---
title: "The Distribution of Distances Between Discrete Events in Fixed Time"
author: "Kristian Hovde Liland"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{The Distribution of Distances Between Discrete Events in Fixed Time}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

# Introduction

This small package contains the Liland distribution for distances between discrete events in fixed time with probability mass, cumulative distribution, quantile function, random number generator, simulation functions and a test for over representation of short distances.

# Example

An example of its use is found in bacterial gene regulation where genes along a chromosome are regulated or not regulated. One may ask if the distances between regulated genes are random or tend to cluster, e.g. as operons. In the following example we have $R=1949$ genes (trials) of which $r=162$ are regulated (success).

```{r}
library(fixedTimeEvents)
R <- 1949; r <- 162
Liland(R, r)
```

\pagebreak

## Probability mass

```{r}
R <- 1949; r <- 162
dL <- dLiland(1:100, R, r)
plot(dL, type = 'l', xlab = "distance", ylab = "probability mass")
```
  
## Testing

A test for over representation of short distances can be performed, e.g. for distances shorter than 2 ($x<2$).

```{r}
Lt <- Liland.test(1:100, 1, R, r)
Liland.crit(1, R, r)
plot(Lt, type='l', xlab='#(x<2)', ylab='p-value')
points(73, Liland.test(73, 1, R, r), col = 2)
```
  
## Simulated distribution

A comparison between distances obtained from sampling from the Bernoulli distribution with
a fixed number of successes and the theoretical values from the Liland distribution follows.

```{r}
sL <- simLiland(5000, 15,5) # 5000 samples, R = 15, r = 5
qqplot(dLiland(1:length(sL),15,5),sL,
       xlab='F(x;15,5)', ylab='Sample (5000)')
abline(0,1, lty=2, col=2)
```