bgam_example1.qmd

---
title: "Bayesian GAM Part1"
author: "Murray Logan"
date: today
date-format: "DD/MM/YYYY"
format: 
  html:
    ## Format
    theme: [default, ../resources/ws-style.scss]
    css: ../resources/ws_style.css
    html-math-method: mathjax
    ## Table of contents
    toc: true
    toc-float: true
    ## Numbering
    number-sections: true
    number-depth: 3
    ## Layout
    page-layout: full
    fig-caption-location: "bottom"
    fig-align: "center"
    fig-width: 4
    fig-height: 4
    fig-dpi: 72
    tbl-cap-location: top
    ## Code
    code-fold: false
    code-tools: true
    code-summary: "Show the code"
    code-line-numbers: true
    code-block-border-left: "#ccc"
    code-copy: true
    highlight-style: atom-one
    ## Execution
    execute:
      echo: true
      cache: true
    ## Rendering
    embed-resources: true
crossref:
  fig-title: '**Figure**'
  fig-labels: arabic
  tbl-title: '**Table**'
  tbl-labels: arabic
engine: knitr
output_dir: "docs"
documentclass: article
fontsize: 12pt
mainfont: Arial
mathfont: LiberationMono
monofont: DejaVu Sans Mono
classoption: a4paper
bibliography: ../resources/references.bib
---

```{r setup, include=FALSE, warnings=FALSE, message=FALSE}
knitr::opts_chunk$set(cache.lazy = FALSE, tidy='styler')
```
 
# Preparations
 
Load the necessary libraries

```{r}
#| label: libraries
#| output: false
#| eval: true
#| warning: false
#| message: false
#| cache: false

library(tidyverse)  #for data wrangling etc
library(cmdstanr)   #for cmdstan
library(brms)       #for fitting models in STAN
library(standist)   #for exploring distributions
library(coda)       #for diagnostics
library(bayesplot)  #for diagnostics
library(ggmcmc)     #for MCMC diagnostics
library(DHARMa)     #for residual diagnostics
library(rstan)      #for interfacing with STAN
library(emmeans)    #for marginal means etc
library(broom)      #for tidying outputs
library(tidybayes)  #for more tidying outputs
library(HDInterval) #for HPD intervals
library(ggeffects)  #for partial plots
library(broom.mixed)#for summarising models
library(posterior)  #for posterior draws
library(ggeffects)  #for partial effects plots
library(patchwork)  #for multi-panel figures
library(bayestestR) #for ROPE
library(see)        #for some plots
library(easystats)     #framework for stats, modelling and visualisation
library(mgcv)
library(gratia)
theme_set(theme_grey()) #put the default ggplot theme back
source('helperFunctions.R')
```



# Scenario

This is an entirely fabricated example (how embarrising).
So here is a picture of some Red Grouse Chicks to compensate..

![Red grouse chicks](../resources/redgrousechicks.jpg){#fig-grouse width="251" height="290"}


x  y
-- --
2  3
4  5
8  6
10 7
14 4

: Format of data.gp.csv data file {#tbl-data.gp .table-condensed}

------    -----------------------------
**x**     - a continuous predictor
**y**     - a continuous response
------    -----------------------------

: Format of data.gp.csv data file {#tbl-data.gp1 .table-condensed}

# Read in the data

```{r readData, results='markdown', eval=TRUE}
data_gam <- read_csv("../data/data_gam.csv", trim_ws = TRUE)
```



# Exploratory data analysis

Model formula:
$$
\begin{align}
y_i &\sim{} \mathcal{N}(\mu_i, \sigma^2)\\
\mu_i &=\beta_0 + f(x_i)\\
f(x_i) &= \sum^k_{j=1}{b_j(x_i)\beta_j}
\end{align}
$$

where $\beta_0$ is the y-intercept, and $f(x)$ indicates an additive smoothing function of $x$. 


# Fit the model


# MCMC sampling diagnostics 

# Model validation 

# Partial effects plots 


# Model investigation 


# Further analyses