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_{i=1}^m \left ( \hat{y}_{i}- y_{i} \right)^2 = \dfrac {1}{2m} \displaystyle \sum _{i=1}^m \left (h_\theta (x_{i}) - y_{i} \right)^2$$ +$$J(\theta_0,\theta_1)=\dfrac{1}{2m}\displaystyle\sum_{i=1}^m\left(\hat{y}_{i}-y_{i} \right)^2=\dfrac{1}{2m}\displaystyle\sum_{i=1}^m\left(h_\theta(x_{i})-y_{i}\right)^2$$ -> 系数 $\frac{1}{2}$ 存在与否都不会影响结果,这里是为了在应用梯度下降时便于求解,平方的导数会抵消掉 $\frac{1}{2}$ 。 +> 系数 $\frac{1}{2}​$ 存在与否都不会影响结果,这里是为了在应用梯度下降时便于求解,平方的导数会抵消掉 $\frac{1}{2}​$ 。 讨论到这里,我们的问题就转化成了**求解 $J\left( \theta_0, \theta_1 \right)$ 的最小值**。 @@ -202,7 +202,7 @@ $$J(\theta_0, \theta_1) = \dfrac {1}{2m} \displaystyle \sum _{i=1}^m \left ( \ha 给定数据集: -![](image\20180106_091307.png) +![](image/20180106_091307.png) 参数在 $\theta_0$ 不恒为 $0$ 时损失函数 $J\left(\theta\right)$ 关于 $\theta_0, \theta_1$ 的3-D图像,图像中的高度为损失函数的值。 diff --git a/notes/week2.md b/notes/week2.md index 0d2e23d..42692f4 100644 --- a/notes/week2.md +++ b/notes/week2.md @@ -1,11 +1,39 @@ [TOC] -# 4 Linear Regression with Multiple Variables +# 4 多变量线性回归(Linear Regression with Multiple Variables) -## 4.1 Multiple Features +## 4.1 多特征(Multiple Features) + +对于一个要度量的对象,一般来说会有不同维度的多个特征。比如之前的房屋价格预测例子中,除了房屋的面积大小,可能还有房屋的年限、房屋的层数等等其他特征: + +![](image/20180107_234509.png) + +这里由于特征不再只有一个,引入一些新的记号 + +> $n$: 特征的总数 +> +> ${x}^{\left( i \right)}$: 代表特征矩阵中第 $i$ 行,也就是第 $i$ 个训练实例。 +> +> ${x}_{j}^{\left( i \right)}$: 代表特征矩阵中第 $i$ 行的第 $j$ 个特征,也就是第 $i$ 个训练实例的第 $j$ 个特征。 + +参照上图,则记号的举例有,${x}^{(2)}\text{=}\begin{bmatrix} 1416\\\ 3\\\ 2\\\ 40 \end{bmatrix}, {x}^{(2)}_{1} = 1416$ + +多变量假设函数 $h$ 表示为:$h_{\theta}\left( x \right)={\theta_{0}}+{\theta_{1}}{x_{1}}+{\theta_{2}}{x_{2}}+...+{\theta_{n}}{x_{n}}$ + +对于 $\theta_0$,和单特征中一样,我们将其看作基础数值。例如,房价的基础价格。 + +参数向量的维度为 $n+1$,在特征向量中添加 $x_{0}$ 后,其维度也变为 $n+1$, 则运用线性代数,可对 $h$ 简化。 + +$h_\theta\left(x\right)=\begin{bmatrix}\theta_0\; \theta_1\; ... \;\theta_n \end{bmatrix}\begin{bmatrix}x_0 \newline x_1 \newline \vdots \newline x_n\end{bmatrix}= \theta^T x$ + +> $\theta^T$: $\theta$ 矩阵的转置 +> +> $x_0$: 为了计算方便我们会假设 $x_0^{(i)} = 1$ ## 4.2 Gradient Descent for Multiple Variables + + ## 4.3 Gradient Descent in Practice I - Feature Scaling ## 4.4 Gradient Descent in Practice II - Learning Rate @@ -19,6 +47,9 @@ ## 4.8 Working on and Submitting Programming Assignments # 5 Octave Matlab Tutorial + +复习时可直接倍速回顾视频,笔记整理暂留。 + ## 5.1 Basic Operations ## 5.2 Moving Data Around @@ -29,4 +60,6 @@ ## 5.5 Control Statements_ for, while, if statement -## 5.6 Vectorization \ No newline at end of file +## 5.6 Vectorization + +## 5.x 常用函数整理 \ No newline at end of file