Time Series Their Analysis by Successive Smoothings

type of trend line is intriguing but baffling.
Many have advocated the S curves: the logistic, the Gompertz, and the
arc-tangent. But it has been suggested by Frischl that the highest or
simplest trend attainable still contains a significant cyclical or periodic
element, and Sombart2 posits a fundamental periodicity in economic
history.
These suggestions leave us in some doubt nhether the fundamental
type of trend is the non-recurrent logistic or a periodic function of the
type of a sine curve. There is sufficient doubt between the two type
forms that neither can, without challenge, be pronounced canonical.
This doubt justifies an attitude of eclecticism, and an empirical rather
than a rigid mathematical procedure.
Practically, it is difficult to fit a sine curve or otlier periodic function,
so that if the problem is attacked as one of fitting a mathematical
trend, it narrows down to the non-recurrent curves. Of the nonrecurrent
trends, the most suitable are probably the S curves, and of
these the logistic may be considered representative. It may be applied
as a simple S curve to a cumulative variable affected by a single major
stimulus, or in more complex generalized forms. However, the generalized
logistic can be fitted only by trial and error, a tedious process.
The empirical method presented below accomplishes much the same
result with simplicity and ease, and is offered as a practical substitute.
Even though the statistician declines to use an empirical curve as the
trend, and plans finally to fit a formal logistic (or other curve), he will
be aided by going first through the rapid empirical procedure, to
crystallize his ideas as to the shape to be sougllt in the formal curve.