From b835ba91e9f210780b4d3582c6a5d4bb809b5a67 Mon Sep 17 00:00:00 2001 From: Will Cornwell Date: Fri, 13 Mar 2026 13:02:33 +1100 Subject: [PATCH 1/3] Add multipanel figures tutorial for ggplot2 Covers facet_wrap, facet_grid, custom strip labels, and the patchwork package for combining independent plots. Also fixes Hugo 0.157 compatibility issues in config.yaml and ghcontributors shortcode, and adds a GitHub Actions deploy-preview workflow. Co-Authored-By: Claude Sonnet 4.6 --- .github/workflows/deploy-preview.yml | 36 +++ config.yaml | 2 +- content/Graphics/ggplot/_index.rmd | 9 +- .../ggplot/ggplot-multipanel/_index.html | 154 ++++++++++++ .../ggplot/ggplot-multipanel/_index.rmd | 226 ++++++++++++++++++ layouts/shortcodes/ghcontributors.html | 16 +- .../figure-html/unnamed-chunk-1-1.png | Bin 0 -> 103614 bytes .../figure-html/unnamed-chunk-10-1.png | Bin 0 -> 70532 bytes .../figure-html/unnamed-chunk-13-1.png | Bin 0 -> 98375 bytes .../figure-html/unnamed-chunk-14-1.png | Bin 0 -> 105032 bytes .../figure-html/unnamed-chunk-15-1.png | Bin 0 -> 156769 bytes .../figure-html/unnamed-chunk-16-1.png | Bin 0 -> 157137 bytes .../figure-html/unnamed-chunk-17-1.png | Bin 0 -> 116489 bytes .../figure-html/unnamed-chunk-18-1.png | Bin 0 -> 121405 bytes .../figure-html/unnamed-chunk-19-1.png | Bin 0 -> 126790 bytes .../figure-html/unnamed-chunk-4-1.png | Bin 0 -> 69887 bytes .../figure-html/unnamed-chunk-5-1.png | Bin 0 -> 83290 bytes .../figure-html/unnamed-chunk-6-1.png | Bin 0 -> 88552 bytes .../figure-html/unnamed-chunk-7-1.png | Bin 0 -> 94031 bytes .../figure-html/unnamed-chunk-8-1.png | Bin 0 -> 65866 bytes .../figure-html/unnamed-chunk-9-1.png | Bin 0 -> 70527 bytes 21 files changed, 432 insertions(+), 11 deletions(-) create mode 100644 .github/workflows/deploy-preview.yml create mode 100644 content/Graphics/ggplot/ggplot-multipanel/_index.html create mode 100644 content/Graphics/ggplot/ggplot-multipanel/_index.rmd create mode 100644 static/Graphics/ggplot/ggplot-multipanel/_index_files/figure-html/unnamed-chunk-1-1.png create mode 100644 static/Graphics/ggplot/ggplot-multipanel/_index_files/figure-html/unnamed-chunk-10-1.png create mode 100644 static/Graphics/ggplot/ggplot-multipanel/_index_files/figure-html/unnamed-chunk-13-1.png create mode 100644 static/Graphics/ggplot/ggplot-multipanel/_index_files/figure-html/unnamed-chunk-14-1.png create mode 100644 static/Graphics/ggplot/ggplot-multipanel/_index_files/figure-html/unnamed-chunk-15-1.png create mode 100644 static/Graphics/ggplot/ggplot-multipanel/_index_files/figure-html/unnamed-chunk-16-1.png create mode 100644 static/Graphics/ggplot/ggplot-multipanel/_index_files/figure-html/unnamed-chunk-17-1.png create mode 100644 static/Graphics/ggplot/ggplot-multipanel/_index_files/figure-html/unnamed-chunk-18-1.png create mode 100644 static/Graphics/ggplot/ggplot-multipanel/_index_files/figure-html/unnamed-chunk-19-1.png create mode 100644 static/Graphics/ggplot/ggplot-multipanel/_index_files/figure-html/unnamed-chunk-4-1.png create mode 100644 static/Graphics/ggplot/ggplot-multipanel/_index_files/figure-html/unnamed-chunk-5-1.png create mode 100644 static/Graphics/ggplot/ggplot-multipanel/_index_files/figure-html/unnamed-chunk-6-1.png create