## Cycles Analysis Methods |
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There are many different techniques available in cycles analysis and this is a short guide to the main ones and when they are appropriate.

**Fourier Analysis** is the standard method for converting any time
series into the sum of a series of sine waves of different frqeuencies.
It is important to understand that any series can be exactly constructed
from sine waves and so it does not prove that real cycles exist. However
if a graph is made of the amplitude versus frequency of the components
then real cycles will stand out as having much higher amplitudes, possibly
on several adjacent frequencies. There are methods to interpolate the frequency
and find the best estimate for each cycle that is considered real.

It is desirable to have certain characteristics in data before it is
analysed for cycles. One of these is that the fluctuations are of similarly
meaningful amplitude of the time period. When a Stock Market Index of 200
years is constructed we find that, partly due to inflation, the index has
grown in a compound manner. For data of this type taking **logarithms**
before looking for cycles is highly desirable. This is true of all measurements
made in currencies and with population bases, especially if they are long
term series. If the result of taking logs is to make the short and medium
term variations more consistent in amplitude over the whole series then
it is probably a good idea.

Another useful prepatory function is to take **differences** of adjacent
data points. This is applicable for price data also, as we then create
a rate of price change series which is generally more interesting. When
it comes to making predictions then we want to know whether it will go
up or down and by how much. When this technique is used the spectrum has
the same peaks but the shorter term ones are emphasised at the expense
of the longer ones.

**Example 1** shows the **Price of Wheat** over a period of more
than 100 years. Because this was last century, not this century, there
is little inflation overall although the price varied considerably. In
A the raw data shows wider variations near the middle of the period when
the price was higher. This is a case where taking logs is recommended and
in B we see that this results in more even short term fluctuations. Finally,
in C the result of taking differences between adjacent values is shown.
The cycle of about 5.5 years is still visible, but the shorter term fluctuations
are now stronger and the longer term ones have all but vanished.

Having prepared our data we are finally ready to do a Fourier analysis. In general cycles of more than 2 time intervals can be detected but there is a danger of "aliasing" for cycles at the very short end. Aliasing means that the wrong period is determined because the measurement interval is too short for the cycle.

**Example 2** shows the **spectra** obtained from the above cases
A, B and C. With quarterly prices it is not possible to detect cycles of
less than 1/2 a year. A cycle of exactly one year is present and it is
especially easy to see in C. Of course a 1 year cycle is just the seasons
and we would expect the price of wheat to vary with the seasons as it is
a seasonal crop and requires storage costs to be available at other times.

There are the same set of cycles which are present in all three graphs in the 5 to 20 year range. The strongest of these is about 5.5 years and is clearly visible in the original data without much imagination. In B this peak is split into 2 while in A and C it is almost split. This indicates that the cycle is modulated on a long term basis. The modulation may be just a result of changing circumstances and have no ongoing basis.

The accurate periods found in C are 1.001, 1.697, 4.20, 4.75, 5.54, 7.3, 9.1, 14.3 years. The strongest ones are the 1 year, the 1.697 year and the 5.54 year cycles. These 3 cycles are almost certainly real and C has correlation coeficients of .225, .198 and .187 respectively with sine waves of the stated periods. The seasonal one would not be disputed.

The period 5.54 years is exactly half of the average **sunspot cycle**
period. There has been debate about the role of sunspots in affecting the
weather and life on earth. However in recent years improved data and methods
have shown that there is no doubt that the earth is affected by variations
in the sun. But why half? Well, if the sunspot cycle does affect the weather
and in turn how well wheat grows then we would expect the variations to
be go beyond optimum condition in two directions (be it too hot and too
cold or too wet and too dry) and therefore in one sunspot cycle there would
be two cycles in wheat growth.

The 1.697 year period is also interesting. There is a weak cycle at
3.39 years which is exactly double that period. The 3.39 year (40.68 month)
cycle is also the dominant cycle in the US stock market and commonly called
the **40 month cycle** and is, according to Dewey, the **most common**ly
reported cycle of all. This finding of cycles that are exactly half as
long as others is a very common occurence. They are called
**harmonics**.
Finding cycles of the same length in different series is also common.

You may wonder why the periods are back to front and all bunched up at one end. This is because Fourier analysis measures frequency not period and we calculate the period as 1/frequency. On a frequency basis it is uniformly spread.