I have a bunch of time series whose power spectra (FFT via R‘s spectrum() function) I’ve been trying to visualize in an intuitive, aesthetically appealing way. At first, I just used lattice’s bwplot, but the spacing of the X-axis here really matters. The spectra’s frequencies aren’t regularly-spaced categories, which is the default of bwplot. If all the series are of the same length, then the coefficients of the are all estimated at the same frequencies, and one can use plyr‘s split-apply-combine magic to compute the distribution of coefficients at each frequency.

I spent some time trying to faithfully reproduce the plotting layout of the figures produced by, e.g. spectrum(rnorm(1e3)) with respect to the axes. This was way more annoying than i expected…

To include a generic example, I started looking around for easily generated, interesting time series. The logistic map is so damned easy to compute and generally pretty interesting, capable of producing a range of frequencies. Still, I was rather surprised by the final output. Playing with these parameters produces a range of serious weirdness — apparent ghosting, interesting asymmetries, etc..

Note that I’ve just used the logistic map for an interesting, self-contained example. You can use the exact same code to generate your own spectrum() box-and-whisker plot by substituting your matrix of time series for mytimeseries below. Do note here that series are by row, which is not exactly standard. For series by column, just change the margin from 1 to 2 in the call to adply below.

## box-and-whisker plot of FFT power spectrum by frequency/period
## vary these parameters -- r as per logistic map
## see http://en.wikipedia.org/wiki/Logistic_map for details
logmap = function(n, r, x0=0.5) { 
    ret = rep(x0,n)
    for (ii in 2:n) {
        ret[ii] = r*ret[ii-1]*(1-ret[ii-1])
    }
    return(ret)
}

## modify these for interesting behavior differences
rfrom = 3.4
rto = 3.7
rsteps=200
seqlen=1e4

require(plyr)
require(ggplot2)

mytimeseries = aaply(seq(from=rfrom, to=rto, length.out=rsteps), 1,
function(x) {
       logmap(seqlen, x)
})

## you can plug in any array here for mytimeseries
## each row is a timeseries
## for series by column, change the margin from 1 to 2, below
logspec = adply( mytimeseries, 1, function(x) {
       ## change spec.pgram parameters as per goals
       ret = spectrum(x, taper=0, demean=T, pad=0, fast=T, plot=F)
       return( data.frame(freq=ret$freq, spec=ret$spec))
})

## boxplot stats at each unique frequency
logspec.bw = ddply(logspec, 'freq', function(x) {
           ret = boxplot.stats(log10(x$spec));
           ## sometimes $out is empty, bind NA to keep dimensions correct
           return(data.frame(t(10^ret$stats), out=c(NA,10^ret$out)))
})

## plot -- outliers are dots
## be careful with spacing of hard returns -- ggplot has odd rules
## as close to default "spectrum" plot as possible
ggplot(logspec.bw, aes(x=1/(freq)))  +  geom_ribbon(aes(ymin=X1,
ymax=X5), alpha=0.35, fill='green')  +
       geom_ribbon(aes(ymin=X2, ymax=X4), alpha=0.5, fill='blue') +
geom_line(aes(y=X3))  +
       geom_point(aes(y=out), color='darkred') +
       scale_x_continuous( trans=c('inverse'), name='Period')  +
       scale_y_continuous(name='Power', trans='log10')

Following my nose further on the logistic map example, I also used the animate package to make a movie that walks through the power spectrum of the map for a range of r values. Maybe one day I’ll set this to music and post it to Youtube! Though I think some strategic speeding up and slowing down in important parameter regions is warranted for a truly epic tour of the logistic map’s power spectrum.

require(animation)
saveVideo({
       ani.options(interval = 1/5, ani.type='png', ani.width=1600, ani.height=1200)
        for (ii in c(seq(from=3.2, to=3.5, length.out=200), seq(from=3.5, to=4, length.out=1200))) {
            spectrum(logmap(1e4, ii), taper=0, demean=T, pad=0, fast=T, sub=round(ii, 3))
        }
},  video.name = "logmap2.mp4", other.opts = "-b 600k")