mode 100644 static/Graphics/ggplot/ggplot-multipanel/_index_files/figure-html/unnamed-chunk-7-1.png create mode 100644 static/Graphics/ggplot/ggplot-multipanel/_index_files/figure-html/unnamed-chunk-8-1.png create mode 100644 static/Graphics/ggplot/ggplot-multipanel/_index_files/figure-html/unnamed-chunk-9-1.png diff --git a/.github/workflows/deploy-preview.yml b/.github/workflows/deploy-preview.yml new file mode 100644 index 0000000..1947b28 --- /dev/null +++ b/.github/workflows/deploy-preview.yml @@ -0,0 +1,36 @@ +name: Deploy Preview + +on: + push: + branches-ignore: + - master + pull_request: + +jobs: + deploy: + runs-on: ubuntu-latest + steps: + - uses: actions/checkout@v4 + + - name: Setup Hugo + uses: peaceiris/actions-hugo@v3 + with: + hugo-version: '0.91.2' + extended: true + + - name: Build + run: hugo --minify + + - name: Deploy to Netlify + id: netlify + uses: nwtgck/actions-netlify@v3 + with: + publish-dir: ./public + production-deploy: false + github-token: ${{ secrets.GITHUB_TOKEN }} + deploy-message: "Deploy preview for ${{ github.ref_name }}" + enable-pull-request-comment: true + enable-commit-comment: true + env: + NETLIFY_AUTH_TOKEN: ${{ secrets.NETLIFY_AUTH_TOKEN }} + NETLIFY_SITE_ID: ${{ secrets.NETLIFY_SITE_ID }} diff --git a/config.yaml b/config.yaml index 36ce842..0543a89 100644 --- a/config.yaml +++ b/config.yaml @@ -4,7 +4,7 @@ defaultContentLanguage: en title: Environmental Computing theme: hugo-theme-learn metaDataFormat: yaml -defaultContentLanguageInSubdir: yes +defaultContentLanguageInSubdir: true googleAnalyticsID: G-KVTJFRWP8M googleAnalytics: UA-77528433-1 params: diff --git a/content/Graphics/ggplot/_index.rmd b/content/Graphics/ggplot/_index.rmd index 572baaf..faa040d 100644 --- a/content/Graphics/ggplot/_index.rmd +++ b/content/Graphics/ggplot/_index.rmd @@ -22,10 +22,11 @@ The package is named after a book called [The Grammar of Graphics](https://books Start with the [basics](/graphics/ggplot/ggplot-basics) to learn the basic syntax of making a graph -Then, visit our other pages to further customise the aesthetics of the graph, including colour and formatting: -* [altering the overall appearance](/graphics/ggplot/ggplot-appearance/) -* [adding titles and axis names](/graphics/ggplot/ggplot-labels/) -* [colours and symbols](/graphics/ggplot/ggplot-colour-shapes/). +Then, visit our other pages to further customise the aesthetics of the graph, including colour and formatting: +* [altering the overall appearance](/graphics/ggplot/ggplot-appearance/) +* [adding titles and axis names](/graphics/ggplot/ggplot-labels/) +* [colours and symbols](/graphics/ggplot/ggplot-colour-shapes/) +* [multipanel figures](/graphics/ggplot/ggplot-multipanel/). Help on all the ggplot functions can be found at the [The master ggplot help site](https://ggplot2.tidyverse.org). diff --git a/content/Graphics/ggplot/ggplot-multipanel/_index.html b/content/Graphics/ggplot/ggplot-multipanel/_index.html new file mode 100644 index 0000000..2beef23 --- /dev/null +++ b/content/Graphics/ggplot/ggplot-multipanel/_index.html @@ -0,0 +1,154 @@ +--- +title: 'Multipanel figures with ggplot2' +weight: 6 +output: html_document +aliases: /plotting-with-ggplot-multipanel-figures/ +--- + + + +

