## Cycles Definition of Terms |
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**Amplitude** is the range of the cycle from low to high measured
in whatever units the cycle is in.

**Amplitude Modulation** means that the amplitude of a cycle varies
over a longer cycle. For example, radio uses AM to transmit an audible
sound of say 400 Hz by varying the amplitude of a radio signal with frequency
100,000 Hz.

**Beats** are produced by two cycles which have similar frequencies
(or periods). For example if two adjacent notes are played on the piano
they might have frequencies of 220 Hz and 233 Hz and they will produce
beats at 13 Hz which is heard as a sort of wa-wa-wa-wa sound. Beats are
equivalent to Amplitude and Frequency modulation at the same time so we
may either look for multiple cycles of similar period or one with a modulating
cycle. These are equivalent.

**Beat Frequency** is the difference between the frequencies of the
two cycles that produce the beats. It is much easier to calculate beats
with frequencies than with periods, as for example, periods of 3 and 4
years will produce beats of 1/(1/3)-(1/4) years or 12 years.

**Calendar.** In modern times the rules are: Thirty days hath September,
April, June and November, all the rest have 31 except February alone which
has 28 and 29 in leap years. Leap years are years which are a multiple
of 4 unless they are a multiple of 100 when they are not leap years unless
they divide by 400 when they are again. I hope that is perfectly clear
:-)

A **Filter** is a device, probably mathematical, that allows some
ranges of frequencies through and not others. It may partially allow some
through. Simple filters can be made by using moving averages.

**Fourier Analysis** is a mathematical technique for turning a time
series into a set of coefficients which are the amplitudes of a set of
sine waves of various frequencies so that those sine waves add up to the
original data.

**Frequency** is the number of cycles per unit time interval, e.g.
7.5 cycles per second (also called Hertz) means that there are 7.5 cycles
in each second of time. Frequency is related to period by: Frequency =
1/Period

**Frequency Modulation** is a variation in the frequency of a cycle
over a longer cycle. For example, radio uses FM to transmit an audible
sound of say 400 Hz by varying the frequency of a radio signal with frequency
100,000,000 Hz.

**Harmonics** of a cycle are other cycles whose frequencies are exact
multiples of the original frequency. Therefore the periods are exact fractions.
For example, a 5.93 year cycle is the 2nd harmonic of an 11.86 year cycle.

**Phase** refers to the timing of the peaks or troughs. For example
we might say that the phase of the 1 year temperature cycle has a maximum
in January (I live in the southern hemisphere).

**Phase Change** refers to a variation in the phase of a cycle. This
is equivalent to saying that the frequency is modulated although may refer
to more extreme cases. Sometimes cycles have a 180 degree phase change,
which means that peaks start happening where troughs were expected and
vice versa.

**Moving Average** is method of smoothing by averaging n terms of
the time series. A moving average may be **centred** in which case it
is plotted at the middle of the time interval which it is the average of,
or **lagged** in which case it is plotted at the last term included.
Centred moving averages are often used for smoothing data. A moving average
which has a number of terms equal to the length of a cycle will remove
that cycle from the data. A moving average with a number of terms which
is half the cycle period will remove shorter cycles and so emphasize the
cycle. The difference between these these last two will also remove longer
cycles and so emphasize cycles of about the selected period only.

**Noise** is any time series which has a spectrum which is continuous
rather than having specific periods present. Noise comes in various types
which are named after colours such as White, Brown and Pink according to
the variation of amplitude with frequency. There is usually noise present
in any time series in addition to any cycles.

**Period** is the time interval between two adjacent peaks or two
adjacent troughs in a cycle, e.g. a cycle period of 4.45 years means that
a full cycle occurs every 4.45 years. Period is related to frequency by:
Period = 1/Frequency

**Smoothing** means removing the smaller random fluctuations so as
to see the trend. Smoothing may be done by a moving average or by various
filters which are equivalent to weighted moving averages.

**Time Series.** Any series of measurements of anything at all that
are usually taken at regular intervals. Sometimes the intervals are not
strictly regular such as calendar months, which are near enough for most
purposes, or trading days which have weekends and public holidays missing
which may cause some problems.

**Weighted Moving Average** is like a moving average but not all
terms are equal. Each moving average term is determined by weighting a
series of consecutive values, for example in the proportions 1:2:3:2:1,
before averaging.