PMFs.Rmd

---
title: "Plotting pmf's and probability densities"
author: "Douglas Bates"
date: "11/12/2014"
output:
  ioslides_presentation:
    keep_md: false
    smaller: yes
    widescreen: yes
---
```{r preliminaries,echo=FALSE,results='hide',cache=FALSE}
library(knitr)
library(ggplot2)
library(reshape2)
opts_chunk$set(fig.align='center',cache=TRUE)
```


# Plotting pmf's
## Use of factors in ggplot2
- We usually plot the probability density function for a continuous
  distribution as a curve or overlaid curves for multiple densities
- A probability mass function for a discrete distibution is usually
  plotted as vertical bars.  This makes showing the pmf for multiple
  distributions a bit more challenging.
- In `qplot` we change from a curve to bars with
  `geom="bar"` *and* changing the x values to a factor.
- The plot on the next slide is for a binomial distribution with
  $n=20$ and $p=0.5$.  It is produced by

```{r bin2005show,eval=FALSE}
library(qqplot2)
xvals <- 0:20          # integer sequence from 0 to 20
qplot(factor(xvals), dbinom(xvals, size=20, prob=0.5), geom="bar", stat="identity", ylab="p(k)", xlab="k")
```

## Pmf of Binomial, n = 20, p = 0.5

```{r bin2005,fig.align='center',echo=FALSE}
xvals <- 0:20
qplot(factor(xvals), dbinom(xvals, size=20, prob=0.5), geom="bar", ylab="p(k)", xlab="k", stat="identity")
```

## Use of xlim to limit the values on the x axis
- Often the pmf will be negligible over some of the possible values of $k$
- If we see that the probability mass is highly concentrated then we
  may wish to restrict the range of x values
- Code for the next two figures is
```{r bin10005show,eval=FALSE}
xvals <- 0:100
p <- qplot(factor(xvals), dbinom(xvals, size=100, prob=0.5), geom="bar", ylab="p(k)", xlab="k", stat="identity")
xvr <- 50+ (-15:15)  # to get a centered range
qplot(factor(xvr), dbinom(xvr, size=100, prob=0.5), geom="bar", ylab="p(k)", xlab="k", stat="identity")
```

## Pmf of Binomial, n = 100, p = 0.5
```{r bin10005,fig.align='center',echo=FALSE}
xvals <- 0:100
(p <- qplot(factor(xvals), dbinom(xvals, size=100, prob=0.5), geom="bar", ylab="p(k)", xlab="k", stat="identity"))
```

## Pmf of Binomial, n = 100, p = 0.5, restricted width
```{r bin10005r,fig.align='center',echo=FALSE}
xvr <- 50+ (-15:15)  # to get a centered range
qplot(factor(xvr), dbinom(xvr, size=100, prob=0.5), geom="bar", ylab="p(k)", xlab="k", stat="identity")
```

## Overlaying pmf's
- It is more difficult to overlay probability mass functions, plotted
  as bars, than to overlay probability density functions, plotted as
  curves.
- You need to specify the position adjustment so that multiple bars
  are visible
- As for the pdf's, you can create multiple evaluations of probability
  functions and melt them to a long shape.
```{r overlayshow,eval=FALSE}
xvals <- 0:20
fr <- melt(variable.name="p", id.vars="k",
           data.frame(k=factor(xvals), 
                      p1=dbinom(xvals,20,0.1),
                      p5=dbinom(xvals,20,0.5), 
                      p9=dbinom(xvals,20,0.9)))
levels(fr$p) <- as.character(c(0.1,0.5,0.9))
(p <- qplot(factor(k), value, data=fr, geom="bar", ylab="p(k)", fill=p, stat="identity"))
```


## Straight overlay
```{r binoverlay,fig.align='center',echo=FALSE}
xvals <- 0:20
fr <- melt(variable.name="p", id.vars="k",
           data.frame(k=factor(xvals), 
                      p1=dbinom(xvals,20,0.1),
                      p5=dbinom(xvals,20,0.5), 
                      p9=dbinom(xvals,20,0.9)))
levels(fr$p) <- as.character(c(0.1,0.5,0.9))
(p <- qplot(factor(k), value, data=fr, geom="bar", ylab="p(k)", fill=p, stat="identity"))
```

## Dodge position
```{r bindodge,fig.align='center',echo=FALSE}
qplot(factor(k), value, data=fr, geom="bar",ylab="p(k)", fill=p, position="dodge",stat="identity")
```

# Evaluating and plotting densities

## Functions to evaluate densities<a id="sec-1-1"></a>

-   Functions to evaluate probability densities in R have names of the
    form `d<dabb>` where `dabb` is the abbreviated distribution name.  For
    example, `norm` for the normal (or Gaussian) density, `unif` for the
    uniform density, `exp` for the exponential density.  A more complete
    list of distributions and their abbreviations is given 
    [here](http://blog.revolutionanalytics.com/2010/08/distributions-in-r.html).

-   One simple way of plotting a theoretical density function is to
    establish a range of x values, evaluate the density (or probability
    mass function) on these values and plot the result.

## Determining the range of x values

-   It is not always straightforward to decide what a reasonable range of
    x values would be.  For example, if I want to plot the exponential
    density for the rate, $\lambda=0.2$, how far out on the right-hand tail
    should I go?  One way to answer this is to find, say, the 0.995
    quantile.
```{r expmax}
(xmax <- qexp(0.995, rate=0.2))
```
    and choose equally spaced values from 0, below which the density is zero, to `xmax`
```{r xvals}
xvals <- seq(0, xmax, length=100)
```


## Creating a plot
```{r densplot,fig.align='center'}
qplot(xvals, dexp(xvals, rate=0.2), geom="line", ylab="density", xlab="x")
```