Direct frequency analysis

Partitioning a significantly large historical time series into components according to
the duration or length of the intervals within the series is one approach to spectral
or frequency analysis. By considering the series to be the sum of many simple
sinusoids with differing amplitudes, wavelengths and starting points, allows for
the combination of a number of the fundamentals to construct an approximating
forecasting function. The Fourier transform uses the knowledge that if an infinite
series of sinusoids are calculated so that they are orthogonal or statistically independent
of one another, the sum will be equal to the original time series itself. The
expression for frequency function F(n) obtained from the time function f (t) is represented
as:
F(n)ó ê
ñê
f (t) cos 2nnt dtñi ê
ñê
f (t) sin 2nnt dt (15.33)
The downside of such an approach for practical usage is that the combination of a
limited number of sin functions results in a relatively smooth curve. Periodicity in
financial markets will more than likely be a spike function of payments made and
not a gradual and uniform inflow/outflow.
In addition, a Fourier transform functions optimally on a metrically based time
series. The calendar is not a metric series and even when weekends and holidays are
removed in an attempt to normalize the series, the length of the business month
remains variable. The result is that any Fourier function will have a phase distortion
over longer periods of time if a temporal correction or time stretching/shrinking
component is not implemented. However, as a means of determining the fundamental
frequencies of a series, the Fourier transform is still a valuable tool, which can be
used for periodicity analysis and as a basis for the construction of a periodic model. 466