Fig. 1 shows the predictions of the future of the US S&P 500 index
performed on Aug.24,2002. The continuous line is the fit and its
extrapolation, using our theory capturing investor herding and
crowd behavior. The theory takes into account the competition between
positive feedback (self-fulfilling sentiment), negative feedbacks
(contrariant behavior and fundamental/value analysis) and inertia
(everything takes time to adjust). Technically, we use what we
call a "super-exponential power-law log-periodic function" derived
from a first order Landau expansion of the logarithm of the price.
The dashed line is the fit and its extrapolation by including in
the function a second log-periodic harmonic. The two fits are performed
using the index data from Aug.9,2000 to Aug.24 2002 that are marked
as black dots. The blue dots show the daily price evolution from
Aug.25,2002 to June 19,2003. The large (respectively small) ticks
in the abscissa correspond to January 1st (respectively to the
first day of each quarter of each year.
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