The BMJ Fractal Analysis Indicator set is a group of indicators based on the concept of the fractal analysis of a time series. All the indicators are based on a common core that takes any time series and analyzes the fractal dimension of the time series. This analysis is based on an algorithm that is designed to estimate the fractal dimension of a time series. It is slightly more complex than the algorithms presented by John Ehlers in Technical Analysis of Stocks and Commodities (October, 2005 or June, 2010) and it is slightly less complex than the algorithm used in the Neuroshell Trader indicator for the calculation of fractal dimension (based on Peters, 1994 and 1996).

The fractal dimension for a time period of a given time series is estimated and converted to the Hurst exponent. The relationship between the fractal dimension and the Hurst exponent is such that the Hurst exponent is equal to two minus the fractal dimension. The fractal dimension varies between 1 and 2 while the Hurst exponent varies between 0 and 1. A time series with a Hurst exponent of 0.5 is said to be random, approximating Brownian motion. A time series with a Hurst exponent greater than 0.5 is said to be persistent, it will tend to travel far, but eventually it will reverse. A time series with a Hurst exponent less than 0.5 is said to be anti-persistent, it will travel less, and it will change direction more frequently. A persistent time series can generally be expected to continue in the direction of the trend, at least until it reverses. An anti-persistent time series will generally not be in a trend and may be expected to reverse direction more frequently. A time series with a Hurst exponent greater than 0.5 would generally be characterized as a trending market, while a time series with a Hurst exponent less than 0.5 would generally be characterized as a trading or cycling market. Any financial instrument and any financial time series can be expected to exhibit both forms of behavior over time.

The theory of the fractal analysis indicators is that by using the fractal dimension and the Hurst exponent we can produce filtered time series that are theoretically stripped of most of the noise in the data. The result is really two foundational indicators upon which all others are developed. The first is the BMJ Hurst indicator which is the calculated Hurst exponent for some look back window in the time series. The second is the BMJ FFilter or the fractal filter. The fractal filter is developed by using the Hurst exponent and converting it to an alpha for an adaptive filter of the time series based on the concept that price variation is directly proportional to some power of the time elapsed (as described by Mandelbrot, 2004 and others).

We might be tempted to call this an adaptive moving average. But it is really much more than that. The fractal filtered time series is so responsive to changes in direction with virtually no lag and no overshoot, that the filtered result can be used as a proxy for the original time series in any traditional technical analysis techniques. The result is a highly responsive smoothing filter of any data series.

The BMJ Fractal Filter can be used as a pre-processor of the time series for any traditional technical analysis. It can also be used as a post-processor to smooth traditional technical analysis techniques or to provide a signal line for cross over triggers. The BMJ Fractal Analysis indicators can also be used in concert with other traditional technical analysis indicators. All three of these approaches are illustrated in the Example Charts.

References:

Ehlers, J. (2005). Fractal adaptive moving averages. Technical Analysis of Stocks & Commodities. V. 23. (10) p. 81-82.

Ehlers, J. (2010). Fractal dimension as a market mode sensor. Technical Analysis of Stocks & Commodities. V. 28. (7) p. 16-20.