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系列专栏#xff1a;机器学习 数据集 ( x ( i ) , y ( i ) ) , i 1 , 2 , ⋯ , m \left(x^{(i)} , … 文章目录 [toc]数据集实际值估计值估计误差代价函数学习率参数更新Python实现导包数据预处理迭代过程数据可视化完整代码 线性拟合结果代价结果 个人主页丷从心
系列专栏机器学习 数据集 ( x ( i ) , y ( i ) ) , i 1 , 2 , ⋯ , m \left(x^{(i)} , y^{(i)}\right) , i 1 , 2 , \cdots , m (x(i),y(i)),i1,2,⋯,m 实际值 y ( i ) y^{(i)} y(i) 估计值 h θ ( x ( i ) ) θ 0 θ 1 x ( i ) h_{\theta}\left(x^{(i)}\right) \theta_{0} \theta_{1} x^{(i)} hθ(x(i))θ0θ1x(i) 估计误差 h θ ( x ( i ) ) − y ( i ) h_{\theta}\left(x^{(i)}\right) - y^{(i)} hθ(x(i))−y(i) 代价函数 J ( θ ) J ( θ 0 , θ 1 ) 1 2 m ∑ i 1 m ( h θ ( x ( i ) ) − y ( i ) ) 2 1 2 m ∑ i 1 m ( θ 0 θ 1 x ( i ) − y ( i ) ) 2 J(\theta) J(\theta_{0} , \theta_{1}) \cfrac{1}{2m} \displaystyle\sum\limits_{i 1}^{m}{\left(h_{\theta}\left(x^{(i)}\right) - y^{(i)}\right)^{2}} \cfrac{1}{2m} \displaystyle\sum\limits_{i 1}^{m}{\left(\theta_{0} \theta_{1} x^{(i)} - y^{(i)}\right)^{2}} J(θ)J(θ0,θ1)2m1i1∑m(hθ(x(i))−y(i))22m1i1∑m(θ0θ1x(i)−y(i))2 学习率 α \alpha α是学习率一个大于 0 0 0的很小的经验值决定代价函数下降的程度 参数更新 Δ θ j ∂ ∂ θ j J ( θ 0 , θ 1 ) \Delta{\theta_{j}} \cfrac{\partial}{\partial{\theta_{j}}} J(\theta_{0} , \theta_{1}) Δθj∂θj∂J(θ0,θ1) θ j : θ j − α Δ θ j θ j − α ∂ ∂ θ j J ( θ 0 , θ 1 ) \theta_{j} : \theta_{j} - \alpha \Delta{\theta_{j}} \theta_{j} - \alpha \cfrac{\partial}{\partial{\theta_{j}}} J(\theta_{0} , \theta_{1}) θj:θj−αΔθjθj−α∂θj∂J(θ0,θ1) [ θ 0 θ 1 ] : [ θ 0 θ 1 ] − α [ ∂ J ( θ 0 , θ 1 ) ∂ θ 0 ∂ J ( θ 0 , θ 1 ) ∂ θ 1 ] \left[ \begin{matrix} \theta_{0} \\ \theta_{1} \end{matrix} \right] : \left[ \begin{matrix} \theta_{0} \\ \theta_{1} \end{matrix} \right] - \alpha \left[ \begin{matrix} \cfrac{\partial{J(\theta_{0} , \theta_{1})}}{\partial{\theta_{0}}} \\ \cfrac{\partial{J(\theta_{0} , \theta_{1})}}{\partial{\theta_{1}}} \end{matrix} \right] [θ0θ1]:[θ0θ1]−α ∂θ0∂J(θ0,θ1)∂θ1∂J(θ0,θ1) [ ∂ J ( θ 0 , θ 1 ) ∂ θ 0 ∂ J ( θ 0 , θ 1 ) ∂ θ 1 ] [ 1 m ∑ i 1 m ( h θ ( x ( i ) ) − y ( i ) ) 1 m ∑ i 1 m ( h θ ( x ( i ) ) − y ( i ) ) x ( i ) ] [ 1 m ∑ i 1 m e ( i ) 1 m ∑ i 1 m e ( i ) x ( i ) ] e ( i ) h θ ( x ( i ) ) − y ( i ) \left[ \begin{matrix} \cfrac{\partial{J(\theta_{0} , \theta_{1})}}{\partial{\theta_{0}}} \\ \cfrac{\partial{J(\theta_{0} , \theta_{1})}}{\partial{\theta_{1}}} \end{matrix} \right] \left[ \begin{matrix} \cfrac{1}{m} \displaystyle\sum\limits_{i 1}^{m}{\left(h_{\theta}\left(x^{(i)}\right) - y^{(i)}\right)} \\ \cfrac{1}{m} \displaystyle\sum\limits_{i 1}^{m}{\left(h_{\theta}\left(x^{(i)}\right) - y^{(i)}\right) x^{(i)}} \end{matrix} \right] \left[ \begin{matrix} \cfrac{1}{m} \displaystyle\sum\limits_{i 1}^{m}{e^{(i)}} \\ \cfrac{1}{m} \displaystyle\sum\limits_{i 