Relation between autocorrelation function and periodogram in time series analysis

Question

I was wondering if anyone could give me some insight on the relation between the ACF and the periodogram of a time series.

I have a bunch of timeseries and their ACF's and periodograms are typically much like the examples below.

For my analysis, I'm mostly interested in periodicity at lag 8 and 16 (for theoretical reasons)

The frequencies 'B' and 'HB' correspond to lag 16 and lag 8 respectively. The time series actually concerns interresponse intervals in musical performance of a piece that consists solely of eighth notes (16 of them in a 4:4 bar so 'B' stand for bar and 'HB' for half bar).

The thing I actually wanted to ask: in my periodograms, I consistently get very large peaks at frequency 0.25 (which corresponds to lag 4). However, the ACF peak at lag 4 is much smaller than those at lag 8 or 16. I was wondering how to interpret this finding. A lot of time series variance can be explained at this frequency even though the lag 4 autocorrelation is quite low?

I hope I was sufficiently clear in my question. If not, don't hesitate to ask me.

Answer 1

The relation between the autocovariance (or autocorrelation) and the spectral density (for which the periodogram is an estimator) is given by the Fourier transform. The two form a so-called Fourier-transform pair meaning the two are time(or space)-domain vs. frequency-domain representations of the same thing. Specifically, if time series $\{X_t\}$ has autovariance function $\gamma(\tau)$ at time lag $\tau$, then the spectral density is defined by \begin{equation} f(\nu)=\sum\limits_{\tau=-\infty}^{\infty} \gamma(\tau)e^{-2i\pi\nu\tau}. \end{equation} In words, the spectral density partitions the autocovariance as energy-per-hertz of a signal. For example, if you have a deterministic signal with period $t=12$, then the series lagged with itself (ACF) at lag 12 will be perfectly correlated (autocorrelation=1). Subsequently, all power in the spectral density will be concentrated at frequency $1/t$.