Spectrum

The autocorrelation plot is not the only tool that we can use to spot periodicities in a time series. The spectrum plot gives information about how power (or variance) in a series is distributed according to frequency.

Figure left: London Measles Spectrum.

The figure left shows the spectrum plot obtained from the first half of the London measles time series. The largest peak occurs at the frequency of 0.5 cycles/year, which corresponds to one cycle every two years or a biennial oscillation. There is also a large peak corresponding to annual oscillation and also a slightly smaller one at three cycles per two years.

Indeed, the spectrum plot can be produced for the other cities. Below we have produced it for Bristol, Liverpool, Manchester and Newcastle and interesting differences emerge. Bristol appears fairly similar to London, whereas the biennial peak seems to smaller relative to the annual peak for Liverpool and Newcastle compared to London and the biennial peak for Manchester seems more pronounced.

The spectrum and the autocorrelation are complementary plots. The spectrum produces information about the frequency content of a time series and the correlogram about internal time-linking. However, one can see that they should be related. For example, a time series with a large negative autocorrelation at lag one means that consecutive values typically do not agree with each other which implies a very fast oscillation which will turn up in the spectrum as a high frequency oscillation. In mathematical terms the autocorrelation and spectrum are linked it is possible to write one in terms of the other and vice versa (they are a Fourier transform pair).

Stationary Models...