标签：sub 线性回归 limit ima lock idea 图片 ble param
Training Set
训练集
| Size in feet2(x) | Price in 1000‘s(y) |
| 2104 | 460 |
| 1416 | 232 |
| 1534 | 315 |
| 852 | 178 |
Hypothesis:
\[{h_\theta }\left( x \right) = {\theta _0} + \theta {x}\]
Notation:
θi‘s: Parameters
θi‘s: 参数
How to choose θi‘s?
如何选择θi‘s？
Idea: Choose θ0, θ1so that h(x) is close to y for our training examples(x, y)
思想：对于训练样本（x, y）来说，选择θ0，θ1 使h(x) 接近y。
minimize（θ0, θ1）\[\sum\limits_{i = 1}^m {{{\left( {{h_\theta }\left( {{x^{(i)}}} \right) - {y^i}} \right)}^2}} \]
选择合适的（θ0, θ1）使得 \[\sum\limits_{i = 1}^m {{{\left( {{h_\theta }\left( {{x^{(i)}}} \right) - {y^i}} \right)}^2}} \] 最小。
为了使公式的数学意义更好，将公式改为 \[\frac{1}{{2m}}\sum\limits_{i = 1}^m {{{\left( {{h_\theta }\left( {{x^{(i)}}} \right) - {y^i}} \right)}^2}} \]
这并不影响 （θ0, θ1）的取值。
定义代价函数（Cost function） \[J\left( {{\theta _0},{\theta _1}} \right) = \frac{1}{{2m}}\sum\limits_{i = 1}^m {{{\left( {{h_\theta }\left( {{x^{(i)}}} \right) - {y^i}} \right)}^2}} \]
目标是 \[\mathop {\min imize}\limits_{{\theta _0},{\theta _1}} J\left( {{\theta _0},{\theta _1}} \right)\]
这个代价函数也称为平方误差代价函数（Squared error function）
总结：
Hypothesis: \[{h_\theta }\left( x \right) = {\theta _0} + \theta {x}\]
Parameters: （θ0, θ1）
Cost Functions: \[J\left( {{\theta _0},{\theta _1}} \right) = \frac{1}{{2m}}\sum\limits_{i = 1}^m {{{\left( {{h_\theta }\left( {{x^{(i)}}} \right) - {y^i}} \right)}^2}} \]
Goal: \[\mathop {\min imize}\limits_{{\theta _0},{\theta _1}} J\left( {{\theta _0},{\theta _1}} \right)\]
例子帮助理解
首先令 θ0=0，则代价函数变为 \[J\left( {{\theta _1}} \right) = \frac{1}{{2m}}\sum\limits_{i = 1}^m {{{\left( {{h_\theta }\left( {{x^{(i)}}} \right) - {y^i}} \right)}^2}} \]
| hθ(x) | J(θ1) |
| 对于给定θ1的情况，它是x的函数 | 是θ1的函数 |
三个训练样本
| x | y |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
当θ1=1时，\[J\left( {{\theta _1}} \right) = \frac{1}{{2m}}\left( {{0^2} + {0^2} + {0^2}} \right) = 0\]
当θ1=0.5时，\[J\left( {0.5} \right) = \frac{1}{{2*3}}\left( {{{\left( {0.5 - 1} \right)}^2} + {{\left( {{\rm{1 - 2}}} \right)}^2} + {{\left( {{\rm{1}}{\rm{.5 - 3}}} \right)}^2}} \right) \approx {\rm{0}}{\rm{.58}}\]
θ1取不同值J(θ1)的值
每一个不同θ1的对应一条直线，我们的目的是找出最合适的θ1（最适合的直线）
标签：sub 线性回归 limit ima lock idea 图片 ble param
原文地址：https://www.cnblogs.com/qkloveslife/p/9824010.html
