Each l -th level swing force fl(t) must have a certain wave form. A number of studies have shown the existence of the log-periodicity in the stock market indexes and prices, thus we consider that fl(t) should include a wave form in log-time

where ?l, Fl are the log-period and phase respectively, and tl a time shift. Considering the existence of power laws and log-periodicity as demonstrated by Sornette’s group and other researchers, we consider pl(t) should have a form which includes a power component and a log-periodic wave component

where Al,Bl,Cl, ß l, ?l,Fl, are unknown parameters pertinent to level l . Sornette et al have also shown the possibility of using more complicated second-order and third-order Landau expansions, however, we consider our use of multilevel log-periodicities could render higher-order Landau expansions unnecessary. Using equation (10), the joint influence of dynamic swings can be expressed as

There are two different ways for actual computation using this model. The first is a general regression of all the unknown parameters of this model using the whole time series data set. The second way is a recursive procedure which successively fits a single level model of (25) to one segment of the time series data corresponding to an Elliott wave in the context of the previous level. However, we recognize that a single logperiodic power law model of (25) can only fit to a single wave, which is what was done by Sornette’s group.

Such a single model cannot predict the regime shift after an antibubble – a correction wave for a bullish trend – has finished. In a truly fractal procedure, on each level, we should fit an elementary fractal made up of two waves: a trending wave followed by a correcting wave. This elementary fractal for an up trending wave and its correcting wave is defined by a pair of log-periodic power laws

In this case, should correspond to the critical point – the time of reversal from the trending wave to the correcting wave. Equations (27) and (28) should be fitted to the up trending wave and the correcting wave respectively. However, we consider there may exist certain geometrical relationships between the two sets of parameters: ( Al,Bl,Cl, ß l, ?l,Fl) versus ( A'l,B'l,C'l, ß'l, ?'l,F'l). This geometry, if existent such as Fibonacci ratios, would substantially reduce the number of unknown parameters. Observations from technical analysis show that this geometry is probabilistic.

Prof. Heping Pan

Next: Multilevel Physical Cycles in Hilbert Transform and A Quantum Space of Price and Time

Summary: Index