Lucilia cuprina (sheep blowfly)
A. J. Nicolson
Life cycle period about 38 days
Adult population count data with ∆t = 2 days
n=?
Guckenheimer and Oster
“Chaos appearing?” nonstationary
Periodogram
Log scale on y-axis
Peak around .05 cycles/unit time
Complex signal Z(t) = V1 (t) + I V_2 (t)
When “at” a period (5) is δk
demod.R
demod<-function(x,freq,span)
{#freq is frequency in cycles/unit time
#span is the length of the smoothing filter. The filter is a cosine taper.
nx<-length(x)
u<-cos(2*pi*freq*c(1:nx))*x
v<-sin(2*pi*freq*c(1:nx))*x
ma<-1-cos(2*pi*c(1:span)/(span+1))
uf<-filter(u,ma)/sum(ma)
vf<-filter(v,ma)/sum(ma)
phase<-atan2(vf,uf)
amp<-.5*log(uf^2+vf^2)
mm<-(span-1)%/%2+1
phase<-c(rep(NA,mm),phase[c(mm:(nx-mm))])
amp<-c(rep(NA,mm),amp[c(mm:(nx-mm))])
bw<-2/span
par(mfrow=c(2,1))
plot(phase,ylim=c(-pi,pi),xlim=c(1,nx),pch=".",main=paste("period",round(1/freq,1)))
plot(amp,xlim=c(1,nx),ylab="log(amp)",pch=".")
return()}
Dynamic spectrum/spectrogram
Commands. Shumway-Stoffer
nobs = length(EXP6) # number of observations
2 wsize = 256 # window size
3 overlap = 128 # overlap
4 ovr = wsize-overlap
5 nseg = floor(nobs/ovr)-1; # number of segments
6 krnl = kernel("daniell", c(1,1)) # kernel
7 ex.spec = matrix(0, wsize/2, nseg)
8 for (k in 1:nseg) {
9 a = ovr*(k-1)+1
10 b = wsize+ovr*(k-1)
11 ex.spec[,k] = spectrum(EXP6[a:b], krnl, taper=.5, plot=F)$spec
}
12 x = seq(0, 10, len = nrow(ex.spec)/2)
13 y = seq(0, ovr*nseg, len = ncol(ex.spec))
14 z = ex.spec[1:(nrow(ex.spec)/2),]
15 filled.contour(x , y, log(z), ylab="time",xlab="frequency
(Hz)",
nlevels=12, col=gray(11:0/11), main="Explosion")
16 persp(x, y, z, zlab="Power",xlab="frequency (Hz)",ylab="time",
ticktype="detailed", theta=25,d=2, main="Explosion") # not shown
The Nicholson equation
http://www.stat.berkeley.edu/~brill/papers04a.html
The Nicholson blowfly experiments: some history and EDA. J. Time Series
Analysis (2012).
Aligning some Nicholson sheep-blowfly data sets with system input
periods Stat (Wileyonlinelibrary.com) DOI:10.1002/stat4.13 (2013)