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\title{Notes on linear regression}
\author{}
\date{}

\begin{document}
\maketitle

A linear model predicts $y$ from $x$ with a straight line, $\hat{y} = \beta_0 + \beta_1 x$.

\section*{Least squares}\label{sec:least-squares}

The coefficients minimise the sum of squared errors:\footnote{Gauss described the method in 1809.}

\[
\hat{\beta}_1 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n} (x_i - \bar{x})^2}
\]

\section*{Checking the fit}\label{sec:checking-the-fit}

\begin{enumerate}
  \item Plot the residuals against $\hat{y}$
  \item Look for curves or a funnel shape
  \item Compare $R^2$ between models
\end{enumerate}

\begin{table}[htbp]
  \centering
  \begin{tabularx}{\linewidth}{>{\raggedright\arraybackslash}X >{\raggedleft\arraybackslash}X >{\raggedleft\arraybackslash}X}
  \toprule
  \textbf{Model} & \textbf{$R^2$} & \textbf{RMSE} \\
  \midrule
  Linear & 0.82 & 3.1 \\
  Quadratic & 0.91 & 2.2 \\
  \bottomrule
  \end{tabularx}
\end{table}

\begin{Verbatim}[breaklines, breakanywhere, fontsize=\small, frame=single, framesep=2mm]
import numpy as np
slope, intercept = np.polyfit(x, y, 1)
\end{Verbatim}

\begin{quote}
A high $R^2$ does not prove that the line is the right shape.
\end{quote}

\end{document}