Assume that we have a time series and we have calculate the corresponding auto-covariance function. Having the auto-covariance function we can calculate the corresponding power spectrum and having the power spectrum we can "restore" the auto-covariance function.

As far as I understood, to obtain the power spectrum we use Fourier transformation and to calculate the auto-covariance function from the power spectrum we use a back Fourier transformation. So, one can say that auto-covariance function and power spectrum are just time and frequency representation of the same function.

The first part of my question is about the interpretation of the power spectrum. For example, for the white-noise the power spectrum is a constant. Does it mean that the white noise can be understood as a sum of sinusoid functions with different frequencies and the same amplitude (the density distribution function of the frequency is uniform)?

I think that the answer is "No". I think that the auto-covariance function should be understood as an infinite sum of sinusoids.

The second part of the question is about use of the power-spectrum. Assume that we have it. Can we use it to predict time series? Or how do we use it?


The power spectrum is the frequency-domain counterpart of the time-domain autocovariance function.

According to the frequency-domain view, a white noise process can be viewed as the sum of an infinite number of cycles with different frequencies where each cycle has the same weight.

The power spectrum is not used to predict a time series. It is used to examine the main characteristics of the series and propose a time series model accordingly. For example, it can be used to detect if seasonality is present in the data, if so, the spectrum will show peaks at the seasonal frequencies.

In principle, the same information can be obtained from both the spectrum and the autocovariance function, but in some contexts it is more convenient to work with the frequency-domain representation. For example, if you are interested in looking at the seasonal frequencies that determine the seasonal pattern observed in the series, the spectrum will show you the cycles that lead that component. If you want to analyze the frequency of some phenomenon such as the business cycle, the spectrum is a straightforward way to get a graphical view to it.

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Spectral density function is an Fourier transform of autocovariance function and there exists a Rietz - Fisher Representation theorem which states that time domain and frequency domain representations for the time series are almost equal.

There are actually methods for estimation on frequency domain which might take into account various frequency bands.


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  • $\begingroup$ (+1) Nice reference. $\endgroup$ – javlacalle Oct 29 '14 at 9:28

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