Testing for periodicity of noisy biological data: periodogram significance?

Question

I'm trying to analyze some noisy time-series data in R. The data are based on the CO2 emission of animals and they show a sort of cyclic periodicity that I'd like to characterize. I'd like to test the hypotheses:

H0: There is no cyclic CO2 emission (i.e. no more than random).

H1: There is a pattern of CO2 emission in cycles or pulses.

So to do this I've imported the data into R, converted it to a time series class, and plotted its periodogram.

t25a <- read.table("data.txt", header=TRUE, sep="\t")
t1 <- ts(t25a$Co2)
plot(t1)
spec.pgram(t1, spans=4, log="no")

Here's what that looks like, with the raw data plotted on top and the periodogram beneath:

In the bottom figure, I can see four or five somewhat-distinct peaks indicating a frequency component in the data. My question is -- are they all equally "important"? Is there any way to test whether the observed peaks are significantly different from each other or from the predictions of the null hypothesis? All I know how to do is find the frequency associated with those peaks, but I'd like a more objective method for determining how many "significant" peaks there really are in the data.

Answer 1

One option would be to simulate data sets under your null hypothesis (don't have the periodicity that you are looking for, but still have the other time series characteristics). If you have a numerical test statistic (number of peaks, or some other measure) then you can compute this for each of many simulated data sets and this will give you the sampling distribution, just compare the test statistic for your actual data to the sampling distribution. If you don't have a straight forward numeric test statistic then you might consider doing a visual test, see:

 Buja, A., Cook, D. Hofmann, H., Lawrence, M. Lee, E.-K., Swayne,
 D.F and Wickham, H. (2009) Statistical Inference for exploratory
 data analysis and model diagnostics Phil. Trans. R. Soc. A 2009
 367, 4361-4383 doi: 10.1098/rsta.2009.0120

The vis.test function in the TeachingDemos package for R helps with implementing this test (but there are other ways as well).