3.7 Equations

Can use $...$ ($$...$$ for blocks) or \(...\) (\[...\] for blocks) to enclose equations. Difference:

  • $...$ provides rendered equation previews in RStudio.
  • \(...\) does not have previews.

Rstudio equation previews do NOT work well with indented equations. \(\rightarrow\) reduce indentation

如果公式缩进，Rstudio 公式预览功能可能不识别。在不影响理解的前提下，减少不必要的缩进以便预览公式。

Multi-case functions using \begin{cases}

\begin{align*}
I_t = 
\begin{cases}
1 & \text{if } r_t>0 \\
0 & \text{if } r_t\leq0
\end{cases}
\end{align*}

will render as

\[\begin{align*} I_t = \begin{cases} 1 & \text{if } r_t>0 \\ 0 & \text{if } r_t\leq0 \end{cases} \end{align*}\]

For equation numbering support in bookdown you need to assign labels.

You may refer to an equation using Eq. \@ref(eq:eq01).

\begin{align} (\#eq:eq01)
\frac{p(x)}{1-p(x)} = \exp (\beta_0+\beta_1 x) \,.
\end{align}

If you want to provide a specific number to the equation, you can use \tag{XX.XX}.

  • With LaTeX

    LaTex allows custom labels.

    \[\begin{align} \label{eq:my-label-latex} \tag{my label latex} \frac{p(x)}{1-p(x)} = \exp (\beta_0+\beta_1 x) \,. \end{align}\]

    My specific label here, see eq \(\eqref{eq:my-label-latex}\) (\eqref{eq:my-label-latex}).

  • With bookdown

    bookdown does NOT support custom tag though.

    \[\begin{align} \frac{p(x)}{1-p(x)} = \exp (\beta_0+\beta_1 x) \,. \tag{3.3} \end{align}\]

    My specific label here, see eq (3.3)

Color eqns using \color{#00CC66}{...}.

But sometime everything follows gets colored. You may want to use {\color{#00CC66} ... } instead.

$$
\color{#008B45}{Y_t} = I_tI_{t-1} + (1-I_t)(1-I_{t-1})
$$

\[\begin{align*} {\color{red}Y_t} = I_tI_{t-1} + (1-I_t)(1-I_{t-1}) \end{align*}\]

  • This only works for color names, not hex codes starting with #, because html requires the # followed by 6 characters to define a color, but LaTeX package xcolor specifically excludes # in color specifications.

  • Here is an (only works for LaTeX).

A workaround: We can write a custom R function to insert the correct syntax depending on the output format using the is_latex_output() and is_html_output() functions in knitr as follows:

colorize <- function(x, color) {
  if (knitr::is_latex_output()) {
    sprintf("\\textcolor{%s}{%s}", color, x)
  } else if (knitr::is_html_output()) {
    sprintf("<span style='color: %s;'>%s</span>", color,
      x)
  } else x
}

We can then use the code in an inline R expression `r colorize("some words in red", "red")`, which will create some words in red, which works for both html and .

Mathjax

https://bookdown.org/yihui/rmarkdown/html-document.html#mathjax-equations

RStudio IDE comes with Mathjax 2.7 bundled. This is used by rmarkdown when local mode is used for example. Path is stored in Sys.getenv("RMARKDOWN_MATHJAX_PATH").

> Sys.getenv("RMARKDOWN_MATHJAX_PATH")
[1] "/Applications/RStudio.app/Contents/Resources/app/resources/mathjax-27"

> rmarkdown:::mathjax_config()
[1] "MathJax.js?config=TeX-AMS-MML_HTMLorMML"

For online mode, rmarkdown uses https://mathjax.rstudio.com/latest/ with config MathJax.js?config=TeX-AMS-MML_HTMLorMML for now.

Default configuration used by the rmarkdown package is given by rmarkdown:::mathjax_config(). As of rmarkdown v2.1, the function returns “MathJax.js?config=TeX-AMS-MML_HTMLorMML”. This configures Mathjax to HTML-CSS.

Change Mathjax configuration to CommonHTML using the following codes.