+

Multipanel figures — where several plots are arranged together into a single figure — are a common way to present related results side by side. In R with ggplot2, there are two main approaches:

+
    +
  1. Faceting, which splits a single dataset into panels based on the values of one or more variables.
  2. +
  3. Combining separate plots using the patchwork package, which lets you arrange completely independent ggplots together.
  4. +
+

Before you get started, read the page on the basics of plotting with ggplot and install the package ggplot2.

+
library(ggplot2)
+

For the examples on this page we will use the iris dataset (already built into R) and the mpg dataset from ggplot2.

+
data(iris)
+data(mpg)
+
+

Faceting with facet_wrap

+

Faceting is ideal when you want to make the same type of plot for subsets of your data. facet_wrap takes a single grouping variable and wraps the panels into rows and columns automatically.

+

The formula syntax ~ variable specifies which variable to split on. For example, to make a separate scatter plot for each iris species:

+
p <- ggplot(iris, aes(Sepal.Length, Petal.Length)) +
+  geom_point() +
+  theme_classic()
+
+p + facet_wrap(~ Species)
+

+

By default facet_wrap tries to make the panel layout as square as possible. You can control the number of rows or columns with nrow or ncol:

+
p + facet_wrap(~ Species, ncol = 1)
+

+

You can add scales = "free" if you want each panel to use its own axis limits, rather than sharing the same scale across all panels. This is useful when the ranges differ greatly between groups, but makes direct visual comparison harder.

+
p + facet_wrap(~ Species, scales = "free")
+

+
+
+

Faceting with facet_grid

+

facet_grid lays panels out in a strict grid defined by two variables — one for rows and one for columns. Use the formula rows ~ cols. To facet only on one dimension, replace the other with a dot (.).

+

Here we use the mpg dataset, splitting by drive type (drv) in rows and number of cylinders (cyl) in columns:

+
ggplot(mpg, aes(displ, hwy)) +
+  geom_point() +
+  theme_classic() +
+  facet_grid(drv ~ cyl)
+

+

facet_grid is useful when you have two categorical variables and want to see every combination. Empty panels appear where no data exist for a given combination.

+

To facet on just one variable across rows with facet_grid, use . ~ variable (or variable ~ . for columns):

+
ggplot(mpg, aes(displ, hwy)) +
+  geom_point() +
+  theme_classic() +
+  facet_grid(. ~ drv)
+

+
+
+

Customising facet labels and strips

+

By default the facet labels are the values of the grouping variable. You can change the strip background and text colour inside a theme() call:

+
p + facet_wrap(~ Species) +
+  theme(
+    strip.background = element_rect(fill = "steelblue"),
+    strip.text = element_text(colour = "white", face = "bold")
+  )
+

+

To rename the labels without changing the underlying data, supply a named character vector to the labeller argument:

+
species_labels <- c(
+  setosa = "I. setosa",
+  versicolor = "I. versicolor",
+  virginica = "I. virginica"
+)
+
+p + facet_wrap(~ Species, labeller = labeller(Species = species_labels)) +
+  theme(strip.text = element_text(face = "italic"))
+

+
+
+

Combining separate plots with patchwork

+

Faceting requires all panels to share the same underlying ggplot call and the same geom type. When you want to combine different plots — perhaps a scatter plot next to a bar plot — use the patchwork package.

+

Install it once with install.packages("patchwork"), then load it:

+
library(patchwork)
+

First, create two separate ggplot objects:

+
p1 <- ggplot(iris, aes(Sepal.Length, Petal.Length, colour = Species)) +
+  geom_point() +
+  theme_classic()
+
+p2 <- ggplot(iris, aes(Species, Sepal.Length, fill = Species)) +
+  geom_boxplot(show.legend = FALSE) +
+  theme_classic()
+

The | operator places plots side by side, and / stacks them vertically:

+
p1 | p2
+

+
p1 / p2
+

+

You can mix | and / and use brackets to control grouping. For example, to put two narrow plots on the right and one tall plot on the left:

+
p3 <- ggplot(iris, aes(Petal.Width, Petal.Length, colour = Species)) +
+  geom_point() +
+  theme_classic()
+
+p1 | (p2 / p3)
+

+
+
+

Adding panel labels with patchwork

+

Publication figures often label each panel with a letter (A, B, C…). patchwork makes this straightforward with plot_annotation:

+
(p1 | (p2 / p3)) +
+  plot_annotation(tag_levels = "A")
+

+

You can also add a shared title across all panels:

+
(p1 | p2) +
+  plot_annotation(
+    title = "Iris sepal and petal measurements",
+    theme = theme(plot.title = element_text(size = 14, face = "bold"))
+  )
+
## Warning: annotation$theme is not a valid theme.
+## Please use `theme()` to construct themes.
+

+
+
+

Collecting shared legends

+

When panels share the same colour or fill aesthetic, patchwork can merge their legends into one with plot_layout(guides = "collect"):

