$$ \text{RMSE}(\bold{X},h)=\sqrt{\dfrac{1}{m}\sum^m_{i=1}\left(h\left(\bold{x}\right)^{(i)}-y^{(i)}\right)^2} $$

• $m$ is number of instances in the dataset.
• $\bold{x}^{(i)}$ is a vector of all the feature values (excluding the label) of the $i^\text{th}$ instance in the dataset.
• $y^{(i)}$ is its label (the desired output value for the instance)
• $\bold{X}$ is a matrix containing all the feature values (excluding labels) of all instances in the datasets.
• $h$ is the prediction function, also called a hypothesis : $\hat{y}^{(i)}= h\left( \bold{x}^{(i)}\right)$.

<aside> 💡 Lowercase italic font for scalar values (such as $m$ or $y^{(i)}$ and function names (such as $h$)

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<aside> 💡 Lowercase bold font for vectors (such as $\bold{x}$).

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<aside> 💡 Uppercase bold font for matrices (such as $\bold{X}$).

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$\bold{x}^{(i)}$는 보통 column vector임.

$$ \bold{x}^{(i)} = \begin{bmatrix}x_1\\x_2\\\vdots\\x_n\end{bmatrix} $$

$\bold{X}$는 $\left(\bold{x}^{(i)}\right)^T$를 row로 가지는 matrix.

$$ \bold{X}=\begin{bmatrix}\left( \bold{x}^{(1)} \right) ^T \\ \left( \bold{x}^{(2)} \right) ^T \\ \vdots \\ \left( \bold{x}^{(m)} \right) ^T \end{bmatrix} $$