Category: Advanced Indicator Set 1

Input parameters

Calculations

Hurst exponent calculations are very complex and cannot be expressed by a single formula. However, in a very simplified form, the Hurst exponent HurstExp is an exponent in the following expression:

R/S = (const * Period) ^ HurstExp,

where:

const is a constant,

Period is the number of points in the look-back window,

R/S = [Max(CumSum) – Min(CumSum)] / StdDev is so called “rescaled range”,

CumSum = Sum[Input[i] – Average(Input, Period)] is the cumulative deviation from the mean over Period points,

StdDev = Sqrt{Sum[(Input[i] – Average(Input, Period))^2]/Period} is the standard deviation of single points from the average across the window.

If Detrend flag=0, the Hurst exponent is computed on the input time series itself without any preprocessing. If Detrend flag=1 (default value), the input time series is first preprocessed with the function Input[0] = Ln(Input[0] / Input[1]). Here Ln is the natural logarithm function, Input[0] is the current bar, and Input[1] is the previous bar. Detrend flag=1 cannot be applied to a time series with zeroes and negative ratios of current/previous bars.

If Hurst Exponent was not calculated for a window (output is blank on the chart), this means that there are too few points for calculations. If you receive way too many blank outputs it is highly recommended that you increase the parameter Period.

Discussion

First introduced by H.Hurst [1] for engineering time series data, and then applied to market time series data [2,3], the Hurst exponent is deemed to be a method of classifying time series data. There are three distinctive cases: (a) 0 < HurstExp < 0.5, (b) HurstExp = 0.5, (c) 0.5 < HurstExp < 1.0.

Case (a) 0 < HurstExp < 0.5 defines an anti-persistent time series. There is a dependency between the current and previous bars. If the system was up in one bar, it will probably go down in the next. If it was down, it will go up. The series tends to reverse its behavior from the previous bar. This type of series differs from a random series by the fact that it is choppier and more volatile than a random series. The closer HurstExp is to zero the more anti-persistent the time series is. The closer HurstExp is to 0.5 the more random the time series is. An anti-persistent time series is rare. One example of such a series is the standard deviation of the price change.

Case (b) HurstExp = 0.5 defines a pure random time series (not chaotic, though!). The current bar does not depend on the previous bars at all. There is no correlation between past and future.

Case (c) 0.5 < HurstExp < 1.0 defines a persistent time series. There is a clear relationship between current and past bars. If the system was up in one bar, it will probably go up in the next. If it was down, it will go down. The series tends to conserve its behavior from previous bar. Trends are clear. The closer HurstExp is  to 1 the more persistent and predictive the time series is. The closer HurstExp is to 0.5 the more random the time series is. Most market time series fall into this category.

The Hurst Exponent is useful for determining how random a price time series is. If the series is very random, then in theory, it will be harder to predict or trade using standard indicators. However, using HurstExp is tricky because the value of 0.5 for a random time series is only true when the window is near 10,000 bars! With less bars than that, a random price series can have HurstExp around 0.6 or 0.7. Therefore, the HurstExp value cannot be evaluated literally in the three cases (ranges) defined above. Since most of the time you will be dealing with windows less than 10,000 bars, you can only see if the series is getting more or less random as the window moves forward.

When you have less than 10,000 bars in the window, the best way to distinguish between random and non-random is to use the Hurst Significance indicator, which tells you the difference between the time series under consideration and a true random time series.

References

1. Hurst, H.E., "Long-Term Storage of Reservoirs ", Transactions of the American Society of Civil Engineers, v.116, 1951.

2. Peters, Edgar E., “Chaos and Order in the Capital Markets”, Second Edition, John Wiley & Sons, 1996.

3. Peters, Edgar E., “Fractal Market Analysis”, John Wiley & Sons, 1994.