---
title: "Trouble with MathJax"
output: 
  html_document:
    mathjax: "https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-AMS_CHTML.js"
    self_contained: false    
---

By default, MathJax scripts are included in HTML documents for rendering LaTeX and MathML equations. You can use the mathjax option to control how MathJax is included:

  • Specify "default" to use an HTTPS URL from a CDN host (currently provided by RStudio).
  • Specify "local" to use a local version of MathJax (which is copied into the output directory). Note that when using "local" you also need to set the self_contained option to false.
  • Specify an alternate URL to load MathJax from another location. To use a self-hosted copy of MathJax.
  • Specify null to exclude MathJax entirely.

Q: Why my eqns are not rendered?

A: MathJax is unlikely to work offline. Check internet connection.

You load MathJax into a web page by including its main JavaScript file into the page. That is done via a <script>tag that links to the MathJax.js file. To do that, place the following line in the <head> section of your document.

For example, if you are using the MathJax distributed network service, the tag might be

<script type="text/javascript"
   src="http://cdn.mathjax.org/mathjax/latest/MathJax.js">
</script>

MathJax is available as a web service from cdn.mathjax.org, so you can obtain MathJax from there without needing to install it on your own server. The CDN is part of a distributed “cloud” network, so it is handled by servers around the world. That means that you should get access to a server geographically near you, for a fast, reliable connection.

The CDN hosts the most current version of MathJax, as well as older versions, so you can either link to a version that stays up-to-date as MathJax is improved, or you can stay with one of the release versions so that your pages always use the same version of MathJax.

RStudio Equation Live Preview

Q: How to define custom macros that can be used in live equation preview in RStudio?

A: Go to /Applications/RStudio.app/Contents/Resources/app/resources/mathjax-27/config/TeX-MML-AM_CHTML.js, add your custom macros as follows.

MathJax.Hub.Config({
  extensions: ['[a11y]/accessibility-menu.js'],
  TeX: {
    Macros: { // define TeX macros
      RR: "{\\bf R}",
      bold: ["{\\bf #1}",1],
      btheta: "\\boldsymbol{\\theta}",
      bAlpha: "\\boldsymbol{\\Alpha}",
    }
  },
});

RStudio functions like a web browser, you can a web inspector to inspect every element of the appearance. Simply right-click pretty much anywhere on RStudio, and hit “Inspect Element” and RStudio DevTools will appear.

The MathJax supporting files can be found in Sources → Page → Electron Isolated Context → mathjax.

You will see MathJax.js?config=TeX-MML-AM_CHTML is loaded to parse and enable live previews of equations.

The user configuration file is located under config/. You will see your custom defined macros here.

For equation numbering support in bookdown::pdf_document2 you need to assign labels. Defualt behavior is not adding numbering.

  • Use \begin{equation}...\end{equations} or \begin{align}...\end{align} environments.

    • Use (\#eq:eq1) or \label{eq:eq1} to add labels.
    • Automatically add numbering.
    • Drawback is that rmd does not have preview of equations.

  • Do NOT enclose the environments in double dollar signs $$. Otherwise, no label is added, but cross-references still show up.

    • $$ do not add numbering automatically.

    • But in bookdown::html_document2, it is ok to use

      $$
\begin{equation} (\#eq:simple-lm)
\hat{\beta}_{\text{OLS}} = \left(\sum_{i=1}^n x_i x_i' \right)^{-1} \left(\sum_{i=1}^n x_i y_i \right) .
\end{equation}
$$

      Then reference with \@ref(eq:simple-lm).

  • Use \@ref(eq:eq1) (note this use parentheses) or the Latex command \eqref{eq:eq1} (this uses curly braces) to cite the equation.

Load the dataset and calculate the monthly return in month $r$ ($r_t$) as

\begin{equation}
r_t = \frac{P_t-P_{t-1}}{P_{t-1}} = \frac{P_t}{P_{t-1}}-1 , 
(\#eq:eq1)
\end{equation}

where $P_t$ is the adjusted price in month $t$. 

Test equation1 \@ref(eq:eq1).

Test equation2 \eqref{eq:eq1}.

eqn numbering in bookdown