+
(p1 | p3) +
+  plot_layout(guides = "collect")
+

+

Combine with plot_annotation to get a clean, publication-ready layout in a few lines of code:

+
(p1 | p3) +
+  plot_layout(guides = "collect") +
+  plot_annotation(
+    tag_levels = "A",
+    title = "Iris petal and sepal relationships"
+  )
+

+
+
+

Further help

+ +

+Pedersen, T L (2024) patchwork: The Composer of Plots. R package. https://patchwork.data-imaginist.com +

+

+Wickham, H (2016) ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York. +

+

Author: Environmental Computing

+

Year: 2025

+

Last updated: Mar 2026

+
diff --git a/content/Graphics/ggplot/ggplot-multipanel/_index.rmd b/content/Graphics/ggplot/ggplot-multipanel/_index.rmd new file mode 100644 index 0000000..b670261 --- /dev/null +++ b/content/Graphics/ggplot/ggplot-multipanel/_index.rmd @@ -0,0 +1,226 @@ +--- +title: 'Multipanel figures with ggplot2' +weight: 6 +output: html_document +aliases: /plotting-with-ggplot-multipanel-figures/ +--- + +```{r, echo=FALSE, warning=FALSE, message=FALSE} +library(ggplot2) +library(patchwork) + +p1 <- ggplot(iris, aes(Sepal.Length, Petal.Length, colour = Species)) + + geom_point() + + theme_classic() + + labs(x = "Sepal length (cm)", y = "Petal length (cm)") + +p2 <- ggplot(iris, aes(Species, Sepal.Length, fill = Species)) + + geom_boxplot(show.legend = FALSE) + + theme_classic() + + labs(x = "Species", y = "Sepal length (cm)") + +(p1 | p2) + plot_layout(guides = "collect") +``` + + +Multipanel figures — where several plots are arranged together into a single figure — are a common way to present related results side by side. In R with ggplot2, there are two main approaches: + +1. **Faceting**, which splits a single dataset into panels based on the values of one or more variables. +2. **Combining separate plots** using the [patchwork](https://patchwork.data-imaginist.com) package, which lets you arrange completely independent ggplots together. + +Before you get started, read the page on the [basics](/graphics/ggplot/ggplot-basics/) of plotting with ggplot and install the package ggplot2. + +```{r, message=FALSE} +library(ggplot2) +``` + +For the examples on this page we will use the `iris` dataset (already built into R) and the `mpg` dataset from ggplot2. + +```{r} +data(iris) +data(mpg) +``` + + +### Faceting with `facet_wrap` + +Faceting is ideal when you want to make the same type of plot for subsets of your data. `facet_wrap` takes a single grouping variable and wraps the panels into rows and columns automatically. + +The formula syntax `~ variable` specifies which variable to split on. For example, to make a separate scatter plot for each iris species: + +```{r, fig.width=9, fig.height=3} +p <- ggplot(iris, aes(Sepal.Length, Petal.Length)) + + geom_point() + + theme_classic() + +p + facet_wrap(~ Species) +``` + +By default `facet_wrap` tries to make the panel layout as square as possible. You can control the number of rows or columns with `nrow` or `ncol`: + +```{r, fig.width=4, fig.height=9} +p + facet_wrap(~ Species, ncol = 1) +``` + +You can add `scales = "free"` if you want each panel to use its own axis limits, rather than sharing the same scale across all panels. This is useful when the ranges differ greatly between groups, but makes direct visual comparison harder. + +```{r, fig.width=9, fig.height=3} +p + facet_wrap(~ Species, scales = "free") +``` + + +### Faceting with `facet_grid` + +`facet_grid` lays panels out in a strict grid defined by two variables — one for rows and one for columns. Use the formula `rows ~ cols`. To facet only on one dimension, replace the other with a dot (`.`). + +Here we use the `mpg` dataset, splitting by drive type (`drv`) in rows and number of cylinders (`cyl`) in columns: + +```{r, fig.width=9, fig.height=6} +ggplot(mpg, aes(displ, hwy)) + + geom_point() + + theme_classic() + + facet_grid(drv ~ cyl) +``` + +`facet_grid` is useful when you have two categorical variables and want to see every combination. Empty panels appear where no data exist for a given combination. + +To facet on just one variable across rows with `facet_grid`, use `. ~ variable` (or `variable ~ .