I recently enjoyed reading O’Hara, R. B., & Kotze, D. J. (2010). Do not log-transform count data. Methods in Ecology and Evolution, 1(2), 118–122. doi:10.1111/j.2041-210X.2010.00021.x.

The article prompted me to think about processes involving discrete events and how these might be presented graphically. I am not talking about counts (which are well represented by a histogram) but the individual events themselves. The problems here being that

1. the data are essentially one dimensional (just a sequence of times at which events occurred) and
2. there may be a great number of events and they can be distributed over a considerable period of time.

Plotting the events as a series of points along a linear axis would therefore make a rather boring plot, possibly with a rather extreme aspect ratio. There had to be a better way! What about wrapping that axis up into an Archimedes’ spiral? Sounds reasonable. Let’s take a look.

First Iteration

Here time runs along the spiral and points indicate the times at which events occurred. In this case I have 21 events occurring at uniform intervals. Although it looks okay, there is one major flaw: the angular separation of the points is uniform but this is not consistent with the idea of a spiral axis. The points should be distributed uniformly in terms of arc length along the spiral!

Revision: Spiral Arc Length

I needed to calculate the arc length along the spiral. Since I was not concerned with the absolute length, I neglected the spiral’s pitch, giving a function which depended only on angle.

spiral.length <- function(phi) {
    phi * sqrt(1 + phi**2) + log(phi + sqrt(1 + phi**2))
}

Then I could interpolate to find the correct location of the events.

Now the events, which are distributed uniformly in time, appear at uniform intervals along the spiral axis. Mission accomplished.

Here is the code to generate the spiral plot:

spiral.plot <- function(t, nturn = 5, colour = "black") {
    npoint = nturn * 720
    #
    curve = data.frame(phi = (0:npoint) / npoint * 2 * pi * nturn, r = (0:npoint) / npoint)
    curve = transform(curve,
                      arclen = spiral.length(phi),
                      x = r* cos(phi),
                      y = r * sin(phi))
    #
    points = data.frame(arclen = t * max(curve$arclen) / max(t))
    points = within(points, {
        phi = approx(curve$arclen, curve$phi, arclen, rule = 2)$y
        r = approx(curve$arclen, curve$r, arclen, rule = 2)$y
        x = r* cos(phi)
        y = r * sin(phi)
    })
    #
    ggplot(curve, aes(x = x, y = y)) + 
        geom_path(colour = "grey") +
        geom_point(data = points, aes(x = x, y = y), size = 3, colour = colour) +
        coord_fixed(ratio = 1) +
        theme(axis.text = element_blank(),
              axis.ticks = element_blank(),
              axis.title = element_blank(),
              panel.background = element_blank(),
              panel.grid.major = element_blank(),
              panel.grid.minor = element_blank())
}

It is unfortunate that I had to transform the data to Cartesian Coordinates in order to plot it. Although ggplot2 does has the capability to generate polar plots, it does not allow polar angles exceeding a single revolution. If anybody has other ideas on how to deal with this more elegantly, I would be very happy to hear from them.

The first enhancement I would apply to this plot would be to find a way of putting tick marks along the spiral. Again, any input would be appreciated.

Practical Application

What about applying it to a more realistic scenario? If we simulate a radioactive decay process using the exponential distribution to yield a series of decay intervals, then these intervals can be accumulated to find the decay times.

> Bq = 5
>
> delay = rexp(2000, Bq)
> 
> decay = data.frame(delay, time = cumsum(delay))
> spiral.plot(decay$time, 20)

As discussed by O’Hara and Kotze, the distribution of events in clumps of varying sizes separated by intervals without events is readily apparent.