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On the Trend, Detrend and the Variability
of Nonlinear and Nonstationary Time Series

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Norden E. Huang
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NASA Goddard Space Flight Center
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Abstract:
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The trend and detrend are frequently encountered terms 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 the
correlation function and spectral analysis, one would have to remove the trend from the
data, or detrend, lest the result would be overwhelmed by the DC terms. Therefore,
detrend is a necessary step before meaningful results can be obtained. As there is a lack
of precise definition for the trend, detrend is also a totally ad hoc operation. In most
cases, trend is taken as the result of a moving mean, a regression analysis, a filtered
operation or simple curve fitting with an a priori 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 detrend operation,
which usually consists of setting the data to 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 have to be adaptive. Second, the trend exists only within a given data
span; therefore, it should be local, and, 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 Hilbert-
Huang Transform, will be given and analysis of Climate data as well as NASDAQ data
will be used as examples to demonstrate the application of the Hilbert-Huang Transform.