Last updated: 2018-05-21

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    Rmd 34e295c stephens999 2018-05-21 workflowr::wflow_publish(“density_est_cell_cycle.Rmd”)
    Rmd 1a00688 stephens999 2018-04-11 add density estimation example

Introduction

We will try estimating the density of the cell cycle values. See (cell_cycle.Rmd) for more on those data.

Histograms

First we try histograms. The histograms here are un-normalized, so not actually densities. We could normalize them easily enough, but I think it is helpful for you to see the counts get smaller.

Let’s start with the default:

d = readRDS("../data/cyclegenes.rds")
hist(d[,1],xlab="theta (cell cycle position)", main="Histogram of theta")

And now try decreasing bin sizes to see the bias/variance trade-off in action.

hist(d[,1],breaks = seq(0,2*pi,length=5),xlab="theta (cell cycle position)", main="Histogram of theta")

hist(d[,1],breaks = seq(0,2*pi,length=20),xlab="theta (cell cycle position)", main="Histogram of theta")

hist(d[,1],breaks = seq(0,2*pi,length=100),xlab="theta (cell cycle position)", main="Histogram of theta")

hist(d[,1],breaks = seq(0,2*pi,length=400),xlab="theta (cell cycle position)", main="Histogram of theta")

hist(d[,1],breaks = seq(0,2*pi,length=1000),xlab="theta (cell cycle position)", main="Histogram of theta")

Density estimation

Let’s try the default density estimation in the R function “density”.

hist(d[,1],breaks = seq(0,2*pi,length=100),xlab="theta (cell cycle position)", main="Histogram of theta",probability=TRUE)
d.dens = density(d[,1])
lines(d.dens$x,d.dens$y,col=2,lwd=2)

The analogue of bin size is “bandwidth”, set using bw. Here we try four very different bandwidths.

hist(d[,1],breaks = seq(0,2*pi,length=100),xlab="theta (cell cycle position)", main="Histogram of theta",probability=TRUE)
d1.dens = density(d[,1],bw=0.01)
d2.dens = density(d[,1],bw=0.1)
d3.dens = density(d[,1],bw=1)
d4.dens = density(d[,1],bw=10)

lines(d1.dens$x,d1.dens$y,col=2,lwd=2)
lines(d2.dens$x,d2.dens$y,col=3,lwd=2)
lines(d3.dens$x,d3.dens$y,col=4,lwd=2)
lines(d4.dens$x,d4.dens$y,col=5,lwd=2)

A common issue with density

Go back to the origin plot.

hist(d[,1],breaks = seq(0,2*pi,length=100),xlab="theta (cell cycle position)", main="Histogram of theta",probability=TRUE)
d.dens = density(d[,1])
lines(d.dens$x,d.dens$y,col=2,lwd=2)

Notice how the density seems too low at both ends of the plot. This is a common problem with density when applied to a finite domain like this. It is because it is estimating the density at 0 and notices there are no observations just to the left of 0, and this biases the estimate of density down.

We can see the problem better if we plot a wider area:

hist(d[,1],breaks = seq(-pi,3*pi,length=100),xlab="theta (cell cycle position)", main="Histogram of theta",probability=TRUE)
d.dens = density(d[,1])
lines(d.dens$x,d.dens$y,col=2,lwd=2)

To deal with this a simple trick is to reflect the data at both ends.

x = d[,1]
x = c(-x,x,2*pi + 2*pi-x)
hist(x,prob=TRUE)

er… yuk! the default histogram was not very nice - have to do it by hand!

hist(x,breaks = seq(-2*pi, 4*pi,length=100),probability=TRUE)
abline(v=0,col=2)
abline(v=2*pi,col=2)

So now we can estimate the density

x.dens = density(x)
hist(x,breaks = seq(-2*pi, 4*pi,length=100),probability=TRUE)
abline(v=0,col=2)
abline(v=2*pi,col=2)
lines(x.dens$x,x.dens$y,col=2)

Also yuk!! I had to experiment with the parameter bw to get a nicer plot….

x.dens = density(x,bw=0.2)
hist(x,breaks = seq(-2*pi, 4*pi,length=100),probability=TRUE)
abline(v=0,col=2)
abline(v=2*pi,col=2)
lines(x.dens$x,x.dens$y,col=3)

I now could get a nice density estimate in the range 0 to 2pi by just truncating and renormalizing to have area 1.

Session information

sessionInfo()
R version 3.3.2 (2016-10-31)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X El Capitan 10.11.6

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] workflowr_1.0.1   Rcpp_0.12.16      digest_0.6.15    
 [4] rprojroot_1.3-2   R.methodsS3_1.7.1 backports_1.1.2  
 [7] git2r_0.21.0      magrittr_1.5      evaluate_0.10.1  
[10] stringi_1.1.7     whisker_0.3-2     R.oo_1.22.0      
[13] R.utils_2.6.0     rmarkdown_1.9     tools_3.3.2      
[16] stringr_1.3.0     yaml_2.1.18       htmltools_0.3.6  
[19] knitr_1.20

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