There are a variety of ways to search for cycles in a time series. Here are some of the more common ones.
1. Fourier analysis (usually FFT or Fast Fourier Transform these days). This is a method of converting a set of regular measurements of something into a set of frequencies that are present and amplitudes and phases of those frequencies. It uses only frequencies that are exact multiples of the lowest detectable frequency. This later feature is a problem for short data sets. If a cycle has modulations then these will appear as complicated structures about a central frequency.
2. Maximum Entropy Spectral Analysis (MESA) is a method designed by Burg which overcomes the exact frequency multiple feature of FFT and also only tries to extract a limited number of cycles, attributing the remaining variations to noise. It is an ideal tool for short data sets or where the frequencies are possibly changing with time and so will appeal to market traders.
3. The difference between two moving averages is a technique that was used by Dewey to highlight a particular period. For example a 9 month cycle might be shown by plotting the data (or perhaps a 3 month centred moving average) less the 9 month centred moving average.
4. A Factor Analysis of a single series repeated multiple times at different time lags will, perhaps surprisingly, tend to extract cycles.
5. Autocorrelations of time series with itself at various lags will tend to show any cycles present, even if the series is very long and the period a little unstable. This is just the situation where FFT is not so useful.
6. ARIMA is a method for determining the various properties of a time series and the best way to treat it such as taking differences, removing seasonal trends and whether it is a random walk or not.
However, before searching for cycles there are sometimes some initial data treatments that are useful in making the analysis more meaningful.
1. For data that does not have a stable mean over time, particularly price data which may go up by large factors over extended periods of time, then it is advisable to take logs of the data first. This means that equal percentage changes in the original data are transformed to equal numerical changes.
2. Take differences between adjacent terms of the data to highlight the rate of change rather than the value itself. This will emphasize the shorter term variations rather than the longer term ones, and for making short term forecasts will often yield more accurate results. Of course this will change a price series into an inflation rate series.
For a great deal more useful information, visit:
Electronic Statistics Textbook is an extremely comprehensive guide to statistics and includes a thorough section on
Time Series Analysis
covering many techniques, although unfortunately not including MESA.
The Maths Archives – Fourier Analysis and Wavelets Links for Java Applets, Fourier Analysis, Harmonic Analysis and other maths archives topics.
The Craft of Economic Modelling
by Clopper Almon, is a tour of how to analyse economic time series. Makes use of G7 program.
Cycles Analysis Methods
by Ray Tomes, similar to this page but with a worked example.
Also, see the other material on the Software and Books page, as many include free manuals for downloading.