This figure shows 9 years of the evolution of the Japanese Nikkei index and more than 7 years of the USA S&P500 index, compared to each other. In previous updates, we applied a simple translation of 11 years between the two indices, paralleling many analysts' procedures. To compare with our new procedure discussed below, this translation can be written mathematically as
(1) time(S&P500) = 1*time(Nikkei) + 11
where the 1 in front of time(Nikkei) means that there is neither a contraction nor a dilation of one time with respect to the other. It is however frustrating to perform such a match by eye-balling and a more rigorous approach is called for, called a cross-correlation described in the next figure caption. Our cross-correlation analysis shows that the best match between the Nikkei and S&P500 indices is obtained by the following mapping between the two times:
(2) time(S&P500) = 0.9375*time(Nikkei) + 16.25.
Notice that the `1' has been replaced by the factor 0.9375, which means that the intrinsic time scale of evolution of the S&P500 is flowing faster than its Nikkei counterpart. As references, this expression matches the time pairs (1984 for Nikkei and 1995 for S&P500) corresponding to a local translation of 11 years and (1988 for Nikkei and 1998.75 for S&P500) corresponding to a local translation of 10.75 years. The shrinking value of the shift expresses the contraction of the US time versus the Japanese time.
When time(Nikkei) = 1990 (that is, Jan-1-1990) which is the very top of the Nikkei bubble, we have time(S&P500) = 2000.6250 (that is, Aug-15-2000) which is the exact onset of the US antibubble found in our papers.
Thus, in this figure , the times of the S&P500 and of the Nikkei here are no more mapped to each other through the 11-year translation as done in previous updates but have in addition a contraction, given by the factor 0.9375 in equation (2). The years are written on the horizontal axis (and marked by a tick on the axis) where January 1 of that year occurs.
This figure illustrates an analogy noted by several observers that our work has made quantitative. The oscillations with decreasing frequency which decorate an overall decrease of the stock markets are observed only in very special stock markets regimes, that we have termed log-periodic "anti-bubbles". (The term antibubble was inspired by the concept of "antiparticle" in physics. Just as an antiparticle is identical to its sister particle except that it carries exactly opposite charges and destroys its sister particle upon encounters, an antibubble is both the same and the opposite of a bubble; it's the same because similar herding patterns occur, but with a bearish vs. bullish slant). By analyzing the mathematical structure of these oscillations, we quantify them into one (or several) mathematical formula(s) that can then be extrapolated to provide the prediction shown in the following figures.