Trends and the process of detrending are common components
in data analysis. Yet, there is no precise mathematical
definition for the trend in a data set, even though in
many applications, such as financial and climatologic
data analyses for example, the trend is precisely the
quantity we want to find. In other applications, such
as in computing correlation functions and in spectral
analysis, one would have to remove the trend from the
data, or detrend the data, lest the result be
overwhelmed by the mean or the trend terms.
Therefore, detrending is a necessary step before
meaningful results can be obtained. As there is
a lack of a precise definition for the trend,
detrending is also a totally ad hoc operation.
In most cases, the trend is taken as the result
of a moving mean, a regression analysis, a
filtering operation or simple curve fitting
with an a priori assumed functional form.
Yet such a trend is determined subjectively
and with certain idealized assumptions.
Furthermore, the trend so determined is
usually different from the quantity taken
away in the detrending operation, which
usually consists of a simple linear fit
of the data as the zero reference.
The real trend should have the following
properties: First, the trend should be an
intrinsic property of the data. In other
words, it should be part of the data, and
driven by the same mechanisms that generate
the observed or measured data. Unfortunately,
most of the available methods define trend by
using an extrinsic approach, such as pre-selected
simple functional forms. Being intrinsic,
therefore, requires that the method used in
defining the trend must be adaptive. Second,
the trend exists only within a given data span;
therefore, it should be local, and, thus should
be associated with a local scale of data length.
Consequently, the trend can only be valid within
that part of data, which should be shorter than
a full local wavelength. Thus, with this
definition, we can avoid the difficulty
encountered by most economists:
``one economist's 'trend' can be another's
'cycle'." New definition of the trend and
variability (or volatility), based on
employing the Empirical Mode Decomposition
Method, will be presented and an analysis
of NASDAQ data will be used as example to
demonstrate the application to financial
data.