` for columns): + +```{r, fig.width=9, fig.height=3} +ggplot(mpg, aes(displ, hwy)) + + geom_point() + + theme_classic() + + facet_grid(. ~ drv) +``` + + +### Customising facet labels and strips + +By default the facet labels are the values of the grouping variable. You can change the strip background and text colour inside a `theme()` call: + +```{r, fig.width=9, fig.height=3} +p + facet_wrap(~ Species) + + theme( + strip.background = element_rect(fill = "steelblue"), + strip.text = element_text(colour = "white", face = "bold") + ) +``` + +To rename the labels without changing the underlying data, supply a named character vector to the `labeller` argument: + +```{r, fig.width=9, fig.height=3} +species_labels <- c( + setosa = "I. setosa", + versicolor = "I. versicolor", + virginica = "I. virginica" +) + +p + facet_wrap(~ Species, labeller = labeller(Species = species_labels)) + + theme(strip.text = element_text(face = "italic")) +``` + + +### Combining separate plots with patchwork + +Faceting requires all panels to share the same underlying `ggplot` call and the same geom type. When you want to combine *different* plots — perhaps a scatter plot next to a bar plot — use the [patchwork](https://patchwork.data-imaginist.com) package. + +Install it once with `install.packages("patchwork")`, then load it: + +```{r, message=FALSE} +library(patchwork) +``` + +First, create two separate ggplot objects: + +```{r} +p1 <- ggplot(iris, aes(Sepal.Length, Petal.Length, colour = Species)) + + geom_point() + + theme_classic() + +p2 <- ggplot(iris, aes(Species, Sepal.Length, fill = Species)) + + geom_boxplot(show.legend = FALSE) + + theme_classic() +``` + +The `|` operator places plots side by side, and `/` stacks them vertically: + +```{r, fig.width=9, fig.height=4} +p1 | p2 +``` + +```{r, fig.width=5, fig.height=8} +p1 / p2 +``` + +You can mix `|` and `/` and use brackets to control grouping. For example, to put two narrow plots on the right and one tall plot on the left: + +```{r, fig.width=10, fig.height=5} +p3 <- ggplot(iris, aes(Petal.Width, Petal.Length, colour = Species)) + + geom_point() + + theme_classic() + +p1 | (p2 / p3) +``` + + +### Adding panel labels with patchwork + +Publication figures often label each panel with a letter (A, B, C…). patchwork makes this straightforward with `plot_annotation`: + +```{r, fig.width=10, fig.height=5} +(p1 | (p2 / p3)) + + plot_annotation(tag_levels = "A") +``` + +You can also add a shared title across all panels: + +```{r, fig.width=10, fig.height=5} +(p1 | p2) + + plot_annotation( + title = "Iris sepal and petal measurements", + theme = theme(plot.title = element_text(size = 14, face = "bold")) + ) +``` + + +### Collecting shared legends + +When panels share the same colour or fill aesthetic, patchwork can merge their legends into one with `plot_layout(guides = "collect")`: + +```{r, fig.width=10, fig.height=4} +(p1 | p3) + + plot_layout(guides = "collect") +``` + +Combine with `plot_annotation` to get a clean, publication-ready layout in a few lines of code: + +```{r, fig.width=10, fig.height=4} +(p1 | p3) + + plot_layout(guides = "collect") + + plot_annotation( + tag_levels = "A", + title = "Iris petal and sepal relationships" + ) +``` + + +### Further help + +* [Facets chapter](https://ggplot2-book.org/facet.html) in the ggplot2 book +* [patchwork documentation](https://patchwork.data-imaginist.com) — including more advanced layouts with `plot_layout()` and `inset_element()` +* Our other ggplot pages: + * [The basics of ggplot2](/graphics/ggplot/ggplot-basics/) + * [Customising your ggplot](/graphics/ggplot/ggplot-appearance/) + * [Adding titles and axis labels](/graphics/ggplot/ggplot-labels/) + * [Colours and shapes](/graphics/ggplot/ggplot-colour-shapes/) + +

Pedersen, T L (2024) *patchwork: The Composer of Plots.* R package. https://patchwork.data-imaginist.com

+ +

Wickham, H (2016) *ggplot2: Elegant Graphics for Data Analysis.* Springer-Verlag New York.

