---
title: "The `bridgedist` Basics"
author: "Bruce J. Swihart"
date: "`r Sys.Date()`"
output: 
  rmarkdown::html_vignette:
    fig_caption: yes
vignette: >
  %\VignetteIndexEntry{The `bridgedist` Basics}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---


```{r, eval=TRUE}
library(bridgedist)
```
<!--
Recreate the plot from Wang and Louis (2003) where the Bridge, Normal, and Logistic all have unit variance and mean 0:

```{r, eval=FALSE}
xaxis <- seq(-3,3,.01) 
plot(xaxis, dbridge(xaxis, phi=1/sqrt(1+3/pi^2)), type="l",
     xlab="x", ylab="density(x)")
lines(xaxis, dnorm(xaxis))
lines(xaxis, dlogis(xaxis, phi=sqrt(3/pi^2)))
```
-->

Recreate the plot from Wang and Louis (2003) where the Bridge, Normal, and Logistic all have unit variance and mean 0 with ggplot2:

```{r, fig.width=6, fig.cap = "Fig. 1. Probability density functions of the Gaussian, logistic and bridge, for logistic, distributions each with zero mean and unit variance."}
library(reshape2)
library(ggplot2)
xaxis = seq(-4,4,.01)
df = data.frame( xaxis,
                 Bridge = dbridge(xaxis, phi=1/sqrt(1+3/pi^2)),
                 Normal = dnorm(xaxis),
                 Logistic = dlogis(xaxis, scale=sqrt(3/pi^2)))
melt.df <- melt(df, id.vars = "xaxis")
colnames(melt.df) <- c("x", "Distribution", "value")
ggplot(melt.df, aes(x, value, color=Distribution)) + 
  geom_line(size=1.05) + 
  ylab("Probability density function") 
```

The implication is that a random variable from a Bridge distribution plus random variable from a standard logistic distribution is a logistic random variable with a scale greater than one (1/phi).  


```{r, fig.width=6, fig.cap = "Fig. 2. 10000 random variates in each panel.  From left to right: the bridge distribution, the logistic with scale=1, the sum of the previous two, and the logistic with scale=1/phi.  Note how similar the third and fourth panel, the application supporting the theory.", warning=FALSE, message=FALSE}
phi <- 0.5
df = data.frame(
                 Bridge = rbridge(1e5, phi=phi),
                 Std_Logistic = rlogis(1e5),
                 BridgePlusStd_Logistic = rbridge(1e5, phi=phi) +  rlogis(1e5),
                 Logistic = rlogis(1e5, scale=1/phi)
)
melt.df <- melt(df)
colnames(melt.df) <- c("Distribution", "value")
ggplot(melt.df, aes(value)) +
  facet_grid(.~Distribution) +
  geom_histogram()
```