000000000005
	 2.0 4.0 5.6000000000000005 2.560000000000002
	 3.0 6.0 8.4 5.760000000000002
MSE 2.986666666666668
w= 2.9000000000000004
	 1.0 2.0 2.9000000000000004 0.8100000000000006
	 2.0 4.0 5.800000000000001 3.2400000000000024
	 3.0 6.0 8.700000000000001 7.290000000000005
MSE 3.780000000000003
w= 3.0
	 1.0 2.0 3.0 1.0
	 2.0 4.0 6.0 4.0
	 3.0 6.0 9.0 9.0
MSE 4.666666666666667
w= 3.1
	 1.0 2.0 3.1 1.2100000000000002
	 2.0 4.0 6.2 4.840000000000001
	 3.0 6.0 9.3 10.890000000000004
MSE 5.646666666666668
w= 3.2
	 1.0 2.0 3.2 1.4400000000000004
	 2.0 4.0 6.4 5.760000000000002
	 3.0 6.0 9.600000000000001 12.96000000000001
MSE 6.720000000000003
w= 3.3000000000000003
	 1.0 2.0 3.3000000000000003 1.6900000000000006
	 2.0 4.0 6.6000000000000005 6.7600000000000025
	 3.0 6.0 9.9 15.210000000000003
MSE 7.886666666666668
w= 3.4000000000000004
	 1.0 2.0 3.4000000000000004 1.960000000000001
	 2.0 4.0 6.800000000000001 7.840000000000004
	 3.0 6.0 10.200000000000001 17.640000000000008
MSE 9.14666666666667
w= 3.5
	 1.0 2.0 3.5 2.25
	 2.0 4.0 7.0 9.0
	 3.0 6.0 10.5 20.25
MSE 10.5

plt.plot(w_list,mse_list)
plt.ylabel('Loss')
plt.xlabel('w')
plt.show()

Linear Model_机器学习_02

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