+ + +**Author**: Environmental Computing + +**Year:** 2025 + +**Last updated:** `r format(Sys.time(), "%b %Y")` diff --git a/layouts/shortcodes/ghcontributors.html b/layouts/shortcodes/ghcontributors.html index 3e8a928..e1e232e 100644 --- a/layouts/shortcodes/ghcontributors.html +++ b/layouts/shortcodes/ghcontributors.html @@ -21,11 +21,15 @@
{{ $url := .Get 0 }} - {{ range getJSON $url }} -
- - - {{.contributions}} commits -
+ {{ with resources.GetRemote $url }} + {{ with .Content | unmarshal }} + {{ range . }} +
+ + + {{.contributions}} commits +
+ {{ end }} + {{ end }} {{ end }}
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Place them in your working directory, then check if you can see the data we'll be using (your working directory should be this file's location) ```{r, eval=FALSE} file.exists("Cairns_Mangroves_30m.tif") file.exists("SST_feb_2013.img") diff --git a/content/datasets/_index.rmd b/content/datasets/_index.rmd index 5e26632..d3057c4 100644 --- a/content/datasets/_index.rmd +++ b/content/datasets/_index.rmd @@ -82,6 +82,10 @@ This page contains the datasets or files that are used throughout the tutorials [Snail_feeding.csv](/datasets/Snail_feeding.csv) +[SST_feb_2013.img](/datasets/SST_feb_2013.img) + +[SST_feb_mean.img](/datasets/SST_feb_mean.img) + ## T [tree_curtis1998.tre](/datasets/tree_curtis1998.tre) diff --git a/static/datasets/SST_feb_2013.img b/static/datasets/SST_feb_2013.img new file mode 100644 index 0000000000000000000000000000000000000000..e2433be0e01a097e5df0d850ffa2e877c0469281 GIT binary patch literal 302021 zcmeF434C5vng0WVKm-8^3MgR{Xt5#?)QTi;Ln)zHXq%Qrh?KNR+CrC?E>K#QQV2^6 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z@#r7@J!H$4A@k8DRNzVgbLrr1`~{-dpnIAPmY!Ut-qdG>&l8zUy#tPjK~7L$05 z8+~)xcPH@d7O~?Qp5x6CTeyGB;&-(8)zBv&*cRk~{F9I35x;SzzZx_C;}b{4;@x@1=o}J}vqp+i{A0%8#|Z)xMwahVcCfz6-$j z1^8YCHS_#N6Kz2sq%qPyS2Nft&+52XYSN9LPD4b0FtH&Vif*IR|nMO#OV_%tLOW%jqHIU~lA%o_qr-gzTWfD>?eE)C>dJO@G%UH{{~p{lI5@ny z>&B7Vny{#wU@WATtmIRTPcsChD8e&&Mz)5E^Ju5cmW#=ySocGzOZ4v-e4pD zu%Tzi_oxn}J} zk9_&e>Az@Nd(n%#Uw`=v*Y0Ra<(}Zj?$ggr_M*-Go{Uo|)|x^4T7t9p9|Mn*O*EiG`>l;eNy5Zr? zbC_hAdm#5f?t$C`xd(C&9DxU(;%kOq zt Date: Fri, 13 Mar 2026 13:23:55 +1100 Subject: [PATCH 3/3] Fix Poisson GAM notation in GAMs intro page --- content/Statistics/gams/_index.rmd | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/Statistics/gams/_index.rmd b/content/Statistics/gams/_index.rmd index 142ad7b..630e798 100644 --- a/content/Statistics/gams/_index.rmd +++ b/content/Statistics/gams/_index.rmd @@ -38,8 +38,8 @@ The function $f$ can be something more funky or kinky - here, we're going to foc You can have combinations of linear and smooth terms in your model, for example $$y = \beta_0 + x_1\beta_1 + f(x_2) + \varepsilon, \quad \varepsilon \sim N(0, \sigma^2)$$ or we can fit generalised distributions and random effects, for example -$$ln(y) = \beta_0 + f(x_1) + \varepsilon, \quad \varepsilon \sim Poisson(\lambda)$$ -$$ln(y) = \beta_0 + f(x_1) + z_1\gamma + \varepsilon, \quad \varepsilon \sim Poisson(\lambda), \quad \gamma \sim N(0,\Sigma)$$ +$$\ln(\lambda) = \beta_0 + f(x_1), \quad y \sim Poisson(\lambda)$$ +$$\ln(\lambda) = \beta_0 + f(x_1) + z_1\gamma, \quad y \sim Poisson(\lambda), \quad \gamma \sim N(0,\Sigma)$$ ### A simple example

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