1}^{m}{e^{(i)} x^{(i)}} \end{matrix} \right] \kern{2em} e^{(i)} h_{\theta}\left(x^{(i)}\right) - y^{(i)} ∂θ0∂J(θ0,θ1)∂θ1∂J(θ0,θ1) m1i1∑m(hθ(x(i))−y(i))m1i1∑m(hθ(x(i))−y(i))x(i) m1i1∑me(i)m1i1∑me(i)x(i) e(i)hθ(x(i))−y(i) [ ∂ J ( θ 0 , θ 1 ) ∂ θ 0 ∂ J ( θ 0 , θ 1 ) ∂ θ 1 ] [ 1 m ∑ i 1 m e ( i ) 1 m ∑ i 1 m e ( i ) x ( i ) ] [ 1 m ( e ( 1 ) e ( 2 ) ⋯ e ( m ) ) 1 m ( e ( 1 ) e ( 2 ) ⋯ e ( m ) ) x ( i ) ] 1 m [ 1 1 ⋯ 1 x ( 1 ) x ( 2 ) ⋯ x ( m ) ] [ e ( 1 ) e ( 2 ) ⋮ e ( m ) ] 1 m X T e 1 m X T ( X θ − y ) \begin{aligned} \left[ \begin{matrix} \cfrac{\partial{J(\theta_{0} , \theta_{1})}}{\partial{\theta_{0}}} \\ \cfrac{\partial{J(\theta_{0} , \theta_{1})}}{\partial{\theta_{1}}} \end{matrix} \right] \left[ \begin{matrix} \cfrac{1}{m} \displaystyle\sum\limits_{i 1}^{m}{e^{(i)}} \\ \cfrac{1}{m} \displaystyle\sum\limits_{i 1}^{m}{e^{(i)} x^{(i)}} \end{matrix} \right] \left[ \begin{matrix} \cfrac{1}{m} \left(e^{(1)} e^{(2)} \cdots e^{(m)}\right) \\ \cfrac{1}{m} \left(e^{(1)} e^{(2)} \cdots e^{(m)}\right) x^{(i)} \end{matrix} \right] \\ \cfrac{1}{m} \left[ \begin{matrix} 1 1 \cdots 1 \\ x^{(1)} x^{(2)} \cdots x^{(m)} \end{matrix} \right] \left[ \begin{matrix} e^{(1)} \\ e^{(2)} \\ \vdots \\ e^{(m)} \end{matrix} \right] \cfrac{1}{m} X^{T} e \cfrac{1}{m} X^{T} (X \theta - y) \end{aligned} ∂θ0∂J(θ0,θ1)∂θ1∂J(θ0,θ1) m1i1∑me(i)m1i1∑me(i)x(i) m1(e(1)e(2)⋯e(m))m1(e(1)e(2)⋯e(m))x(i) m1[1x(1)1x(2)⋯⋯1x(m)] e(1)e(2)⋮e(m) m1XTem1XT(Xθ−y)
由上述推导得 Δ θ 1 m X T e \Delta{\theta} \cfrac{1}{m} X^{T} e Δθm1XTe θ : θ − α Δ θ θ − α 1 m X T e \theta : \theta - \alpha \Delta{\theta} \theta - \alpha \cfrac{1}{m} X^{T} e θ:θ−αΔθθ−αm1XTe Python实现
导包
import numpy as np
import matplotlib.pyplot as plt数据预处理
x np.array([4, 3, 3, 4, 2, 2, 0, 1, 2, 5, 1, 2, 5, 1, 3])
y np.array([8, 6, 6, 7, 4, 4, 2, 4, 5, 9, 3, 4, 8, 3, 6])m len(x)x np.c_[np.ones([m, 1]), x]
y y.reshape(m, 1)
theta np.zeros([2, 1])迭代过程
alpha 0.01
iter_cnt 1000 # 迭代次数
cost np.zeros([iter_cnt]) # 代价数据for i in range(iter_cnt):h x.dot(theta) # 估计值error h - y # 误差值cost[i] 1 / (2 * m) * error.T.dot(error) # 代价值# 更新参数delta_theta 1 / m * x.T.dot(error)theta - alpha * delta_theta数据可视化
# 回归结果
plt.scatter(x[:, 1], y, cblue)
plt.plot(x[:, 1], h, r-)
plt.show()# 代价结果
plt.plot(cost)
plt.show()完整代码
import numpy as np
import matplotlib.pyplot as pltx np.array([4, 3, 3, 4, 2, 2, 0, 1, 2, 5, 1, 2, 5, 1, 3])
y np.array([8, 6, 6, 7, 4, 4, 2, 4, 5, 9, 3, 4, 8, 3, 6])m len(x)x np.c_[np.ones([m, 1]), x]
y y.reshape(m, 1)
theta np.zeros([2, 1])alpha 0.01
iter_cnt 1000 # 迭代次数
cost np.zeros([iter_cnt]) # 代价数据for i in range(iter_cnt):h x.dot(theta) # 估计值error h - y # 误差值cost[i] 1 / (2 * m) * error.T.dot(error) # 代价值# 更新参数delta_theta 1 / m * x.T.dot(error)theta - alpha * delta_theta# 线性拟合结果
plt.scatter(x[:, 1], y, cblue)
plt.plot(x[:, 1], h, r-)
plt.show()# 代价结果
plt.plot(cost)
plt.show()线性拟合结果 代价结果
