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

Unlike the dynamic swings characterized by log-periodicity, physical cycles have linear periodicity and linear phase. Therefore, we can use the well-established signal processing techniques for detecting physical cycles including the periodicity and phase shift. Hilbert transform of a function f(t) is defined for all t by

Hilbert transform is orthogonal to the original function f(t), so it can be used to create an analytical signal from a real signal

are the instantaneous amplitude and the instantaneous phase. Therefore, Hilbert transform provides a basic means for modeling physical cycles of linear periodicity and phase shift of the market price time series (taken as real signals). Ehlers (2001) developed a set of practical algorithms for calculating the dominant period and instantaneous phase from price time series data. However, his approach is limited to a single level. Our model of physical cycles generalizes to multilevels.

In essence, we first generalize a simple Hilbert transform to a Hilbert wavelet, and then the time series data are analyzed through a multiscale wavelet transform into a wavelet pyramid. On each level of this pyramid, we detect the dominant periodicity and calculate the instantaneous phase from the Hilbert transform on that level. The outcome of this multilevel Hilbert transform and detection procedure will be a complete description of multilevel periodicities and instantaneous phases. Pan (1996) conducted a comparative performance study for a number of complex wavelets.

A Quantum Space of Price and Time

Elliott wave theory suggests that the price, more often than not, traces back to certain Fibonacci ratios such as 38.2%, 50%, 61.8%, 100%, etc. The support or resistance levels have a quantum nature (or at least geometrical). Gann was probably the first in discovering the quantum structure of the price-time space. Gann theory of price-time cycles (Pan 2003) suggests that there is a significant symmetry between the price and the time, and time is often more important than price.

When certain time zones and price levels coincide, the reversal is imminent. On the other hands, the log-periodic power law models show clear geometrical properties that are consistent with the Gann angles developed in technical analysis. The dynamic swings and physical cycles identified and modeled from the historical index time series will most likely define a quantum space of price and time.

This space is divided by important price levels and time zones defined by the previous dynamic swings and physical cycles. The market will most likely travel from one significant price level to another or from one significant time zone to another. The actual path is not only determined by dynamic swing phase and physical cycle phase, but also by the possible news impact.

Prof. Heping Pan

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