## ----setup, include = FALSE---------------------------------------------------
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  dpi = 300,
  dev = "png"
)
library(gridmicrotex)
library(grid)

## ----body---------------------------------------------------------------------
body <- r"(\section{Methods}
We fit a straight line to $n$ observations $(x_i, y_i)$ by least
squares, which chooses the intercept and slope that minimise the sum
of squared residuals
\[ S(\beta_0, \beta_1) = \sum_{i=1}^{n} (y_i - \beta_0 - \beta_1 x_i)^2. \]
Setting both partial derivatives to zero gives the estimates
\begin{align}
  \hat\beta_1 &= \frac{\sum_i (x_i - \bar x)(y_i - \bar y)}{\sum_i (x_i - \bar x)^2}, \label{slope} \\
  \hat\beta_0 &= \bar y - \hat\beta_1 \bar x. \label{intercept}
\end{align}

The fitted values are $\hat y_i = \hat\beta_0 + \hat\beta_1 x_i$, and
the residuals $e_i = y_i - \hat y_i$ are what is left over. Together
\eqref{slope} and \eqref{intercept} make them sum to zero.

\subsection{Assumptions}
The errors are taken to be
\begin{itemize}
  \item independent of one another,
  \item of constant variance $\sigma^2$, and
  \item normally distributed --- which matters only for the tests.
\end{itemize}
)"

## ----draw-body, fig.width = 6, fig.height = 5.2, out.width = "90%"------------
grid.newpage()
grid.latex(body, input_mode = "document", max_width = 5.6 * 72,
           x = 0.03, y = 0.97, hjust = 0, vjust = 1,
           gp = gpar(fontsize = 11))

## ----theorem, fig.width = 6, fig.height = 1.9, out.width = "90%"--------------
thm <- r"(\newtheorem{thm}{Theorem}
\begin{thm}[Gauss--Markov]\label{gm}
With uncorrelated errors of equal variance, least squares has the least
variance among the linear unbiased estimators.
\end{thm}
\begin{proof}
Write any other such estimator as least squares plus a correction, and
show that the correction can only add variance.
\end{proof}
Theorem~\ref{gm} is why \verb|lm(y ~ x)| is the default.)"
grid.newpage()
grid.latex(thm, input_mode = "document", max_width = 5.6 * 72,
           x = 0.03, y = 0.97, hjust = 0, vjust = 1,
           gp = gpar(fontsize = 11))

## ----justify, fig.width = 6, fig.height = 1.05, out.width = "90%"-------------
para <- r"(Least squares has a closed form, which is why it was the
method of choice long before computers made iterative fitting cheap. It
is also optimal among unbiased linear estimators when the errors are
uncorrelated with equal variance: the Gauss--Markov theorem.)"
grid.newpage()
grid.latex(para, input_mode = "document", max_width = 5.6 * 72,
           justify = TRUE, line_break = "optimal",
           x = 0.03, y = 0.95, hjust = 0, vjust = 1,
           gp = gpar(fontsize = 11))

## ----pdf, eval = FALSE--------------------------------------------------------
# d <- latex_dims(body, input_mode = "document", max_width = 5.5 * 72,
#                 gp = gpar(fontsize = 11))
# height <- as.numeric(d$height) / 72 + 1   # inches, with margins
# 
# cairo_pdf("methods.pdf", width = 6.5, height = height)
# grid.latex(body, input_mode = "document", max_width = 5.5 * 72,
#            x = unit(0.5, "in"), y = unit(1, "npc") - unit(0.5, "in"),
#            hjust = 0, vjust = 1, gp = gpar(fontsize = 11))
# dev.off()